2019/01/05 by Javanmard, Adel, Mondelli, Marco, Montanari, Andrea · 1 citation
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.1901.01375
Fitting a function by using linear combinations of a large number N of `simple' components is one of the most fruitful ideas in statistical learning. This idea lies at the core of a variety of methods, from two-layer neural networks to kernel regression, to boosting. In general, the resulting risk minimization problem is non-convex and is solved by gradient descent or its variants. Unfortunately, little is known about global convergence properties of these approaches. Here we consider the problem of learning a concave function f on a compact convex domain Ω⊆ \mathbb Rd, using linear combinations of `bump-like' components (neurons). The parameters to be fitted are the centers of N bumps, and the resulting empirical risk minimization problem is highly non-convex. We prove that, in the limit in which the number of neurons diverges, the evolution of gradient descent converges to a Wasserstein gradient flow in the space of probability distributions over Ω. Further, when the bump width δ tends to 0, this gradient flow has a limit which is a viscous porous medium equation. Remarkably, the cost function optimized by this gradient flow exhibits a special property known as displacement convexity, which implies exponential convergence rates for N→∞, δ→ 0. Surprisingly, this asymptotic theory appears to capture well the behavior for moderate values of δ, N. Explaining this phenomenon, and understanding the dependence on δ,N in a quantitative manner remains an outstanding challenge.