📅 Date: Wednesday, July 15, 2026
🕚 Time: 11:00 – 12:00
Location: Heinzel Seminar Room, Office Building West
Speaker: Basil Saeed (Stanford)
Title: High-dimensional asymptotics of empirical risk beyond convexity
Abstract:
I will discuss a Kac–Rice approach to studying the high-dimensional asymptotics of empirical risk minimization in the proportional regime under Gaussian covariates. The Kac–Rice formula yields an exponential bound on the expected number of local minima with prescribed macroscopic statistics. Combined with Markov’s inequality, this bound localizes minimizers and gives sharp asymptotics for estimation error, prediction error, and related observables whenever a “rate trivialization” phenomenon occurs. I will first explain the general framework and its consequences in the convex, multi-index setting, including the sharp characterization obtained in [AMS25]. I will then describe recent progress for non-convex losses, where rate trivialization can still hold above a sufficiently large sample-to-dimension ratio, leading again to sharp predictions [MS26]. (A = Kiana Asgari, M = Andrea Montanari)