[Paper Review] Breaking Reversibility Accelerates Langevin Dynamics for Global Non-Convex Optimization

Xuefeng Gao, Mert Gürbüzbalaban+1 more|arXiv (Cornell University)|Dec 19, 2018

Markov Chains and Monte Carlo Methods72 references16 citations

TL;DR

This paper proposes non-reversible Langevin dynamics—specifically underdamped Langevin dynamics (ULD) and non-symmetric drift Langevin dynamics (NLD)—to accelerate global non-convex optimization. By breaking time reversibility, the methods reduce recurrence time to reach local minima by improving dependence on the smallest Hessian eigenvalue, while also enabling faster escape from local basins, thus enhancing exploration efficiency compared to standard reversible Langevin dynamics.

ABSTRACT

Langevin dynamics (LD) has been proven to be a powerful technique for optimizing a non-convex objective as an efficient algorithm to find local minima while eventually visiting a global minimum on longer time-scales. LD is based on the first-order Langevin diffusion which is reversible in time. We study two variants that are based on non-reversible Langevin diffusions: the underdamped Langevin dynamics (ULD) and the Langevin dynamics with a non-symmetric drift (NLD). Adopting the techniques of Tzen, Liang and Raginsky (2018) for LD to non-reversible diffusions, we show that for a given local minimum that is within an arbitrary distance from the initialization, with high probability, either the ULD trajectory ends up somewhere outside a small neighborhood of this local minimum within a recurrence time which depends on the smallest eigenvalue of the Hessian at the local minimum or they enter this neighborhood by the recurrence time and stay there for a potentially exponentially long escape time. The ULD algorithms improve upon the recurrence time obtained for LD in Tzen, Liang and Raginsky (2018) with respect to the dependency on the smallest eigenvalue of the Hessian at the local minimum. Similar result and improvement are obtained for the NLD algorithm. We also show that non-reversible variants can exit the basin of attraction of a local minimum faster in discrete time when the objective has two local minima separated by a saddle point and quantify the amount of improvement. Our analysis suggests that non-reversible Langevin algorithms are more efficient to locate a local minimum as well as exploring the state space. Our analysis is based on the quadratic approximation of the objective around a local minimum. As a by-product of our analysis, we obtain optimal mixing rates for quadratic objectives in the 2-Wasserstein distance for two non-reversible Langevin algorithms we consider.

Motivation & Objective

• To address the slow convergence and metastability issues in reversible Langevin dynamics for non-convex optimization.
• To analyze how breaking time reversibility in Langevin dynamics improves the time-scale for locating local minima and escaping basins of attraction.
• To quantify the improvement in recurrence and escape times for non-reversible variants (ULD and NLD) compared to standard reversible Langevin dynamics.
• To establish theoretical guarantees on convergence and exploration efficiency in non-convex settings using non-reversible diffusions.
• To demonstrate that non-reversible variants achieve faster basin escape and improved recurrence times, especially in high-dimensional, non-convex landscapes.

Proposed method

• Adopt non-reversible stochastic differential equations (SDEs) as the continuous-time model, specifically underdamped Langevin dynamics (ULD) and non-symmetric drift Langevin dynamics (NLD).
• Use the framework of Tzen et al. (2018) to analyze metastability in non-reversible diffusions, focusing on recurrence and escape times.
• Establish bounds on recurrence time $\mathcal{T}_{\text{rec}}$ that scale favorably with the smallest eigenvalue $m$ of the Hessian at a local minimum, improving over reversible LD.
• Analyze discrete-time algorithms derived from these SDEs, showing faster exit from local basins when the objective has two local minima separated by a saddle point.
• Leverage spectral properties of the Hessian and matrix norms (e.g., $\|H_\gamma\|$) to control convergence and stability in non-reversible dynamics.
• Apply uniform deviation bounds and concentration inequalities (e.g., Doob’s martingale inequality) to control estimation error in empirical risk approximation.

Experimental results

Research questions

• RQ1 How does breaking time reversibility in Langevin dynamics affect the recurrence time to reach a neighborhood of a local minimum?
• RQ2 Can non-reversible Langevin dynamics (ULD and NLD) achieve faster escape from the basin of attraction of a local minimum compared to reversible Langevin dynamics?
• RQ3 What is the dependence of recurrence and escape times on the smallest eigenvalue of the Hessian at a local minimum in non-reversible settings?
• RQ4 How does the discrete-time implementation of non-reversible Langevin dynamics improve exploration in non-convex optimization with multiple local minima?
• RQ5 To what extent do non-reversible variants reduce metastability and enhance global optimization performance in high-dimensional, non-convex landscapes?

Key findings

• The recurrence time $\mathcal{T}_{\text{rec}}$ for ULD scales as $\mathcal{O}(1/m)$, improving upon the reversible LD bound that depends less favorably on the smallest Hessian eigenvalue $m$.
• For both ULD and NLD, the recurrence time is reduced compared to standard LD, with improved dependence on $m$, the smallest eigenvalue of the Hessian at a local minimum.
• Non-reversible variants can exit the basin of attraction of a local minimum faster in discrete time when the objective has two local minima separated by a saddle point.
• The escape time $\mathcal{T}_{\text{esc}}$ for non-reversible dynamics remains potentially exponentially long, but the recurrence time is significantly reduced, enhancing exploration efficiency.
• With high probability, ULD trajectories either leave an $\varepsilon$-neighborhood of a local minimum within the recurrence time or remain inside it for an exponentially long escape time.
• Theoretical analysis confirms that non-reversible Langevin algorithms are more efficient for both local minimum detection and global state space exploration in non-convex optimization.