Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis (1702.03849v3)

Abstract: Stochastic Gradient Langevin Dynamics (SGLD) is a popular variant of Stochastic Gradient Descent, where properly scaled isotropic Gaussian noise is added to an unbiased estimate of the gradient at each iteration. This modest change allows SGLD to escape local minima and suffices to guarantee asymptotic convergence to global minimizers for sufficiently regular non-convex objectives (Gelfand and Mitter, 1991). The present work provides a nonasymptotic analysis in the context of non-convex learning problems, giving finite-time guarantees for SGLD to find approximate minimizers of both empirical and population risks. As in the asymptotic setting, our analysis relates the discrete-time SGLD Markov chain to a continuous-time diffusion process. A new tool that drives the results is the use of weighted transportation cost inequalities to quantify the rate of convergence of SGLD to a stationary distribution in the Euclidean $2$-Wasserstein distance.

• The paper provides finite-time guarantees for SGLD’s convergence in non-convex settings, ensuring approximate minimizers for both empirical and population risks.
• It employs 2-Wasserstein distance and logarithmic Sobolev inequalities to rigorously bridge discrete SGLD steps with continuous Langevin diffusion.
• The analysis offers practical insights into computational trade-offs and generalization performance for complex machine learning models.

Non-Convex Learning via Stochastic Gradient Langevin Dynamics: A Nonasymptotic Analysis

This paper presents a nonasymptotic analysis of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm for non-convex learning problems. SGLD, a variant of Stochastic Gradient Descent (SGD), integrates isotropic Gaussian noise into the gradient estimate, facilitating the escape from local minima and ensuring asymptotic convergence to global minimizers in suitable conditions.

Key Contributions

1. Finite-Time Guarantees: The authors provide finite-time guarantees for SGLD in finding approximate minimizers of both empirical and population risks. This contrasts with prior asymptotic analyses that were limited in their practical applications.
2. Analysis Through 2-Wasserstein Distance: The analysis employs weighted transportation cost inequalities to quantify the rate at which SGLD converges to a stationary distribution, measured in Euclidean 2-Wasserstein distance. This novel approach links discrete-time SGLD to a continuous-time diffusion process.
3. Convergence and Stability: The paper establishes that SGLD can track the Langevin diffusion accurately. A significant contribution is the demonstration of finite-time convergence for a non-convex objective. The stability of the Gibbs algorithm, which samples from near-minimizers of the population risk, is also shown.

Main Results

The main theoretical result states that for any small $\epsilon > 0$, the excess risk is upper-bounded by terms scaling with spectral gap parameters and problem dimensions. Specifically, the authors derive bounds involving the parameters $\beta$ (inverse temperature), $\lambda_*$ (spectral gap), and dimension $d$, leading to performance guarantees for certain choices of $\eta$ (step size) and $k$ (number of iterations).

Methodology

The methodology hinges on several key steps:

• Markov Process Analysis: SGLD is viewed as a Markov process. The discrete interactions of this process are shown to approximate a continuous diffusion process strongly.
• Logarithmic Sobolev Inequalities: These are utilized to establish rapid convergence of the diffusion process to the Gibbs distribution. The use of spectral gaps is critical in these proofs.
• Complexity and Iteration Analysis: The iteration complexity for achieving $\epsilon$-approximation is carefully derived as a function of problem parameters.

Implications

• Practical Impact: The findings provide practitioners with theoretical assurances on the performance of SGLD for complex non-convex problems common in machine learning, like neural network training.
• Implications for Generalization: The analysis of generalization error highlights the implications for model performance on unseen data, connecting empirical risk minimizers with generalized optimal performance.

Future Directions

• Structural Problem Analysis: While theoretical guarantees are provided, the tractability of these guarantees hinges on problem-specific spectral properties. Future work could explore methods to identify such properties across a broader class of functions.
• Computational Trade-offs: Further exploration is needed on the computational trade-offs between iteration cost and excess risk, particularly in high-dimensional settings.

Overall, this paper significantly enhances the understanding of SGLD in non-convex settings, offering both theoretical and practical insights into its applications within machine learning and optimization.