Scalable Gradients for Stochastic Differential Equations (2001.01328v6)

Abstract: The adjoint sensitivity method scalably computes gradients of solutions to ordinary differential equations. We generalize this method to stochastic differential equations, allowing time-efficient and constant-memory computation of gradients with high-order adaptive solvers. Specifically, we derive a stochastic differential equation whose solution is the gradient, a memory-efficient algorithm for caching noise, and conditions under which numerical solutions converge. In addition, we combine our method with gradient-based stochastic variational inference for latent stochastic differential equations. We use our method to fit stochastic dynamics defined by neural networks, achieving competitive performance on a 50-dimensional motion capture dataset.

• The paper generalizes the adjoint sensitivity method from ODEs to SDEs using a backward Stratonovich formulation, enabling scalable gradient computation.
• The proposed algorithm leverages Brownian bridge sampling and splittable pseudorandom generators to reduce memory usage and time complexity.
• The approach enhances numerical stability and integrates with gradient-based variational inference, achieving competitive performance on high-dimensional datasets.

Scalable Gradients for Stochastic Differential Equations: An Expert Overview

This paper presents a significant advancement in the efficient computation of gradients for stochastic differential equations (SDEs), extending the adjoint sensitivity method traditionally used for ordinary differential equations (ODEs). By leveraging the adjoint sensitivity framework, the authors introduce a stochastic adjoint sensitivity method, facilitating scalable and memory-efficient computation of gradients in SDEs. The proposed method's time complexity is shown to improve upon existing approaches, with the potential to significantly benefit stochastic optimal control, machine learning, and other domains where SDEs are prevalent.

Key Contributions

• Generalization of the Adjoint Sensitivity Method: The paper extends the adjoint sensitivity method to cover SDE contexts, a non-trivial challenge due to the intrinsic stochasticity and continuous nature of these processes. The work derives a backward Stratonovich stochastic differential equation whose solution yields the desired gradients.
• Efficient Algorithm Design: The authors present a novel algorithm that circumvents the need to store noise or intermediate states during forward simulation, utilizing a Brownian bridge sampling technique and splittable pseudorandom number generators for efficient noise querying during the backward pass.
• Numerical Stability and Efficiency: The method accommodates high-order adaptive solvers, potentially enhancing numerical stability and enabling more complex SDEs to be handled with modest computational resources.
• Integration with Variational Inference: The stochastic adjoint approach is integrated with gradient-based stochastic variational inference, allowing it to fit first-order SDEs defined by neural networks. The method shows competitive performance in modeling complex systems, exemplified by applications to a high-dimensional motion capture dataset.

Numerical Results and Claims

The numerical experiments presented demonstrate the method's utility in fitting high-dimensional SDEs effectively. For example, the approach achieved competitive results on a 50-dimensional motion capture dataset, highlighting its potential for applications in dynamical systems with numerous interacting variables. The authors provide an asymptotic complexity analysis (see Table 1 in the original paper), indicating the scalability benefits of their method compared to pathwise approaches and backpropagation through solver operations, characterized by constant memory usage and logarithmic time complexity with respect to the number of solver steps.

Theoretical and Practical Implications

Theoretical implications of this work include a new understanding of backward stochastic calculus and the applicability of the adjoint method to SDEs. Practically, this work could transform how complex stochastic systems are modeled, especially in fields requiring real-time simulation and optimization under uncertainty, such as finance, biology, and autonomous systems.

Future Directions

Future research inspired by this work could focus on:

• Extending the approach to cover non-diagonal diffusion matrix scenarios more efficiently.
• Exploring the integration of control variates or antithetic paths to reduce the variance of gradient estimates further.
• Detailed theoretical analysis under the framework of rough paths, offering potentially stronger convergence and error bounds.

This paper offers the research community a robust method for efficiently solving SDEs, enabling scalable machine learning models where differential equations with stochastic components play a critical role. As such, it forms a foundation upon which further advancements in scalable stochastic optimization could be developed, influencing both theoretical explorations and practical computational tools.