Recently, the anchor acceleration, an acceleration mechanism distinct from Nesterov's, has been discovered for minimax optimization and fixed-point problems, but its mechanism is not understood well, much less so than Nesterov acceleration. In this work, we analyze continuous-time models of anchor acceleration. We provide tight, unified analyses for characterizing the convergence rate as a function of the anchor coefficient $\beta(t)$, thereby providing insight into the anchor acceleration mechanism and its accelerated $\mathcal{O}(1/k^2)$-convergence rate. Finally, we present an adaptive method inspired by the continuous-time analyses and establish its effectiveness through theoretical analyses and experiments.

Since Polyak's pioneering work, heavy ball (HB) momentum has been widely studied in minimization. However, its role in min-max games remains largely unexplored. As a key component of practical min-max algorithms like Adam, this gap limits their effectiveness. In this paper, we present a continuous-time analysis for HB with simultaneous and alternating update schemes in min-max games. Locally, we prove smaller momentum enhances algorithmic stability by enabling local convergence across a wider range of step sizes, with alternating updates generally converging faster. Globally, we study the implicit regularization of HB, and find smaller momentum guides algorithms trajectories towards shallower slope regions of the loss landscapes, with alternating updates amplifying this effect. Surprisingly, all these phenomena differ from those observed in minimization, where larger momentum yields similar effects. Our results reveal fundamental differences between HB in min-max games and minimization, and numerical experiments further validate our theoretical results.

Federated averaging (FedAvg) is a popular algorithm for horizontal federated learning (FL), where samples are gathered across different clients and are not shared with each other or a central server. Extensive convergence analysis of FedAvg exists for the discrete iteration setting, guaranteeing convergence for a range of loss functions and varying levels of data heterogeneity. We extend this analysis to the continuous-time setting where the global weights evolve according to a multivariate stochastic differential equation (SDE), which is the first time FedAvg has been studied from the continuous-time perspective. We use techniques from stochastic processes to establish convergence guarantees under different loss functions, some of which are more general than existing work in the discrete setting. We also provide conditions for which FedAvg updates to the server weights can be approximated as normal random variables. Finally, we use the continuous-time formulation to reveal generalization properties of FedAvg.

Recently, the anchor acceleration, an acceleration mechanism distinct from Nesterov's, has been discovered for minimax optimization and fixed-point problems, but its mechanism is not understood well, much less so than Nesterov acceleration. In this work, we analyze continuous-time models of anchor acceleration. We provide tight, unified analyses for characterizing the convergence rate as a function of the anchor coefficient \beta(t), thereby providing insight into the anchor acceleration mechanism and its accelerated \mathcal{O}(1/k 2) -convergence rate. Finally, we present an adaptive method inspired by the continuous-time analyses and establish its effectiveness through theoretical analyses and experiments.

Algorithms that solve zero-sum games, multi-objective agent objectives, or, more generally, variational inequality (VI) problems are notoriously unstable on general problems. Owing to the increasing need for solving such problems in machine learning, this instability has been highlighted in recent years as a significant research challenge. In this paper, we provide an overview of recent progress in the use of continuous-time perspectives in the analysis and design of methods targeting the broad VI problem class. Our presentation draws parallels between single-objective problems and multi-objective problems, highlighting the challenges of the latter. We also formulate various desiderata for algorithms that apply to general VIs and we argue that achieving these desiderata may profit from an understanding of the associated continuous-time dynamics.

Algorithms that solve zero-sum games, multi-objective agent objectives, or, more generally, variational inequality (VI) problems are notoriously unstable on general problems. Owing to the increasing need for solving such problems in machine learning, this instability has been highlighted in recent years as a significant research challenge. In this paper, we provide an overview of recent progress in the use of continuous-time perspectives in the analysis and design of methods targeting the broad VI problem class. Our presentation draws parallels between single-objective problems and multi-objective problems, highlighting the challenges of the latter. We also formulate various desiderata for algorithms that apply to general VIs and we argue that achieving these desiderata may profit from an understanding of the associated continuous-time dynamics.