Unifying Entropy Regularization in Optimal Control: From and Back to Classical Objectives via Iterated Soft Policies and Path Integral Solutions

This paper develops a unified perspective on several optimal control formulations through the lens of Kullback-Leibler (KL) regularization. We propose a central problem that separates the KL penalties on policies and transitions with independent weights, thus generalizing the standard trajectory-level KL-regularization used in probabilistic optimal control. This umbrella formulation recovers various control problems: the classical Stochastic Optimal Control (SOC), Risk-Sensitive Stochastic Optimal Control (RSOC), and their policy-based KL-regularized counterparts, termed soft-policy SOC and RSOC, which yield tractable surrogates. Beyond being regularized variants, these soft-policy formulations majorize the original SOC and RSOC, thus, iterating their solutions recovers the original objectives. We further identify a synchronized case of soft-policy RSOC where the policy and transition KL weights coincide, yielding a linear Bellman operator, path-integral solution, and compositionality -- extending these computationally favourable properties to a broad class of control problems.