Adaptive Feature Selection: Computationally Efficient Online Sparse Linear Regression under RIP

Abstract: Online sparse linear regression is an online problem where an algorithm repeatedly chooses a subset of coordinates to observe in an adversarially chosen feature vector, makes a real-valued prediction, receives the true label, and incurs the squared loss. The goal is to design an online learning algorithm with sublinear regret to the best sparse linear predictor in hindsight. Without any assumptions, this problem is known to be computationally intractable. In this paper, we make the assumption that data matrix satisfies restricted isometry property, and show that this assumption leads to computationally efficient algorithms with sublinear regret for two variants of the problem. In the first variant, the true label is generated according to a sparse linear model with additive Gaussian noise. In the second, the true label is chosen adversarially.