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Unifying Entropy Regularization in Optimal Control: From and Back to Classical Objectives via Iterated Soft Policies and Path Integral Solutions
Published 5 Dec 2025 in math.OC, cs.LG, cs.RO, and eess.SY | (2512.06109v1)
Abstract: This paper develops a unified perspective on several stochastic optimal control formulations through the lens of Kullback-Leibler regularization. We propose a central problem that separates the KL penalties on policies and transitions, assigning them independent weights, thereby generalizing the standard trajectory-level KL-regularization commonly used in probabilistic and KL-regularized control. This generalized formulation acts as a generative structure allowing to recover various control problems. These include the classical Stochastic Optimal Control (SOC), Risk-Sensitive Optimal Control (RSOC), and their policy-based KL-regularized counterparts. The latter we refer to as soft-policy SOC and RSOC, facilitating alternative problems with tractable solutions. Beyond serving as regularized variants, we show that these soft-policy formulations majorize the original SOC and RSOC problem. This means that the regularized solution can be iterated to retrieve the original solution. Furthermore, we identify a structurally synchronized case of the risk-seeking soft-policy RSOC formulation, wherein the policy and transition KL-regularization weights coincide. Remarkably, this specific setting gives rise to several powerful properties such as a linear Bellman equation, path integral solution, and, compositionality, thereby extending these computationally favourable properties to a broad class of control problems.
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Practical Applications
Immediate Applications
Below is a set of concrete, deployable use cases that can be implemented now using the paper’s unified KL-regularized optimal control framework, soft-policy variants, path-integral solutions, and MM-based algorithms.
- Safe policy regularization toward trusted baselines (SP-SOC)
- Sector: robotics, autonomous vehicles, industrial automation
- What: Regularize new controllers toward a stabilizing or certified baseline policy to improve performance while preserving safety.
- Workflow/product: KL-regularized policy iteration (MM) that solves a soft subproblem each iteration; “baseline-aware” MPC with a tunable λP to control deviation from the baseline.
- Assumptions/dependencies: A baseline policy ρ exists and is safe; system transitions ι are reasonably known or simulatable; finite-horizon setup; proper tuning of λP.
- Sampling-based control via path integrals (SRS-SP-RSOC)
- Sector: drones, mobile manipulation, legged robots, autonomous driving
- What: Use the linear Bellman operator and path-integral solution to estimate desirability z via forward trajectories under baseline dynamics/policy; reweight to obtain optimal soft policies.
- Workflow/product: A parallel forward-simulation engine that computes z_t = E[e{-λ cost}] under (ρ, ι); integration into MPC/PI2-like planners for real-time control.
- Assumptions/dependencies: Ability to simulate baseline dynamics; positive λ with synchronized weights (λP = λS); well-specified cost; sufficient compute for parallel sampling.
- Compositional policy synthesis (SRS-SP-RSOC)
- Sector: robotics (task blending), autonomous driving (multi-objective planning), human-robot interaction
- What: Compose multiple sub-objectives (e.g., safety, efficiency, comfort) by linear combination of terminal desirabilities, yielding mix-of-experts policies with interpretable weights.
- Workflow/product: A “policy composer” that builds z_t from components z_tn with weights γ_n and synthesizes mixture policies π_t = Σ w_tn π_tn.
- Assumptions/dependencies: Terminal desirability decomposes as z_T = Σ γ_n e{-λ c_Tn}; synchronized weights (λP = λS > 0); cost shaping is meaningful and measurable.
- Risk-aware control under model uncertainty (RSOC/SP-RSOC)
- Sector: energy (microgrid dispatch), operations (inventory/supply chains), finance (portfolio rebalancing), process control
- What: Encode risk-seeking (λS > 0) or risk-averse (λS < 0) attitudes by optimizing over auxiliary transition models τ under KL penalties—hedging against favorable/worst-case evolutions.
- Workflow/product: A “risk knob” in MPC/optimal control that tunes λS to navigate optimism/pessimism; scenario-aware controllers that explicitly trade off cost and KL deviation.
- Assumptions/dependencies: Baseline transitions ι are defined; absolute continuity constraints hold; mapping from cost to exponential utility is calibrated; availability of scenario generation.
- Distributionally robust approximations using KL ambiguity (DRO linkage)
- Sector: process control, energy, logistics
- What: Implement controllers that hedge against model mismatches by penalizing deviations via KL terms; exploit equivalence to risk-averse control in certain regimes.
- Workflow/product: A DRO-inspired controller that sets ambiguity via λ and solves SP-RSOC/SOC surrogates; integration with existing robust MPC toolchains.
- Assumptions/dependencies: Validity of DRO–RSOC equivalence in the chosen regime; meaningful baseline dynamics; ability to estimate or bound KL ambiguity.
- Data-driven RL with Control-as-Inference connections
- Sector: software, reinforcement learning, robotics RL
- What: Use EM on the PGM (optimality variables) to solve risk-seeking RSOC; perform density matching (I- or M-projection) to obtain soft policies or risk-sensitive solutions.
- Workflow/product: An “inference-driven RL” module that alternates E-steps (posterior over trajectories) and M-steps (policy update), and a policy-reweighting tool compatible with SAC-like pipelines.
