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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf,len=2048
+page_content='TRAINING WITH MIXED-PRECISION FLOATING-POINT ASSIGNMENTS  Wonyeol Lee 1 Rahul Sharma 2 Alex Aiken 1  ABSTRACT  When training deep neural networks, keeping all tensors in high precision (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 32-bit or even 16-bit floats)  is often wasteful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' However, keeping all tensors in low precision (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 8-bit floats) can lead to unacceptable  accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Hence, it is important to use a precision assignment—a mapping from all tensors (arising in  training) to precision levels (high or low)—that keeps most of the tensors in low precision and leads to sufficiently  accurate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We provide a technique that explores this memory-accuracy tradeoff by generating precision  assignments that (i) use less memory and (ii) lead to more accurate models at the same time, compared to the  precision assignments considered by prior work in low-precision floating-point training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Our method typically  provides > 2× memory reduction over a baseline precision assignment while preserving training accuracy, and  gives further reductions by trading off accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Compared to other baselines which sometimes cause training  to diverge, our method provides similar or better memory reduction while avoiding divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  1  INTRODUCTION  In deep neural network training, floating-point formats are  usually used to represent tensors and it is worthwhile to use  the smallest bitwidth format that gives acceptable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  For example, it is common to replace tensors using 32-bit  floats with tensors that use 16-bit floats (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Kalamkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The benefits are easy to under-  stand: computations using lower-precision floats not only  use less memory but are also faster (due to improved vec-  tor parallelism, locality, and reduced data movement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The  downside is that there is generally some loss of training accu-  racy, and in the worst case training may not even converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  For such low-precision floating-point training, the most  common approaches use two floating-point formats—one  for lower-precision floats (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 8-bit floats) and the other  for higher-precision floats (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 16-bit floats)—and assign  one of the two formats to each tensor (including weights,  activations, and their gradients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The precision assignments  studied in previous work fall into one of two assignment  schemes (which both have several variants): the uniform  assignment uses low precision for almost all tensors  (often excepting those in the first and/or last few layers)  (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018), while the operator-based  assignment limits low precision to the input tensors of  certain operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', convolutions) (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Prior work has shown that both precision assignment  schemes (with well-chosen low-bitwidth floating-point  formats) can match the accuracy of 32-bit-float training  1Stanford University, USA  2Microsoft Research, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Correspondence to: Wonyeol Lee <wonyeol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='cs@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='com>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Kalamkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Chmiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Drumond  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Cambier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  There is an important limitation in all prior approaches  to low-precision floating-point training: they use very  few precision assignments (most often just one) for a  given set of models, but there are some other models and  inputs where the chosen precision assignment (i) results  in noticeably worse accuracy than 32-bit-float training,  (ii) causes training to even diverge, or (iii) admits a more  efficient assignment that achieves similar training accuracy  (see Figures 1, 3, and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  In this paper, we present a new, automated method for choos-  ing precision assignments that removes the limitations de-  scribed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' To do so, we formally introduce the memory-  accuracy tradeoff problem: given a dataset, a model, and  two floating-point precision levels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', bitwidths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' high and  low), find a mixed precision assignment (a mapping from  all tensors arising in training to high/low precision) for the  model that maximizes training accuracy subject to a given  upper bound on the model aggregate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', the total number  of bits of all tensors appearing in training).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The model aggre-  gate is a proxy for the memory and time required for training,  as it is roughly proportional to memory footprint and also  well-correlated with training time (since training is often  dominated by data movement) (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We prove that the memory-accuracy tradeoff problem  is theoretically difficult (namely NP-hard) partly due  to the exponential number of possible mixed precision  assignments (which we often refer to simply as precision  assignments for brevity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The large number of possible  assignments makes the problem difficult in practice as  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='13464v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='LG]  31 Jan 2023  Training with Mixed-Precision Floating-Point Assignments  (a) SqueezeNet  (b) ShuffleNet-v2  (c) MobileNet-v2  Figure 1: Training trajectory of various models on CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Colors denote precision assignments: all-32-bit πfp32 (red),  uniform πunif (yellow), and operator-based πop (blue) (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' the latter two use the 8-bit (and 16-bit) floats in (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  2019) as low (and high) precision numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Markers denote the “width multiplier” of a model, which controls the capacity  of the model (see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='3): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='0 (•), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='5 (■), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='25 (▲), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 (  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Some lines of πunif are missing as they converge to small  values or diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Observe that neither πunif nor πop works best for all models: in some models, πop has a similar accuracy  to πfp32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' but in other (and all) models, the accuracy drop of πop (and πunif) from πfp32 are noticeably large (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', >1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  well: there is no known analytical method for predicting  the training accuracy of a given precision assignment, and  for any practical model there are far too many precision  assignments to simply test them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We propose a simple (heuristic) approach to the tradeoff  problem that prioritizes tensors for low-precision formats  based on the tensor’s size (with an additional step described  below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' More specifically, our algorithm takes as input  a single parameter giving a desired upper bound on the  model aggregate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Starting with the largest tensor in the  model, tensors are assigned low precision in size order  (from largest to smallest) until the model aggregate falls  below the given upper bound;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' all remaining tensors are  assigned high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Our main result is that this method  discovers mixed precision assignments that use less memory  while achieving higher training accuracy than previous  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' While we cannot show that our method finds  Pareto-optimal memory-accuracy tradeoffs, we do show that  our results are closer to Pareto-optimal than prior methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Some precision assignments initially generated by our  algorithm cause training to diverge due to an excessive  number of overflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' To address this issue, we propose an  overflow handling technique that promotes tensors causing  too many overflows from low precision to high precision  during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  In our experiments, these promotions  consume only a small amount of additional memory (< 3%  of the maximum model aggregate) and prevent training  from diverging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The overflow handling technique is not  specific to our algorithm and can be applied to other  precision assignment methods as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We evaluate a PyTorch implementation of our method using  experiments on standard image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We first  demonstrate that the precision assignments computed by our  method alleviate the limitations of existing methods: they  indeed explore the tradeoff between memory and accuracy  and exhibit a better tradeoff than the uniform and operator-  based assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We then show the two main components  of our method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', precision demotion of larger tensors  and precision promotion of overflowing tensors) are both  important to produce competitive precision assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We also provide some guidance on how users may apply  our method to navigate the memory-accuracy tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  To summarize, this work makes four main contributions:  We formally introduce the memory-accuracy tradeoff  problem to explore better mixed precision assignments  for low-precision floating-point training and prove the  NP-hardness of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We present a novel precision assignment technique, as a  heuristic solution to the tradeoff problem, that proposes  assignments based on a single parameter denoting a  desired upper bound on the model aggregate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We present a novel technique that can handle an exces-  sive number of overflows arising in training while using  a small amount of additional memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The technique can  be applied to any (not just our) precision assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We demonstrate that the mixed precision assignments  found by our method do explore the tradeoff between  memory and training accuracy, and outperform existing  precision assignment methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We remark that this work focuses on low-precision  floating-point training, not fixed-point training (which uses  fixed-point formats), since we want to target upcoming  hardware (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', (Andersch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2022)) with native support  for low-precision floats (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 8-bit floats) and their oper-  ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Also, this work focuses on low-precision training  (which trains a model from scratch), not inference (which  assumes a pre-trained model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' More discussion is in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Our precision assignment method typically provides > 2×  memory reduction over the operator-based assignment  while maintaining similar training accuracy and gives  further reductions by trading off accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Our method  also provides similar memory reduction to the uniform  assignment, while avoiding the divergence of training often  caused by a uniform assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  70  accuracy (%)  60  50  test  40  0  50  100  150  200  epoch75  70  test accuracy (%)  65  60  55  50  45  0  50  100  150  200  epoch75  test accuracy (%)  70  65  60  55  50  45  0  50  100  150  200  epochTraining with Mixed-Precision Floating-Point Assignments  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  After discussing  related work (§2), we define the memory-accuracy tradeoff  problem and study its hardness (§3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We then describe our  algorithm for the problem (§4) and our evaluation (§5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We  conclude with limitations and future work (§6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  2  RELATED WORK  Low-precision floating-point training has been exten-  sively studied since the work of (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  One active research direction is to select appropriate floating-  point formats (or their variants) for low- and high-precision  numbers in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Various floating-point formats have  been proposed, including FP16 (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018),  BF16 (Kalamkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019), FP8 (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018),  HFP8 (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019), and FP6 (Chmiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2021),  along with some variants such as HBFP (Drumond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  2018), S2FP8 (Cambier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2020), and BM (Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Recently, the problem of automatically selecting  such floating-point formats has been considered: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', (Yang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Another research direction is to develop algo-  rithmic techniques that improve training accuracy under low  precision: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', (Sa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Zamirai  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Björck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Our work is orthogonal  and complementary to all these prior works: they consider  various floating-point formats or training algorithms but use  a fixed precision assignment, which is either the uniform  or operator-based assignment (or their variants);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' our work  explores various precision assignments once floating-point  formats and training algorithms are fixed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', based on  the prior works).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The tradeoff between memory and accu-  racy in training is also considered in (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2022),  but the work differs from ours: they vary floating-point for-  mats when a precision assignment is fixed, while we vary  precision assignments when floating-point formats are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Low-precision fixed-point training uses fixed-point for-  mats as a low-precision representation instead of a floating-  point format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Some works use fixed-point formats for for-  ward tensors and floating-point formats for backward ten-  sors: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', (Courbariaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Choi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Other  works use only fixed-point formats for all tensors: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Das  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Banner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Sakr & Shanbhag, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Rajagopal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Among all these  works, some consider various mixed precision assignments  with different bitwidth (fixed-point) formats (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', (Sakr &  Shanbhag, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2020));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' but they are not ap-  plicable to our context (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', floating-point training) since  they rely on some properties of fixed-point formats that do  not hold for floating-point formats (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', all numbers in a  given format are equally distributed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The approach taken in  (Rajagopal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2020) is orthogonal and complementary  to ours: they use only the uniform precision assignment, but  change the underlying low-precision formats during train-  ing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' we consider various mixed precision assignments, but  fix the underlying low-precision formats during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Low-precision inference, often called neural network quan-  tization (in a narrow sense), aims at reducing the latency  or memory of neural network inference (instead of train-  ing) by using low-precision numbers (Nagel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Existing approaches typically assume a pre-trained model  and try to find low-precision formats for each part of  the inference computation, either by retraining the model  (called quantization-aware training) or without any retrain-  ing (called post-training quantization);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', (Gholami  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2022) for surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Some works on  inference consider various mixed precision assignments, but  they are not applicable to our context: they focus on making  inference more efficient and usually assume a pre-trained  model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' we focus on making training more efficient and aim  at learning a model from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Floating-point tuning is another related topic, which con-  siders the following problem: given a program, assign ap-  propriate formats (among given candidates) to the program’s  floating-point variables such that the program’s output has  an error smaller than a given threshold for all given inputs,  while also maximizing performance (Rubio-González et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Chiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Guo & Rubio-González,  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Menon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' This problem is different from  the problem we consider: the former is concerned with the  floating-point error after a single run of a program, while  we are concerned with the training accuracy after a large  number of runs of a program (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', a gradient computation)  where each run affects the next run;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' further, the former con-  siders general-purpose programs, while we consider deep  learning programs and exploit their unique features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  3  PROBLEM  In this section, we first provide background on low-  precision floating-point training (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1), based on which  the memory-accuracy tradeoff problem is introduced (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We then prove the NP-hardness of the problem (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Our  approach in §3–4 is more formal than most related works  for two reasons: (i) we show the problem is NP-hard, which  has not been considered in prior work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' and (ii) to clearly  describe the precision assignments to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 Background: Low-Precision Floating-Point Training  Let T be the set of real-valued tensors and let [n] denote the  set {1, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For a supervised learning task, we usually  consider a model network M = (f1, · · · , fn) parameter-  ized by θ = (θ1, · · · , θn) ∈ Tn, and a loss network L =  (fn+1, · · · , fm), where fi : T2 → T is a primitive operator  on tensors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', convolution, batch normalization, maxpool,  and softmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Given an input-output pair (x, y) ∈ T2, the  Training with Mixed-Precision Floating-Point Assignments  𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  𝑑𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  𝑑𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',#  𝑓$%!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  𝑑𝑓$%!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  ⋯  𝑓$  𝑑𝑓$,#  𝑑𝑓$,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  𝑓&  𝑑𝑓&,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  ⋯  ⋯  ⋯  #  𝑑𝑣&  %𝑣#  &𝜃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  #  𝑑𝑣#  #  𝑑𝜃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  %𝑣$%!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  %𝑣$%#  #  𝑑𝑣$%!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  #  𝑑𝑣$%#  &𝜃$  #  𝑑𝜃$  %𝑣&%!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  #  𝑑𝑣&%!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  𝑦  : forward computation  : backward computation  %𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  %𝑣$  %𝑣&  #  𝑑𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  #  𝑑𝑣$  Figure 2: A diagram showing the tensors and operators  used in a gradient computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (1) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For  brevity, rounding functions rndπ(·) are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  model M computes a predicted output y′ of x by iteratively  applying fi(·, θi) to x (i ∈ [n]), and L computes a loss  from y′ by iteratively applying fi′(·, y) to y′ (i′ ∈ [m]\\[n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  A standard way to train M is to minimize the loss value  using the gradient descent algorithm: iteratively update θ  by following the gradient of the loss with respect to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Floating-point training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In practice, we perform a gradient  computation usually with tensors represented in floating-  point formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Let π : TS → FP be a precision assignment  giving the floating-point format of each tensor, where TS =  ∆  {vi, dv i, θj, dθj | i ∈ [m + 1], j ∈ [n]} is the set of tensors  arising in a gradient computation (explained below), and  FP =  ∆ {fp(e, m, b) | e, m ∈ N, b ∈ Z} is the set of floating-  point formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Here fp(e, m, b) denotes a floating-point  format that consists of a 1-bit sign, an e-bit exponent, and  an m-bit mantissa, and has an (additional) exponent bias  of b ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' A common choice of π is πfp32(t) =  ∆ fp32 for  all t ∈ TS, where fp32 =  ∆ fp(8, 23, 0) is the standard 32-bit  floating-point format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Given a precision assignment π, a gradient computation is  typically performed by the backpropagation algorithm: with  ˆv1 = rndπ(v1)(x) and ˆdv m+1 = rndπ(dv m+1)(1), compute  ˆvi+1 = rndπ(vi+1)(fi(ˆvi, ˆui)),  ˆθj = rndπ(θj)(θj),  ˆdv i = rndπ(dv i)(df i,1( ˆdv i+1, ˆvi, ˆui)),  ˆ  dθj = rndπ(dθj)(df j,2( ˆdv j+1, ˆvj, ˆθj)),  (1)  for i ∈ [m] and j ∈ [n];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' see Figure 2 for a diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Here  rnd : FP × T → T is a function rounding a given input  to a given floating-point format, df i,1, df i,2 : T3 → T are  the backward operators of fi with respect to its first and  second arguments, respectively, and ˆui = ˆθi if i ∈ [n] and  y otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We call vi and θj the forward tensors, and  dv i and dθj the backward tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We put a hat over each  tensor to emphasize that its value is the output of a rounding  function to a possibly low-precision format;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' remark that  such a rounding function is not used within fi, df i,1, and  df i,2, since they typically use large bitwidth floats (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  fp32) and no low-precision floats internally (Kalamkar  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Cambier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' After the computation,  ˆ  dθj stores the gradient of the loss value with respect to θj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  The overall picture of floating-point training is now de-  scribed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In each iteration of the gradient descent  algorithm, we compute ˆ  dθj via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (1) using a given preci-  sion assignment π, training data (x, y), and current weights  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We then update each θj by θj ← rndfp32(θj − η · ˆ  dθj)  given a learning rate η > 0, and proceed to the next iteration  until the training ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Here we use fp32 to represent θj  by following the convention in low-precision floating-point  training: a “master copy” of weights (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', θj) is stored sep-  arately from the weight values (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', ˆθj) used in a gradient  computation, and is usually represented by fp32 (Micike-  vicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Kalamkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Cambier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The