What is the long-run distribution of stochastic gradient descent? A large deviations analysis (2406.09241v2)

Abstract: In this paper, we examine the long-run distribution of stochastic gradient descent (SGD) in general, non-convex problems. Specifically, we seek to understand which regions of the problem's state space are more likely to be visited by SGD, and by how much. Using an approach based on the theory of large deviations and randomly perturbed dynamical systems, we show that the long-run distribution of SGD resembles the Boltzmann-Gibbs distribution of equilibrium thermodynamics with temperature equal to the method's step-size and energy levels determined by the problem's objective and the statistics of the noise. In particular, we show that, in the long run, (a) the problem's critical region is visited exponentially more often than any non-critical region; (b) the iterates of SGD are exponentially concentrated around the problem's minimum energy state (which does not always coincide with the global minimum of the objective); (c) all other connected components of critical points are visited with frequency that is exponentially proportional to their energy level; and, finally (d) any component of local maximizers or saddle points is "dominated" by a component of local minimizers which is visited exponentially more often.

• The paper reveals that SGD’s long-run behavior converges to a Boltzmann-Gibbs distribution, linking step-size to an effective temperature.
• It employs a cost matrix and transition graph based on large deviations theory to quantify transition probabilities among critical points.
• The study shows that SGD iterates concentrate exponentially near local minimizers while largely avoiding saddle points and local maxima.

Overview of the Long-Run Distribution of Stochastic Gradient Descent

The paper "What is the Long-Run Distribution of Stochastic Gradient Descent? A Large Deviations Analysis" explores the stochastic gradient descent (SGD) algorithm and its behavior in the context of non-convex optimization problems. The authors provide a comprehensive analysis of the long-term behavior of SGD using large deviations theory and demonstrate that the distribution of SGD's iterates in the long run resembles a Boltzmann-Gibbs distribution.

Key Contributions

1. Boltzmann-Gibbs Distribution Underlying SGD: The paper establishes that in the presence of stochastic noise, the long-term distribution of SGD mimics the Boltzmann-Gibbs distribution. This distribution, well-known in statistical physics, plays a crucial role in equilibrium thermodynamics. The authors link the SGD's step-size to the "temperature" of this system, with the energy levels determined by the problem's objective function and the noise statistics.
2. Cost Matrix and Transition Graph: The paper introduces a cost matrix to quantify the probable transitions between different critical components of the problem's state space. A complete weighted directed graph, termed the transition graph, is used to encode these transitions. This graph facilitates understanding how SGD iterates move between local minimizers, saddle points, and maxima.
3. Exponential Concentration Near Critical Regions: It is shown that in the asymptotic sense, SGD iterates are exponentially likely to be located close to the critical points of the objective function, with a pronounced bias towards regions minimizing a certain energy functional. Notably, this does not always coincide with the global minimum.
4. Avoidance of Non-Minimizing Critical Points: The work further establishes that regions corresponding to local maxima or saddle points are exponentially less likely to be explored by SGD compared to local minimizers.
5. Potential Function and Invariant Measure: Building on large deviations theory, the analysis frames a potential function that aids in quantifying the likelihood of transitions. The authors provide conditions under which invariant measures exist and characterize these measures in the context of subsampled SGD dynamics.

Methodological Approach

• Large Deviations Principle (LDP): The authors employed LDP to assess the likelihood of rare events in SGD, such as moving against the gradient flow for extended periods. By setting up a framework akin to Hamiltonian mechanics, the paper provides a principled approach to identifying low-action trajectories, which are more likely to be observed under SGD.
• Quasi-potential and Transition Costs: The quasi-potential, a key construct from the theory of randomly perturbed systems, is used to understand the energy landscape of SGD. The authors define specific transition costs that provide insights into the system's metastability and transitions between critical areas.

Theoretical and Practical Implications

• Theoretically, the work bridges a gap between optimization on non-convex landscapes and ideas from statistical physics, offering a novel interpretation of SGD's long-term behavior in terms of thermodynamic ensembles. This provides a deeper understanding of the conditions under which SGD gravitates toward certain solutions over others.
• Practically, this insight is pivotal for tasks like hyperparameter tuning in machine learning, where understanding the dynamics of training algorithms helps in better stabilizing and enhancing model convergence, especially in large networks encountered in deep learning.

Future Directions

The paper invites several intriguing directions for future research:

• Noise Models and Convergence Analysis: Extending the results to investigate different noise models and their convergence rates could yield broader insights into SGD's robustness and effectiveness in diverse settings.
• Quantitative Generalization Insights: Understanding how energy landscapes influence generalization could lead to new approaches for model design in neural networks, especially concerning SGD's exploration-exploitation trade-offs.
• Adapting to Unbounded Domains: Further refinement is needed to address potential scalability issues when dealing with unbounded domains, which frequently appear in real-world problems.

This comprehensive examination of SGD's long-term dynamics underscores the intricacies underlying non-convex optimization and highlights fundamental principles guiding iterative algorithm behavior in noisy environments. The marriage of thermodynamics with optimization presents a compelling framework that merits further exploration in the frontiers of algorithmic design and analysis.