Gradient descent with adaptive stepsize converges (nearly) linearly under fourth-order growth (2409.19791v1)

Abstract: A prevalent belief among optimization specialists is that linear convergence of gradient descent is contingent on the function growing quadratically away from its minimizers. In this work, we argue that this belief is inaccurate. We show that gradient descent with an adaptive stepsize converges at a local (nearly) linear rate on any smooth function that merely exhibits fourth-order growth away from its minimizer. The adaptive stepsize we propose arises from an intriguing decomposition theorem: any such function admits a smooth manifold around the optimal solution -- which we call the ravine -- so that the function grows at least quadratically away from the ravine and has constant order growth along it. The ravine allows one to interlace many short gradient steps with a single long Polyak gradient step, which together ensure rapid convergence to the minimizer. We illustrate the theory and algorithm on the problems of matrix sensing and factorization and learning a single neuron in the overparameterized regime.

• The paper demonstrates that an adaptive stepsize strategy achieves nearly linear convergence even when functions exhibit fourth-order growth.
• It introduces a decomposition of the gradient along a smooth manifold, interlacing multiple short steps with a long Polyak step to enhance convergence.
• The findings extend linear convergence guarantees to high-dimensional optimization tasks, with applications in matrix sensing and neural network training.

Analysis of Gradient Descent with Adaptive Stepsize under Fourth-Order Growth

The paper "Gradient descent with adaptive stepsize converges (nearly) linearly under fourth-order growth" challenges conventional wisdom regarding the linear convergence of gradient descent (GD). Traditional optimization theory posits that GD achieves linear convergence on smooth convex functions that exhibit quadratic growth away from their minimizers. The authors present a contrasting viewpoint by demonstrating that GD with an adaptive stepsize can achieve a local (nearly) linear convergence rate even when the function exhibits fourth-order growth.

Overview of Key Contributions

The core contribution of the paper lies in its theoretical and empirical demonstration that adaptive stepsize GD converges at a local linear rate under a significantly milder growth condition than previously assumed. Key elements of the paper include:

1. Adaptive Stepsize Gradient Descent:
   • The authors develop an adaptive stepsize scheme hinging on a decomposition theorem. This decomposition reveals the existence of a smooth manifold (referred to as the "ravine") around the optimal solution, allowing the function to grow quadratically away from the ravine and exhibit constant order growth along it.
   • The algorithm interlaces multiple short gradient steps with a single long Polyak step to ensure rapid convergence.
2. Analytic Properties of Ravines:
   • The authors elucidate the properties of the ravine and demonstrate how the gradient can be decomposed into tangent and normal components relative to the ravine.
   • Crucially, they show that the normal component of the gradient exhibits well-behaved regularity properties up to an error term controlled by the gradient’s tangent component.
3. Smooth Manifold Definition and Existence:
   • The paper introduces the concept of a ravine and demonstrates that any smooth function admits such a manifold due to principles akin to the Morse lemma with parameters.

Theoretical Implications

The findings have several theoretical implications:

• General Convergence Guidance: By extending the conditions for linear convergence from quadratic to fourth-order growth, the results significantly broaden the applicability of GD with adaptive stepsizes.
• Manifold-Based Decomposition: The notion of decomposing the gradient along a manifold tangential to the null space of the Hessian offers a nuanced understanding of function behavior and optimization.

Practical Implications and Future Directions

The practical implications are substantial:

• Algorithm Design: The epoch-based adaptive stepsize algorithm proposed may pave the way for more efficient optimization in high-dimensional, over-parameterized settings common in modern machine learning.
• Application to Specific Problems: The empirical examples showcase potential applications for matrix sensing, matrix factorization, and neural network training in the overparameterized regime.
• Theoretical Extensions: Future work may explore extending the results to non-smooth functions, more complex manifold structures, and other sophisticated optimization methods.

Numerical Results and Insights

The paper presents compelling numerical evidence to support the theoretical findings. Notably, in examples like matrix sensing and learning single neurons:

• Matrix Sensing: Empirical results align with theoretical predictions, showing nearly linear convergence for overparameterized setups.
• Neural Network Training: The adaptive stepsize methodology yields significantly better performance compared to fixed stepsize approaches, reflecting in the learning speed and achieving a fine balance between tangent and normal gradient components.

Conclusion

This paper makes a substantial contribution to the understanding of gradient descent in settings where traditional quadratic growth conditions are relaxed to fourth-order growth. By leveraging an adaptive stepsize strategy intertwined with manifold decomposition, the authors offer a robust framework for improving convergence rates. The practical and theoretical advancements proposed could serve as a critical foundation for future research in optimization and machine learning. The results underscore the potential for adaptive stepsizes to enhance GD’s efficiency in both theoretical formulations and practical applications, opening up new avenues for exploration in high-dimensional and complex optimization landscapes.