Adaptive Proximal Gradient Methods for Convex Optimization: A Review

In this article, we delve into a new framework for adaptive proximal gradient methods that can efficiently minimize convex functions with Lipschitz gradients. This approach combines the advantages of proximal gradient methods with the flexibility of adaptive stepsizes. By leveraging the properties of a special class of combinatorial sequences, we devise a two-parameter algorithm that converges to the optimal solution in a finite number of iterations. Our key insight is that the restriction on the parameter r allows for a more aggressive recovery, but at the cost of a controlled number of iterations.
The article begins by introducing the general framework of adaptive proximal gradient methods and their convergence properties. We then delve into a specific class of combinatorial sequences that enable the development of a two-parameter algorithm. The proof of the main theorem is provided in the subsequent section, followed by numerical simulations to validate the theoretical findings. Finally, we conclude with some remarks on the significance of our results and potential applications.
The core idea behind this framework is to use adaptive stepsizes that adjust according to the progress of the algorithm. By doing so, the method can efficiently minimize a convex function while avoiding the need for expensive iterative methods like gradient descent. The key challenge is finding the right balance between aggressiveness and conservativeness in the adaptive stepsize, which we address through our special class of combinatorial sequences.
At its core, this approach involves using a combination of proximal gradient method and adaptive stepsizes to minimize a convex function. Proximal gradient methods are efficient when dealing with non-differentiable functions, but they can be slow for smooth functions with Lipschitz gradients. By combining these two methods, we create an adaptive framework that can handle both scenarios efficiently.
Our main contribution is the development of a two-parameter algorithm that converges to the optimal solution in a finite number of iterations. This algorithm combines the advantages of proximal gradient methods with the flexibility of adaptive stepsizes, making it more efficient and practical for solving real-world optimization problems.
In summary, this article presents a new framework for adaptive proximal gradient methods that can efficiently minimize convex functions with Lipschitz gradients. By leveraging the properties of a special class of combinatorial sequences, we devise a two-parameter algorithm that converges to the optimal solution in a finite number of iterations. Our results provide a more aggressive recovery while maintaining control over the number of iterations, making this approach more practical and efficient for solving real-world optimization problems.