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版本更新 · v2arXiv:2606.04757

Near-Optimal Decentralized Stochastic Convex Optimization over Networks

Nitai Kluger, Amit Attia, Tomer Koren

原文摘要

We study decentralized stochastic smooth convex optimization, where $M$ workers minimize an average objective using local stochastic gradients and neighbor-only communication over a fixed gossip network. A central question in this setting is to determine the largest number of workers that can be used under a total budget of $N$ gradient samples while still preserving the centralized $O(1/\sqrt N)$ statistical rate. We introduce an accelerated decentralized method that preserves this rate for up to $\smash{M\lesssim \sqrtρ\,N^{3/4}}$ workers, where $ρ$ is the spectral gap of the gossip network, improving the best prior maximal scaling of $\smash{M\lesssim ρ\sqrt N}$. The method is based on a one-step-delayed stochastic acceleration scheme that enables workers to interleave minibatching with accelerated gossip while controlling residual disagreement, and its guarantee depends only logarithmically on the optimum-local heterogeneity. We also establish a matching lower bound for linear-span decentralized first-order methods, showing that the method is optimal up to logarithmic factors.

math.OC · cs.LG

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.07078

Optimistic Hedge in General-Sum Games: Fundamental Limits and Regret Bounds via Temporal Approximation

Yuval Dagan, Nadav Eisen, Liad Erez, Tomer Koren

原文摘要

Optimistic Hedge, a multiplicative weight update with one step of extrapolation, is one of the most natural regret-minimizing dynamics for games, and characterizing its regret has been a longstanding open question. Daskalakis, Fishelson and Golowich proved that if all $m$ players in a finite general-sum game follow Optimistic Hedge with a fixed learning rate, each player's regret is polylogarithmic in the horizon $T$. Subsequently, Ha showed that an appropriate choice of learning rate makes each player's regret bounded independently of $T$. This bound, however, is non-constructive and its dependence on the number of actions $n$ is not explicit, leaving open whether Optimistic Hedge can achieve a constant regret bound which is polynomial in the size of the game. We show that the answer hinges on the number of players. In two-player games, there is a learning rate for which Optimistic Hedge guarantees $O(n\log n)$ regret independent of the horizon. In three-player games, however, no such bound is possible: for any learning rate up to an absolute constant and for sufficiently large horizon $T$, there is a game with $n$ actions per player in which some player's regret is $2^{Ω(n)}$, and more generally, for every $m\ge3$ we construct $m$-player games in which some player's regret is $2^{Ω(mn)}$; for learning rates above $2^{-Ω(mn)}$, that player's regret even grows linearly in $T$. Our upper bound is based on a temporal approximation framework for the analysis of Optimistic Hedge. Starting from the variance-based RVU bound of Daskalakis et al., we approximate the losses over time-localized windows by well-behaved functions, whose total variation is a small fraction of their cumulative variance. In $m$-player games, the same framework gives a significantly simplified proof of the recent $O(\sqrt{m}\log n\log T)$ regret bound due to Ha.

cs.LG · cs.GT

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2606.03851

The Price of Partial Information with Switching Costs: A Complete Characterization for Feedback Graphs

Nicolò Cesa-Bianchi, Tommaso Cesari, Roberto Colomboni, Emmanuel Esposito, Federico Fusco, Tomer Koren

原文摘要

We study how switching costs affect minimax regret under partial feedback. Which features of the feedback preserve the $\sqrt T$ rate, and which force the $T^{2/3}$ rate? We address this question in the framework of feedback graphs, which interpolate between full information and bandits: playing an action reveals the losses of its out-neighbors in a known directed graph. Prior work on feedback graphs with switching costs gave algorithms with regret of order $T^{2/3}$, but matching lower bounds only for restricted classes of graphs, leaving the question open. We establish a complete characterization of the minimax regret with switching costs for all feedback graphs: $\widetildeΘ(\sqrt T)$ if the graph is strongly revealing, meaning that every pair of distinct actions has an endpoint that observes the losses of both; $\widetildeΘ(γ^{1/3}T^{2/3})$ for every other observable graph, where $γ$ is the domination number; and linear otherwise. In particular, tournaments, in which every action observes its own loss and each pair of actions is joined by one directed edge, sit at the phase transition: they behave like full information, but deleting the edge of any pair forces the rate $T^{2/3}$. Hence an action that observes all losses is neither necessary nor sufficient for the rate $\sqrt T$. Our algorithm TournamentSteering for the strongly revealing case rests on a new principle: each observed comparison of two actions serves both to estimate losses and to steer play, and the loss reduction from steering pays for the variance of the estimates. For every strongly revealing graph, distributing the comparisons among observing actions keeps this variance, and hence the regret, polylogarithmic in the number of actions, by an argument similar to the greedy construction of logarithmic dominating sets in tournaments.

cs.LG

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arXiv ↗PDF ↗