memory overhead of this master copy is very  small compared to the memory required to store other ten-  sors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', activation tensors vi) (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Low-precision floating-point training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In low-precision  training, we use a precision assignment π where some  tensors have a smaller bitwidth than fp32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Particularly  well-studied are π that use two predetermined floating-  point bitwidths (which are different) and optionally vary  the rest of the format from tensor to tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We call  C : TS × {lo, hi} → FP a precision-candidate assignment  if C(t, lo) has the same bitwidth for all t ∈ TS, the same  holds for hi, and the bitwidth for lo is smaller than that for hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We also define Π(C) =  ∆ {π : TS → FP | ∀t ∈ TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' π(t) ∈  {C(t, lo), C(t, hi)}} to be the set of precision assignments  that conform to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Among various precision assignments in Π(C), two have  received the most attention:  the uniform assignment  πunif,C (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018) and the operator-based  assignment πop,C (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The former assigns  low-precision formats to all tensors uniformly1, and the  latter to (most of) the input tensors of GEMM operators (in  both forward and backward passes):  πunif,C(t) =  ∆ C(t, lo) for all t ∈ TS,  πop,C(t) =  ∆  �  �  �  �  �  �  �  C(t, lo) if t ∈ {vi, θi, dv i+1} for some i  and fi is a GEMM operator  (but not the first/last one)  C(t, hi) otherwise,  (2)  where a GEMM operator refers to a general matrix multi-  plication operator which arises in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', fully-connected or  convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' A particular variant πop′,C of πop,C  has received much attention as well (Kalamkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  PyTorch, 2022), which assigns low-precision formats to  (most of) the input and output tensors of GEMM operators:  it is defined as πop,C except that {vi, θi, dv i+1} in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (2)  is replaced by {vi, θi, vi+1, dv i, dθi, dv i+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For several  choices of C, these assignments have been shown to produce  training accuracy similar to that by πfp32 on many datasets  and models (see §1–2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  1For simplicity we define πunif,C without the common  exceptions for tensors near v1 and/or vm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Training with Mixed-Precision Floating-Point Assignments  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2  Memory-Accuracy Tradeoff Problem  We now introduce the following problem based on §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1,  to address the limitation of existing approaches for  low-precision floating-point training discussed in §1:  Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 (Memory-accuracy tradeoff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Given training  data {(xi, yi)}, a model and loss network M and L, a  precision-candidate assignment C, and a lower bound  r ∈ [0, 1] on the low-precision ratio, find π ∈ Π(C) that  maximizes acc(π) subject to ratiolo(π) ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Here acc(π) denotes the accuracy of the model M when  trained with π on {(xi, yi)}, and ratiolo(π) denotes the  low-precision ratio of π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', the portion of the tensors  represented in low-precision under π, among all tensors  appearing in a gradient computation:2  ratiolo(π) =  ∆ size({t ∈ TS | π(t) = C(t, lo)})  size(TS)  ∈ [0, 1]  where size(T) =  ∆  �  t∈T size(t) denotes the total size  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', number of elements) of all tensors in T ⊆ TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For  instance, ratiolo(πhi) = 0 and ratiolo(πlo) = 1 for the  all-high-precision assignment πhi and the all-low-precision  assignment πlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The problem asks for a precision assign-  ment that maximizes training accuracy under a memory  constraint, which is expressed as a fraction of the memory  required to train the model using πhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='3  NP-Hardness of the Problem  We prove that the memory-accuracy tradeoff problem from  §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2 is NP-hard by showing that there is a polynomial-time  reduction from the knapsack problem to this problem:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Recall the knapsack problem: given n items  with weights wi ∈ N and profits pi ∈ N (i ∈ [n]), find a  subset of the items that maximizes the total profit while its  total weight does not exceed a given threshold W ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Given an instance (w, p, W) of the knapsack problem, we  construct an instance of Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 such that we get the  following (informal) correspondence between the two: wi  corresponds to the size of the parameter tensor θi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' pi to the  i-th component of the input data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' W to the lower bound r  on the low-precision ratio (in an inverse way);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' and selecting  the i-th item corresponds to assigning a high-precision  format to the tensor θi (and related tensors), which roughly  decreases the low-precision ratio by wi while increasing  the accuracy of the model (after training) by pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Based  2As explained in §1, the low-precision ratio is a proxy for  the reduction in memory as well as training time (because the  low-precision ratio increases as the model aggregate decreases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Note that it is not always possible to simply measure training time,  as some floating-point bitwidths of interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 8-bit) are not  supported natively by current hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  on this informal correspondence, we formally prove that  an optimal solution to the above instance of Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1  can be converted in linear time to an optimal solution to  the given knapsack problem (w, p, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' That is, we have  a linear-time reduction from the knapsack problem (which  is NP-hard (Karp, 1972)) to Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 which is therefore  NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For a detailed proof, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Intuitively, the proof relies on two aspects of Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1:  the size of the search space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', |Π(C)|) is exponential in  the size of the problem (especially |TS|), and some values  representable in a high-precision format underflow to 0 in  a lower-precision format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Note that underflows are relevant  in low-precision training: they frequently arise in practice,  degrading the results of training (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  The NP-hardness result indicates that it is unlikely any  polynomial-time algorithm solves the problem exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  4  ALGORITHM  In this section, we propose a novel (heuristic) algorithm for  the memory-accuracy tradeoff problem (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1), and a new  technique to handle overflows arising in training (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1  Precision Demotion for Saving Memory  Consider an input to the memory-accuracy trade-off prob-  lem (Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1): a model and loss network M = (f1,  · · , fn) and L = (fn+1, · · · , fm), a precision-candidate  assignment C, and a lower bound r on the low-precision  ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Given the input, our algorithm works in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Tensor grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We first group tensors in TS such that  each group consists of all the tensors between two “adjacent”  GEMM operators (see below for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' This grouping re-  duces the search space over precision assignments, from all  of Π(C) to a subset in which the same precision is assigned  to the tensors in the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' This specific grouping  strategy is based on two observations: a majority of floating-  point operations are carried out in GEMM operators, and it  is standard (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', in PyTorch) to use the same precision for a  forward tensor and its corresponding backward tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Formally, we group tensors as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Let fk and fk′  (k < k′) be GEMM operators that are “adjacent”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', there  is no GEMM operator in {fk+1, · · · , fk′−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For each such  (fk, fk′), we create a group {vi, dv i, θj, dθj | i ∈ (k, k′] ∩  [m + 1], j ∈ (k, k′] ∩ [n]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' After that, we create two more  groups for the remaining tensors: one for the tensors near v1  and the other for tensors near vm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' As a result, we obtain  a set of disjoint groups of tensors {T1, T2, · · · } ⊆ 2TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Precision demotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Given the groups of tensors, T1, T2,  · · , we construct a precision assignment π as follows: ini-  tialize π to the all-high-precision assignment and update  π by demoting the precision of all tensors in a group to  low precision, one group at a time, until the low-precision  Training with Mixed-Precision Floating-Point Assignments  ratio of π becomes greater than r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We demote the pre-  cision of groups in decreasing order of their sizes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  the total number of elements in tensors);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' that is, the pre-  cision of a larger size group is demoted earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Formally,  let {T ′  1, T ′  2, · · · } be the reordering of {T1, T2, · · · } such  that size(T ′  1) ≥ size(T ′  2) ≥ · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' After initializing π by  π(t) = C(t, hi) for all t, we iterate over i ∈ N and update π  to π(t) = C(t, lo) for all t ∈ T ′  i, until ratiolo(π) ≥ r is first  satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The resulting π is the output of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  The intuition behind using group size as the priority order  for precision demotion is based on the fact that it is actually  optimal in a very simplified setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Suppose that an input  x to the model M stores a quantity of information I and  the forward computation of M is nothing but a process of  extracting the information in the input into a small number  of values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', the tensor vn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Assume that passing through  each group Oi = {fk+1, · · · , fk′} of operators (corre-  sponding to the group Ti of tensors) reduces the amount of  information by a factor αi ∈ (0, 1), and using low precision  on the group Ti further reduces the amount of information  by a constant factor β ∈ (0, 1) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Then, the amount  of information left in vn+1 becomes I × (α1α2 · · · ) × βl,  where l is the number of groups in low precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In this  simplified setting, maximizing the amount of information in  vn+1 is equivalent to minimizing the number of groups in  low precision, which is achieved precisely by demoting the  largest groups first (when there is a constraint on the low-  precision ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We show empirically (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='4) that using the  decreasing size order in precision demotion indeed produces  better precision assignments than using other orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2  Precision Promotion for Handling Overflows  While our algorithm in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 exerts a constraint on memory  usage, it places no explicit constraint on training accuracy,  and so not surprisingly for some models and datasets  the resulting precision assignment causes training to  diverge—accuracy decreases significantly and remains low  after some point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We observe that when training begins to  diverge (and a bit before that), many overflows occur in the  rounding function of some tensors ˆvi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', an