Mert Gürbüzbalaban

Studies accelerated methods when gradient estimates are biased as well as noisy. Computes the risk-sensitive index of generalized momentum methods via a Riccati-equation reduction, characterizes when it blows up, and derives a large deviation principle whose rate function is the convex conjugate of that index — turning rare-event behavior of the running suboptimality into a designable quantity, with finite-time high-probability guarantees beyond quadratics.

Builds a unified convergence and risk analysis for the generalized momentum family (covering gradient descent, heavy ball, and Nesterov acceleration) under stochastic gradient errors, bounding the entropic risk and entropic value-at-risk of suboptimality. The bounds support risk-averse parameter selection (RA-GMM): choosing step and momentum on the rate–risk Pareto frontier rather than for expected performance alone.

Treats first-order algorithms as feedback dynamical systems subject to worst-case, finite-energy gradient errors and computes their H∞ norm in closed form on quadratics — together with the explicit worst-case noise achieving it (a decaying cosine). Introduces robustly stable variants (RS-GD, RS-HB) with near-optimal L₂ gain, quantifies real stability radii of momentum methods, and extends the guarantees beyond quadratics through matrix-inequality certificates.

Provides non-asymptotic global convergence guarantees for stochastic gradient Hamiltonian Monte Carlo on non-convex problems, quantifying when and how momentum accelerates Langevin-based optimization.

Introduces decentralized versions of stochastic gradient Langevin dynamics and Hamiltonian Monte Carlo for Bayesian learning over networks of agents, with non-asymptotic guarantees on sampling accuracy.

Shows that SGD can produce heavy-tailed, power-law iterate fluctuations even from light-tailed data: multiplicative gradient noise drives a Kesten-type random recursion whose stationary law has a tail index controlled by the stepsize-to-batch-size ratio — connecting algorithm hyperparameters to the tail behavior, and thereby the generalization properties, of the learned solutions.