348 research outputs found
Survival of the strictest: Stable and unstable equilibria under regularized learning with partial information
In this paper, we examine the Nash equilibrium convergence properties of
no-regret learning in general N-player games. For concreteness, we focus on the
archetypal follow the regularized leader (FTRL) family of algorithms, and we
consider the full spectrum of uncertainty that the players may encounter - from
noisy, oracle-based feedback, to bandit, payoff-based information. In this
general context, we establish a comprehensive equivalence between the stability
of a Nash equilibrium and its support: a Nash equilibrium is stable and
attracting with arbitrarily high probability if and only if it is strict (i.e.,
each equilibrium strategy has a unique best response). This equivalence extends
existing continuous-time versions of the folk theorem of evolutionary game
theory to a bona fide algorithmic learning setting, and it provides a clear
refinement criterion for the prediction of the day-to-day behavior of no-regret
learning in game
The Computational Complexity of Multi-player Concave Games and Kakutani Fixed Points
Introduced by the celebrated works of Debreu and Rosen in the 1950s and 60s,
concave -person games have found many important applications in Economics
and Game Theory. We characterize the computational complexity of finding an
equilibrium in such games. We show that a general formulation of this problem
belongs to PPAD, and that finding an equilibrium is PPAD-hard even for a rather
restricted games of this kind: strongly-convex utilities that can be expressed
as multivariate polynomials of a constant degree with axis aligned box
constraints. Along the way we provide a general computational formulation of
Kakutani's Fixed Point Theorem, previously formulated in a special case that is
too restrictive to be useful in reductions, and prove it PPAD-complete
Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds
Numerous applications in machine learning and data analytics can be
formulated as equilibrium computation over Riemannian manifolds. Despite the
extensive investigation of their Euclidean counterparts, the performance of
Riemannian gradient-based algorithms remain opaque and poorly understood. We
revisit the original scheme of Riemannian gradient descent (RGD) and analyze it
under a geodesic monotonicity assumption, which includes the well-studied
geodesically convex-concave min-max optimization problem as a special case. Our
main contribution is to show that, despite the phenomenon of distance
distortion, the RGD scheme, with a step size that is agnostic to the manifold's
curvature, achieves a curvature-independent and linear last-iterate convergence
rate in the geodesically strongly monotone setting. To the best of our
knowledge, the possibility of curvature-independent rates and/or last-iterate
convergence in the Riemannian setting has not been considered before
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