input tensor to  the function rndπ(vi)(·) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (1) contains many elements  whose magnitude is larger than the maximum representable  number of the format π(vi) (Figure 6(a-b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' This  rapid increase in overflows in individual tensors is a signal  that training may diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Precision promotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Based on this observation, after each  gradient computation we update the current precision as-  signment π by promoting to high precision (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', C(t, hi))  any forward tensor t whose overflow ratio is greater than a  given threshold Θ ∈ (0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' this updated precision assign-  ment is used in the next gradient computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Here the  overflow ratio of t ∈ TS denotes the number of overflows  arising in the rounding function of ˆt in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (1), divided by  the number of elements in ˆt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We show empirically (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='4)  that training always converges using this technique and the  additional memory cost of promotion is small (in our exper-  iments, < 3% of the maximum model aggregate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For the  experiments, we use Θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' in fact we found that a wide  range of values for Θ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='01, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='001) all work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Note that this technique is not specific to our algorithm and  can also be applied to other precision assignment methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We apply precision promotion only to forward tensors for  two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' First, dynamic loss scaling (Micikevicius  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Nvidia, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' PyTorch, 2022)  already handles overflows in backward tensors, but not in  forward tensors: loss scaling multiplies the backward loss  tensor dv m+1 by a constant before performing backward  computation, to scale up all backward tensors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' the dynamic  version adjusts the constant during training in a way that  avoids overflows in backward tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Note that dynamic  loss scaling does not affect forward tensors at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Second,  we cannot use a similar idea to handle overflows in forward  tensors, because forward tensors are not linear in the input  tensor v1 whereas backward tensors are linear in the back-  ward loss tensor dv m+1 (by the linearity of differentiation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Precision promotion incurs little if any computational over-  head: checking whether a single rounding operation over-  flows is cheap, and we only apply rounding functions to the  output tensor of an arithmetic-intensive operator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', con-  volution and batch normalization), amortizing the cost of the  overflow checks over a large number of other operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  5  EXPERIMENTS  In this section, we evaluate our precision assignment  technique (developed in §4) on standard training tasks to  answer three research questions:  Does our technique explore the tradeoff between  memory and accuracy and achieve a better tradeoff than  existing (fixed) precision assignments (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='3)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Are the two main components of our technique,  precision demotion/promotion of larger/overflowing  tensors, important for good performance (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='4)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  How can we choose the parameter r in our technique  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', a lower bound on the low-precision ratio) (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='5)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1  Implementation  We have implemented our precision assignment technique  using PyTorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Given a model and loss  network, and a dataset, our implementation takes as param-  eters a precision-candidate assignment C and a lower bound  r on the low-precision ratio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' it then automatically assigns  precisions to tensors (appearing in training) according to  our technique and uses those assigned precisions in gradient  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' To make these procedures automatic, our  implementation works as follows:  Training with Mixed-Precision Floating-Point Assignments  For each PyTorch class for a primitive operator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  torch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='Conv2d), our implementation provides its  wrapped version (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', mpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='Conv2d) which records  auxiliary information for our technique (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', floating-  point format of input/output tensors) and applies proper  rounding functions in forward/backward computations  based on the auxiliary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Models should now  use the wrapped classes instead of the original ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Our implementation first constructs a computation graph  (of a given model and loss network) dynamically by  running a forward computation on a minibatch of input  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The computation graph and other information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  each tensor’s size) are recorded in the wrapped classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Using the auxiliary information just recorded, our  implementation then constructs a precision assignment  according to §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1, uses it in gradient computations, and  updates it after each gradient computation according  to §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We record the precision assignment also in the  wrapped classes to automatically apply proper rounding  functions in gradient computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  As no current hardware natively supports low-precision  formats used in the experiments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 8-bit floats) and  their operations, we simulate them with 32-bit floats  and 32-bit operations followed by rounding functions as  described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We implement the rounding functions  based on the QPyTorch library (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' a few  extensions are required though, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', to support exponent  bias and signal overflows for dynamic loss scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We  automatically apply these rounding functions after each  primitive operator, by using PyTorch’s hook feature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='register_*hook).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2  Experiment Setups  Datasets and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' As benchmarks for our experiments,  we use the image classification task and three datasets for the  task: CIFAR-10 and CIFAR-100 (Krizhevsky, 2009), and  ImageNet (Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' we choose them since  they have been widely used in recent works on low-precision  training as a standard choice (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Sakr &  Shanbhag, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Rajagopal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Chmiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  For the task and datasets, we use four well-known models:  SqueezeNet (Iandola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2016), ShuffleNet-v2 (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  2018), MobileNet-v2 (Sandler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018), and ResNet-  18 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' they are chosen since models with  relatively few weights, such as these, are generally known  to be more difficult to train with low precision than those  with more weights (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We considered other  tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', language modeling) and related models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  RNN/transformer-based models) but did not include them  in our experiments because substantial additional implemen-  tation effort orthogonal to our main contributions would be  required: these models use some PyTorch operators that do  not support per-tensor precision assignments, so applying  our technique to these models requires significant modifica-  tions to PyTorch internals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Precision-candidate and precision assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For the  experiments, we use the precision-candidate assignment  C studied in (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019), which uses 16-bit (and  8-bit) floats for high (and low) precision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' in particular,  C(t, hi) = fp(6, 9, 0) for all (forward/backward) tensors  t, and C(t, lo) = fp(4, 3, 4) for all forward tensors t and  fp(5, 2, 0) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We choose this particular C because it  uses sub-32-bit floating-point formats for both low and high  precision and the precision assignment πop,C was shown to  achieve accuracy comparable to 32-bit training (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The three floating-point formats used in C allow  subnormals but no infinities and NaNs, which are rounded  to the largest or smallest representable numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' While no  current hardware is available for the latter two 8-bit formats,  they will be supported natively on NVIDIA’s forthcoming  H100 GPU (Andersch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Because our technique  is parameterized by a precision-candidate assignment, it is  easily applied to other assignments as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We evaluate our technique by varying its parameter r  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', a lower bound on low-precision ratio) over deciles  r ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2, · · · , 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We write πours,r to denote the  precision assignment chosen by our technique (described  in §4) for a given r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', πours,0 is the all-high-precision  assignment, and πours,1 is the all-low-precision assignment  equipped with our precision promotion technique (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  By following (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019), all precision assignments  (including πours,r) in our experiments use high precision  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 16 bits) for all backward weight tensors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', ˆ  dθj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Other setups and compute time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' All experiments were  performed on NVIDIA V100 GPUs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' total compute time for  all experiments was 1,008 GPU-days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We train all models in  a standard way: we apply dynamic loss scaling (a standard  technique used in low-precision floating-point training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' see  §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2 for details) except for 32-bit training, and use standard  settings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', learning rate);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' see Appendix B for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Due to random variations in training, we perform four runs  of training for each configuration and report the average and  the range of measured quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='3  Comparison with Existing Precision Assignments  To compare our technique with existing precision assign-  ments for floating-point training, we train each model  with the following precision assignments: all-32-bit πfp32,  uniform πunif (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018), operator-based  πop (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019), its variant πop′ (Kalamkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' PyTorch, 2022), and ours πours,r (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 and §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2  for their definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We choose πunif, πop, and πop′  as baselines because existing precision assignments for  floating-point