Many problems of contemporary interest in signal processing and machine learning involve highly non-convex optimization problems. While nonconvex problems are known to be intractable in general, simple local search heuristics such as (stochastic) gradient descent are often surprisingly effective at finding global optima on real or randomly generated data. In this talk I will discuss some results explaining the success of these heuristics focusing on two problems.

The first problem is about learning the optimal weights of the shallowest of neural networks consisting of a single Rectified Linear Unit (ReLU). I will discuss this problem in the high-dimensional regime where the number of observations are fewer than the ReLU weights. I will show that projected gradient descent on a natural least-squares objective, when initialization at 0, converges at a linear rate to globally optimal weights with a number of samples that is optimal up to numerical constants. The second problem focuses on maximizing continuous submodular functions that emerge in a variety of areas in machine learning, including utility functions in matrix approximations and network inference. Despite the apparent lack of convexity/concavity in such functions, I will show that stochastic projected gradient methods can provide strong approximation guarantees for maximizing continuous submodular functions with convex constraints. In particular, this result allows us to approximately maximize discrete, monotone submodular optimization problems via projected gradient descent on a continuous relaxation, directly connecting the discrete and continuous domains.

Parts of this talk is based on joint work with Hamed Hassani and Amin Karbasi.