Making Gradient Descent Optimal for Strongly Convex Stochastic Optimization (1109.5647v7)

Published 26 Sep 2011 in cs.LG and math.OC

Abstract: Stochastic gradient descent (SGD) is a simple and popular method to solve stochastic optimization problems which arise in machine learning. For strongly convex problems, its convergence rate was known to be O(\log(T)/T), by running SGD for T iterations and returning the average point. However, recent results showed that using a different algorithm, one can get an optimal O(1/T) rate. This might lead one to believe that standard SGD is suboptimal, and maybe should even be replaced as a method of choice. In this paper, we investigate the optimality of SGD in a stochastic setting. We show that for smooth problems, the algorithm attains the optimal O(1/T) rate. However, for non-smooth problems, the convergence rate with averaging might really be \Omega(\log(T)/T), and this is not just an artifact of the analysis. On the flip side, we show that a simple modification of the averaging step suffices to recover the O(1/T) rate, and no other change of the algorithm is necessary. We also present experimental results which support our findings, and point out open problems.

Authors (3)

Alexander Rakhlin (100 papers)
Ohad Shamir (110 papers)
Karthik Sridharan (58 papers)

Citations (746)

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Summary

Making Gradient Descent Optimal for Strongly Convex Stochastic Optimization

The paper "Making Gradient Descent Optimal for Strongly Convex Stochastic Optimization" by Alexander Rakhlin, Ohad Shamir, and Karthik Sridharan provides a rigorous investigation into the optimality of Stochastic Gradient Descent (SGD) under strongly convex stochastic settings. The authors focus on establishing and dissecting the convergence properties of SGD, specifically comparing its efficacy to more complex alternatives in the context of both smooth and non-smooth optimization problems.

Key Contributions

The primary contributions presented in this paper are:

Extension to Smooth Problems: The authors demonstrate that for smooth, strongly convex functions, standard SGD, both with and without averaging, achieves the optimal $O(1/T)$ convergence rate where $T$ represents the number of iterations.
Non-Smooth Problem Insight: For non-smooth problems, it is shown that the convergence rate of standard SGD with averaging degrades to $\Omega(\log(T)/T)$ . This indicates that the $O(\log(T)/T)$ rate is not merely an artifact of existing analyses but an intrinsic characteristic of the algorithm under these conditions.
Modification for Optimality: A significant insight is provided on modifying the averaging process in SGD. By averaging only the latter $\alpha T$ points (where $\alpha \in (0, 1)$ ), the optimal $O(1/T)$ rate can be restored even for non-smooth problems without necessitating complicated algorithmic overhauls.
Experimental Validation: Empirical studies bolster the theoretical findings, indicating that the proposed modification enhances performance on both artificial and real-world datasets.

Theoretical Foundations and Results

SGD is analyzed in the field of stochastic convex optimization. The baseline assumption considers SGD initialized at $w_1 = \mathbf{0}$ , iteratively refined using unbiased gradient estimates from the stochastic gradient oracle. The work particularly focuses on functions characterized by strong convexity and establishes enhanced theoretical guarantees for smooth functions via direct stochastic analysis:

Smooth Functions: When the objective function $F$ is both $\lambda$ -strongly convex and $\mu$ -smooth, it was established that the last iterate $w_T$ and the average of iterates $\bar{w}_T$ both exhibit the optimal $O(1/T)$ convergence rate for SGD. This is formally proven leveraging the properties of strong convexity and smoothness.
Non-Smooth Functions: The treatment of non-smooth functions reveals an intriguing duality: while traditional SGD with full averaging returns a suboptimal $\Omega(\log(T)/T)$ rate, a simple modification suffices to restore optimality. The $\alpha$ -suffix averaging strategy empirically and theoretically ensures that the convergence rate aligns with the optimal $O(1/T)$ . The alteration underscores practical relevance, maintaining simplicity without sacrificing performance.

Practical and Theoretical Implications

The results have compelling implications:

Algorithm Selection: Practitioners can continue to leverage SGD for its simplicity but with the nuanced modification proposed for instances involving non-smooth convex problems. The findings argue against the need for more complex algorithms under many practical circumstances, provided the proper averaging technique is employed.
High-Probability Bounds: The analysis goes beyond traditional expected bounds, supplying high-probability convergence assurances. While introducing an additional $\log(\log(T)/\delta)$ factor, it provides robust performance guarantees essential for real-world applications.
Future Developments: These insights prompt further research into adaptive averaging strategies and their theoretical underpinning across a broader array of optimization landscapes.

In conclusion, this work substantiates the efficacy of SGD in strongly convex settings, emphasizing smooth problem contexts while offering a practical remedy for non-smooth scenarios. The adjustable averaging approach pioneered here renders SGD both a theoretically sound and practically attractive option, affirming its continued relevance in the optimization toolkit. The fusion of comprehensive theoretical analysis with empirical validation marks this paper as a critical reference point in convex optimization research. Future work can build on these findings, exploring finer adaptations and broader classes of optimization problems.

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