Tempered Sequential Monte Carlo for Trajectory and Policy Optimization with Differentiable Dynamics

Stochastic trajectory optimization methods like STOMP enable planning with non-differentiable costs, offering substantial flexibility over gradient-based approaches. We show that STOMP implicitly minimizes the KL divergence from a Boltzmann trajectory distribution, revealing an elegant Variational Inference (VI) structure underlying its updates. Building on this insight, we propose the \textit{Proximal Inference for Stochastic Trajectory Optimization} (PISTO) algorithm that stabilizes the updates by augmenting the objective with a KL regularization between successive Gaussian proposals. This proximal formulation admits a trust-region interpretation and yields closed-form mean updates computable as expectations under a surrogate distribution. We estimate these expectations via importance-weighted Monte Carlo sampling, producing a simple, derivative-free algorithm that inherits STOMP's ability to handle non-differentiable and discontinuous costs without modification. On robot arm motion planning benchmarks, PISTO achieves an 89% success rate -- outperforming CHOMP (63%) and STOMP (68%) -- while producing shorter, smoother paths at twice the speed of competing stochastic methods. We further validate PISTO on contact-rich MuJoCo locomotion and manipulation tasks, where it consistently outperforms both CEM and MPPI baselines in reward.

Trajectory Optimization (TO) solvers exploit known system dynamics to compute locally optimal trajectories through iterative improvements. A downside is that each new problem instance is solved independently; therefore, convergence speed and quality of the solution found depend on the initial trajectory proposed. To improve efficiency, a natural approach is to warm-start TO with initial guesses produced by a learned policy trained on trajectories previously generated by the solver. Diffusion-based policies have recently emerged as expressive imitation learning models, making them promising candidates for this role. Yet, a counterintuitive challenge comes from the local optimality of TO demonstrations: when a policy is rolled out, small non-optimal deviations may push it into situations not represented in the training data, triggering compounding errors over long horizons. In this work, we focus on learning-based warm-starting for gradient-based TO solvers that also provide feedback gains. Exploiting this specificity, we derive a first-order loss for Sobolev learning of diffusion-based policies using both trajectories and feedback gains. Through comprehensive experiments, we demonstrate that the resulting policy avoids compounding errors, and so can learn from very few trajectories to provide initial guesses reducing solving time by $2\times$ to $20 \times$. Incorporating first-order information enables predictions with fewer diffusion steps, reducing inference latency.

We introduce trajectory-regularized stochastic optimal control (TRSOC), which augments standard stochastic optimal control (SOC) with a Kullback--Leibler (KL) divergence between controlled and reference trajectory distributions. Using Girsanov's theorem, the trajectory KL reduces to a quadratic drift mismatch penalty, yielding a modified running cost that preserves the dynamic programming (DP) structure. We derive the corresponding Hamilton--Jacobi--Bellman (HJB) equation and characterize the optimal policy. In the linear-quadratic (LQ) setting, the formulation admits a closed-form solution with an augmented control cost. Experiments show that the regularization parameter induces a trade-off between performance-driven and reference-preserving behavior, including cases with reference dynamics learned from offline data.