This paper proposes a novel non-parametric multidimensional convex regression estimator which is designed to be robust to adversarial perturbations in the empirical measure. We minimize over convex functions the maximum (over Wasserstein perturbations of the empirical measure) of the absolute regression errors. The inner maximization is solved in closed form resulting in a regularization penalty involves the norm of the gradient.

We consider a game-theoretical multi-agent learning problem where the feedback information can be lost during the learning process and rewards are given by a broad class of games known as variationally stable games. We propose a simple variant of the classical online gradient descent algorithm, called reweighted online gradient descent (ROGD) and show that in variationally stable games, if each agent adopts ROGD, then almost sure convergence to the set of Nash equilibria is guaranteed, even when the feedback loss is asynchronous and arbitrarily corrrelated among agents. We then extend the framework to deal with unknown feedback loss probabilities by using an estimator (constructed from past data) in its replacement. Finally, we further extend the framework to accomodate both asynchronous loss and stochastic rewards and establish that multi-agent ROGD learning still converges to the set of Nash equilibria in such settings. Together, these results contribute to the broad lanscape of multi-agent online learning by significantly relaxing the feedback information that is required to achieve desirable outcomes.

In this paper, we examine a class of non-convex stochastic optimization problems which we call variationally coherent, and which properly includes pseudo-/quasiconvex and star-convex optimization problems. To solve such problems, we focus on the widely used stochastic mirror descent (SMD) family of algorithms (which contains stochastic gradient descent as a special case), and we show that the last iterate of SMD converges to the problem's solution set with probability 1. This result contributes to the landscape of non-convex stochastic optimization by clarifying that neither pseudo-/quasi-convexity nor star-convexity is essential for (almost sure) global convergence; rather, variational coherence, a much weaker requirement, suffices. Characterization of convergence rates for the subclass of strongly variationally coherent optimization problems as well as simulation results are also presented.

We consider a model of game-theoretic learning based on online mirror descent (OMD) with asynchronous and delayed feedback information. Instead of focusing on specific games, we consider a broad class of continuous games defined by the general equilibrium stability notion, which we call λ-variational stability. Our first contribution is that, in this class of games, the actual sequence of play induced by OMD-based learning converges to Nash equilibria provided that the feedback delays faced by the players are synchronous and bounded. Subsequently, to tackle fully decentralized, asynchronous environments with (possibly) unbounded delays between actions and feedback, we propose a variant of OMD which we call delayed mirror descent (DMD), and which relies on the repeated leveraging of past information. With this modification, the algorithm converges to Nash equilibria with no feedback synchronicity assumptions and even when the delays grow superlinearly relative to the horizon of play.

We consider a game-theoretical multi-agent learning problem where the feedback information can be lost during the learning process and rewards are given by a broad class of games known as variationally stable games. We propose a simple variant of the classical online gradient descent algorithm, called reweighted online gradient descent (ROGD) and show that in variationally stable games, if each agent adopts ROGD, then almost sure convergence to the set of Nash equilibria is guaranteed, even when the feedback loss is asynchronous and arbitrarily corrrelated among agents. We then extend the framework to deal with unknown feedback loss probabilities by using an estimator (constructed from past data) in its replacement. Finally, we further extend the framework to accomodate both asynchronous loss and stochastic rewards and establish that multi-agent ROGD learning still converges to the set of Nash equilibria in such settings. Together, these results contribute to the broad lanscape of multi-agent online learning by significantly relaxing the feedback information that is required to achieve desirable outcomes.

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