Shifted Composition IV: Underdamped Langevin and Numerical Discretizations with Partial Acceleration

Abstract: Quantifying the convergence rate of the underdamped Langevin dynamics (ULD) is a classical topic, in large part due to the possibility for diffusive-to-ballistic speedups -- as was recently established for the continuous-time dynamics via space-time Poincare inequalities. A central challenge for analyzing ULD is that its degeneracy necessitates the development of new analysis approaches, e.g., the theory of hypocoercivity. In this paper, we give a new coupling-based framework for analyzing ULD and its numerical discretizations. First, in the continuous-time setting, we use this framework to establish new parabolic Harnack inequalities for ULD. These are the first Harnack inequalities that decay to zero in contractive settings, thereby reflecting the convergence properties of ULD in addition to just its regularity properties. Second, we build upon these Harnack inequalities to develop a local error framework for analyzing discretizations of ULD in KL divergence. This extends our framework in part III from uniformly elliptic diffusions to degenerate diffusions, and shares its virtues: the framework is user-friendly, applies to sophisticated discretization schemes, and does not require contractivity. Applying this framework to the randomized midpoint discretization of ULD establishes (i) the first ballistic acceleration result for log-concave sampling (i.e., sublinear dependence on the condition number), and (ii) the first $d^{1/3}$ iteration complexity guarantee for sampling to constant total variation error in dimension $d$.