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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)} )$. |
