High-Order Langevin Monte Carlo Algorithms

Langevin algorithms are popular Markov chain Monte Carlo (MCMC) methods for large-scale sampling problems that often arise in data science. We propose Monte Carlo algorithms based on the discretizations of $P$-th order Langevin dynamics for any $P\geq 3$. Our design of $P$-th order Langevin Monte Carlo (LMC) algorithms is by combining splitting and accurate integration methods. We obtain Wasserstein convergence guarantees for sampling from distributions with log-concave and smooth densities. Specifically, the mixing time of the $P$-th order LMC algorithm scales as $O\left(d^{\frac{1}{R}}/\epsilon^{\frac{1}{2R}}\right)$ for $R=4\cdot 1_{\{ P=3\}}+ (2P-1)\cdot 1_{\{ P\geq 4\}}$, which has a better dependence on the dimension $d$ and the accuracy level $\epsilon$ as $P$ grows. Numerical experiments illustrate the efficiency of our proposed algorithms.

Algorithmic Stability of Stochastic Gradient Descent with Momentum under Heavy-Tailed Noise

Understanding the generalization properties of optimization algorithms under heavy-tailed noise has gained growing attention. However, the existing theoretical results mainly focus on stochastic gradient descent (SGD) and the analysis of heavy-tailed optimizers beyond SGD is still missing. In this work, we establish generalization bounds for SGD with momentum (SGDm) under heavy-tailed gradient noise. We first consider the continuous-time limit of SGDm, i.e., a Levy-driven stochastic differential equation (SDE), and establish quantitative Wasserstein algorithmic stability bounds for a class of potentially non-convex loss functions. Our bounds reveal a remarkable observation: For quadratic loss functions, we show that SGDm admits a worse generalization bound in the presence of heavy-tailed noise, indicating that the interaction of momentum and heavy tails can be harmful for generalization. We then extend our analysis to discrete-time and develop a uniform-in-time discretization error bound, which, to our knowledge, is the first result of its kind for SDEs with degenerate noise. This result shows that, with appropriately chosen step-sizes, the discrete dynamics retain the generalization properties of the limiting SDE. We illustrate our theory on both synthetic quadratic problems and neural networks.