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Stochasticity in Neural ODEs: An Empirical Study
Viktor Oganesyan, Alexandra Volokhova, Dmitry Vetrov
TL;DR
Continuous-time neural models have potential benefits but lack empirical design guidance, and neural ODEs typically use deterministic dynamics. This study compares neural ODEs and SDEs on image-classification tasks; stochasticity helps without augmentation, but augmentation makes their performance similar.
Problem
Continuous-time neural models offer theoretical benefits but lack empirical analysis and design guidelines for their development and practical use.
Source: §1 · p. 1Method
Results
Without augmentation, neural SDEs generalize better than neural ODEs; with augmentation, they perform similarly, and augmented neural ODEs outperform unaugmented neural SDEs.
Source: §3.2 · p. 4Takeaways & Limitations
For image classification, stochasticity alone is not a sufficiently powerful neural ODE regularizer, though neural SDEs may help where data augmentation is hard to use.
Source: §4 · p. 5
Abstract
from arXiv · showhide
Stochastic regularization of neural networks (e.g. dropout) is a wide-spread technique in deep learning that allows for better generalization. Despite its success, continuous-time models, such as neural ordinary differential equation (ODE), usually rely on a completely deterministic feed-forward operation. This work provides an empirical study of stochastically regularized neural ODE on several image-classification tasks (CIFAR-10, CIFAR-100, TinyImageNet). Building upon the formalism of stochastic differential equations (SDEs), we demonstrate that neural SDE is able to outperform its deterministic counterpart. Further, we show that data augmentation during the training improves the performance of both deterministic and stochastic versions of the same model. However, the improvements obtained by the data augmentation completely eliminate the empirical gains of the stochastic regularization, making the difference in the performance of neural ODE and neural SDE negligible.
1 Introduction
The introduction frames stochastic regularization as a possible way to improve neural ODE generalization and reports that data augmentation can outweigh its gains. It also identifies limited empirical analysis and design guidance for continuous-time models as a research gap.
1 Introduction
- Neural networks can overfit because of their expressive power, motivating regularization methods such as l2-regularization, dropout, early stopping, and batch normalization.The introduction cites [3] for these techniques.Source: §1 · p. 1
- Continuous-time models lack empirical analysis and design guidelines despite theoretical benefits, hindering their development and practical application.The introduction notes potential parameter-efficiency and adaptive computational time for supervised learning.Source: §1 · p. 1
- The paper studies stochastic regularization by extending neural ODEs to neural SDEs, where noisy trajectories are intended to encourage neighboring inputs to produce similar outputs.The authors describe this as a way to reduce divergence and support robustness to small perturbations.Source: §1 · p. 1
- Across three image-classification tasks—CIFAR-10, CIFAR-100, and TinyImageNet —common data augmentation allows neural ODEs to achieve better generalization than introducing stochasticity, which can regularize them.Source: §1 · p. 2
2 Considered models
The paper presents ResNet, neural ODE, and neural SDE models as differential-equation integration systems. Their main distinction is the integration block, with stochastic noise added for the SDE.
2 Considered models
- A residual block adds a neural-network function of its input, corresponding to one Euler step of numerical integration.The neural network has parameters θ; zin and zout denote the residual block's input and output.Source: §2 · p. 2
- Neural ODEs extend the residual-block idea by allowing any numerical integration scheme over the integration interval.The interval is bounded by tbegin and tend, and an ODE solver performs the integration.Source: §2 · p. 2
- Neural SDEs add a stochastic term to the differential equation, with dW as a vector Wiener process and σ as the scalar noise magnitude.An SDE numerical method performs the integration.Source: §2 · p. 2
- All three model types use sequences of integration blocks separated by down-sampling blocks, with otherwise similar architectures and dynamic functions.The integration blocks implement the residual, ODE, or SDE mappings.Source: §2 · p. 2
3 Experiments
The experiments compare deterministic and stochastic continuous models across a toy task and three image-classification tasks. Stochasticity improves results without augmentation, but augmentation makes neural SDE and neural ODE performance similar.
