Stability of Stochastic Gradient Descent on Nonsmooth Convex Losses (2006.06914v1)

Abstract: Uniform stability is a notion of algorithmic stability that bounds the worst case change in the model output by the algorithm when a single data point in the dataset is replaced. An influential work of Hardt et al. (2016) provides strong upper bounds on the uniform stability of the stochastic gradient descent (SGD) algorithm on sufficiently smooth convex losses. These results led to important progress in understanding of the generalization properties of SGD and several applications to differentially private convex optimization for smooth losses. Our work is the first to address uniform stability of SGD on {\em nonsmooth} convex losses. Specifically, we provide sharp upper and lower bounds for several forms of SGD and full-batch GD on arbitrary Lipschitz nonsmooth convex losses. Our lower bounds show that, in the nonsmooth case, (S)GD can be inherently less stable than in the smooth case. On the other hand, our upper bounds show that (S)GD is sufficiently stable for deriving new and useful bounds on generalization error. Most notably, we obtain the first dimension-independent generalization bounds for multi-pass SGD in the nonsmooth case. In addition, our bounds allow us to derive a new algorithm for differentially private nonsmooth stochastic convex optimization with optimal excess population risk. Our algorithm is simpler and more efficient than the best known algorithm for the nonsmooth case Feldman et al. (2020).

Stability of Stochastic Gradient Descent on Nonsmooth Convex Losses

The paper addresses the uniform stability of stochastic gradient descent (SGD) in the context of nonsmooth convex losses, marking a significant extension of prior work focused on smooth losses. Uniform stability is a key concept in understanding the generalization properties of learning algorithms; it provides a bound on the worst-case change in the output of a model when a single data point in the dataset is replaced. This paper on nonsmooth convex losses is essential as many practical loss functions exhibit nonsmooth characteristics, such as hinge loss in support vector machines.

Key Contributions

1. Upper and Lower Stability Bounds: The authors provide comprehensive analysis leading to upper and lower bounds on uniform stability for various forms of SGD and full-batch GD under arbitrary Lipschitz nonsmooth convex losses. They specifically highlight that SGD exhibits inherently different stability properties in nonsmooth scenarios compared to smooth ones.
2. Dimension-Independent Generalization Bounds: The findings include the first dimension-independent generalization bounds for multi-pass SGD in nonsmooth settings. This is a crucial advancement, enabling us to draw meaningful insights into the generalization capabilities of SGD without encountering the curse of dimensionality.

Implications

The paper's results have significant implications for theoretical understanding and practical application in machine learning and optimization:

• Nonsmooth vs. Smooth Losses: The paper emphasizes that nonsmooth losses inherently challenge the stability and generalization of SGD. This necessitates careful consideration in algorithm design, such as adjusting step size and increasing iteration counts to optimize the trade-off between stability and empirical risk.
• Algorithm Design for Privacy: Beyond generalization, the findings are relevant for differential privacy applications. Stability bounds are closely tied to differential privacy guarantees, leading to improved designs of privacy-preserving algorithms in stochastic convex optimization contexts.

Future Directions

The insights from this paper open several avenues for further exploration:

• Algorithmic Innovations: Developing new variations of SGD that leverage nonsmooth loss properties while maintaining computational efficiency and strong privacy guarantees.
• Theoretical Extensions: Expanding the theoretical groundwork to encompass more classes of nonsmooth losses, potentially improving assumptions and bounds on stability and generalization.
• Empirical Validation: Conducting empirical studies to verify theoretical predictions, exploring different real-world scenarios where nonsmooth convex losses are prevalent.

In conclusion, by bridging the gap between smooth and nonsmooth loss functions in the analysis of uniform stability of SGD, this paper contributes vital understanding necessary for both theory and application in modern machine learning contexts, including privacy-preserving optimization.