Symmetric Linear Bandits with Hidden Symmetry
Nam Phuong Tran, The Anh Ta, Debmalya Mandal, Long Tran-Thanh
TL;DR
This work studies high-dimensional linear stochastic bandits with hidden symmetry, where the reward is invariant under a hidden subgroup $\mathcal{G}\le\mathcal{S}_d$ acting by coordinate permutations. It shows that merely knowing the existence of a symmetry offers no benefit unless structure is restricted, and then casts learning the symmetry as model selection over a structured collection of low-dimensional fixed-point subspaces. The proposed Explore-Models-then-Commit (EMC) algorithm achieves regret $O\big(d_0^{2/3}T^{2/3}\log d\big)$, and, under a well-separated partition assumption, improves to $O\big(d_0\sqrt{T}\log d\big)$, by leveraging a subspace lattice and restricted isometry properties. Experiments demonstrate competitive performance across sparsity and partition-structured regimes, highlighting the approach’s ability to break the curse of dimensionality when hidden symmetries can be learned online. The results provide a new perspective on leveraging symmetry as an inductive bias in sequential decision-making, with potential impact on large-scale, structured learning problems.
Abstract
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^{2/3} 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)} )$.