3 Experiments
- Experiments compare residual, ODE, and SDE models on a toy task and three image datasets, including stochastic training, trajectory averaging, and deterministic test-time evaluation.The SDE noise parameter σ is grid-searched, and training uses backpropagation rather than the adjoint method because of numerical instability.Source: §3 · p. 3, §3.2 · p. 3
3.1 Toy dataset
- On the toy task, SDENet reaches an almost optimal result, whereas ResNet and ODENet achieve nearly the same accuracy.The table reports test accuracy over five runs and identifies the theoretical optimum.Source: §3.1 · p. 3, Table 1 · p. 3
3.2 CIFAR-10, CIFAR-100, and Tiny ImageNet
- ResNet and ODENet perform similarly, with occasional ResNet superiority; the authors conclude precise integration does not increase model expressiveness [17].The comparison contrasts ResNet's Euler integration with ODENet's fourth-order Runge-Kutta integration.Source: Figure 2 · p. 3, §3.2 · p. 3
- Without augmentation, SDENet consistently beats ODENet; with augmentation they perform similarly, while augmented ODENet outperforms unaugmented SDENet.Figure 3 distinguishes augmentation settings and test-time modes; the paper reports that stochasticity adds no material regularization when training uses augmentation.Source: Figure 3 · p. 4, §3.2 · p. 4
- Averaging stochastic trajectories improves predictions, while deterministic test-time evaluation mostly matches averaging but loses quality on augmented TinyImageNet.The authors attribute the exception to bias from mapping the expected trajectory through a nonlinear prediction function and recommend caution with deterministic evaluation.Source: §3.2 · p. 4, Figure 3 · p. 4
- ODENet with batch normalization occasionally performs significantly worse than ODENet without it, a result the authors say merits further investigation.Figure 4 compares accuracy differences across three tasks.Source: Figure 4 · p. 4, §3.2 · p. 4
4 Conclusion
The conclusion finds that neural SDE stochasticity improves generalization without augmentation, but adds no quality when data augmentation is used. It also reports that ResNet can match or outperform continuous models on image classification, with less computation.
4 Conclusion
- Neural SDE stochasticity improves generalization without augmentation, but does not increase quality when training uses data augmentation.The authors conclude stochasticity is not a powerful enough regularizer for neural ODEs in image classification; they suggest neural SDEs may help on tasks where augmentation is hard to handle.Source: §4 · p. 5
- ResNet occasionally outperforms neural ODEs on image classification, while more precise integration does not increase expressiveness and requires more computation.The authors conclude residual networks manage these image-classification tasks better and more efficiently than continuous models.Source: §4 · p. 5
Appendix
The appendix provides experiment details intended to support reproduction of the CIFAR-10, CIFAR-100, and TinyImageNet results.
Appendix
- The appendix provides details of experiments on CIFAR-10, CIFAR-100, and TinyImageNet for possible reproduction of the results.Source: Appendix · p. 7
Detailed results of the main experiments
The section presents test accuracies for the considered models across three classification tasks and augmentation conditions; Figure 5 visualizes Table 2 values using means and standard deviations.
Detailed results of the main experiments
- Table 2 reports test accuracies of the considered models on three classification tasks with and without data augmentation.Source: Table 2 · p. 7, Detailed results of the main experiments · p. 7
- Figure 5 represents Table 2 values with mean line segments and half-transparent rectangles for standard deviations, omitting outlying values.Source: Figure 5 · p. 7
Hyperparameters and training procedure
Hyperparameters were selected using validation accuracy, after which the divided data were joined for final training. Optimizer settings varied by experiment, and TinyImageNet's validation set served as the final evaluation set.
Hyperparameters and training procedure
- The authors selected σ, learning rate, and batch size by validation accuracy, then joined the train and validation parts to train the final model.Source: Hyperparameters and training procedure · p. 7
- Training used SGD with Nesterov momentum 0.9 and a ReduceLROnPlateau scheduler; learning rate, warmup steps, and weight decay varied by experiment.Tables 3 and 4 list optimizer parameters for experiments without and with augmentation, respectively.Source: Hyperparameters and training procedure · pp. 7–8, Table 3 · p. 8, Table 4 · p. 8
- Table 5 lists other hyperparameters, and its number of steps denotes the number of numerical-integration steps.Source: Table 5 · p. 8
- TinyImageNet models were trained on its train set and evaluated on its labeled validation set because its test set has no labels.Source: Hyperparameters and training procedure · p. 8
Architectures
The architectures are designed to make residual and continuous models comparable while varying their integration blocks. The section also specifies implementation choices for stochastic dynamics, normalization, and time dependence.
Architectures
- The models are designed for adequate test accuracy and 100% training accuracy, enabling study of regularizers in models able to overfit.Source: Architectures · p. 8
- ResNet differs from ODENet and SDENet only in the integration block: ResBlock versus ODEBlock.Figures 6 and 7 depict the model architectures; × n marks a block repeated n times.Source: Architectures · p. 8, Figure 7 · p. 9, Figure 6 · p. 9
- SDENet and ODENet share an architecture, but SDENet adds the stochastic term in equation 3 and uses a different solver.The passage states that these models differ also in their neural differential equation solvers.Source: Architectures · p. 8
- Batch normalization is omitted from f in ODENet and SDENet, while ODENet+BN adds it to f as in ResBlock.Source: Architectures · p. 8
- The dynamic function f depends on time t, implemented by adding a time-valued channel to z(t).Accordingly, ODEBlock convolution layers use input_channels = output_channels + 1.Source: Architectures · p. 8