Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization (1707.06618v3)

Abstract: We present a unified framework to analyze the global convergence of Langevin dynamics based algorithms for nonconvex finite-sum optimization with $n$ component functions. At the core of our analysis is a direct analysis of the ergodicity of the numerical approximations to Langevin dynamics, which leads to faster convergence rates. Specifically, we show that gradient Langevin dynamics (GLD) and stochastic gradient Langevin dynamics (SGLD) converge to the almost minimizer within $\tilde O\big(nd/(\lambda\epsilon) \big)$ and $\tilde O\big(d^7/(\lambda^5\epsilon^5) \big)$ stochastic gradient evaluations respectively, where $d$ is the problem dimension, and $\lambda$ is the spectral gap of the Markov chain generated by GLD. Both results improve upon the best known gradient complexity results (Raginsky et al., 2017). Furthermore, for the first time we prove the global convergence guarantee for variance reduced stochastic gradient Langevin dynamics (SVRG-LD) to the almost minimizer within $\tilde O\big(\sqrt{n}d^5/(\lambda^4\epsilon^{5/2})\big)$ stochastic gradient evaluations, which outperforms the gradient complexities of GLD and SGLD in a wide regime. Our theoretical analyses shed some light on using Langevin dynamics based algorithms for nonconvex optimization with provable guarantees.

• The paper establishes global convergence guarantees for GLD, SGLD, and SVRG-LD with precise iteration and gradient evaluation bounds.
• It leverages the ergodicity of Markov chains in Langevin dynamics to efficiently escape local minima and approach global optima.
• The analysis highlights that variance reduction in SVRG-LD offers superior performance for nonconvex problems compared to traditional methods.

Overview of Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization

The paper "Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization" presents a detailed analytical framework to establish the global convergence properties of Langevin dynamics-based algorithms applied to nonconvex finite-sum optimization problems. The authors focus on optimizing functions composed of $n$ nonconvex components, a task typically considered computationally challenging due to the NP-hard nature of nonconvex optimization.

Algorithmic Analysis

The paper introduces three primary algorithms: Gradient Langevin Dynamics (GLD), Stochastic Gradient Langevin Dynamics (SGLD), and Stochastic Variance Reduced Gradient Langevin Dynamics (SVRG-LD). All these algorithms incorporate Langevin dynamics, which is a stochastic process leveraging the gradient of the objective function and additional noise (in the form of Brownian motion), underpinning their approach to escape local minima and potentially reach global solutions. The convergence analysis is centered on the ergodicity of the Markov chains generated by numerical approximations to Langevin dynamics, fundamentally distinct from traditional gradient descent approaches.

Convergence Results

The paper provides precise convergence rate analyses, establishing:

1. GLD: The algorithm converges to an almost minimizer within $\tilde O\big(nd/(\lambda\epsilon) \big)$ iterations, surpassing previous known results in gradient complexity.
2. SGLD: Achieves convergence within $\tilde O\big(d^7/(\lambda^5\epsilon^5) \big)$ stochastic gradient evaluations, further improving upon prior analytical bounds.
3. SVRG-LD: For the first time globally analyzed within this context, converges under $$\tilde O\big(\sqrt{n}d^5/(\lambda^4\epsilon^{5/2})\big)$$ evaluations, offering superior performance in scenarios of moderate component functions count and precision requirements.

Numerical Insights

The convergence is intricately tied to the spectral gap $\lambda$ of the Markov chain generated by the algorithms, with the spectral gap scaling being costly in dimension $d$ due to its exponential dependence. However, this analysis asserts improvements over the state-of-the-art by circumventing the discretization error associated with continuous Markov processes.

Practical and Theoretical Implications

Practically, these findings guide the application of Langevin dynamics in machine learning tasks requiring nonconvex optimization, such as deep learning model training, highlighting scenarios where variance reduction techniques like SVRG-LD can be more advantageous compared to GLD and SGLD. The theoretical implications extend to understanding ergodic behavior in approximate sampling methodologies, which are intimately connected to optimization processes.

Speculative Future Work

Future research can focus on enhancing spectral gap analyses, as improving this could lead toward more scalable solutions in higher dimensions. Moreover, delving deeper into the ergodic properties and numerical approximation methods might open pathways for faster convergence rates, essential for practical implementations. Adaptive schemes that optimize the inverse temperature parameter $\beta$ could also yield further insights into dynamically tailoring algorithms to specific problem structures.

In conclusion, this paper significantly advances the analytical understanding of Langevin dynamics-based algorithms, positioning them as promising methods for tackling complex nonconvex optimization challenges.