A Slingshot Approach to Learning in Monotone Games

In this paper, we address the problem of computing equilibria in monotone games. The traditional Follow the Regularized Leader algorithms fail to converge to an equilibrium even in two-player zero-sum games. Although optimistic versions of these algorithms have been proposed with last-iterate convergence guarantees, they require noiseless gradient feedback. To overcome this limitation, we present a novel framework that achieves last-iterate convergence even in the presence of noise. Our key idea involves perturbing or regularizing the payoffs or utilities of the games. This perturbation serves to pull the current strategy to an anchored strategy, which we refer to as a {\it slingshot} strategy. First, we establish the convergence rates of our framework to a stationary point near an equilibrium, regardless of the presence or absence of noise. Next, we introduce an approach to periodically update the slingshot strategy with the current strategy. We interpret this approach as a proximal point method and demonstrate its last-iterate convergence. Our framework is comprehensive, incorporating existing payoff-regularized algorithms and enabling the development of new algorithms with last-iterate convergence properties. Finally, we show that our algorithms, based on this framework, empirically exhibit faster convergence.

Last-Iterate Convergence with Full- and Noisy-Information Feedback in Two-Player Zero-Sum Games

The theory of learning in games is prominent in the AI community, motivated by several rising applications such as multi-agent reinforcement learning and Generative Adversarial Networks. We propose Mutation-driven Multiplicative Weights Update (M2WU) for learning an equilibrium in two-player zero-sum normal-form games and prove that it exhibits the last-iterate convergence property in both full- and noisy-information feedback settings. In the full-information feedback setting, the players observe their exact gradient vectors of the utility functions. On the other hand, in the noisy-information feedback setting, they can only observe the noisy gradient vectors. Existing algorithms, including the well-known Multiplicative Weights Update (MWU) and Optimistic MWU (OMWU) algorithms, fail to converge to a Nash equilibrium with noisy-information feedback. In contrast, M2WU exhibits the last-iterate convergence to a stationary point near a Nash equilibrium in both of the feedback settings. We then prove that it converges to an exact Nash equilibrium by adapting the mutation term iteratively. We empirically confirm that M2WU outperforms MWU and OMWU in exploitability and convergence rates.

Mutation-Driven Follow the Regularized Leader for Last-Iterate Convergence in Zero-Sum Games

In this study, we consider a variant of the Follow the Regularized Leader (FTRL) dynamics in two-player zero-sum games. FTRL is guaranteed to converge to a Nash equilibrium when time-averaging the strategies, while a lot of variants suffer from the issue of limit cycling behavior, i.e., lack the last-iterate convergence guarantee. To this end, we propose mutant FTRL (M-FTRL), an algorithm that introduces mutation for the perturbation of action probabilities. We then investigate the continuous-time dynamics of M-FTRL and provide the strong convergence guarantees toward stationary points that approximate Nash equilibria under full-information feedback. Furthermore, our simulation demonstrates that M-FTRL can enjoy faster convergence rates than FTRL and optimistic FTRL under full-information feedback and surprisingly exhibits clear convergence under bandit feedback.

Automated Segmentation of Hip and Thigh Muscles in Metal Artifact-Contaminated CT using Convolutional Neural Network-Enhanced Normalized Metal Artifact Reduction

In total hip arthroplasty, analysis of postoperative medical images is important to evaluate surgical outcome. Since Computed Tomography (CT) is most prevalent modality in orthopedic surgery, we aimed at the analysis of CT image. In this work, we focus on the metal artifact in postoperative CT caused by the metallic implant, which reduces the accuracy of segmentation especially in the vicinity of the implant. Our goal was to develop an automated segmentation method of the bones and muscles in the postoperative CT images. We propose a method that combines Normalized Metal Artifact Reduction (NMAR), which is one of the state-of-the-art metal artifact reduction methods, and a Convolutional Neural Network-based segmentation using two U-net architectures. The first U-net refines the result of NMAR and the muscle segmentation is performed by the second U-net. We conducted experiments using simulated images of 20 patients and real images of three patients to evaluate the segmentation accuracy of 19 muscles. In simulation study, the proposed method showed statistically significant improvement (p<0.05) in the average symmetric surface distance (ASD) metric for 14 muscles out of 19 muscles and the average ASD of all muscles from 1.17 +/- 0.543 mm (mean +/- std over all patients) to 1.10 +/- 0.509 mm over our previous method. The real image study using the manual trace of gluteus maximus and medius muscles showed ASD of 1.32 +/- 0.25 mm. Our future work includes training of a network in an end-to-end manner for both the metal artifact reduction and muscle segmentation.