Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit (1905.09883v2)

Abstract: In deep latent Gaussian models, the latent variable is generated by a time-inhomogeneous Markov chain, where at each time step we pass the current state through a parametric nonlinear map, such as a feedforward neural net, and add a small independent Gaussian perturbation. This work considers the diffusion limit of such models, where the number of layers tends to infinity, while the step size and the noise variance tend to zero. The limiting latent object is an It^o diffusion process that solves a stochastic differential equation (SDE) whose drift and diffusion coefficient are implemented by neural nets. We develop a variational inference framework for these \textit{neural SDEs} via stochastic automatic differentiation in Wiener space, where the variational approximations to the posterior are obtained by Girsanov (mean-shift) transformation of the standard Wiener process and the computation of gradients is based on the theory of stochastic flows. This permits the use of black-box SDE solvers and automatic differentiation for end-to-end inference. Experimental results with synthetic data are provided.

• The paper introduces a novel framework that transforms deep latent Gaussian models into continuous-time neural SDEs via the diffusion limit.
• It develops a tailored variational inference method employing stochastic automatic differentiation and a Girsanov transformation to approximate posterior distributions.
• Experiments on synthetic data reveal that while fine discretization improves convergence, gains become incremental beyond a certain mesh size, highlighting computational trade-offs.

Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit

The paper by Belinda Tzen and Maxim Raginsky investigates the interfacing of deep latent Gaussian models (DLGMs) with the framework of stochastic differential equations (SDEs), specifically focusing on the diffusion limit. This work extends the concept of neural ODEs, introduced by Chen et al., into the stochastic domain, offering a new perspective on continuous-time probabilistic generative models.

At the core of Tzen and Raginsky's exploration is the transformation of DLGMs into a continuous-time model. In a DLGM, a latent variable evolves through discrete time-steps via a Markov chain process, subject to nonlinear transformations and Gaussian perturbations. As the number of layers grows infinitely while step sizes and noise variances diminish, the model approaches a diffusion process governed by an Itô SDE. This manifests a compelling parallel with the neural ODE paradigm by embodying neural-network-based drift and diffusion coefficients, effectively framing these as neural SDEs.

A crucial contribution of this work is the development of a variational inference framework specially tailored for neural SDEs. The authors employ stochastic automatic differentiation on the Wiener space to achieve this. They utilize a Girsanov transformation to approximate the posterior distributions and compute gradients utilising stochastic flow theory. Such a method allows the integration of black-box SDE solvers and automatic differentiation techniques, facilitating end-to-end inference within these models.

The experimental validation, conducted on synthetic datasets, demonstrates the feasibility of the proposed framework. The experimental convergence is dependent on the discretization mesh size and sample size, reflecting the computational intricacy intrinsic to neural SDEs. Interestingly, it was observed that gains from exceedingly fine discretization steps remain incremental beyond a certain mesh size, suggesting that neural SDEs do not necessarily exhibit significant practical advantages over DLGMs in specific regimes.

The practical implications of this research are profound, particularly for systems where the stochastic nature is indispensable and cannot be effectively approximated by deterministic models. Neural SDEs introduce a methodologically novel avenue by embedding randomness directly into the neural architecture, expanding its utility in modeling continuous-time phenomena where uncertainty plays a critical role. This is prominently evident in financial modeling, meteorological predictions, and complex biological systems.

The theoretical implications are equally substantial, as the introduction of randomness into neural architectures via SDEs presents a unique paradigm for examining the expressive power of neural generative models. Tzen and Raginsky's analysis offers a deeper mechanistic understanding of how neural networks can be constructed to mimic continuous stochastic processes, thereby providing a theoretical foundation for future explorations of deeper and more complex probabilistic architectures.

This foundational work elicits several directions for future research. One potential development would be the exploration of adaptive SDE solvers within this framework to potentially streamline computational efficiency. Moreover, the intersection of neural SDEs with probabilistic numerical integration methods—where uncertainty can be directly quantified—presents promising prospects for evolution into more robust and informative modeling frameworks.

In conclusion, Tzen and Raginsky's examination of neural stochastic differential equations as deep latent Gaussian models in the diffusion limit represents a significant step in the convergence of stochastic processes and deep learning. By marrying the expressive capabilities of neural architectures with the mathematical robustness of SDEs, this framework not only augments theoretical understandings but carves a path for advancements in practical applications where uncertainty and continuous-time modeling need to be intrinsically captured.