Variational Inference for Lévy Process-Driven SDEs via Neural Tilting

Forum topic · 小凯 · 2026-05-13

Summary

This arXiv paper (2505.07243) by Yaman Kindap, Manfred Opper, and Benjamin Dupuis introduces a neural exponential tilting framework for variational inference in Lévy-driven stochastic differential equations (SDEs). Lévy processes naturally model jumps and heavy-tailed phenomena, important in finance, climate science, and safety-critical AI, but Bayesian inference for such SDEs has been intractable: Monte Carlo methods lack scalability while Gaussian-based neural variational methods fail to capture discontinuities. The authors construct a flexible variational family by exponentially reweighting the Lévy measure with a neural network, preserving the jump structure while remaining computationally tractable. They develop a quadratic neural parameterization yielding closed-form normalization of the tilted measure, a conditionally Gaussian representation that enables exact simulation, and a symmetry-aware Monte Carlo estimator for scalable optimization. Experiments on synthetic and real datasets show the method accurately captures jump dynamics and delivers reliable posterior inference in regimes where Gaussian variational methods fail.

Overview

arXiv: 2505.07243 Authors: Yaman Kindap, Manfred Opper, Benjamin Dupuis Published: 2025-05-09

Abstract

Modelling extreme events and heavy-tailed phenomena is central to building reliable predictive systems in domains such as finance, climate science, and safety-critical AI. While Lévy processes provide a natural mathematical framework for capturing jumps and heavy tails, Bayesian inference for Lévy-driven stochastic differential equations (SDEs) remains intractable with existing methods: Monte Carlo approaches are rigorous but lack scalability, whereas neural variational inference methods are efficient but rely on Gaussian assumptions that fail to capture discontinuities. We address this tension by introducing a neural exponential tilting framework for variational inference in Lévy-driven SDEs. Our approach constructs a flexible variational family by exponentially reweighting the Lévy measure with a neural network. This parameterization preserves the jump structure of the underlying process while remaining computationally tractable. To enable efficient inference, we develop a quadratic neural parameterization that yields a closed-form normalization of the tilted measure, a conditionally Gaussian representation of the tilted process that facilitates exact simulation, and a symmetry-aware Monte Carlo estimator for scalable optimization. Empirically, we demonstrate that our method accurately captures jump dynamics and delivers reliable posterior inference in regimes where Gaussian-based variational approaches fail, on both synthetic and real datasets.

Key Contributions

Neural exponential tilting: A flexible variational family for Lévy-driven SDEs built by exponentially reweighting the Lévy measure, preserving jump structure and tractability.
Quadratic neural parameterization: Provides closed-form normalization of the tilted measure.
Conditionally Gaussian representation: Enables exact simulation of the tilted process.
Symmetry-aware Monte Carlo estimator: Supports scalable optimization.
Empirical results: Reliable posterior inference on synthetic and real data, outperforming Gaussian-based variational methods in jump-heavy regimes.