Convergence of Stochastic Gradient Langevin Dynamics in the Lazy Training Regime
This provides theoretical guarantees for SGLD in deep learning optimization, but it is incremental as it builds on existing lazy training concepts.
The paper tackled the convergence analysis of stochastic gradient Langevin dynamics (SGLD) in the lazy training regime, showing that it achieves exponential convergence to the empirical risk minimizer with finite-time and finite-width bounds on the optimality gap.
Continuous-time models provide important insights into the training dynamics of optimization algorithms in deep learning. In this work, we establish a non-asymptotic convergence analysis of stochastic gradient Langevin dynamics (SGLD), which is an Itô stochastic differential equation (SDE) approximation of stochastic gradient descent in continuous time, in the lazy training regime. We show that, under regularity conditions on the Hessian of the loss function, SGLD with multiplicative and state-dependent noise (i) yields a non-degenerate kernel throughout the training process with high probability, and (ii) achieves exponential convergence to the empirical risk minimizer in expectation, and we establish finite-time and finite-width bounds on the optimality gap. We corroborate our theoretical findings with numerical examples in the regression setting.