Symmetric Linear Bandits with Hidden Symmetry

High-dimensional linear bandits with low-dimensional structure have received considerable attention in recent studies due to their practical significance. The most common structure in the literature is sparsity. However, it may not be available in practice. Symmetry, where the reward is invariant under certain groups of transformations on the set of arms, is another important inductive bias in the high-dimensional case that covers many standard structures, including sparsity. In this work, we study high-dimensional symmetric linear bandits where the symmetry is hidden from the learner, and the correct symmetry needs to be learned in an online setting. We examine the structure of a collection of hidden symmetry and provide a method based on model selection within the collection of low-dimensional subspaces. Our algorithm achieves a regret bound of $ O(d_0^{1/3} T^{2/3} \log(d))$, where $d$ is the ambient dimension which is potentially very large, and $d_0$ is the dimension of the true low-dimensional subspace such that $d_0 \ll d$. With an extra assumption on well-separated models, we can further improve the regret to $ O(d_0\sqrt{T\log(d)} )$.

Learning the Expected Core of Strictly Convex Stochastic Cooperative Games

Reward allocation, also known as the credit assignment problem, has been an important topic in economics, engineering, and machine learning. An important concept in credit assignment is the core, which is the set of stable allocations where no agent has the motivation to deviate from the grand coalition. In this paper, we consider the stable allocation learning problem of stochastic cooperative games, where the reward function is characterised as a random variable with an unknown distribution. Given an oracle that returns a stochastic reward for an enquired coalition each round, our goal is to learn the expected core, that is, the set of allocations that are stable in expectation. Within the class of strictly convex games, we present an algorithm named \texttt{Common-Points-Picking} that returns a stable allocation given a polynomial number of samples, with high probability. The analysis of our algorithm involves the development of several new results in convex geometry, including an extension of the separation hyperplane theorem for multiple convex sets, and may be of independent interest.