Langevin Monte Carlo Beyond Lipschitz Gradient Continuity
This work addresses a limitation in sampling algorithms for researchers in computational statistics and machine learning, offering a more general and robust method, though it appears incremental as it builds on existing LMC frameworks.
The paper tackles the problem of extending Langevin Monte Carlo (LMC) methods to handle potentials beyond the standard Lipschitz gradient assumption, such as those with convex, strongly convex tails, and polynomial growth, by introducing the Inexact Proximal Langevin Algorithm (IPLA), which achieves improved convergence rates and provides bounds on all moments of the Markov chain.
We present a significant advancement in the field of Langevin Monte Carlo (LMC) methods by introducing the Inexact Proximal Langevin Algorithm (IPLA). This novel algorithm broadens the scope of problems that LMC can effectively address while maintaining controlled computational costs. IPLA extends LMC's applicability to potentials that are convex, strongly convex in the tails, and exhibit polynomial growth, beyond the conventional $L$-smoothness assumption. Moreover, we extend LMC's applicability to super-quadratic potentials and offer improved convergence rates over existing algorithms. Additionally, we provide bounds on all moments of the Markov chain generated by IPLA, enhancing its analytical robustness.