On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes
(1805.08114v3)
Published 21 May 2018 in stat.ML, cs.LG, and math.OC
Abstract: Stochastic gradient descent is the method of choice for large scale optimization of machine learning objective functions. Yet, its performance is greatly variable and heavily depends on the choice of the stepsizes. This has motivated a large body of research on adaptive stepsizes. However, there is currently a gap in our theoretical understanding of these methods, especially in the non-convex setting. In this paper, we start closing this gap: we theoretically analyze in the convex and non-convex settings a generalized version of the AdaGrad stepsizes. We show sufficient conditions for these stepsizes to achieve almost sure asymptotic convergence of the gradients to zero, proving the first guarantee for generalized AdaGrad stepsizes in the non-convex setting. Moreover, we show that these stepsizes allow to automatically adapt to the level of noise of the stochastic gradients in both the convex and non-convex settings, interpolating between $O(1/T)$ and $O(1/\sqrt{T})$, up to logarithmic terms.
On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes
The paper "On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes" by Xiaoyu Li and Francesco Orabona addresses an important issue in the optimization algorithms prominently used in machine learning: the convergence and performance variability of Stochastic Gradient Descent (SGD) owing to the choice of stepsizes. The authors focus on adaptive stepsizes that have gained popularity for their intuitive advantage in minimizing manual tuning of hyperparameters, yet lack comprehensive theoretical support, especially in the context of non-convex optimization problems.
Overview of Contributions
Asymptotic Convergence in Non-Convex Settings: The paper extends the theoretical understanding of adaptive stepsizes by demonstrating almost sure asymptotic convergence of gradients to zero for SGD using adaptive stepsizes akin to AdaGrad. This is claimed to be the first guarantee for such stepsizes in non-convex settings. The methodology does not require projections onto bounded sets, which aligns with real-world machine learning applications where such constraints do not typically apply.
Adaptive Convergence Rates: The authors analyze generalized AdaGrad stepsizes that adapt dynamically to the noise level present in stochastic gradients in both convex and non-convex scenarios. They show that these stepsizes allow SGD to interpolate between the convergence rates O(1/T) seen in deterministic gradient descent and O(1/T) typical in stochastic settings. The adaptive behavior is achieved without prior knowledge of the noise variance, which could reduce the parameter tuning efforts significantly.
Comprehensive Theoretical Guarantees: The paper provides both theoretical and probabilistic bounds that highlight the smooth transition between different convergence regimes depending on the noise characteristics. Through rigorous mathematical proofs, Li and Orabona establish conditions under which SGD with adaptive stepsizes will have better convergence rates, thus providing a solid foundation for further exploration and exploitation of these methods in practical applications.
Methodological Insights
The paper bases its analysis on a specific class of adaptive stepsizes, namely, a generalized version of the AdaGrad stepsizes. Two primary forms are considered: global and coordinate-wise step sizes, which adjust based on accumulated past gradient information. The authors introduce an additional parameter, epsilon, that allows fine-tuning the decrement rate of the stepsizes, critical for obtaining almost sure convergence results.
Practical and Theoretical Implications
The findings of this research have several implications on both the practical algorithms used for training machine learning models and the theoretical exploration of optimization methods:
Theory-Practice Conformance: By closing the theoretical gap on the convergence properties of adaptive stepsizes, the research provides a credible theoretical backing for the observed empirical efficacy of such methods in training large-scale deep learning models.
Reduction in Hyperparameter Tuning: The adaptive nature reduces reliance on exhaustive hyperparameter tuning, simplifying practical implementations and potentially accelerating the troubleshooting phases of machine learning model development.
Foundation for Future Research: The theoretical insights offered could direct future explorations related to parameter-free optimization, non-convex stochastic optimization, and even potential modifications or enhancements to existing algorithms.
Future Directions
The authors acknowledge the necessity to further refine their analysis, especially concerning obtaining bounds that depend logarithmically on the confidence level rather than polynomially. Addressing these challenges could further enhance the usability of adaptive algorithms in scenarios requiring stringent convergence guarantees. Another potential avenue of continued research is relaxing some of the assumptions, such as noise boundedness, which may not always hold in more complex real-world scenarios.
In conclusion, this paper robustly addresses the convergence characteristics of SGD with adaptive stepsizes in non-convex settings, thereby substantially contributing to the field of optimization in machine learning. It sets the stage for eliminating manual tuning challenges and enhancing algorithmic robustness through an adaptive methodology.