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The design and performance analysis of bandit algorithms in the presence of stage-wise safety or reliability constraints has recently garnered significant interest. In this work, we consider the linear stochastic bandit problem under additional linear safety constraints that need to be satisfied at each round. We provide a new safe algorithm based on linear Thompson Sampling (TS) for this problem and show a frequentist regret of order O(d 3/2 log 1/2 d • T 1/2 log 3/2 T ), which remarkably matches the results provided by [Abeille et al., 2017] for the standard linear TS algorithm in the absence of safety constraints. We compare the performance of our algorithm with a UCB-based safe algorithm and highlight how the inherently randomized nature of TS leads to a superior performance in expanding the set of safe actions the algorithm has access to at each round.

We study the problem of recovering an unknown signal x given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator xL and a spectral estimator xs . The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine xL and xs . At the heart of our analysis is the exact characterization of the empirical joint distribution of (x, xL , xs ) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of xL and xs , given the limiting distribution of the signal x. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form θ xL + xs and we derive the optimal combination coefficient. In order to establish the limiting distribution of (x, xL , xs ), we design and analyze an Approximate Message Passing (AMP) algorithm whose iterates give xL and approach xs . Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.

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