training fall into one of the three assignments  (or their variants) (see §1–2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Training with Mixed-Precision Floating-Point Assignments  Figure 3: Results of training ShuffleNet-v2 on ImageNet with πfp32, πunif (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018), πop (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  2019), πop′ (Kalamkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019), and πours,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Left: Each line shows the average training trajectory for each precision  assignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' πours,r is colored from navy to yellow (darker for smaller r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Right: Each point shows the memory-accuracy  tradeoff of each precision assignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' a red-dashed line shows the accuracy of πfp32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' and shaded areas show the variation  among four training runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In the right figure, top-right points are better than bottom-left ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Observe that there are •s  above and to the right of  and  , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ⋆ is missing as its y-value is too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  (a) CIFAR-10, SqueezeNet  (b) CIFAR-100, SqueezeNet  (c) CIFAR-100, SqueezeNet†  (d) CIFAR-10, ShuffleNet-v2  (e) CIFAR-100, ShuffleNet-v2  (f) CIFAR-100, ShuffleNet-v2†  (g) CIFAR-10, MobileNet-v2  (h) CIFAR-100, MobileNet-v2  (i) CIFAR-100, MobileNet-v2†  (j) CIFAR-10, ResNet-18  (k) CIFAR-100, ResNet-18  (l) CIFAR-100, ResNet-18†  Figure 4: Memory-accuracy tradeoffs of πunif (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018), πop (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019), πop′ (Kalamkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  2019), and πours,r for four models and their smaller variants on CIFAR-10 and CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The variant models have width  multiplier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='25 and are marked by †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Top-right points are better than bottom-left ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In all but three plots, there are •s  above and to the right of  and  , respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' even in the three plots (g,h,k), •s have almost the same tradeoffs to  and  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In half of all plots, ⋆ has much smaller y-values than other points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The training trajectories for the above plots  and the results of other smaller models are in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  92  90  88  fp32  op  86  op  unif  80  ours  78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=' ratio72  70  68  fp32  op  66  op  unif  2  ours  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=' ratio66  test accuracy (%)  64  X  62  fp32  60  op  op  unif  2  ours  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=' ratio94  (%)  92  test accuracy (  90  fp32  88  op  op  unif  80  ours  78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  low-prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ratio72  (%)  70  test accuracy (  68  66  fp32  op  op  unif  2  ours  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  low-prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ratio56  test accuracy (%)  54  52  fp32  op  50  op  unif  2  ours  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  low-prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ratioTraining with Mixed-Precision Floating-Point Assignments  We train the four models mentioned in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2 on CIFAR-10  and CIFAR-100, and ShuffleNet-v2 on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We also  train smaller variants of the four models (which are more  difficult to train with low precision) on CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We  obtain these variant models by following (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', by applying a well-known approach for model reduction  that uses a parameter called the width multiplier (Howard  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2017): each variant model reduces the number of  channels in most tensors by a width multiplier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' we use three  values {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1} for the width multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We train  just one model on ImageNet due to the large amount of  computation involved: for each model, 44 training runs (11  choices for r and 4 runs for each choice) are required for  πours,r and each run on ImageNet takes nearly a half day  with 16 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We use ShuffleNet-v2 for ImageNet since  the model shows interesting memory-accuracy tradeoffs  when trained on the (smaller) CIFAR datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Figure 3 presents training results of ShuffleNet-  v2 on ImageNet: its left graph plots the average training  trajectory for each precision assignment, and its right graph  shows how each precision assignment trades off between  memory and accuracy, where memory is represented (in-  versely) by the low-precision ratio of the assignment and ac-  curacy is the best test accuracy of the model during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Each point in the right graph shows the average accuracy  of four runs of training, while the shaded area shows the  variation in accuracy among those four training runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Figure 3 shows three points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  First, as the parameter r  increases, the average accuracy drop of πours,r from πfp32  increases (up to 5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In contrast, πunif and πop′ have a  much larger average accuracy drop (more than 30%), as  some training runs diverge when πunif and πop′ are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Second, the tradeoff given by πours,r is better (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', closer  to Pareto-optimal) than by πop: πours,r for r ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='4}  has both higher accuracy and larger low-precision ratio (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  memory reduction) than πop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In particular, πours,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='4 has  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='6× the memory reduction of πop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Third, πours,r provides  options that πop cannot (which has an accuracy drop of  >1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' If we want accuracy closer to πfp32, say within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='5%,  we can use πours,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2 with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='6% more memory than πop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' If  we can tolerate a larger accuracy loss, say ≈ 3%, then we  can use πours,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='7 with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='9× the memory reduction of πop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  CIFAR-10/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Figure 4 presents the memory-accuracy  tradeoffs of precision assignments for the four models on  CIFAR-10 and CIFAR-100, and their smaller variants (with  width multiplier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='25) on CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The results for other  smaller variants are similar and included in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  The conclusions from Figure 3 hold for Figure 4: πours,r  provides a range of options by varying r and exhibits a  better tradeoff than πunif, πop, and πop′ in almost all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We give a detailed comparison as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' First, in half of  all 12 plots, πunif shows a similar tradeoff to πours,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' But  in the remaining half, πunif has an accuracy drop much  larger than all other precision assignments including πours,r,  since using πunif often makes training diverge while using,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', πours,1 does not do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Second, in all but two plots,  πours,r shows a strictly better tradeoff than πop: πours,r has  noticeably larger (> 2×) memory reduction than πop while  maintaining similar accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Even in the two plots, πours,r  has a tradeoff very close to πop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Note that in three plots,  πop has an accuracy drop of >1% while πours,r provides  several options that have smaller accuracy drops and more  memory savings at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Third, πours,r shows a  strictly better (or similar) tradeoff than πop′ in all but two  (or two) plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Note that πop′ has accuracy smaller than πop  in all but one plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Also it has an accuracy drop of >1% in  half of all plots, and sometimes makes training even diverge  (in one plot here and three other plots in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='4 Ablation Study: Precision Demotion and Promotion  Precision demotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  To evaluate the decision to use  precision demotion in decreasing-size order, we train  the four models on CIFAR-100 with πours,r, πours[inc],r  (which demotes tensor groups in increasing-size order) and  πours[rand],r (which demotes tensor groups in random or-  der).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For πours[rand], three different random orders are used  in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The results, presented in Figure 5 (and Ap-  pendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2), show that the order of precision demotion has a  significant impact on the resulting memory-accuracy trade-  off, and that decreasing order provides the best results in  nearly all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Increasing order consistently shows the  worst results, suggesting our intuition (given in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1) for  choosing decreasing order has some basis in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Precision promotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' To understand whether precision pro-  motion of overflowing tensors is important to our technique,  we train ShuffleNet-v2 on ImageNet using πours[no-promo],r  which does not promote tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The results, presented in  Figure 6(a), show that several training trajectories diverge  in early epochs and fail to recover afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Figure 6(b)  plots the top-5 tensor overflow ratios for the highlighted  trajectory in Figure 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The overflow ratios first spike  about when divergence occurs around epoch 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' A closer  look shows that the spike in overflow ratio occurs shortly  before divergence, and starts first in a few tensors and then  propagates to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' These observations indicate that an  excessive number of overflows in a few tensors are the cause  of the training divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Finally, Figure 6(c-d) shows that precision promotion is  effective at preventing the divergence of training while  sacrificing only a small amount of memory reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The  figure shows ShuffleNet-v2 on ImageNet trained using our  technique with and without precision promotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Figure 6(c)  shows that without precision promotion large accuracy  drops occur due to divergence, whereas with precision pro-  motion training converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Figure 6(d) shows that the total  Training with Mixed-Precision Floating-Point Assignments  (a) SqueezeNet  (b) ShuffleNet-v2  (c) MobileNet-v2  Figure 5: Memory-accuracy tradeoffs of πours,r, πours[inc],r, and πours[rand],r for three models on CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Observe  that •s are above and to the right of other points in nearly all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The results of ResNet-18 are in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  (a)  (b)  (c)  (d)  Figure 6: Training ShuffleNet-v2 on ImageNet with πours,r and πours[no-promo],r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (a) Training trajectories of πours[no-promo],r  for different r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' colors denote r values (darker for smaller r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (b) Top-5 overflow ratios of tensors at each epoch, for the  highlighted trajectory in (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' the largest ratio is blue and the fifth largest