Soft MPCritic: Value-Based Reinforcement Learning for Non-Differentiable Predictive Control

Abstract

Reinforcement learning (RL) and model predictive control (MPC) offer complementary strengths, yet existing MPC-RL frameworks commonly rely on gradient-based optimization and therefore cannot handle the non-differentiable dynamics and objectives common in RL. We propose Soft MPCritic, an MPC-RL framework that learns a soft value function and uses sampling-based planning both for online control and for generating value targets. Specifically, Soft MPCritic instantiates MPC with model predictive path integral control (MPPI) and learns a terminal Q-function through fitted value iteration. Because the value targets are computed by the same planner used for control, the learned value function is aligned with the planner and implicitly extends its effective horizon. To keep target generation tractable, we introduce an amortized warm-start strategy that reuses the open-loop action sequences planned during online control to initialize batched MPPI computations of value targets, reducing computational cost without sacrificing solution quality. For robustness to model error, Soft MPCritic plans over scenarios drawn from an ensemble of dynamics models trained for next-step prediction. On both classical and constrained control tasks, Soft MPCritic learns effective policies with non-differentiable objectives, particularly where finite-horizon MPPI alone falls short. Soft MPCritic thus offers a practical blueprint for synthesizing MPC policies directly from black-box objectives, without differentiable surrogates.