Global Non-convex Optimization with Discretized Diffusions
This work addresses global optimization challenges in machine learning and related fields, offering a more flexible framework than existing Langevin-based methods, though it appears incremental in extending diffusion-based optimization.
The paper tackles the problem of global optimization for convex and non-convex functions by showing that discretized diffusions beyond Langevin can converge to global minimizers, enabling the design of diffusions for broader function classes. It provides non-asymptotic error bounds based on objective properties and introduces new Stein factor bounds for Poisson equations.
An Euler discretization of the Langevin diffusion is known to converge to the global minimizers of certain convex and non-convex optimization problems. We show that this property holds for any suitably smooth diffusion and that different diffusions are suitable for optimizing different classes of convex and non-convex functions. This allows us to design diffusions suitable for globally optimizing convex and non-convex functions not covered by the existing Langevin theory. Our non-asymptotic analysis delivers computable optimization and integration error bounds based on easily accessed properties of the objective and chosen diffusion. Central to our approach are new explicit Stein factor bounds on the solutions of Poisson equations. We complement these results with improved optimization guarantees for targets other than the standard Gibbs measure.