Equivalence Between Continuous-Time Risk-Sensitive Control and Rényi Divergence Minimization
Abstract
In this study, we show that a continuous-time risk-sensitive control problem is equivalent to a Rényi divergence minimization problem over trajectory path measures. Reformulating stochastic optimal control as probabilistic inference via Kullback-Leibler (KL) divergence minimization avoids the computational intractability of the Hamilton-Jacobi-Bellman equation. However, standard KL control is inherently risk-neutral, and recent minimax extensions remain restricted to risk-averse settings. Our equivalence result resolves this limitation by offering a unified probabilistic framework for arbitrary risk attitudes in continuous-time nonlinear systems. Based on Girsanov theorem, we explicitly map the risk sensitivity to the Rényi divergence order, deriving a noise-dependent control penalty scaled by risk preference. This formulation seamlessly modulates tail-weighting behaviors, interpolating between zero-forcing for risk-averse policies and mass-covering for risk-seeking policies. These findings bridge stochastic control and information-theoretic inference, providing a foundation for sampling-based control algorithms.
Related
- Explaining f-Divergence-Based Regularization via Local Curvature and Sharpness-Aware Minimization
- A restart-free accelerated algorithm for nonconvex minimization: continuous- and discrete-time analysis
- Bridging persistence of discrete and continuous 1D maps: Total variation, permutation equivalence, and diagonal affine transformations
- On free minimal constant speedups violating continuous orbit equivalence in p-adic ℤd-odometers of adding type
- On the Equivalence of Stochastic Control and Path Space Formulations for Schrödinger Bridges over Compact Connected Lie Groups
BibTeX
@misc{kataoka2026equivalence,
title = {{Equivalence Between Continuous-Time Risk-Sensitive Control and Rényi Divergence Minimization}},
author = {Shinji Kataoka and Kaoru Teranishi and Yasumasa Fujisaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07045},
eprint = {2609.07045},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07045},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Equivalence Between Continuous-Time Risk-Sensitive Control and Rényi Divergence Minimization
AU - Shinji Kataoka
AU - Kaoru Teranishi
AU - Yasumasa Fujisaki
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07045
UR - https://arxiv.org/abs/2609.07045
AB - In this study, we show that a continuous-time risk-sensitive control problem is equivalent to a Rényi divergence minimization problem over trajectory path measures. Reformulating stochastic optimal control as probabilistic inference via Kullback-Leibler (KL) divergence minimization avoids the computational intractability of the Hamilton-Jacobi-Bellman equation. However, standard KL control is inherently risk-neutral, and recent minimax extensions remain restricted to risk-averse settings. Our equivalence result resolves this limitation by offering a unified probabilistic framework for arbitrary risk attitudes in continuous-time nonlinear systems. Based on Girsanov theorem, we explicitly map the risk sensitivity to the Rényi divergence order, deriving a noise-dependent control penalty scaled by risk preference. This formulation seamlessly modulates tail-weighting behaviors, interpolating between zero-forcing for risk-averse policies and mass-covering for risk-seeking policies. These findings bridge stochastic control and information-theoretic inference, providing a foundation for sampling-based control algorithms.
ER -