Langevin diffusions are rapidly convergent under appropriate functional inequality assumptions. Hence, it is natural to expect that with additional smoothness conditions to handle the discretization errors, their discretizations like the Langevin Monte Carlo (LMC) converge in a similar fashion. This research program was initiated by Vempala and Wibisono (2019), who established results under log-Sobolev inequalities. Chewi et al. (2022) extended the results to handle the case of Poincar\'e inequalities. In this paper, we go beyond Poincar\'e inequalities, and push this research program to its limit. We do so by establishing upper and lower bounds for Langevin diffusions and LMC under weak Poincar\'e inequalities that are satisfied by a large class of densities including polynomially-decaying heavy-tailed densities (i.e., Cauchy-type). Our results explicitly quantify the effect of the initializer on the performance of the LMC algorithm. In particular, we show that as the tail goes from sub-Gaussian, to sub-exponential, and finally to Cauchy-like, the dependency on the initial error goes from being logarithmic, to polynomial, and then finally to being exponential. This three-step phase transition is in particular unavoidable as demonstrated by our lower bounds, clearly defining the boundaries of LMC.

We establish disintegrated PAC-Bayesian generalisation bounds for models trained with gradient descent methods or continuous gradient flows. Contrary to standard practice in the PAC-Bayesian setting, our result applies to optimisation algorithms that are deterministic, without requiring any de-randomisation step. Our bounds are fully computable, depending on the density of the initial distribution and the Hessian of the training objective over the trajectory. We show that our framework can be applied to a variety of iterative optimisation algorithms, including stochastic gradient descent (SGD), momentum-based schemes, and damped Hamiltonian dynamics.

We analyze the complexity of sampling from a class of heavy-tailed distributions by discretizing a natural class of It\^o diffusions associated with weighted Poincar\'e inequalities. Based on a mean-square analysis, we establish the iteration complexity for obtaining a sample whose distribution is $\epsilon$ close to the target distribution in the Wasserstein-2 metric. In this paper, our results take the mean-square analysis to its limits, i.e., we invariably only require that the target density has finite variance, the minimal requirement for a mean-square analysis. To obtain explicit estimates, we compute upper bounds on certain moments associated with heavy-tailed targets under various assumptions. We also provide similar iteration complexity results for the case where only function evaluations of the unnormalized target density are available by estimating the gradients using a Gaussian smoothing technique. We provide illustrative examples based on the multivariate $t$-distribution.

We establish generalization error bounds for stochastic gradient Langevin dynamics (SGLD) with constant learning rate under the assumptions of dissipativity and smoothness, a setting that has received increased attention in the sampling/optimization literature. Unlike existing bounds for SGLD in non-convex settings, ours are time-independent and decay to zero as the sample size increases. Using the framework of uniform stability, we establish time-independent bounds by exploiting the Wasserstein contraction property of the Langevin diffusion, which also allows us to circumvent the need to bound gradients using Lipschitz-like assumptions. Our analysis also supports variants of SGLD that use different discretization methods, incorporate Euclidean projections, or use non-isotropic noise.