Modified Bryson-Frazier Smoothing and Hyperparameter Learning for Temporal Gaussian Process Regression

One-dimensional Gaussian processes with stationary, integrable kernel functions admit exact or arbitrarily accurate state-space representations, enabling linear-time inference through Kalman filtering and Rauch-Tung-Striebel (RTS) smoothing. However, the RTS smoother requires inversion of predicted state covariance matrices, which can become ill-conditioned and may therefore lead to numerical instabilities. In this work, we revisit the modified Bryson-Frazier (MBF) smoother as an alternative to the RTS smoother for Gaussian process regression in its state-space representation. In addition to reducing computational cost and memory requirements, the MBF smoother computes the same posterior distributions as the RTS smoother while avoiding the problematic covariance matrix inversion and the associated numerical instabilities. Furthermore, we demonstrate that the intermediate quantities computed by the MBF smoother can be reused to compute gradients of the negative log marginal likelihood, enabling kernel hyperparameter learning with minimal additional cost. Together, these results establish the MBF smoother as a unified and numerically robust approach to inference and kernel hyperparameter learning for one-dimensional Gaussian process regression.

Deep-Unfolded Coordination

Distributed optimization is a highly scalable and structurally transparent technique to solve multi-agent robotics problems; however, such methods often suffer from the need for highly-specialized, problem-specific hyperparameter tunings. In this work, we propose Deep Coordinator, a deep-unfolding framework that learns to dynamically adjust the hyperparameters of ADMM-DDP, a popular distributed solver for robotics tasks, at solve-time in response to optimizer performance. Our architecture consists of unrolling a fixed number of ADMM-DDP iterations into a neural network with learnable functions between layers mapping the optimizer state to the next hyperparameters. To the best of our knowledge, Deep Coordinator is the first deep-unfolding framework to adapt the penalty parameters of a non-convex optimizer at solve-time; we show that the mainstream supervised approach can yield degenerate solutions when training such models, and propose an unsupervised learning scheme. On simulations with fleets of cars and quadrotors, Deep Coordinator produces trajectories of comparable quality 6.18-9.44x faster than conventional solvers. Furthermore, Deep Coordinator retains its performance benefits when deployed to systems up to 8x larger than trained on.