Finding an approximate second-order stationary point (SOSP) is a well-studied and fundamental problem in stochastic nonconvex optimization with many applications in machine learning.

We study the problem of learning general (i.e., not necessarily homogeneous) halfspaces with Random Classification Noise under the Gaussian distribution. We establish nearly-matching algorithmic and Statistical Query (SQ) lower bound results revealing a surprising information-computation gap for this basic problem. Specifically, the sample complexity of this learning problem is $\widetilde{\Theta}(d/\epsilon)$, where $d$ is the dimension and $\epsilon$ is the excess error. Our positive result is a computationally efficient learning algorithm with sample complexity$\tilde{O}(d/\epsilon + d/\max(p, \epsilon))^2)$, where $p$ quantifies the bias of the target halfspace. On the lower bound side, we show that any efficient SQ algorithm (or low-degree test)for the problem requires sample complexity at least $\Omega(d^{1/2}/(\max(p, \epsilon))^2)$. Our lower bound suggests that this quadratic dependence on $1/\epsilon$ is inherent for efficient algorithms.

Finding an approximate second-order stationary point (SOSP) is a well-studied and fundamental problem in stochastic nonconvex optimization with many applications in machine learning.However, this problem is poorly understood in the presence of outliers, limiting the use of existing nonconvex algorithms in adversarial settings.In this paper, we study the problem of finding SOSPs in the strong contamination model, where a constant fraction of datapoints are arbitrarily corrupted.We introduce a general framework for efficiently finding an approximate SOSP with \emph{dimension-independent} accuracy guarantees, using $\widetilde{O}({D^2}/{\epsilon})$ samples where $D$ is the ambient dimension and $\epsilon$ is the fraction of corrupted datapoints.As a concrete application of our framework, we apply it to the problem of low rank matrix sensing, developing efficient and provably robust algorithms that can tolerate corruptions in both the sensing matrices and the measurements.In addition, we establish a Statistical Query lower bound providing evidence that the quadratic dependence on $D$ in the sample complexity is necessary for computationally efficient algorithms.

Is there a communication-efficient learning algorithm that runs in polynomial time?

We study the problem of generalized uniformity testing of a discrete probability distribution: Given samples from a probability distribution p over an unknown size discrete domain Ω, we want to distinguish, with probability at least 2/ 3, between the case that p is uniform on some subset of Ω versus null -far, in total variation distance, from any such uniform distribution. We establish tight bounds on the sample complexity of generalized uniformity testing. In more detail, we present a computationally efficient tester whose sample complexity is optimal, within constant factors, and a matching worst-case information-theoretic lower bound.

Given a set of samples from a collection of probability distributions, can we determine whether these distributions satisfy a certain property?

In this work, we study the fully observable discrete case where the structure of the network is given.

In this paper, we study the same problem in the Massart noise model that we now define.

In fact, without such assumptions, this problem is NP-hard [Sim02; MR18].