Stochastic Langevin Monte Carlo for (weakly) log-concave posterior distributions
This work addresses computational efficiency in Bayesian inference for machine learning, providing theoretical guarantees for a broader class of distributions, though it is incremental as it extends prior methods to weakly convex settings.
The paper tackles the problem of sampling from weakly log-concave posterior distributions using a stochastic Langevin Monte Carlo method, establishing a computational cost bound of order (d log(n)^2)^(1+r)^2 [log^2(ε^{-1}) + n^2 d^{2(1+r)} log^{4(1+r)}(n)] for an ε approximation in entropy.
In this paper, we investigate a continuous time version of the Stochastic Langevin Monte Carlo method, introduced in [WT11], that incorporates a stochastic sampling step inside the traditional over-damped Langevin diffusion. This method is popular in machine learning for sampling posterior distribution. We will pay specific attention in our work to the computational cost in terms of $n$ (the number of observations that produces the posterior distribution), and $d$ (the dimension of the ambient space where the parameter of interest is living). We derive our analysis in the weakly convex framework, which is parameterized with the help of the Kurdyka-Łojasiewicz (KL) inequality, that permits to handle a vanishing curvature settings, which is far less restrictive when compared to the simple strongly convex case. We establish that the final horizon of simulation to obtain an $\varepsilon$ approximation (in terms of entropy) is of the order $( d \log(n)^2 )^{(1+r)^2} [\log^2(\varepsilon^{-1}) + n^2 d^{2(1+r)} \log^{4(1+r)}(n) ]$ with a Poissonian subsampling of parameter $\left(n ( d \log^2(n))^{1+r}\right)^{-1}$, where the parameter $r$ is involved in the KL inequality and varies between $0$ (strongly convex case) and $1$ (limiting Laplace situation).