To address this gap, we study deterministic/stochastic versions of SAM with practical configurations (i.e., constant

However, the cross-client edges naturally exist among clients.

The sample-averaged gradient of the CSO objective is biased due to its nested structure, and therefore requires a high sample complexity for convergence. We introduce a general stochastic extrapolation technique that effectively reduces the bias.

Reaching consensus and convergence to equilibrium are two major challenges of multi-agent systems. Although each has attracted significant attention, relatively few studies address both challenges at the same time. This paper examines the connection between the notions of consensus and equilibrium in a multi-agent system where multiple interacting sub-populations coexist. We argue that consensus can be seen as an intricate component of intra-population stability, whereas equilibrium can be seen as encoding inter-population stability. We show that smooth fictitious play, a well-known learning model in game theory, can achieve both consensus and convergence to equilibrium in diverse multi-agent settings. Moreover, we show that the consensus formation process plays a crucial role in the seminal thorny problem of equilibrium selection in multi-agent learning.

Suppose h ( )=e x p ( ) . M -matrix (see [ 50 ]).

Our framework applies to a wide range of SMF-type problems for multi-class classification with auxiliary features.