Group-averaged Markov chains II: tuning of group action in finite state space

We study group-averaged Markov chains obtained by augmenting a $\pi$-stationary transition kernel $P$ with a group action on the state space via orbit kernels. Given a group $\mathcal{G}$ with orbits $(\mathcal{O}_i)_{i=1}^k$, we analyse three canonical orbit kernels: namely the Gibbs $(G)$, Metropolis-Hastings $(M)$, and Barker $(B)$ kernels, as well as their multiplicative sandwiches $QPQ$ and the additive mixtures $\frac{1}{2}(P+Q)$ where $Q\in\{G,M,B\}$. We show that $M^t, B^t \to G$ blockwise as $t \to \infty$ under suitable conditions, that the projection chains induced by $(\mathcal{O}_i)_{i=1}^k$ coincide for $GPG$ and $P$, and that orbit averaging never deteriorates the absolute spectral gap or asymptotic variance when $P$ is reversible. We give a direct and simple proof of Pythagorean identity under the Kullback-Leibler (KL) divergence, showing that $GPG$ arises naturally as an information projection of $P$ onto the set of $G$-invariant transition matrices. For a given $P$, we characterise the optimal choice of $G$ with a fixed number of orbits that minimises the one-step KL divergence to stationarity. Analogously, for a given $G$, we characterise the optimal choice of $P$ and give sufficient conditions under which $GPG = \Pi$. We further show that alternating projections over multiple group actions converge at a rate governed by the singular values of an overlap matrix, and that in structured cases, this yields exact sampling where the number of group actions grows logarithmically with the size of the state space. Based on the theory, we propose two heuristics to tune $G$ in practice. We also illustrate the results on discrete uniform and multimodal examples, including the Curie-Weiss model where $GPG$ achieves polynomial (in inverse temperature and dimension) mixing while Glauber dynamics remains exponentially slow.