A Other related work

In addition to the work on noisy convex optimization, the current paper is also thematically related to works in learning theory and complexity where the goal is to reconstruct simple classes of functions under outlier noise. This includes work on reconstruction of low-degree polynomials [4, 14, 15]. In particular, [15] gave an efficient algorithm whose error tolerance matches the information theoretic limits. In addition, recently, [9] achieved similar algorithmic guarantees for functions which are sparse in the Fourier space. While similar in spirit, the model in these works differ from the current paper in one crucial way - namely, while we only put a bound on the volume of the outlier locations, they, in addition, assume that the outlier locations are also uniformly distributed in the domain. At a more technical level, the results in [4, 14, 15, 9] crucially rely on techniques originating from coding theory such as the Goldreich-Levin theorem [13] and the Berlekamp-Welch algorithm [6].