Poisson equations underpin average-reward reinforcement learning, but beyond ergodicity they can be ill-posed, meaning that solutions are non-unique and standard fixed point iterations can oscillate on reducible or periodic chains. We study finite-state Markov chains with $n$ states and transition matrix $P$. We show that all non-decaying modes are captured by a real peripheral invariant subspace $\mathcal{K}(P)$, and that the induced operator on the quotient space $\mathbb{R}^n/\mathcal{K}(P)$ is strictly contractive, yielding a unique quotient solution. Building on this viewpoint, we develop an end-to-end pipeline that learns the chain structure, estimates an anchor based gauge map, and runs projected stochastic approximation to estimate a gauge-fixed representative together with an associated peripheral residual. We prove $\widetilde{O}(T^{-1/2})$ convergence up to projection estimation error, enabling stable Poisson equation learning for multichain and periodic regimes with applications to performance evaluation of average-reward reinforcement learning beyond ergodicity.

Bayes' rule is used to produce a parameter estimate, which then decides the policy

We do not foresee direct negative societal impact from the current work. Also, one may argue that our work is catalytic to enhancing the'disclosure risk' of individuals, i.e. an adversary might be able to make accurate Granted, no existing privacy frameworks can guard against this. We prove its ergodicity in Theorem S-3.1, which implies Theorem 3.3 . The model is such that the set { x: f ( x |) > 0 } does not depend on . The Metropolis-within-Gibbs sampler is aperiodic by construction, since some proposals can be rejected.

In contrast to MCMC approaches like Hamiltonian Monte Carlo, no stochastic step is required. Instead, the proposed deterministic dynamics in an extended state space exactly sample the target distribution, specified by an energy function, under an assumption of ergodicity. Alternatively, the dynamics can be interpreted as a normalizing flow that samples a specified energy model without training.

Multi-proposal MCMC algorithms have recently gained attention for their potential to improve performance, especially through parallel implementation on modern hardware. This paper introduces a novel family of gradient-free MCMC algorithms that combine the multi-try Metropolis (MTM) with stereographic MCMC framework, specifically designed for efficient sampling from heavy-tailed targets. The proposed stereographic multi-try Metropolis (SMTM) algorithm not only outperforms traditional Euclidean MTM and existing stereographic random-walk Metropolis methods, but also avoids the pathological convergence behavior often observed in MTM and demonstrates strong robustness to tuning. These properties are supported by scaling analysis and extensive simulation studies.

Ergodic exploration has spawned a lot of interest in mobile robotics due to its ability to design time trajectories that match desired spatial coverage statistics. However, current ergodic approaches are for continuous spaces, which require detailed sensory information at each point and can lead to fractal-like trajectories that cannot be tracked easily. This paper presents a new ergodic approach for graph-based discretization of continuous spaces. It also introduces a new time-discounted ergodicity metric, wherein early visitations of information-rich nodes are weighted more than late visitations. A Markov chain synthesized using a convex program is shown to converge more rapidly to time-discounted ergodicity than the traditional fastest mixing Markov chain. The resultant ergodic traversal method is used within a hierarchical framework for active inspection of confined spaces with the goal of detecting anomalies robustly using SLAM-driven Bayesian hypothesis testing. Both simulation and physical experiments on a ground robot show the advantages of this framework over greedy and random exploration methods for left-behind foreign object debris detection in a ballast tank.

Robotic search and rescue, exploration, and inspection require trajectory planning across a variety of domains. A popular approach to trajectory planning for these types of missions is ergodic search, which biases a trajectory to spend time in parts of the exploration domain that are believed to contain more information. Most prior work on ergodic search has been limited to searching simple surfaces, like a 2D Euclidean plane or a sphere, as they rely on projecting functions defined on the exploration domain onto analytically obtained Fourier basis functions. In this paper, we extend ergodic search to any surface that can be approximated by a triangle mesh. The basis functions are approximated through finite element methods on a triangle mesh of the domain. We formally prove that this approximation converges to the continuous case as the mesh approximation converges to the true domain. We demonstrate that on domains where analytical basis functions are available (plane, sphere), the proposed method obtains equivalent results, and while on other domains (torus, bunny, wind turbine), the approach is versatile enough to still search effectively. Lastly, we also compare with an existing ergodic search technique that can handle complex domains and show that our method results in a higher quality exploration.