Rigorous dynamical mean field theory for stochastic gradient descent methods

The evolution of deep learning has driven intense interest in understanding the dynamics of large-scale optimization algorithms. Prior works, such as [20], applied continuous-time DMFT from statistical physics to analyze gradient flow, providing qualitative insights. However, the discrete-time nature of practical algorithms like SGD and momentum methods remains less rigorously understood. Existing studies often rely on heuristic or approximate methods, with limited scope for non-Gaussian data or complex regularizations. Bridging this gap requires developing rigorous discrete-time theories capable of handling realistic data distributions and algorithmic variations.

The core challenge is to derive exact asymptotic equations describing the high-dimensional behavior of stochastic gradient algorithms in discrete time, considering realistic data covariance structures, mini-batch sampling, and non-linear regularizations. Existing approaches lack rigorous proofs for these settings, limiting their predictive power and theoretical grounding. Accurately modeling the influence of stochastic noise, memory effects, and data correlations remains a fundamental obstacle, impeding systematic optimization and understanding of convergence and generalization in deep learning.

This paper's key innovations include: 1) a rigorous derivation of discrete-time DMFT equations for a broad class of gradient algorithms, 2) incorporation of arbitrary positive definite data covariance matrices, 3) handling of non-separable regularizations and non-linear update functions, 4) use of iterative Gaussian conditioning to obtain explicit formulas, and 5) validation through extensive numerical experiments. These advances extend the physics-inspired continuous-time models to practical discrete algorithms, enabling precise analysis of training dynamics under realistic conditions, and providing a solid theoretical foundation for future research.

• �� Model the gradient algorithms as recursive equations involving memory kernels and stochastic effects.
• �� Use iterative Gaussian conditioning to transform high-dimensional dynamics into low-dimensional Gaussian processes with explicit covariance structures.
• �� Incorporate mini-batch sampling by modeling stochastic gradients as Gaussian noise with controlled variance.
• �� Derive recursive equations for the state variables, including iterates, preactivations, and auxiliary fields, accounting for data covariance and regularization.
• �� Prove convergence of these equations in high dimensions by establishing concentration bounds and stability properties.
• �� Validate formulas via numerical simulations on synthetic Gaussian datasets, varying batch sizes, learning rates, and covariance structures.

Simulations involved synthetic Gaussian datasets with controlled covariance matrices, testing various batch sizes (e.g., 0.1, 0.5, 1), learning rates (e.g., 0.01, 0.1), and regularization parameters. The algorithms included vanilla SGD, momentum, and Langevin dynamics. Results compared the empirical distribution of iterates and preactivations with theoretical predictions, showing errors decreasing as 1/√d. Additional experiments with non-identity covariance matrices confirmed the formulas' robustness. Ablation studies examined the impact of memory kernels and stochastic noise, highlighting their roles in convergence and stability.

The derived equations precisely predict the evolution of the algorithm's parameters in high dimensions, matching empirical data within statistical error bounds. They reveal how mini-batch size, learning rate, and data covariance influence convergence speed and noise accumulation. The formulas also elucidate the buildup of memory effects, explaining phenomena observed in deep neural network training, such as delayed convergence and stability issues. These results provide a rigorous theoretical basis for hyperparameter tuning and algorithm design in large-scale machine learning.

The framework can guide practitioners in selecting optimal batch sizes, learning rates, and regularization schemes for training large neural networks. It also offers insights into the stability and generalization properties of stochastic algorithms, aiding in the development of more robust training protocols. Furthermore, the formulas can be integrated into automated hyperparameter optimization tools, accelerating the deployment of deep learning models in industry. Long-term, this work paves the way for a unified theoretical understanding of high-dimensional optimization in AI systems.

The analysis assumes Gaussian data, which may not hold in real-world scenarios with complex distributions. Computational costs of solving the recursive equations increase with model size, limiting scalability. The current model does not incorporate non-stationary data or adaptive hyperparameters, which are common in practice. Extending the theory to these settings remains an open challenge, requiring further methodological innovations.