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We consider the problem of sequentially learning to estimate, in the meansquared error (MSE) sense, a Gaussian $K$-vector of unknown covariance byobserving only $m < K$ of its entries in each round. We first establish aconcentration bound for MSE estimation. We then frame the estimation problemwith bandit feedback, and propose a variant of the successive eliminationalgorithm. We also derive a minimax lower bound to understand the fundamentallimit on the sample complexity of this problem.

Adaptive Estimation of Random Vectors with Bandit Feedback: A mean-squared error viewpoint