red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (c) Memory-accuracy tradeoffs of πours,r  and πours[no-promo],r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (d) Low-precision ratio when training ends vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' when training starts, for πours,r and πours[no-promo],r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  The results on CIFAR-10 are in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  size of tensors promoted to high precision is small for all r  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2 for similar results for CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='5  Choosing the value of r  The time and space savings of our method are most signif-  icant when a model is regularly retrained, which commonly  occurs when new data is periodically incorporated into  an existing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Assuming that new data has a similar  distribution to existing data, we can choose a single r (a  parameter in our method) by conducting one set of exper-  iments where we train with πfp32 and πours,r for different r  and then choose the r value that maximizes model aggregate  savings while still having an acceptable drop in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  To simulate this scenario, we create five datasets ImageNet-  200-i (i ∈ [5]) as follows, so that each of them contains  different but similar data: randomly select 1/5 of the classes  in ImageNet (which has 1000 classes in total), and split the  training data of each class evenly into five new datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  For each ImageNet-200-i, we train ShuffleNet-v2 with  πfp32 and πours,r and present the results in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Based  on the tradeoff results of πours,r, we can choose r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='4 if  we desire an average of < 1% accuracy drop from πfp32, and  we can choose r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='9 if an average ≈ 3% accuracy drop  is tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We make two more observations: the tradeoff  result of πours,r is similar across all five datasets even  though each dataset is different, and for each r the variance  Figure 7:  Memory-accuracy tradeoffs of πours,r for  ShuffleNet-v2 on ImageNet-200-i (i ∈ [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  in the accuracy of πours,r from different datasets and runs of  training is similar to that of πfp32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Thus we expect that on  a new but similar dataset, πours,r would have an accuracy  drop similar to Figure 7 with acceptable variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  6  LIMITATIONS AND FUTURE WORK  Our work has the same limitation present in prior works  on low-precision floating-point training: as 8-bit floats and  operations are not handled natively in hardware, but rather  simulated in software, we cannot directly measure the poten-  tial speedup of our method, though we do expect speedups  to be proportional to the reduction in the model aggregate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We leave it as future work to perform such experiments  on future hardware (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', NVIDIA’s H100) that natively  supports more low-precision formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Another direction  for future work is to integrate our method into systems for  automatically optimizing deep learning computations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  (Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Unger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2022)) to accelerate training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  72  accuracy (%)  70  68  test  ours[inc]  ours[rand]  66  ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=' ratio74-  test accuracy (%)  72  70  ours[inc]  68  ours[rand]  ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=' ratio74  (%)  accuracy(  72  70  test  ours[inc]  ours[rand]  68  ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content='2  ours[no-promo]  ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content='0  low-prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ratio (start)64  62  茶  60  58  fp32  ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=' ratioTraining with Mixed-Precision Floating-Point Assignments  REFERENCES  Andersch,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  Palmer,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  Krashinsky,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  Stam,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  Mehta,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  Brito,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  and  Ramaswamy,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  NVIDIA  Hopper  Architecture  In-Depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='nvidia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='com/blog/  nvidia-hopper-architecture-in-depth/,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Banner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Hubara, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Hoffer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', and Soudry, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Scalable  methods for 8-bit training of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  In  NeurIPS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' 5151–5159, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Björck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', De Sa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Gomes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', and  Weinberger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Low-Precision Reinforcement Learning:  Running Soft Actor-Critic in Half Precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In ICML,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' 980–991, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Cambier, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Bhiwandiwalla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Gong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Elibol, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  Nekuii, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', and Tang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Shifted and Squeezed 8-bit  Floating Point format for Low-Precision Training of  Deep Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In ICLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Chiang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Baranowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Briggs, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Solovyev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  Gopalakrishnan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', and Rakamaric, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Rigorous  floating-point mixed-precision tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=', Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Baines, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Narvaez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=', McCormick, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  Mohd-Yusof, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Luo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Mudigere, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Smelyan-  skiy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', and Aiken, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Unity: Accelerating DNN Train-  ing Through Joint Optimization of Algebraic Transforma-  tions and Parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In OSDI, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' 267–284, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=', Brand, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=', and Gopalakrish-  nan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Training Deep Neural Networks with 8-bit Float-  ing Point Numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In NeurIPS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' 7686–7695, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Chen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', and Shi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Training and Inference  with Integers in Deep Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In ICLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Chee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Sa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', and Udell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  How Low Can We Go: Trading Memory for Error in  Low-Precision Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In ICLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Yang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Kirichenko, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Bai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Wilson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  and De Sa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' SWALP : Stochastic Weight Averaging in  Low Precision Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In ICML, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' 7015–7024, 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Shen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Xing, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Tian, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Deng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',  Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', and Hua, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='-s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Quantization Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In  CVPR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' 7308–7316, 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Zamirai, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Aberger, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', and Sa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Revisiting BFloat16 Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='06192, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Yang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', and Sa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' QPyTorch:  A Low-Precision Arithmetic Simulation Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='04540, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Huang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Zhou,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Guo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Guo, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Du, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Zhi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', and Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Fixed-Point Back-Propagation Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In CVPR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  2327–2335, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Training with Mixed-Precision Floating-Point Assignments  Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Ni, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Wen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', and Zou,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  DoReFa-Net:  Training Low Bitwidth Convolu-  tional Neural Networks with Low Bitwidth Gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='06160, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Training with Mixed-Precision Floating-Point Assignments  A  PROBLEM: DEFERRED PROOF  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We prove the NP-hardness of Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 (the  memory-accuracy tradeoff problem) by reducing the  knapsack problem (which is NP-hard) to the tradeoff  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  More precisely, we prove that the knapsack  problem can be solved in polynomial time if we assume  an oracle for the tradeoff problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Recall the knapsack problem: given n items with weights  wi ∈ N and profits pi ∈ N (i ∈ [n]), and given a threshold  W ∈ N, decide which items to choose such that the total  profit of the chosen items is maximized while their total  weight does not exceed W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' That is, find α ∈ {0, 1}n that  maximizes �  i∈[n] αipi subject to �  i∈[n] αiwi ≤ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' This  problem is well-known to be NP-hard (Karp, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Given an instance of the knapsack problem (w, p, W), we  construct an instance of the tradeoff problem as follows:  Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  The following construct uses a constant  k ∈ N and floating-point formats fphi, fplo ∈ FP (one  for high precision and the other for low precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Below we will specify the conditions they should satisfy,  and show that some k, fphi, and fplo indeed satisfy the  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We write rndhi(·) and rndlo(·) as shorthand  for rndfphi(·) and rndfplo(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Training setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We consider a very simple setting for  training: the gradient descent algorithm with a learning  rate η = 2−l (l ∈ N) is applied for just one