Online Coordinate Descent Algorithms for Time-Varying Convex Optimization Problems

Konsep Inti

This paper extends coordinate descent algorithms to the online setting where the objective function varies over time. It provides a thorough regret analysis for both random and deterministic online coordinate descent algorithms under convex and strongly convex settings.

Abstrak

The paper considers the problem of online convex optimization where the objective function is time-varying. It extends coordinate descent algorithms to the online case, where the objective function changes after a finite number of iterations. Instead of solving the problem exactly at each time step, the algorithm only applies a finite number of coordinate descent iterations.

The key highlights and insights are:

1. The paper provides regret analysis for both random and deterministic online coordinate descent algorithms under convex and strongly convex settings. It derives static and dynamic regret bounds for these algorithms.

2. For random coordinate descent, the static regret bounds are O(√T) for convex functions and O(log T) for strongly convex functions. The dynamic regret bounds are O(√CT T) for convex functions and O(CT) for strongly convex functions, where CT captures the variation of the optimal solutions over time.

3. For deterministic coordinate descent algorithms, the regret bounds are shown to be comparable to those of online gradient descent algorithms under similar assumptions. The paper relates the regret of the coordinate descent algorithms to the regret of the online projected gradient descent algorithm.

4. The regret bounds achieved by the online coordinate descent algorithms are consistent with the existing literature on centralized full-gradient based online optimization algorithms.

5. The paper demonstrates that coordinate descent algorithms can be effectively extended to the online optimization setting and provide competitive performance compared to gradient-based methods.

Statistik

None.

Kutipan

None.

Wawasan Utama Disaring Dari

Online Convex Optimization Using Coordinate Descent Algorithms

by Yank... pada arxiv.org 04-26-2024

https://arxiv.org/pdf/2201.10017.pdf

Pertanyaan yang Lebih Dalam

How can the analysis be extended to handle the case where multiple components of the decision variable are allowed to be updated simultaneously in each iteration

To extend the analysis to handle the case where multiple components of the decision variable are updated simultaneously in each iteration, we would need to modify the update step of the online coordinate descent algorithm. Instead of updating a single component at a time, we would update a subset of components together. This would involve redefining the update rule to incorporate the gradients of the selected components and adjusting the projection step to ensure the updated decision variable remains within the feasible set. The analysis would need to consider the interactions between the updated components and how they affect the overall optimization process. By updating multiple components simultaneously, the algorithm may converge faster and provide more efficient solutions, especially in high-dimensional optimization problems.

What are the potential applications and practical benefits of using online coordinate descent algorithms compared to online gradient-based methods

Online coordinate descent algorithms offer several advantages and practical benefits compared to online gradient-based methods. One key benefit is their ability to handle large-scale optimization problems efficiently. By updating only a subset of components in each iteration, coordinate descent algorithms can be computationally less expensive than gradient-based methods, especially when the dimensionality of the problem is high. This makes them suitable for applications in machine learning, distributed optimization, and other domains where computational resources are limited. Additionally, coordinate descent algorithms are well-suited for problems with sparse data or when the objective function has a block structure, as they allow for parallel and distributed implementations. They also offer flexibility in terms of the selection of updating rules, such as random or deterministic strategies, providing adaptability to different problem settings and constraints.

Can the online coordinate descent framework be further generalized to handle time-varying constraints or stochastic constraints

The online coordinate descent framework can be further generalized to handle time-varying constraints or stochastic constraints by incorporating additional terms or constraints in the optimization problem. For time-varying constraints, the constraint set Θ can be updated at each iteration based on the changing constraints, and the projection step in the algorithm can be modified accordingly to ensure feasibility. Stochastic constraints can be integrated by considering probabilistic constraints or incorporating uncertainty in the optimization process. This may involve using techniques from stochastic optimization or robust optimization to account for the variability in the constraints. By extending the framework to handle these types of constraints, the online coordinate descent algorithms can be applied to a wider range of dynamic and uncertain optimization problems, enhancing their applicability in real-world scenarios.