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Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise
Xuanqing Liu, Tesi Xiao, Si Si, Qin Cao, Sanjiv Kumar, Cho-Jui Hsieh
TL;DR
Neural ODE lacks the stochastic regularization mechanisms commonly used in discrete networks, motivating a continuous stochastic alternative. The paper introduces Neural SDE, analyzes its stability and efficient training, and reports better generalization and robustness than Neural ODE. The paper also notes assumptions underlying its stability analysis.
Problem
Neural ODE is deterministic and therefore does not directly incorporate regularization mechanisms such as dropout and Gaussian noise used in discrete networks.
Method
Neural SDE injects stochastic noise into Neural ODE dynamics through SDEs, supports multiple noise types, and uses an efficient adjoint-style backpropagation method.
Results
Neural SDE achieves better generalization and robustness than Neural ODE, including 84.55% versus 81.63% test accuracy on CIFAR-10 and better performance in 8 out of 10 non-adversarial corruption cases.
Takeaways & Limitations
Stochastic noise can stabilize Neural ODE predictions while improving robustness to adversarial and non-adversarial noise.
Takeaways & Limitations
The stability analysis assumes measurable drift and diffusion functions and an implicit relationship between the Brownian motions used in the analysis.
Abstract
from arXiv · showhide
Neural Ordinary Differential Equation (Neural ODE) has been proposed as a continuous approximation to the ResNet architecture. Some commonly used regularization mechanisms in discrete neural networks (e.g. dropout, Gaussian noise) are missing in current Neural ODE networks. In this paper, we propose a new continuous neural network framework called Neural Stochastic Differential Equation (Neural SDE) network, which naturally incorporates various commonly used regularization mechanisms based on random noise injection. Our framework can model various types of noise injection frequently used in discrete networks for regularization purpose, such as dropout and additive/multiplicative noise in each block. We provide theoretical analysis explaining the improved robustness of Neural SDE models against input perturbations/adversarial attacks. Furthermore, we demonstrate that the Neural SDE network can achieve better generalization than the Neural ODE and is more resistant to adversarial and non-adversarial input perturbations.
1 Introduction
Neural ODE provides an efficient continuous approximation to ResNet but lacks stochastic regularization mechanisms used in discrete networks. Neural SDE addresses this gap by injecting noise into continuous dynamics to improve generalization and robustness.
- Neural ODE: Neural ODE models hidden-state evolution continuously and offers memory efficiency, parameter efficiency, and explicit control of numerical error.A standard ODE solver computes hidden and final states from the network input.
- Motivation: Neural ODE lacks dropout and other stochastic regularization mechanisms because it is a deterministic system.These mechanisms are commonly used to reduce generalization errors and improve robustness to adversarial attacks.
- Neural SDE: Neural SDE incorporates stochastic noise injection into continuous neural networks through stochastic differential equations.The framework can model regularization mechanisms such as Bernoulli dropout and Gaussian noise.
- Results: 84.55% test accuracy was achieved by Neural SDE versus 81.63% for Neural ODE on CIFAR-10 with other factors unchanged.The comparison is reported as evidence of improved generalization.
- Training: The framework develops an efficient scalable backpropagation method for training Neural SDE networks.The method is rooted in stochastic control theory.
2 Related work
Related work connects continuous neural networks to ResNet and numerical ODE solvers, while surveying noise-injection methods used to regularize discrete networks and improve robustness.
- Neural ODE: Neural ODE directly models hidden-state dynamics with an ODE solver rather than using independent weights for each layer.Earlier work linked deep architectures with discretized numerical ODE solvers and established parameter-efficient designs.
- Noise injection: Dropout randomly disables neurons and can be viewed as multiplying hidden states by Bernoulli random variables.The method is used to avoid overfitting.
- Noise injection: Stochastic depth, Shake-Shake, and DropBlock inject different structured forms of randomness into residual or convolutional networks.They respectively drop residual blocks, switch between residual blocks, or remove continuous regions of hidden states.
- Robustness: Random self-ensemble adds Gaussian noise to hidden states during both training and testing to improve robustness to perturbations or adversarial attacks.This differs from regularizers whose noise is used only during training and whose networks are fixed at test time.
3 Neural Stochastic Differential Equation
Neural SDE extends continuous neural networks with stochastic noise injection, providing a framework for modeling dropout and Gaussian noise while analyzing robustness to input perturbations. The framework also introduces memory-efficient gradient computation and theoretical conditions under which noise stabilizes perturbations.
- 3.1 Modeling randomness in neural networks: Neural SDE adds a diffusion term to continuous neural networks, enabling stochastic noise injection through a general SDE formulation.The framework can represent multiple randomness-injection mechanisms through the transformation G(h_t, t; v).
