Adaptive, Doubly Optimal No-Regret Learning Algorithms for Strongly Monotone Games and Exp-Concave Games with Gradient Feedback

Core Concepts

The paper presents feasible variants of online gradient descent (AdaOGD) and online Newton step (AdaONS) that achieve optimal regret in the single-agent setting and optimal last-iterate convergence to the unique Nash equilibrium in the multi-agent setting, without requiring any prior knowledge of problem parameters.

Abstract

The paper focuses on the problem of online learning with gradient feedback, where an agent interfaces with an environment by choosing an action at each period and receives a cost function and gradient feedback. The standard metric for judging the performance of an online learning algorithm is regret, which measures the difference between the total cost incurred by the algorithm and the total cost incurred by the best fixed action in hindsight.
The paper presents two main contributions:

Feasible Variant of OGD (AdaOGD):

AdaOGD is a variant of online gradient descent (OGD) that does not require knowing the strong convexity or strong monotonicity parameters.
In the single-agent setting with strongly convex cost functions, AdaOGD achieves a near-optimal regret of O(log^2(T)).
In the multi-agent setting of strongly monotone games, if each agent employs AdaOGD, the joint action converges to the unique Nash equilibrium at a near-optimal last-iterate rate of O(log^3(T)/T).

Feasible Variant of ONS (AdaONS):

AdaONS is a variant of online Newton step (ONS) that does not require knowing the exp-concavity parameter.
In the single-agent setting with exp-concave cost functions, AdaONS achieves a near-optimal regret of O(d log^2(T)).
The paper also introduces a new class of exp-concave games and shows that if each agent employs AdaONS, the time-average of the joint action converges to the unique Nash equilibrium at a near-optimal rate of O(d log^2(T)/T).
The key to the adaptivity of both AdaOGD and AdaONS is a simple and unifying randomized strategy that selects the step size based on a set of independent and identically distributed geometric random variables. This allows the algorithms to be feasible and doubly optimal, in contrast to previous work that required knowing the problem parameters.

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Key Insights Distilled From

Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback

by Michael I. J... at arxiv.org 04-01-2024

https://arxiv.org/pdf/2310.14085.pdf

Deeper Inquiries

How can the adaptive techniques developed in this paper be extended to other online learning settings, such as online learning with dynamic regret or adaptive regret

The adaptive techniques developed in this paper can be extended to other online learning settings, such as online learning with dynamic regret or adaptive regret, by incorporating similar randomization strategies for step size selection. For dynamic regret settings, where the environment is non-stationary and the best action is drifting over time, adapting the step size selection based on random variables, as demonstrated in the paper, can help in achieving adaptivity to changing conditions. Similarly, for adaptive regret settings, where the regret minimization needs to be responsive to varying sequences and intervals, the randomized step size selection approach can be applied to adjust learning rates dynamically. By leveraging the principles of adaptivity and randomization in step size selection, these techniques can be extended to a broader range of online learning scenarios with different regret criteria.

What are the potential applications of the new class of exp-concave games introduced in the paper, and how can the analysis be further generalized to other classes of games

The new class of exp-concave games introduced in the paper has various potential applications across different domains. Exp-concave functions are a generalization of strongly convex functions and have found applications in optimization, game theory, and machine learning. The analysis presented in the paper can be further generalized to other classes of games by adapting the algorithms and techniques to accommodate different cost functions and game structures. The exp-concave games can be applied in settings where the cost functions exhibit exponential concavity, such as in pricing strategies, resource allocation, and network optimization problems. By extending the analysis to more general classes of games, the results can be applied to a wider range of scenarios, providing insights into optimal learning strategies in diverse environments.

Can the ideas behind the randomized step size selection be combined with other online learning algorithms, such as Adam or AdaGrad, to achieve similar adaptivity and double optimality results

The ideas behind the randomized step size selection, as demonstrated in the paper, can be combined with other online learning algorithms, such as Adam or AdaGrad, to achieve similar adaptivity and double optimality results. By incorporating randomization in the step size selection process, these algorithms can adapt to varying conditions and unknown parameters without requiring prior knowledge. The randomized step size selection can enhance the adaptivity of algorithms like Adam or AdaGrad, making them more robust in dynamic environments and ensuring optimal performance without the need for explicit parameter tuning. By integrating the principles of adaptivity and randomization, these algorithms can achieve double optimality in a wider range of online learning settings.