Learning and 1-bit Compressed Sensing under Asymmetric Noise

We study the approximate recovery problem under noise: Given corrupted 1-bit measurements of the form sign(w∗ · xi), recover a vector w with a small 0/1 loss w.r.t. w∗ ∈ R. In learning theory, this is known as the problem of learning halfspaces with noise, and in signal processing, as 1bit compressed sensing, in which there is an additional assumption that w∗ is t-sparse. Direct formulations of the approximate recovery problem are non-convex and are NP-hard to optimize. In this paper, we propose adaptively solving a sequence of convex optimizations to mitigate the issue. Our algorithms output solutions with error as small as the information-theoretic limit under bounded and adversarial noise models. We also show that the usual one-shot approach of minimizing a convex surrogate fails to achieve this goal for a large family of loss functions.