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61:, a coarse integral guess can be made and refined over many iterations until the error in precision makes it such that the smallest addition or subtraction to the guess is still too coarse to be an acceptable answer. When this happens, the precision can be increased to something more precise, which allows for smaller increments to be used for the approximation.
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Abdelfattah, Ahmad; Anzt, Hartwig; Boman, Erik G.; Carson, Erin; Cojean, Terry; Dongarra, Jack; Gates, Mark; Grützmacher, Thomas; Higham, Nicholas J.; Li, Sherry; Lindquist, Neil; Liu, Yang; Loe, Jennifer; Luszczek, Piotr; Nayak, Pratik; Pranesh, Sri; Rajamanickam, Siva; Ribizel, Tobias; Smith,
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is the scaling factor. Within the optimizer update, the scaled gradient is cast to a higher precision before it is scaled down (no longer underflowing, as it is in a higher precision) to update the weights.
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Micikevicius, Paulius; Narang, Sharan; Alben, Jonah; Diamos, Gregory; Elsen, Erich; Garcia, David; Ginsburg, Boris; Houston, Michael; Kuchaiev, Oleksii (2018-02-15). "Mixed
Precision Training".
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algorithms can use coarse and efficient half-precision floats for certain tasks, but can be more accurate if they use more precise but slower single-precision floats. Some platforms, including
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A common usage of mixed-precision arithmetic is for operating on inaccurate numbers with a small width and expanding them to a larger, more accurate representation. For example, two
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A floating-point number is typically packed into a single bit-string, as the sign bit, the exponent field, and the significand or mantissa, from left to right. As an example, a
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update. This is done to prevent the gradients from underflowing to zero when using low-precision data types like FP16. Mathematically, if the unscaled gradient is
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can often be performed in FP16 without loss of accuracy, even if the master copy weights are stored in FP32. Low-precision weights are used during forward pass.
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CPUs and GPUs, provide mixed-precision arithmetic for this purpose, using coarse floats when possible, but expanding them to higher precision when necessary.
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to automatically adjust the scale factor for loss scaling. That is, it periodically increase the scale factor. Whenever the gradients contain a
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means automatically converting a floating-point number between different precisions, such as from FP32 to FP16, during training. For example,
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582:{\displaystyle {\frac {\partial (k{\mathcal {L}})}{\partial \mathbf {w} }}=k{\frac {\partial {\mathcal {L}}}{\partial \mathbf {w} }}}
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utilize mixed-precision arithmetic to be more efficient with regards to memory and processing time, as well as power consumption.
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implements automatic mixed-precision (AMP), which performs autocasting, gradient scaling, and loss scaling.
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Barry; Swirydowicz, Kasia; Thomas, Stephen; Tomov, Stanimire; Tsai, Yaohung M.; Yamazaki, Ichitaro;
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is the scaling factor. Since gradient scaling and loss scaling are mathematically equivalent by
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means multiplying the loss function by a constant factor during training, typically before
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601:(indicating overflow), the weight update is skipped, and the scale factor is decreased.
39:(16-bit) floating-point numbers may be multiplied together to result in a more accurate
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57:) are good candidates for mixed-precision arithmetic. In an iterative algorithm like
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standard 32-bit float ("FP32", "float32", or "binary32") is packed as follows:
657:(2020). "A Survey of Numerical Methods Utilizing Mixed Precision Arithmetic".
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775:"What Every User Should Know About Mixed Precision Training in PyTorch"
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43:(32-bit) float. In this way, mixed-precision arithmetic approximates
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624:"Difference Between Single-, Double-, Multi-, Mixed-Precision"
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that uses numbers with varying widths in a single operation.
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by a constant factor during training, typically before the
680:"The US again has the world's most powerful supercomputer"
589:, loss scaling is an implementation of gradient scaling.
751:"Mixed Precision — PyTorch Training Performance Guide"
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292:Mixed-precision arithmetic is used in the field of
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726:"Mixed-Precision Training of Deep Neural Networks"
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478:{\displaystyle k{\mathcal {L}}}
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451:{\displaystyle {\mathcal {L}}}
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397:{\displaystyle s\mathbf {g} }
372:{\displaystyle \mathbf {g} }
329:The weights are stored in a
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678:Holt, Kris (8 June 2018).
17:Mixed-precision arithmetic
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21:floating-point arithmetic
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630:. 15 November 2019
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732:. 2017-10-11
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133:Significand
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634:30 December
628:NVIDIA Blog
337:Autocasting
331:master copy
59:square root
799:Categories
784:2024-09-10
760:2024-09-10
736:2024-09-10
711:1710.03740
664:2007.06674
605:References
119:Number of
115:precision
569:∂
557:∂
537:∂
516:∂
351:gradients
130:Exponent
107:Exponent
684:Engadget
296:, since
81:IEEE 754
67:such as
37:bfloat16
27:Overview
779:PyTorch
689:20 July
323:PyTorch
485:where
404:where
308:, and
302:Nvidia
281:~34.0
275:16383
253:~19.2
247:16383
225:~15.9
203:Double
175:Single
136:Total
69:Summit
53:(like
706:arXiv
659:arXiv
306:Intel
219:1023
197:~7.2
169:~3.3
127:Sign
113:Bits
109:bias
102:Bits
99:Type
691:2018
636:2020
278:113
272:128
269:112
259:Quad
191:127
142:Half
599:NaN
310:AMD
266:15
250:64
244:80
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222:53
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35:or
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