Trading Regret for Efficiency: Online Convex Optimization with Long Term Constraints (1111.6082v3)

Abstract: In this paper we propose a framework for solving constrained online convex optimization problem. Our motivation stems from the observation that most algorithms proposed for online convex optimization require a projection onto the convex set $\mathcal{K}$ from which the decisions are made. While for simple shapes (e.g. Euclidean ball) the projection is straightforward, for arbitrary complex sets this is the main computational challenge and may be inefficient in practice. In this paper, we consider an alternative online convex optimization problem. Instead of requiring decisions belong to $\mathcal{K}$ for all rounds, we only require that the constraints which define the set $\mathcal{K}$ be satisfied in the long run. We show that our framework can be utilized to solve a relaxed version of online learning with side constraints addressed in \cite{DBLP:conf/colt/MannorT06} and \cite{DBLP:conf/aaai/KvetonYTM08}. By turning the problem into an online convex-concave optimization problem, we propose an efficient algorithm which achieves $\tilde{\mathcal{O}}(\sqrt{T})$ regret bound and $\tilde{\mathcal{O}}(T^{3/4})$ bound for the violation of constraints. Then we modify the algorithm in order to guarantee that the constraints are satisfied in the long run. This gain is achieved at the price of getting $\tilde{\mathcal{O}}(T^{3/4})$ regret bound. Our second algorithm is based on the Mirror Prox method \citep{nemirovski-2005-prox} to solve variational inequalities which achieves $\tilde{\mathcal{\mathcal{O}}}(T^{2/3})$ bound for both regret and the violation of constraints when the domain $\K$ can be described by a finite number of linear constraints. Finally, we extend the result to the setting where we only have partial access to the convex set $\mathcal{K}$ and propose a multipoint bandit feedback algorithm with the same bounds in expectation as our first algorithm.

• The paper introduces a novel framework that relaxes per-iteration constraint satisfaction, achieving sub-linear regret and controlled cumulative violations.
• It develops an algorithm and adapts the mirror prox method to reduce expensive projection operations in complex convex sets.
• The approach extends to bandit settings with partial information, offering rigorous theoretical guarantees and practical trade-offs between regret and constraint violations.

Online Convex Optimization with Long Term Constraints

The paper "Trading Regret for Efficiency: Online Convex Optimization with Long Term Constraints" by Mehrdad Mahdavi, Rong Jin, and Tianbao Yang addresses a significant issue in the field of online convex optimization: the computational inefficiency of projection operations onto complex convex sets during iterative decision-making processes. The authors propose a new framework for solving online convex optimization problems, which relaxes the requirement of constraint satisfaction in each iteration, ensuring only that the cumulative constraints are satisfied over the long term.

This research is motivated by the observation that although projection operations onto simple sets, such as the Euclidean ball, can be computationally trivial, they may become prohibitive for arbitrary complex sets in practical scenarios. Consequently, the paper introduces an innovative approach by formulating the problem as an online convex-concave optimization task, thus enabling the use of gradient descent methods in a modified form.

The contributions of this paper are multifold:

1. Algorithm Development: The authors propose a novel algorithm that achieves a sub-linear regret bound of $O(\sqrt{T})$ while providing a constraint violation bound of $O(T^{3/4})$. The algorithm leverages online gradient descent with the insight that constraints need only be satisfied cumulatively, rather than at each iteration. This results in a significant reduction in computational complexity.
2. Mirror Prox Method: For convex sets that can be described by a finite number of linear constraints, the authors adapt the mirror prox method to achieve improved bounds of $O(T^{2/3})$ for both regret and constraint violation. This method capitalizes on the variational inequalities framework, offering a more efficient solution than direct projection methods.
3. Partial Information Setting: The paper extends the algorithm to a bandit setting where full knowledge of the convex domain is unavailable, only permitting access to a limited number of oracle evaluations. Even in this scenario, the authors demonstrate that similar performance bounds can be achieved, thus broadening the applicability of the proposed framework.
4. Theoretical Guarantees: The paper provides rigorous theoretical analysis to support the proposed methods. The authors derive bounds for regret and constraint violations and discuss the conditions under which these bounds hold. In particular, they show how adjusting algorithm parameters can balance the trade-off between regret minimization and constraint satisfaction.

The practical implications of this research are far-reaching. The proposed framework can be applied to various domains where online learning algorithms interact with constraints, such as recommendation systems where user preferences need long-term satisfaction, or financial applications where budget constraints are considered. Moreover, the framework's adaptability to partial information scenarios makes it relevant for real-world applications where full data availability cannot be assumed.

Theoretically, this paper opens several avenues for future work. It raises the question of whether the derived bounds on regret and constraint violations are optimal or if further improvements can be achieved. Additionally, exploring other types of loss functions, such as strongly convex ones, or extending the framework to include stochastic constraints, may yield further insights and improvements.

In conclusion, this paper makes substantial contributions to online convex optimization by reformulating constraint satisfaction as a long-term goal, which significantly reduces computational demand and enhances algorithm efficiency. The innovative approach and the comprehensive theoretical foundation provided lay a groundwork for both advancing the field and for practical implementations in computationally intensive environments.