epoch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' all  parameters are initialized to 0 and their master copies  are represented in fphi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' and the negative loss of a model  on training data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', −L(fθ(x), y) using notations to  be described below) is used as the accuracy of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Here l ∈ N can be any natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Model and loss networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' A model network M and  a loss network L are given as Figure 8, where M has n  parameter tensors θi ∈ Rwi of size wi (i ∈ [n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For an  input-output pair (x, y) ∈ Rn × R, M and L compute a  predicted output fθ(x) ∈ R and a loss L(fθ(x), y) ∈ R  as follows (assuming that no rounding functions are  applied):  fθ(x) =  ∆ �  i∈[n]  �  j∈[wi]  θi,jxi,  L(fθ(x), y) =  ∆ 2−k|fθ(x) − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Roughly speaking, M is (a variant of) a linear classifier  and L is a ℓ1-loss (scaled by 2−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Training data consists of a single input-  output pair (x, y) ∈ Rn × R that satisfies the following:  xi = rndlo(  �  pi/wi),  y < −2−(k+l) �  i∈[n]  wix2  i  for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Here y can take any value as long as  it satisfies the above inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Note that xi can be  different from  �  pi/wi since the latter value may not  be representable in fplo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Precision-candidate  assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  A  precision-  candidate assignment C : TS×{hi, lo} → FP is given as:  C(t, hi) =  ∆ fphi,  C(t, lo) =  ∆ fplo  for all t ∈ TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  That is, for all tensors, fphi is used as a high-precision  format and fphi as a low-precision format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Here fphi and  fplo should satisfy the following:  ehi ≥ elo,  mhi ≥ mlo,  (3)  |rndlo(s) − s| < |s| · err  for all s ∈ S1,  (4)  rndlo(s) = 0  for all s ∈ S2,  (5)  rndhi(s) = s  for all s ∈ S2 ∪ S3,  (6)  where ehi and mhi (and elo and mlo) denote the number  of exponent bits and mantissa bits of fphi (and fplo), and  err =  ∆ 1/(6n · maxi∈[n]pi),  S1 =  ∆ {  �  pi/wi | i ∈ [n]},  S2 =  ∆ {2−k} ∪ {2−kxi | i ∈ [n]},  S3 =  ∆ {2−(k+l)xi | i ∈ [n]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (4) says that the relative error of representing each  s ∈ S1 in fplo should be less than err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (5) says that  each s ∈ S2 should underflow to 0 when represented  in fplo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (6) says that each s ∈ S2 ∪ S3 should be  representable in fphi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Low-precision ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' A lower bound r ∈ [0, 1] on the  low-precision ratio is given as:  r =  ∆ max  �  0, 1 − 2W + 1  size(TS)  �  ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  So r decreases linearly as W increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We make three points on the above construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  First, each part of the knapsack problem (w, p, W) is  used in the following parts of the construction: wi is  used mainly in the size of the parameter tensor θi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' pi  in the input xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' and W in the lower bound r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Second, there exist k ∈ N and fphi, fplo ∈ FP that sat-  isfy Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (3)–(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' This can be shown as follows: first,  by taking sufficiently many exponent and mantissa bits  for fplo, we can make Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (4) satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' next, by taking a  sufficiently large k, we can make Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (5) satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' finally,  by taking sufficiently many exponent and mantissa bits  for fphi, we can make Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (6) satisfied (since  xi is representable in fplo and 2−(k+l) is a power of two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Third, some well-known models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', ShuffleNet-v2)  have a similar structure to M in that they apply the fol-  lowing operations as a subroutine: split a tensor into mul-  tiple tensors, apply some operators to each split tensor,  and combine the resulting tensors into a single tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Training with Mixed-Precision Floating-Point Assignments  ⫶  𝑥!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='∈ #  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='∑!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='∈ # ∑$∈ %!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' 𝑣#&!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=',$.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  2() 𝑣*#&+ − 𝑦  𝑣,,+  𝜃+,$ ⋅ 𝑣+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='$∈ %"  𝑣,,#  𝜃+,$ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='$∈ %"  𝜃#,$ ⋅ 𝑣# .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='$∈ %#  𝜃#,$ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='$∈ %#  𝑦  𝑣,∈ℝ#  𝑣+∈ℝ  𝑣#∈ℝ  𝜃#∈ℝ%#  𝜃+∈ℝ%"  𝑣*#∈ℝ%#  𝑣#&+∈ℝ%"  𝑣*#&+∈ℝ  𝑣*#&*∈ℝ  𝑦∈ℝ  ⫶  ⫶  split  conv  conv  sum  loss  ℳ  ℒ  Figure 8: The model network M and the loss network L used in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We now prove that the knapsack problem (w, p, W) can  be solved in polynomial time, if an oracle to the above  tradeoff problem is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Suppose that π ∈ Π(C) is an  optimal solution to the above tradeoff problem (given by  the oracle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Define an item selection α ∈ {0, 1}n for the  knapsack problem as:  αi =  ∆  �  1  if π(dθi) = π(dv n+i) = π(dv 2n+1) = fphi  0  otherwise  for each i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Note that α can be constructed from π in  linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Thus, it suffices to show that α is an optimal  solution to the knapsack problem (w, p, W), which is  equivalent to the following two claims:  Claim 1: We have �  i∈[n] αiwi ≤ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Claim 2: For any α′ ∈ {0, 1}n with �  i∈[n] α′  iwi ≤ W,  we have �  i∈[n] α′  ipi ≤ �  i∈[n] αipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  We now prove each claim as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Proof of Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' If α = (0, · · · , 0), then the claim clearly  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Suppose that α ̸= (0, · · · , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Then, we have  1 −  1 + 2 �  i∈[n] αiwi  size(TS)  ≥ ratiolo(π)  ≥ r ≥ 1 − 1 + 2W  size(TS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Here the first inequality uses α ̸= (0, · · · , 0) and the defini-  tion of α and M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' the second inequality uses the fact that π is  a valid solution to the above tradeoff problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' and the third  inequality uses the definition of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Hence, the claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Suppose that the claim does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Then, there exists α′ ∈ {0, 1}n such that  �  i∈[n]  α′  iwi ≤ W,  �  i∈[n]  α′  ipi >  �  i∈[n]  αipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Define a precision assignment π′ ∈ Π(C) as:  π′(dv 2n+1) =  ∆ fphi,  π′(dθi) =  ∆ π′(dv n+i) =  ∆ fphi  for all i ∈ [n] with α′  i = 1,  π′(t) =  ∆ fplo  for all other t ∈ TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Then, we have ratiolo(π′) ≥ r by �  i∈[n] α′  iwi ≤ W and  the definition of π′, M, and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Hence, it suffices to show  acc(π) < acc(π′), because this would contradict to the fact  that π is an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  To show acc(π) < acc(π′), we prove the following two  lemmas: the first lemma gives a closed form of acc(π) and  acc(π′), and the second lemma shows that �  i∈[n] βiwix2  i  is close to �  i∈[n] βipi (where the former summation  appears in acc(π) and acc(π′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The following hold:  acc(π) = 2−ky + 2−(2k+l) �  i∈[n]  αiwix2  i ,  acc(π′) = 2−ky + 2−(2k+l) �  i∈[n]  α′  iwix2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We prove the equation for acc(π) only, since the  equation for acc(π′) can be proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  First, we show that for all i ∈ [n] and j ∈ [wi],  ˆ  dθi,j = αi · 2−kxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  (7)  Pick any i ∈ [n] and j ∈ [wi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Note that by the definition  of M, we have  ˆ  dθi,j = rndπ(dθi)  �  rndπ(dv n+i)(rndπ(dv 2n+1)(2−k))  rndvi(rndv0(xi))  �  = rndπ(dθi)  �  rndπ(dv n+i)(rndπ(dv 2n+1)(2−k)) · xi  �  ,  where the second equality uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (3) and that xi is  representable in fplo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (7) by case analysis  on αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Suppose αi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Then, by the definition of αi,  Training with Mixed-Precision Floating-Point Assignments  π(dθi) = π(dv n+i) = π(dv 2n+1) = fphi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' From this, we  get the desired equation:  ˆ  dθi,j = rndhi  �  rndhi(rndhi(2−k)) · xi  �  = rndhi(2−k · xi) = 2−kxi,  where the last two equalities use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Suppose now  αi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Then, by the definition of αi, at least one of π(dθi),  π(dv n+i), and π(dv 2n+1) is fplo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' If π(dv n+i) = fplo or  π(dv 2n+1) = fplo, we get the desired equation:  ˆ  dθi,j = rndπ(dθi)  �  rndlo(2−k) · xi  �  = rndπ(dθi)(0 · xi) = 0,  where the first equation uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (6), and the  second equation uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The remaining case is when  π(dv n+i) = π(dv 2n+1) = fphi and π(dθi) = fplo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We get  the desired equation in this case as well:  ˆ  dθi,j = rndlo  �  rndhi(rndhi(2−k)) · xi  �  = rndlo(2−k · xi) = 0,  where the second equality uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (6), and the last equality  uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Hence, we have proved Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Next, let θi be the i-th parameter tensor before training  starts, and θ′  i be the corresponding tensor after training ends  (i ∈ [n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Then, by the definition of the tradeoff problem  constructed above, we have θi,j = 0 and  θ′  i,j = θi,j − rndhi(2−l · ˆ  dθi,j)  = 0 − rndhi(2−l · (αi · 2−kxi))  = αi · (−2−(k+l)xi),  where the second equality uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (7) and the third equality  uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Using this equation, we finally obtain the  conclusion of this lemma:  acc(π) = −L(fθ′(x), y)  = −2−k���y −  �  i∈[n]  �  j∈[wi]  θ′  i,jxi  ���  = −2−k���y −  �  i∈[n]  �  j∈[wi]  αi · (−2−(k+l)xi) · xi  ���  = −2−k���y +  �  i∈[n]  αi · 2−(k+l)wix2  i  ���  = 2−k�  y +  �  i∈[n]  αi · 2−(k+l)wix2  i  �  = 2−ky + 2−(2k+l) �  i∈[n]  αiwix2  i ,  where the first two equalities use the definition of accuracy,  and the second last equality uses the definition of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' This  concludes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  ■  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For any β ∈ {0, 1}n,  ���  �  i∈[n]  βiwix2  i −  �  i∈[n]  βipi  ��� < 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' We first show that for any i ∈ [n],  |wix2  i − pi| < 1  2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Pick any i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' (4) and the definition of xi, we have  ���xi −  � pi  wi  ��� <  � pi  wi 1  6n · maxj∈[n] pj  ≤  � pi  wi 1  6npi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  From this, we have  � pi  wi  �  1 −  1  6npi  �  < xi <  � pi  wi  �  1 +  1  6npi  �  ,  pi  wi  �  1 −  1  6npi  �2  < x2  i < pi  wi  �  1 +  1  6npi  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  From this, we obtain the desired result:  |wix2  i − pi| < pi  ��  1 +  1  6npi  �2  − 1  �  = pi  �  1  3npi  +  1  (6npi)2  �  < pi  �  1  3npi  +  1  6npi  �  = pi ·  1  2npi  = 1  2n,  where the second inequality uses 6npi > 1 (as n, pi ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Using this result, we can show the conclusion as follows:  ���  �  i∈[n]  βiwix2  i −  �  i∈[n]  βipi  ��� =  ���  �  i∈[n]  βi(wix2  i − pi)  ���  ≤  �  i∈[n]  |βi| · |wix2  i − pi|  <  �  i∈[n]  1  2n = 1  2,  where the last inequality uses |βi| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' This completes the  proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  ■  Using the two lemmas, we prove acc(π) < acc(π′) as