Improved Complexity for Smooth Nonconvex Optimization: A Two-Level Online Learning Approach with Quasi-Newton Methods (2412.02175v1)

Abstract: We study the problem of finding an $\epsilon$-first-order stationary point (FOSP) of a smooth function, given access only to gradient information. The best-known gradient query complexity for this task, assuming both the gradient and Hessian of the objective function are Lipschitz continuous, is ${O}(\epsilon^{-7/4})$. In this work, we propose a method with a gradient complexity of ${O}(d^{1/4}\epsilon^{-13/8})$, where $d$ is the problem dimension, leading to an improved complexity when $d = {O}(\epsilon^{-1/2})$. To achieve this result, we design an optimization algorithm that, underneath, involves solving two online learning problems. Specifically, we first reformulate the task of finding a stationary point for a nonconvex problem as minimizing the regret in an online convex optimization problem, where the loss is determined by the gradient of the objective function. Then, we introduce a novel optimistic quasi-Newton method to solve this online learning problem, with the Hessian approximation update itself framed as an online learning problem in the space of matrices. Beyond improving the complexity bound for achieving an $\epsilon$-FOSP using a gradient oracle, our result provides the first guarantee suggesting that quasi-Newton methods can potentially outperform gradient descent-type methods in nonconvex settings.

• The paper’s main contribution is a two-level online learning approach that achieves a gradient complexity of O(d^(1/4) ε^(-13/8)) for first-order stationary points.
• It reformulates nonconvex optimization as an online convex problem, enabling efficient quasi-Newton Hessian updates using only gradient information.
• The method outperforms traditional bounds and advances quasi-Newton techniques, offering both practical benefits and strong theoretical guarantees.

Improved Complexity for Smooth Nonconvex Optimization: A Two-Level Online Learning Approach with Quasi-Newton Methods

The paper at hand introduces a novel optimization method aimed at enhancing the gradient complexity for achieving an ε-first-order stationary point (FOSP) in smooth nonconvex optimization problems. By leveraging a two-level online learning framework alongside a quasi-Newton method, this research breaks the previously established complexity barrier, demonstrating significant theoretical and computational advancements.

Summary of Contributions

The authors present a method that reformulates the problem of finding a stationary point into minimizing regret in an online convex optimization setting. The innovative aspect of this approach is the integration of a quasi-Newton method that optimistically updates Hessian approximations, relying solely on first-order (gradient) information. A core achievement of this work is the derivation of a gradient complexity of O(d^1/4 ε^-13/8), providing guarantees that outperform established bounds such as O(ε^-7/4) under certain dimensional constraints (e.g., when d = O(ε^-1/2)).

This contribution not only enhances the complexity bounds for nonconvex optimization using gradient oracles but also places quasi-Newton methods in favorable light, suggesting their potential superiority over gradient descent-based approaches in nonconvex contexts. This is particularly noteworthy, as prior understanding of quasi-Newton methods did not yet showcase provable advantages in nonconvex settings.

Methodological Insight

The crux of the proposed method is its two-level online learning scheme. The process involves first reframing the problem of finding a first-order stationary point as an online convex optimization problem where the loss functions are governed by the gradients of the objective function. Following this, the authors employ a novel optimistic quasi-Newton method to tackle this problem. The optimism here is embedded in the quasi-Newton update mechanism, which anticipates future gradients based on past errors.

Additionally, a second online learning problem is constructed concerning the updating of the Hessian approximation in the form of matrices. This dual layering in online learning allows the method to achieve notable improvements in complexity without requiring second-order information such as Hessian-vector products.

Numerical and Theoretical Implications

This paper’s results highlight the potential efficacy of quasi-Newton methods in specific nonconvex settings, opening new avenues for inquiry into their applications. The work delivers theoretical guarantees through rigorous complexity analysis and convergence proofs, reinforcing the prospects for quasi-Newton methods in overcoming traditional barriers associated with large-scale nonconvex optimization.

The implications extend to practical settings where large dimension and effective iterations are of concern. In environments where only gradient information is accessible, the proposed method's resilience and efficiency present a competitive edge. With respect to computational costs, the research intricately addresses the challenges of solving trust-region subproblems through inventive matrix-vector product bounds, thereby achieving practicality in addition to theoretical rigor.

Speculative Future Work

Future research could explore extending these findings to even broader classes of nonconvex functions, potentially incorporating stochastic elements or exploring settings with weaker smoothness assumptions. Another potential direction is refining the method to fully utilize advanced quasi-Newton strategies which might provide even tighter complexity bounds and adaptability to real-world applications in machine learning and beyond.

In summary, this paper offers a substantial contribution to the theoretical and practical understanding of optimizing nonconvex functions by ingeniously employing a two-level online learning approach refined with optimistic quasi-Newton methodology, fostering novel complexity achievements and plotting a direction for future explorations.