Faster Convergence of Stochastic Accelerated Gradient Descent under Interpolation (2404.02378v1)

Abstract: We prove new convergence rates for a generalized version of stochastic Nesterov acceleration under interpolation conditions. Unlike previous analyses, our approach accelerates any stochastic gradient method which makes sufficient progress in expectation. The proof, which proceeds using the estimating sequences framework, applies to both convex and strongly convex functions and is easily specialized to accelerated SGD under the strong growth condition. In this special case, our analysis reduces the dependence on the strong growth constant from $\rho$ to $\sqrt{\rho}$ as compared to prior work. This improvement is comparable to a square-root of the condition number in the worst case and address criticism that guarantees for stochastic acceleration could be worse than those for SGD.

Faster Convergence Rates for Stochastic Nesterov Acceleration in Deep Learning

Convergence Under Interpolation Conditions

Recent advancements in deep learning have highlighted the efficacy of over-parameterized models that are capable of interpolating the training data perfectly. This paper, authored by Mishkin, Pilanci, and Schmidt, focuses on enhancing our understanding of stochastic acceleration techniques within this context, particularly examining Stochastic Accelerated Gradient Descent (SAGD). The authors propose an improved analysis for SAGD under the interpolation condition—a scenario where the learning models perfectly fit the training data. The paper makes a significant contribution by demonstrating that stochastic algorithms can achieve accelerated convergence rates, similar to deterministic counterparts, under this framework.

Theoretical Insights

The authors base their work on the premises of the interpolation condition, extending the reach of Nesterov's accelerated methods to stochastic gradient methods. The key innovation lies in proving accelerated convergence rates using a generalized analysis framework that applies to both convex and strongly convex functions. A novel aspect of this analysis is the reduction of dependency on the strong growth constant in deriving convergence rates, which signifies an improvement over previous works.

Numerical Results and Speculation

Although the paper primarily explores theoretical analyses, the implications of these findings strongly suggest potential practical advancements in training deep neural networks. By improving the efficiency of stochastic gradient descent methods in the over-parameterized regime, this work paves the way for faster and more computationally efficient training processes for large-scale models. Future research may expand on the practical applications and effectiveness of these theoretical improvements in stochastic acceleration.

Comparison with Previous Work

The paper highlights how existing analyses under strong growth conditions exhibit linear dependence on the strong growth constant, potentially rendering stochastic acceleration slower than straightforward SGD. The authors' approach distinguishes itself by offering a squared-root dependence on the growth constant, outperforming previous bounds and ensuring that stochastic accelerated methods offer genuine acceleration. This comparison with existing literature not only underscores the novelty of the paper's contribution but also sets a new benchmark for analyzing stochastic acceleration methods.

Acceleration with Preconditioning

An interesting extension discussed is the applicability of the proposed method to stochastic AGD with full matrix preconditioning. Preconditioning, an approach to modify the geometry of the optimization problem, can further enhance the convergence rates if the preconditioner is appropriately chosen. This insight opens myriad possibilities for further exploration, particularly in the context of optimizing deep neural network training processes.

Conclusion and Future Directions

Mishkin, Pilanci, and Schmidt have made a compelling case for the accelerated convergence of stochastic gradient methods under interpolation. This work not only refines our theoretical understanding but also holds promise for substantial practical impacts on training deep learning models. Looking ahead, several avenues for future research emerge, including the exploration of stochastic AGD under relaxed conditions and the development of adaptive methods to leverage the findings in a broader range of applications.

In summary, this paper makes a significant theoretical advancement by establishing improved convergence rates for stochastic accelerated gradient methods under interpolation, offering potential pathways to more efficient algorithmic frameworks in machine learning.