On the numerical reliability of nonsmooth autodiff: a MaxPool case study

TL;DR

It is suggested that nonsmooth MaxPool Jacobians with lower norms help maintain stable and efficient test accuracy, whereas those with higher norms can result in instability and decreased performance.

Abstract

This paper considers the reliability of automatic differentiation (AD) for neural networks involving the nonsmooth MaxPool operation. We investigate the behavior of AD across different precision levels (16, 32, 64 bits) and convolutional architectures (LeNet, VGG, and ResNet) on various datasets (MNIST, CIFAR10, SVHN, and ImageNet). Although AD can be incorrect, recent research has shown that it coincides with the derivative almost everywhere, even in the presence of nonsmooth operations (such as MaxPool and ReLU). On the other hand, in practice, AD operates with floating-point numbers (not real numbers), and there is, therefore, a need to explore subsets on which AD can be numerically incorrect. These subsets include a bifurcation zone (where AD is incorrect over reals) and a compensation zone (where AD is incorrect over floating-point numbers but correct over reals). Using SGD for the training process, we study the impact of different choices of the nonsmooth Jacobian for the MaxPool function on the precision of 16 and 32 bits. These findings suggest that nonsmooth MaxPool Jacobians with lower norms help maintain stable and efficient test accuracy, whereas those with higher norms can result in instability and decreased performance. We also observe that the influence of MaxPool's nonsmooth Jacobians on learning can be reduced by using batch normalization, Adam-like optimizers, or increasing the precision level.

Figures (14)

• Figure 1: Histogram of $\mathrm{backprop}$ variation $D_{m,q}$ for LeNet-5 on MNIST (128 mini-batch size) at 32-bit precision, comparing $P$ with $\tilde{P}$ and $P$ with $Q$ over $M = 1000$ experiments.
• Figure 2: Histogram of $\mathrm{backprop}$ variation under nondeterministic GPU operations, where $f$ is a LeNet-5 network on MNIST with batch size 128 for $M =1000$ experiments.
• Figure 3: Histogram of $\mathrm{backprop}$ variation with $\mathrm{ReLU}$-derived programs, where $f$ is a LeNet-5 network on MNIST with batch size 128 for $M = 1000$ experiments.
• Figure 4: Impact of different size parameters on the proportion of affected mini-batches (see Equation (\ref{['eq:proportionbatch']}) using CIFAR10 dataset. First: Different VGG network sizes. Second: VGG11 with varying mini-batch sizes. Third: VGG11 with and without batch normalization.
• Figure 5: Training a VGG network on CIFAR10 with SGD. We performed ten random initializations for each experiment, depicted by the boxplots and the filled contours (standard deviation).

Theorems & Definitions (16)

• Definition 1: Calculus model, programs and nonsmooth AD
• Remark 1
• Example 1
• Definition 2: Backprop set
• Remark 2
• Remark 3
• Definition 3: Compensation, bifurcation and regular zones
• Proposition 1
• Remark 4: Backprop returns a gradient a.e.
• Definition 4: Clarke Jacobian of matrix’s maximum function