Deep Learning Method for Stationary Distribution of Reflected Brownian Motion

Deep Bayesian reinforcement learning adapts to unseen tasks by inferring latent transition and reward models, but existing methods typically rely on variational posteriors and evidence lower bounds, introducing approximation error and unstable task representations. We introduce GLiBRL, a deep Bayesian RL framework that combines generalised linear task models with learnable non-linear basis functions. GLiBRL features conjugate Bayesian inference, yielding exact, sequential posterior updates over task parameters and model noise, together with a closed-form marginal likelihood that eliminates variational inference. The update is naturally permutation-invariant, allowing GLiBRL to integrate with both off- and on-policy algorithms. GLiBRL also learns task representation admitting an exact kernel identity, relating distances between task representations to kernel discrepancies over the task contexts. Compared against eight representative or recent meta reinforcement learning methods, GLiBRL achieves the highest aggregate zero-shot test performance on both the MuJoCo locomotion and MetaWorld manipulation benchmarks.

Recent advances in generative modeling have enabled the efficient computation of Schrödinger bridges (SB) in high-dimensional settings by leveraging partially simulation-free training methods inspired by flow matching. However, these have not covered SBs with reflecting dynamics, a useful model choice with built-in guarantees that generated samples stay in the data domain. Existing alternatives for reflected SBs instead rely on more complex training based on forward--backward SDE theory, requiring expensive higher-order derivatives and sampling entire paths during training. In this article, we introduce a partially simulation-free framework that allows reflected SBs to be trained similarly to flow matching, using a new sampling method and regression target. We demonstrate our results by coupling pairs of well-known high-dimensional image datasets. Using reflected dynamics incurs negligible additional wall-clock time during both training and inference while maintaining or slightly improving generative performance.

The minima of modern neural network loss functions are typically not isolated, rather they form connected components of reparameterization invariant solutions on the training data. Analytically characterizing these solutions is a hard problem, but sampling approaches are feasible. By construction, existing methods either spread over low-loss regions, and thus do not sample reparameterization invariant solutions exactly, or are inherently local, which limits exploration of other minima valleys. We propose sampling such reparameterization invariant models using a dynamical system based on kinetic energy, subject to a gravitational pull and a friction term that dissipates energy from the system. Our proposed sampler, DiMS, is guaranteed to sample exactly from the minimum level sets and depends on physically motivated hyperparameters which allows control over the exploration capabilities of the sampler. We consider uncertainty quantification in Bayesian inference as the motivating problem and observe improved performance compared to previously proposed approaches.