Adaptive Gradient Descent without Descent (1910.09529v2)

Abstract: We present a strikingly simple proof that two rules are sufficient to automate gradient descent: 1) don't increase the stepsize too fast and 2) don't overstep the local curvature. No need for functional values, no line search, no information about the function except for the gradients. By following these rules, you get a method adaptive to the local geometry, with convergence guarantees depending only on the smoothness in a neighborhood of a solution. Given that the problem is convex, our method converges even if the global smoothness constant is infinity. As an illustration, it can minimize arbitrary continuously twice-differentiable convex function. We examine its performance on a range of convex and nonconvex problems, including logistic regression and matrix factorization.

• The paper introduces an adaptive gradient descent method that requires only gradient information and ensures convergence for continuously twice-differentiable convex functions without needing line search or global constant knowledge.
• The method adapts stepsizes based on approximate inverse local Lipschitz constants, allowing it to navigate varying curvatures more efficiently than fixed-step or global-constant methods.
• This adaptive approach demonstrates practical viability across convex and non-convex problems like logistic regression and matrix factorization, potentially streamlining optimization in varied machine learning tasks.

Analyzing Adaptive Gradient Descent Methods: A Rigorous Examination

The paper investigates adaptive gradient descent (GD) methods, a topic of substantial significance in optimization and machine learning. Specifically, it focuses on a method that eschews traditional requirements such as explicitly function values, linesearch procedures, or comprehensive knowledge of the objective function, relying solely on gradient information.

A key assertion made is that two simple rules suffice to guide effective gradient descent: maintaining cautious adaptability in increasing the stepsize and adhering closely to the local curvature without overshooting. The proposed method stands out by adapting to local geometry with theoretical convergence assurances, especially critical in convex scenarios where global smoothness constant requisites typically hamper traditional methods.

Methodological Departures and Performance Analysis

The paper presents a method capable of minimizing any continuously twice-differentiable convex function, asserting convergence even when the global Lipschitz constant transcends feasible computation. It explores applications extending from logistic regression to matrix factorization, thereby benchmarking performance across both convex and nonconvex landscapes. This breadth of applicability reflects the method’s potential to advance areas traditionally bounded by specific smoothness assumptions.

The theoretical underpinnings leverage simple yet profound stepsize adaptations through iterative calculations of approximate inverse local Lipschitz constants. This stands in stark contrast to fixed-size approaches, adaptive methods reliant on global constants, or oft-unweildy line search techniques. Unlike classical methods, the proposed approach does not mandate stringent monotonicity assumptions, liberating it from convergence limitations inherent to methods anchored in global smoothness and strong convexity.

Broader Impacts and Future Directions

The research proposes an intriguing adaptive algorithm that potentially mitigates prevalent difficulties in GD methods concerning the determination of suitable stepsizes. While empirical results demonstrate its efficiency, particularly in faster convergence rates compared to conventional methods, the paper itself remains conservative in attributing revolutionary claims.

Practically, the proposed methodology could streamline optimization tasks across machine learning applications, notably in scenarios characterized by strong variance in curvature or unpredictable landscapes, extending potentially to hyperparameter tuning or neural network training where gradient instability presents formidable challenges.

The theoretical implications pivot on improving composite optimization, where integrating this approach could marry convex and nonconvex regimes more efficiently. While an intriguing foundation for adaptive methods in stochastic or momentum-based settings is implied, significant theoretical challenges persist, particularly in nonconvex scenarios where robust adaptive strategies remain elusive.

Conclusion

The paper provides rigorous advancements in understanding and applying adaptive gradient descent, presenting a method that is both theoretically sound and practically viable. It emphasizes improved adaptability to local problem structures while ensuring convergence without dependency on exhaustive parameter tuning or global estimations, thus offering a compelling direction for further exploration and refinement. While the method’s direct implications are empirical, suggesting broad efficiency gains, the broader applicability and theoretical integration remain fertile ground for future research innovations.