fol-  lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' First, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2 and �  i∈[n] αipi < �  i∈[n] α′  ipi,  we have  �  i∈[n]  αiwix2  i <  �  i∈[n]  αipi + 1  2  ≤  �  i∈[n]  α′  ipi − 1  2 <  �  i∈[n]  α′  iwix2  i ,  where the second inequality comes from αi, α′  i ∈ {0, 1}  and pi ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' From this, and by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1, we obtain  acc(π) < acc(π′) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' This concludes the proof of  Claim 2, thereby finishing the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Training with Mixed-Precision Floating-Point Assignments  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2, we proved  the NP-hardness of Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 by making use of only a  few limited aspects of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For instance, we used  the fact that some values representable in a high-precision  format round to zero in a low-precision format;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' on the other  hand, many other values representable in a high-precision  format round to non-zero values in a low-precision format,  and this indeed occurs in practical training (even more  frequently than underflows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Also, we used a simple setting  for training in which a gradient descent algorithm is applied  for one epoch, training data consist of one input-output pair,  and test data is the same as training data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' on the other hand,  in practical training, a gradient descent algorithm is applied  for many epochs, training data consists of many input-output  pairs, and test data is different from training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 is general enough so that it embraces all the  aforementioned aspects of floating-points and training,  including those that are not considered in the proof of  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Since those aspects are likely to make the  problem even more difficult, we conjecture that the problem  would be more intractable than being NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  B  EXPERIMENTS: DEFERRED DETAILS  The datasets we use have the following licenses:  CIFAR-10 and CIFAR-100: These datasets are under  the MIT license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  ImageNet: This dataset can be used “only for non-  commercial research and educational purposes.” For  more details, see its webpage (Stanford Vision Lab,  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  The implementations of models we use have the following  licenses:  SqueezeNet for CIFAR-10 and CIFAR-100: We adapt  an implementation of the model in a public GitHub  repository (Pathak, 2020), whose license information  is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  ShuffleNet-v2,  MobileNet-v2,  and ResNet-18 for  CIFAR-10 and CIFAR-100: We adapt an implementation  of these models in a public GitHub repository (kuangliu,  2021), which is under the MIT license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  ShuffleNet-v2 for ImageNet and ImageNet-200-i: We  adapt an implementation of the model in the torchvision  library (PyTorch, 2022b), which is under the BSD  3-Clause license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  The details of how we train models are as follows:  Four models on CIFAR-10 and CIFAR-100: We train  the four models with a standard setup (kuangliu, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  In particular, we run the (non-Nesterov) SGD optimizer  for 200 epochs with minibatch size of 128 (over 1  GPU), learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1, momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='9, weight  decay of 5 × 10−4, and the cosine annealing scheduler  for learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For dynamic loss scaling, we use  initial scale of 216, growth factor of 2, back-off factor  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='5, and growth interval of 1 epoch, as suggested in  PyTorch (PyTorch, 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  ShuffleNet-v2 on ImageNet: We train the model with the  default setup given in PyTorch’s GitHub repository (Py-  Torch, 2022c), except that we use larger minibatch size  and learning rate as in (Kalamkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' PyTorch,  2022d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Krizhevsky, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2017) to reduce  the wall-clock time of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In particular, we run  the (non-Nesterov) SGD optimizer for 90 epochs with  minibatch size of 1024 (over 16 GPUs), learning rate  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='4, momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='9, weight decay of 10−4, and  the cosine annealing scheduler for learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' For  dynamic loss scale, we use initial scale of 216, growth  factor of 2, back-off factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='5, and growth interval  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='5 epoch, as suggested in PyTorch (PyTorch, 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  ShuffleNet-v2 on ImageNet-200-i: We train the model  with the same settings for ImageNet except that we use  the default values for minibatch size and learning rate  given in (PyTorch, 2022c), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', minibatch size of 256  (over 4 GPUs) and learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  C  EXPERIMENTS: DEFERRED RESULTS  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1  Comparison with Existing Precision Assignments  Figure 9 presents results omitted in Figure 4: training  results of smaller variant models (which have width  multiplier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='5 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1) on CIFAR-100 with πfp32, πunif,  πop, πop′, and πours,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The figure shows similar results  to Figure 4: the results for the variant models with width  multiplier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='5 (and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1) are similar to those for the original  models (and the variant models with width multiplier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Figures 10 and 11 show the average training trajectories  for the configurations presented in Figures 4 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='2 Ablation Study: Precision Demotion and Promotion  Figure 12 presents results omitted in Figure 5: training re-  sults of ResNet-18 on CIFAR-100 with πours,r, πours[inc],r,  and πours[rand],r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  The figure shows similar results to  Figure 5 except that it shows smaller differences in memory-  accuracy tradeoff between the three precision assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Figure 13 presents results omitted in Figure 6: training  results of four models on CIFAR-10 with πours,r and  πours[no-promo],r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  The figure shows similar results to  Figure 6 except that the training of ResNet-18 on CIFAR-10  does not diverge even with πours[no-promo],r for all r values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  Training with Mixed-Precision Floating-Point Assignments  (a) CIFAR-100, SqueezeNet‡  (b) CIFAR-100, SqueezeNet¶  (c) CIFAR-100, ShuffleNet-v2‡  (d) CIFAR-100, ShuffleNet-v2¶  (e) CIFAR-100, MobileNet-v2‡  (f) CIFAR-100, MobileNet-v2¶  (g) CIFAR-100, ResNet-18‡  (h) CIFAR-100, ResNet-18¶  Figure 9: Continued from Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Memory-accuracy tradeoffs of πunif (Micikevicius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2018), πop (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019),  πop′ (Kalamkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=', 2019), and πours,r for smaller variants of four models on CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' The variant models have width  multiplier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='5 (marked by ‡) or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='1 (marked by ¶).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Top-right points are better than bottom-left ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In all but one plots,  there are •s above and to the right of  and  , respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' even in the one plot (g), •s have almost the same tradeoffs to  and  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' In three of all plots, ⋆ has much smaller y-values than other points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ⋆ is missing in (h) as its y-value is too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  66  test accuracy (%)  64  62  fp32  60  op  op  unif  2  ours  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  low-prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ratio44  accuracy (%)  40  36  fp32  32  op  test a  op  unif  24  ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  low-prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ratio70  test accuracy (%)  68  66  fp32  64  op  op  unif  2  ours  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  low-prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ratio56  52  X  48  44  fp32  op  op  unif  24  ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  low-prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ratio72  test accuracy (%)  70  68  fp32  66  op  op  unif  2  ours  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content='00  low-prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ratio56  test accuracy (%)  52  48  fp32  44  op  op  unif  36  ours  32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content='00  low-prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ratio76  test accuracy (%)  74  72  fp32  op  70  op  unif  2  ours  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content='00  low-prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ratio60  test accuracy (%)  56  52  fp32  op  48  op  unif  40  +  ours  36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=' ratioTraining with Mixed-Precision Floating-Point Assignments  (a) CIFAR-10, SqueezeNet  (b) CIFAR-100, SqueezeNet  (c) CIFAR-100, SqueezeNet†  (d) CIFAR-10, ShuffleNet-v2  (e) CIFAR-100, ShuffleNet-v2  (f) CIFAR-100, ShuffleNet-v2†  (g) CIFAR-10, MobileNet-v2  (h) CIFAR-100, MobileNet-v2  (i) CIFAR-100, MobileNet-v2†  (j) CIFAR-10, ResNet-18  (k) CIFAR-100, ResNet-18  (l) CIFAR-100, ResNet-18†  Figure 10: Training trajectories for the configurations shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Each line shows the average training trajectory  for each precision assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' πours,r is colored from navy to yellow (darker for smaller r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content='90  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='test accuracy (  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=' MobileNet-v2‡  (f) CIFAR-100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' MobileNet-v2¶  (g) CIFAR-100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ResNet-18‡  (h) CIFAR-100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' ResNet-18¶  Figure 11: Training trajectories for the configurations shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Each line shows the average training trajectory  for each precision assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content="do  unif  ours  40  0  50  100  150  200  epoch70  test accuracy (%)  60  50  fp32  op  40  op'  unif  ours  30  0  50  100  150  200  epochTraining with Mixed-Precision Floating-Point Assignments  (a) ResNet-18  Figure 12: Continued from Figure 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Memory-accuracy tradeoffs of πours,r, πours[inc],r, and πours[rand],r for ResNet-18  on CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Observe that •s are above and to the right of other points in nearly all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  (a) CIFAR-10, SqueezeNet  (b) CIFAR-10, ShuffleNet-v2  (c) CIFAR-10, MobileNet-v2  (d) CIFAR-10, ResNet-18  Figure 13: Continued from Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Training four models on CIFAR-10 with πours,r and πours[no-promo],r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Column 1:  Training trajectories of πours[no-promo],r for different r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' colors denote r values (darker for smaller r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Column 2: Top-5  overflow ratios of tensors at each epoch, for the highlighted trajectory in (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' the largest ratio is blue and the fifth largest  red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Column 3: Memory-accuracy tradeoffs of πours,r and πours[no-promo],r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' Column 4: Low-precision ratio when training  ends vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content=' when training starts, for πours,r and πours[no-promo],r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
+page_content='  82  (%)  80  test accuracy (  78  76  ours[inc]  ours[rand]  ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfBDfd/content/2301.13464v1.pdf'}
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