- Assumptions/dependencies: A probabilistic graphical model with optimality variables and cost encoding; a baseline policy and transitions; accurate trajectory sampling and likelihood computation.
- Offline policy improvement from demonstrations (SP-SOC)
- Sector: robotics, autonomous driving, assistive devices
- What: Use expert demonstrations to define ρ, then derive soft-policy controllers that match expert behaviors while improving with explicit cost and KL regularization.
- Workflow/product: “Demo-to-controller” pipeline combining behavior cloning with KL-regularized control updates toward lower cost trajectories.
- Assumptions/dependencies: Quality and coverage of expert demonstrations; cost function alignment with desired performance; stable baseline.
- Deterministic dynamics special case (DOC/SP-DOC)
- Sector: motion planning, CNC/robotic manufacturing, precise actuation systems
- What: In deterministic baseline dynamics, RSOC/SP-RSOC collapse to SOC/SP-SOC, simplifying implementation while preserving regularization benefits.
- Workflow/product: Lightweight soft-policy planner for deterministic systems; reduced complexity controllers with guaranteed descent via MM surrogate.
- Assumptions/dependencies: Deterministic or nearly deterministic transitions; finite-horizon planning; existence of baseline ρ.
- Safety-first exploration and constraint adherence (SP-RSOC with λS < 0)
- Sector: healthcare devices, autonomous vehicles, industrial robots
- What: Use risk-averse soft policies to limit exploration inside safe envelopes while optimizing performance; KL penalties ensure proximity to certified behaviors.
- Workflow/product: “Safety-mode” controllers that enforce tight regularization and pessimistic transition modeling; tunable safety margins through λP and λS.
- Assumptions/dependencies: Certified baseline controller; safety constraints encoded in costs; known operational envelope and dynamics.
Long-Term Applications
These applications require further methodological development, scaling, theoretical extension, or domain integration before widespread deployment.
- Unified control software toolkit implementing the central KL-regularized problem (C-KLR-CP)
- Sector: software tools for control/RL, robotics platforms
- What: A general-purpose library exposing policy and transition KL penalties with independent weights, supporting soft-policy SOC/RSOC, synchronized settings, MM iterations, and CaI links.
- Dependencies: Robust numerical methods, stability analysis for large-scale systems, standardized interfaces to simulators and hardware.
- Time-varying regularization schedules (λP_t, λS_t)
- Sector: robotics, autonomous driving, industrial control
- What: Adaptive schedules that modulate baseline adherence and risk attitude across a horizon (e.g., conservative at start, bolder near the end).
- Dependencies: Theory and tooling for time-varying weights; safeguards against nonstationary behavior; identification routines.
- Multi-agent extensions and linearly solvable Markov games
- Sector: swarms, traffic systems, multi-robot coordination
- What: Extend synchronized SP-RSOC properties (linearity, compositionality) to multi-agent settings for tractable coordination and compositional strategy mixture.
- Dependencies: Game-theoretic generalizations; communication constraints; convergence guarantees; shared baselines across agents.
- Formal verification and certification of risk-sensitive soft policies
- Sector: aerospace, medical robotics, safety-critical automation
- What: Develop certifiable pipelines ensuring monotonic descent via MM, bounded KL deviations, and safe risk attitudes for regulatory compliance.
- Dependencies: Verified numerical solvers, interpretability of KL weights and risk parameters, integration with formal methods.
- Adaptive DRO mapping for nonstationary environments
- Sector: energy grids, supply chains, financial markets
- What: Online estimation of KL ambiguity sets and risk parameters under drifting models; controllers that re-tune λS to maintain robustness over time.
- Dependencies: Drift detection, online learning of baselines, guarantees under model updates.
- Partial observability and belief-space extensions (POMDPs)
- Sector: healthcare devices, autonomous navigation in uncertain environments
- What: Extend the framework to operate in belief space, preserving linearity/compositionality where possible; derive soft policies over beliefs.
- Dependencies: Theoretical generalization to POMDPs; practical filters/estimators; scalability.
- Human-in-the-loop and assistive systems with compositional objectives
- Sector: neuroprosthetics, rehabilitation robotics, assistive driving
- What: Blend objectives (comfort, effort, safety) via compositional desirability; adjust risk and KL weights to reflect user preferences or clinician guidance.
- Dependencies: Human modeling fidelity, preference estimation, closed-loop adaptability, clinical validation.
- Real-time compositional planners for autonomous driving
- Sector: transportation
- What: Real-time mixing of sub-policies (lane-keeping, obstacle avoidance, comfort) with explainable weights; improved responsiveness via path-integral sampling.
- Dependencies: Efficient hardware acceleration; robust perception-to-cost mapping; safety case development.
- Integration with learning-based cost shaping and inverse RL
- Sector: robotics, RL research
- What: Learn cost functions from data and use KL-regularized control for safe, risk-aware deployment; unify inverse RL with soft-policy SOC/RSOC.
- Dependencies: Reliable cost learning, generalization guarantees, mechanisms to prevent reward hacking.
- Policy guidance and governance for safe exploration in AI systems
- Sector: policy/regulation, corporate governance
- What: Codify KL-regularized proximity to certified behaviors and explicit risk parameters as governance levers; provide operational guidance on λ selection and monitoring.
- Dependencies: Cross-sector consensus, metrics for proximity and risk, auditing and reporting frameworks.
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