Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling
This provides theoretical guarantees for annealing techniques in MCMC, addressing a gap for practitioners in computational statistics and machine learning dealing with complex distributions.
The paper tackles the problem of sampling from non-log-concave, multimodal densities by analyzing annealed Langevin Monte Carlo, establishing a non-asymptotic oracle complexity of $\widetilde{O}\left( rac{deta^2\mathcal{A}^2}{\varepsilon^6} ight)$ to achieve $\varepsilon^2$ accuracy in KL divergence.
We consider the outstanding problem of sampling from an unnormalized density that may be non-log-concave and multimodal. To enhance the performance of simple Markov chain Monte Carlo (MCMC) methods, techniques of annealing type have been widely used. However, quantitative theoretical guarantees of these techniques are under-explored. This study takes a first step toward providing a non-asymptotic analysis of annealed MCMC. Specifically, we establish, for the first time, an oracle complexity of $\widetilde{O}\left(\frac{dβ^2{\cal A}^2}{\varepsilon^6}\right)$ for the simple annealed Langevin Monte Carlo algorithm to achieve $\varepsilon^2$ accuracy in Kullback-Leibler divergence to the target distribution $π\propto{\rm e}^{-V}$ on $\mathbb{R}^d$ with $β$-smooth potential $V$. Here, ${\cal A}$ represents the action of a curve of probability measures interpolating the target distribution $π$ and a readily sampleable distribution.