- 3.1 Modeling randomness in neural networks: The framework models additive and multiplicative Gaussian noise, with Gaussian noise after residual blocks serving as a continuous regularization analogue.The noise variance is controlled by the diagonal matrix Σ(t), while discrete residual blocks inject analogous Gaussian perturbations.
- 3.1 Modeling randomness in neural networks: Neural SDE also incorporates dropout, shake-shake regularization, and stochastic depth as stochastic components of continuous networks.Dropout is represented through a Bernoulli-based construction, and the framework identifies other stochastic layers with similar regularization roles.
- 3.2 Back-propagating through SDE integral: An adjoint-like backpropagation method computes Neural SDE gradients iteratively with O(1) memory instead of O(N) memory for autograd through an N-step solver.The method jointly solves the gradient dynamics and original SDE without storing intermediate states.
- 3.3 Robustness of Neural SDE: The zero perturbation solution exists, but nonzero perturbations can become arbitrarily large unless the diffusion term is chosen to constrain their evolution.The stability analysis extends Lyapunov-style reasoning to multidimensional stochastic systems through the drift and diffusion dynamics.
4 Experimental Results
Experiments evaluate Neural SDE variants for generalization and robustness under non-adversarial and adversarial perturbations. Across these settings, Neural SDE improves on Neural ODE, while dropout-style noise suppresses hidden-state perturbation accumulation.
- Experimental setup: Neural SDE models include multiplicative and dropout noise, with diffusion choices defined as alternatives to the Neural ODE baseline.The experiments also consider additive noise, while Neural ODE sets the diffusion term to zero.
- Generalization Performance: Neural SDE consistently outperforms ODE across CIFAR-10, STL-10, and Tiny-ImageNet2 in the generalization experiment.Keeping testing-time noise and ensembling forward outputs is reported to produce further improvements.
- Improved non-adversarial robustness: Neural SDE performs better than Neural ODE in 8 out of 10 non-adversarial corruption cases, while the remaining two cases are very close.Both models are trained only on clean data, without seeing or augmenting with the test corruptions.
- Improved adversarial robustness: Both Neural SDE variants are more resistant to adversarial attack than Neural ODE, and dropout noise outperforms multiplicative noise.The comparison uses multi-step ℓ∞-PGD attacks, with results shown across CIFAR-10, STL-10, and Tiny-ImageNet.
- Visualizing the perturbations of hidden states: With dropout-style diffusion, hidden-state perturbation error accumulates much slower than in the ordinary Neural ODE model.The experiment tracks εt over time after a PGD-perturbed input is propagated through the network.
5 Conclusion
The paper introduces Neural SDE to stabilize Neural ODE predictions through stochastic noise. It reports better generalization and improved robustness to adversarial and non-adversarial noise.
- Neural SDE stabilizes Neural ODE predictions by injecting stochastic noise.
- Neural SDE achieves better generalization and improves robustness to adversarial and non-adversarial noises.
A Proofs
The appendix establishes existence, uniqueness, and non-vanishing properties for the perturbation SDE under stated assumptions. These results support the subsequent stability analysis.
- Under the stated assumptions, the drift and diffusion satisfy conditions that guarantee a unique solution of the perturbation SDE.
- When the initial perturbation is nonzero, it remains nonzero for all t ≥ 0 almost surely.
- The proof bounds the perturbation using a stopping time that keeps its norm within a prescribed interval before time T.
- The probability of the perturbation reaching zero is bounded by δ∥ε0∥^-1e^{Kθ(1+Kθ)T}, which tends to zero as δ → 0.
Proof of Theorem 3.2
The proof of Theorem 3.2 applies Itô calculus and martingale bounds to derive the perturbation stability result. It handles the zero and nonzero initial-perturbation cases separately.
- For ε0 = 0, the perturbation remains identically zero, while the nonzero case is analyzed using the preceding lemma.
- Applying Itô’s formula to V(εt,t) and bounding its martingale term yields a logarithmic stability inequality.
- The exponential martingale inequality and Borel–Cantelli lemma convert the stochastic bound into an almost-sure long-time estimate.
Proof of Corollary 3.2.1
The proof applies a stability theorem to the stochastic differential equation under Lipschitz and structural assumptions. A Lyapunov-function bound supplies the theorem’s constants and completes the argument.
- The drift f(h_t, t; w) is L-Lipschitz in h_t, while the diffusion is G(h_t, t; v) = σh_t with m = 1.
- Under these conditions, equation (16) has a unique solution, with f∆ and G∆ satisfying the stated assumptions.
- The Lyapunov derivative satisfies LV(ε, t) ≤ (2L + σ2)V(ε, t), establishing the required growth bound.
- The theorem is applied with c1 = 1, p = 2, c2 = 2L + σ2, and c3 = 4σ2 to complete the proof.