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User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient (1710.00095v4)

Published 29 Sep 2017 in math.ST, cs.LG, math.PR, stat.CO, stat.ML, and stat.TH

Abstract: In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We provide nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.

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Authors (2)
  1. Arnak S. Dalalyan (26 papers)
  2. Avetik G. Karagulyan (1 paper)
Citations (284)
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Summary

  • The paper introduces a refined error bound for the first-order LMC algorithm using horizon-free varying step-size, improving precision logarithmically.
  • The paper establishes new upper bounds for LMC algorithms with both deterministic and stochastic inaccurate gradient evaluations, quantifying their impact on sampling error.
  • The paper demonstrates that second-order LMC methods yield non-asymptotic guarantees and enhanced performance in poorly conditioned, high-dimensional settings.

Overview of User-friendly Guarantees for Langevin Monte Carlo with Inaccurate Gradient

The paper presents an analytical investigation into the performance of Langevin Monte Carlo (LMC) algorithms, particularly focusing on scenarios where the gradient evaluations are noisy. The authors tackled the problem of sampling from a smooth and strongly log-concave probability density function by analyzing discretized versions of Langevin diffusion. The paper provides significant improvements and extensions on the error bounds and convergence rates, especially when measured in the Wasserstein-2 distance.

Key Contributions

  1. Error Bounds with Varying Step-size: The paper introduces a refined upper bound for the first-order LMC algorithm by optimizing the step-size variation. This approach is horizon-free, allowing the sampling precision to improve by a logarithmic factor compared to a constant step-size setup.
  2. Handling Inaccurate Gradient Evaluations: The authors extend their analysis to situations where the gradient of the log-density cannot be accurately evaluated. They consider both deterministic and stochastic approximations of the gradient, providing new upper bounds that quantify how inaccuracies in gradient evaluation impact the sampling error of the LMC.
  3. Second-order LMC Analysis: Furthermore, the paper establishes upper bounds for enhanced versions of LMC algorithms that utilize the Hessian of the log-density (second-order LMC). Non-asymptotic guarantees indicate improved performance of these second-order methods in poorly conditioned settings.

Analytical Insights

  • Wasserstein Distance: The paper employs the Wasserstein-2 distance as the primary metric for assessing sampling error, arguing its suitability over other metrics like the total variation or Kullback-Leibler divergence due to its ability to directly guarantee the accuracy of approximating first and second-order moments.
  • Recursive Inequalities and Convergence Rates: Several lemmas establish recursive inequalities for the error terms, which are pivotal in deriving the convergence rates for the LMC algorithms both with accurate and noisy gradients. These recursive formulas are instrumental in providing actionable and simplified sampling guarantees.

Implications and Future Directions

The authors' work has immediate implications in the field of sampling from high-dimensional log-concave distributions, providing tools and guidelines that are especially important when handling high dimensions and considering computational constraints. The advanced bounds in scenarios of inaccurate gradients pave the way for practical algorithms in settings where full gradient information is either impossible or computationally costly.

In theoretical terms, the paper enriches the understanding of how Langevin-based algorithms perform under uncertainty. Practically, these methods have direct applications in machine learning, statistics, and any domain reliant on efficient posterior sampling or approximation.

The future research direction might involve exploring more complex log-concave structures, including those involving non-smooth elements, or expanding these guarantees to scalable, distributed computational environments. Additionally, the exploration of lower bounds for sampling would further enhance our understanding of the efficiency and limits of these algorithms.

The paper is an exemplary demonstration of how theoretical advances can align closely with practical applicability, providing both refined methods and profound insights into the field of stochastic processes and their applications in machine learning and beyond.

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