SGD: General Analysis and Improved Rates (1901.09401v4)

Abstract: We propose a general yet simple theorem describing the convergence of SGD under the arbitrary sampling paradigm. Our theorem describes the convergence of an infinite array of variants of SGD, each of which is associated with a specific probability law governing the data selection rule used to form mini-batches. This is the first time such an analysis is performed, and most of our variants of SGD were never explicitly considered in the literature before. Our analysis relies on the recently introduced notion of expected smoothness and does not rely on a uniform bound on the variance of the stochastic gradients. By specializing our theorem to different mini-batching strategies, such as sampling with replacement and independent sampling, we derive exact expressions for the stepsize as a function of the mini-batch size. With this we can also determine the mini-batch size that optimizes the total complexity, and show explicitly that as the variance of the stochastic gradient evaluated at the minimum grows, so does the optimal mini-batch size. For zero variance, the optimal mini-batch size is one. Moreover, we prove insightful stepsize-switching rules which describe when one should switch from a constant to a decreasing stepsize regime.

Analyzing Stochastic Gradient Descent: Convergence and Optimal Sampling Strategies

The paper "SGD: General Analysis and Improved Rates" contributes to the theoretical understanding of Stochastic Gradient Descent (SGD) by developing a generalized framework to analyze its convergence properties under diverse sampling paradigms. Specifically, it establishes formulations that provide insight into the role of different data sampling schemes when conducting mini-batch SGD and investigates the expected smoothness, which influences the algorithm's convergence rates.

Key Contributions

The authors propose a unified theorem for the convergence of SGD that accommodates arbitrary sampling schemes. This approach allows the examination of a broad spectrum of mini-batch strategies, revealing how different sampling probabilities influence the stepsize and total computational complexity. Importantly, the paper derives closed-form expressions for optimal stepsizes and mini-batch sizes for various sampling strategies, such as independent sampling and $\tau$-nice sampling. Notable findings include the result that as the variance of stochastic gradients at the optimum increases, so does the optimal mini-batch size, emphasizing the importance of considering gradient variance in designing efficient SGD methodologies.

A central innovation of this work is the introduction of "expected smoothness", a concept that relaxes some of the traditional assumptions about uniform boundedness of gradient variances. This assumption makes the analysis more robust and applicable to a wider set of practical scenarios, especially when individual component functions may not be convex. Additionally, the paper shows that using expected smoothness leads to linear convergence results without assuming strong convexity in terms of the original optimization problem.

Theoretical Implications

The theoretical implications of this paper are significant, as they expand the general understanding of how sampling affects SGD's performance. By deriving precise expressions for both expected smoothness and gradient noise under various sampling methods, this analysis provides a toolkit for predicting SGD's convergence behavior in a given problem context. The identification of optimal switching from constant to decreasing stepsize provides further practical guidance on effectively managing learning rates throughout the optimization process.

In contexts where SGD is employed to solve over-parameterized models, as seen in many contemporary deep learning applications, the derived results indicate that the optimal mini-batch size tends to be smaller, potentially converging to a size of one when distributed gradient noise is zero, reflecting conditions similar to deterministic gradient descent.

Future Directions

The research opens several pathways for future exploration. Understanding the role of expected smoothness in non-convex settings can produce further insights into the nuances of mini-batch size selection and adaptive learning schedules. Additionally, exploring the generalization properties of models trained with sampling methods optimized according to this framework remains an intriguing domain yet to be fully investigated. Emerging trends in distributed and decentralized training further necessitate extensions of this analysis to scalable architectures, potentially uncovering new paradigms for coordinating SGD across multiple nodes efficiently.

Overall, this paper provides strong analytical foundations for the development of more nuanced and context-aware SGD implementations that can be calibrated for specific datasets and computational environments, thereby extending the versatility of SGD in tackling large-scale optimization problems.