Gradient Methods with Online Scaling (2411.01803v2)

Abstract: We introduce a framework to accelerate the convergence of gradient-based methods with online learning. The framework learns to scale the gradient at each iteration through an online learning algorithm and provably accelerates gradient-based methods asymptotically. In contrast with previous literature, where convergence is established based on worst-case analysis, our framework provides a strong convergence guarantee with respect to the optimal scaling matrix for the iteration trajectory. For smooth strongly convex optimization, our results provide an $O(\kappa^\star \log(1/\varepsilon)$) complexity result, where $\kappa^\star$ is the condition number achievable by the optimal preconditioner, improving on the previous $O(\sqrt{n}\kappa^\star \log(1/\varepsilon))$ result. In particular, a variant of our method achieves superlinear convergence on convex quadratics. For smooth convex optimization, we show for the first time that the widely-used hypergradient descent heuristic improves on the convergence of gradient descent.

• The paper presents an adaptive gradient method that dynamically scales gradients via online learning, achieving convergence rates of O(κ log(1/ε)).
• It demonstrates superlinear convergence for strongly convex quadratic problems and outperforms traditional methods like Nesterov's acceleration in experiments.
• Empirical validations on least squares and logistic regression tasks confirm the framework's effectiveness in mitigating poorly conditioned problems.

An Overview of "Gradient Methods with Online Scaling"

The paper "Gradient Methods with Online Scaling" introduces a novel framework aimed at enhancing the convergence rates of gradient-based optimization methods by incorporating online learning techniques. The approach leverages an online learning algorithm to adaptively scale the gradient vectors at each iteration, thereby offering a strong trajectory-based convergence guarantee in relation to the optimal scaling matrix suited for the iterative trajectory itself.

Key Contributions

The contributions of this paper are multi-fold:

• The framework provides an adaptive first-order gradient method with theoretical guarantees on convergence rates that match those achievable by optimal preconditioning methods, such as an O(κ log(1/ε)) complexity for smooth strongly convex optimization. This is a notable improvement over the previously best-known O(√nκ log(1/ε)) result, which incorporates dimension dependence.
• A variant of this method demonstrates superlinear convergence for strongly convex quadratics through first-order information.
• The paper is the first to provide a theoretical proof that the hypergradient descent heuristic, popular in empirical settings, can indeed improve the convergence rate of standard gradient descent methods in certain scenarios.

Methodology

The framework revolves around the concept of scaling the gradient via a learned scaling matrix. This is achieved by viewing the problem as an instance of online convex optimization, where the choice of the scaling matrix serves as the decision variable. The surrogate loss function, which is convex and possibly Lipschitz, allows the application of online learning algorithms to adjust the scaling matrix dynamically, promising convergence that is on par with any fixed scaling matrix optimized for the iterative trajectory.

Numerical Results

Significant numerical results are presented for a range of optimization problems, including least squares and regularized logistic regression tasks. The empirical validations demonstrate that the proposed methods can effectively mitigate the effects of poorly conditioned problems, often outperforming accelerated gradient techniques such as Nesterov's Accelerated Gradient Descent (SAGD) and adaptive methods like AdaGrad. In particular, on quadratic problems, the framework rivals the theoretical improvements suggested by optimal preconditioners, achieving superlinear convergence rates under specific settings.

Practical Implications and Limitations

From a practical standpoint, the choice of the set P of candidate scaling matrices considerably affects the efficiency of the algorithm, as does the choice of the online learning algorithm A. The paper suggests simple yet effective structures such as diagonal or low-rank matrices for P to balance computational efficiency and convergence speed. Additionally, the use of advanced online algorithms like AdaGrad can further improve practical performance.

The methodology's dependence on knowing the optimal objective value or a tight lower bound highlights a potential limitation, though the authors propose several strategies to address this issue.

Implications for Future Research

This work opens several avenues for future research. Extending trajectory-based convergence guarantees to other gradient-based techniques like accelerated and stochastic gradient descent would be a promising direction. Additionally, investigating the potential of the framework in non-convex, non-smooth, or constrained optimization problems can further elucidate the versatility and robustness of the proposed methods. Moreover, the insights into the hypergradient heuristic introduce interesting challenges and opportunities for enhancing the theoretical understanding and practical applications of hypergradient methods.

In conclusion, the paper presents a well-rounded theoretical advancement, supplemented by empirical validation, in the optimization domain. It provides a pathway to improving gradient-based methods' efficiency and effectiveness by leveraging adaptive scaling through online learning, contributing to the broader understanding of both theoretical and practical optimization landscapes.