Projected gradient methods for nonconvex and stochastic optimization: new complexities and auto-conditioned stepsizes (2412.14291v1)

Abstract: We present a novel class of projected gradient (PG) methods for minimizing a smooth but not necessarily convex function over a convex compact set. We first provide a novel analysis of the "vanilla" PG method, achieving the best-known iteration complexity for finding an approximate stationary point of the problem. We then develop an "auto-conditioned" projected gradient (AC-PG) variant that achieves the same iteration complexity without requiring the input of the Lipschitz constant of the gradient or any line search procedure. The key idea is to estimate the Lipschitz constant using first-order information gathered from the previous iterations, and to show that the error caused by underestimating the Lipschitz constant can be properly controlled. We then generalize the PG methods to the stochastic setting, by proposing a stochastic projected gradient (SPG) method and a variance-reduced stochastic gradient (VR-SPG) method, achieving new complexity bounds in different oracle settings. We also present auto-conditioned stepsize policies for both stochastic PG methods and establish comparable convergence guarantees.

• The paper presents a unified analysis of projected gradient methods that achieves iteration complexity bounds (O(LD_X/ε + LlD_X²/ε²)) for nonconvex problems.
• The paper introduces auto-conditioned variants that dynamically adjust stepsizes without requiring prior curvature information, significantly easing parameter tuning.
• The paper extends these techniques to stochastic settings by incorporating variance reduction methods, lowering sample complexity and enhancing high-probability guarantees.

Projected Gradient Methods for Nonconvex and Stochastic Optimization: New Complexities and Auto-Conditioned Stepsizes

The paper "Projected Gradient Methods for Nonconvex and Stochastic Optimization: New Complexities and Auto-Conditioned Stepsizes" authored by Guanghui Lan, Tianjiao Li, and Yangyang Xu proposes a novel class of projected gradient (PG) methods tailored for minimizing smooth, nonconvex functions over convex compact sets. The authors present substantial developments in both deterministic and stochastic settings, providing new insights into the convergence complexities and introducing auto-conditioned variants that are parameter-free and do not require knowledge of critical problem parameters.

Summary of Contributions

1. Novel Analysis of Projected Gradient Methods: The authors establish a new complexity bound for the vanilla PG method that incorporates both upper and lower curvature information, achieving a unified treatment for convex and nonconvex optimization. This analysis demonstrates that the PG method achieves an iteration complexity of $\mathcal{O}(LD_X/\epsilon + LlD_X^2/\epsilon^2)$, matching the best-known bounds provided by more sophisticated methods.
2. Auto-Conditioned Projected Gradient (AC-PG) Methods: The paper advances an auto-conditioned variant of the PG method which adapts dynamically without requiring prior knowledge of the Lipschitz constant or lower curvature. The AC-PG approach estimates the Lipschitz constant using first-order information from previous iterations and manages errors resulting from underestimated constants. This method achieves iteration complexity comparable to that of the standard PG method.
3. Stochastic Projected Gradient (SPG) Methods: In the stochastic setting, the authors propose an SPG method that appropriately adapts batch sizes to maintain efficient convergence, achieving a sample complexity of $\mathcal{O}(1/\epsilon^4)$ under certain assumptions. Furthermore, the introduction of the auto-conditioned variant, AC-SPG, provides a robust framework for scenarios where the upper curvature is unknown, achieving complexity bounds that incorporate this uncertainty.
4. Variance Reduction Techniques: To further accelerate convergence, the variance-reduced stochastic projected gradient (VR-SPG) method is introduced, which reduces the sample complexity to $\mathcal{O}(1/\epsilon^3)$, leveraging variance-reduction techniques. This advance is significant as it provides a more efficient solution technique under the assumption of Lipschitz continuity of the stochastic gradient.
5. Two-Phase Approach for High Probability Guarantees: The authors develop a two-phase AC-SPG approach to improve high-probability guarantees for obtaining $(\epsilon, \delta)$-stationary solutions, mitigating the limitations of prior methods which only provide expectation-based results.

Implications and Future Research Directions

The developments in this paper have substantial implications for various applications in machine learning, reinforcement learning, and other domains that require efficient nonconvex optimization solutions. The unified theory for convex and nonconvex structures opens new pathways for developing optimization algorithms with fewer stringent requirements on problem-specific parameters.

Practically, the introduction of auto-conditioning mechanisms can significantly reduce the tuning effort associated with optimization algorithms, as they adaptively adjust to the problem's curvature. The exploration of variance reduction within the PG framework marks a potential advancement towards closing the gap between theoretical complexity and practical performance.

Future research could explore extending these methodologies to more complex problem structures, such as non-smooth or highly constrained environments. Scaling these techniques for very high-dimensional settings and distributed computation frameworks could push the boundaries of applicability in real-world scenarios. Additionally, exploring the interplay between variance reduction methods and adaptive step-size policies might yield even more efficient algorithms tailored for complex, dynamic data-driven environments.