Accelerating trajectory optimization with Sobolev-trained diffusion policies

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.

SVL: Goal-Conditioned Reinforcement Learning as Survival Learning

Standard approaches to goal-conditioned reinforcement learning (GCRL) that rely on temporal-difference learning can be unstable and sample-inefficient due to bootstrapping. While recent work has explored contrastive and supervised formulations to improve stability, we present a probabilistic alternative, called survival value learning (SVL), that reframes GCRL as a survival learning problem by modeling the time-to-goal from each state as a probability distribution. This structured distributional Monte Carlo perspective yields a closed-form identity that expresses the goal-conditioned value function as a discounted sum of survival probabilities, enabling value estimation via a hazard model trained via maximum likelihood on both event and right-censored trajectories. We introduce three practical value estimators, including finite-horizon truncation and two binned infinite-horizon approximations to capture long-horizon objectives. Experiments on offline GCRL benchmarks show that SVL combined with hierarchical actors matches or surpasses strong hierarchical TD and Monte Carlo baselines, excelling on complex, long-horizon tasks.

Variance-Reduced Model Predictive Path Integral via Quadratic Model Approximation

Sampling-based controllers, such as Model Predictive Path Integral (MPPI) methods, offer substantial flexibility but often suffer from high variance and low sample efficiency. To address these challenges, we introduce a hybrid variance-reduced MPPI framework that integrates a prior model into the sampling process. Our key insight is to decompose the objective function into a known approximate model and a residual term. Since the residual captures only the discrepancy between the model and the objective, it typically exhibits a smaller magnitude and lower variance than the original objective. Although this principle applies to general modeling choices, we demonstrate that adopting a quadratic approximation enables the derivation of a closed-form, model-guided prior that effectively concentrates samples in informative regions. Crucially, the framework is agnostic to the source of geometric information, allowing the quadratic model to be constructed from exact derivatives, structural approximations (e.g., Gauss- or Quasi-Newton), or gradient-free randomized smoothing. We validate the approach on standard optimization benchmarks, a nonlinear, underactuated cart-pole control task, and a contact-rich manipulation problem with non-smooth dynamics. Across these domains, we achieve faster convergence and superior performance in low-sample regimes compared to standard MPPI. These results suggest that the method can make sample-based control strategies more practical in scenarios where obtaining samples is expensive or limited.