Gradient Descent Converges to Minimizers (1602.04915v2)

Abstract: We show that gradient descent converges to a local minimizer, almost surely with random initialization. This is proved by applying the Stable Manifold Theorem from dynamical systems theory.

• The paper proves that gradient descent almost surely converges to local minimizers under the strict saddle property using the Stable Manifold Theorem.
• It employs random initialization and a constant step size below the Lipschitz threshold to avoid saddle points in non-convex optimization.
• The findings imply that simple gradient descent can reliably yield high-quality solutions, reducing the need for complex optimization modifications.

An Insightful Overview of "Gradient Descent Converges to Minimizers"

The paper "Gradient Descent Converges to Minimizers" by Jason D. Lee, Max Simchowitz, Michael I. Jordan, and Benjamin Recht addresses an important problem in optimization: the entrapment of gradient descent algorithms in saddle points. The authors present a theoretical analysis demonstrating that gradient descent, under specific conditions, almost surely converges to local minimizers when initialized randomly. This conclusion has substantial implications for the application and understanding of optimization algorithms in both theory and practice.

Summary of the Paper

The authors' central contribution lies in proving that gradient descent avoids settling at saddle points under a mild theoretical framework using the Stable Manifold Theorem from dynamical systems. The main conditions include that the objective function $f: \mathbb{R}^d \to \mathbb{R}$ should be twice continuously differentiable and satisfy the "strict saddle property." Specifically, a critical point of $f$ must either be a local minimizer or a strict saddle, where the Hessian at a strict saddle point has at least one negative eigenvalue.

By employing tools from dynamical systems theory, the paper demonstrates that with random initialization and an appropriately small constant step size (less than the reciprocal of the gradient's Lipschitz constant), gradient descent converges to local minima or diverges to negative infinity almost surely. This result is formalized through Proposition \ref{thm:main}, articulating that the algorithm avoids saddle points of measure zero in the optimization space.

Theoretical and Practical Implications

Theoretically, the paper provides a robust framework that builds on earlier work indicating the difficulty of locating non-convex local minimizers due to NP-completeness in the worst-case scenario. By reframing the problem with mild regularity conditions, the authors reveal that saddle points need not obstruct gradient descent except in highly contrived cases with specific initializations.

Practically, this insight strengthens the empirical observation that simple optimization algorithms often yield high-quality solutions, despite the absence of explicit mechanisms to circumvent saddle points. Traditional methods overcome such difficulties through modifications like added noise or curvature-based techniques, which can be computationally prohibitive. This work suggests that, in many contexts, such complexity may be unnecessary. Future algorithm designs and analyses might leverage these insights to develop more efficient optimization methods without heavy reliance on complex initializations.

Future Developments and Considerations

The authors conclude by acknowledging areas for further research, such as relaxing the condition on the strict saddle property and exploring broader algorithm classes under this framework. Additionally, they hint at investigating the applicability of their results to more intricate optimization procedures like ADMM or coordinate descent, emphasizing the potential for considerable generalization.

The paper does leave open questions about the strict saddle property and whether it universally applies to realistic objective functions encountered in practical machine learning contexts. There is also the intriguing possibility of spontaneous emergence of chaos or complex dynamics when these conditions are breached, an area ripe for exploration.

By bridging gaps in theoretical understanding where worst-case scenarios often dominate the narrative, this work contributes to a more nuanced comprehension of gradient-based optimization, fostering continued innovation in algorithm development across numerous domains in computer science and beyond.