7 research outputs found
On the convergence of mirror descent beyond stochastic convex programming
In this paper, we examine the convergence of mirror descent in a class of
stochastic optimization problems that are not necessarily convex (or even
quasi-convex), and which we call variationally coherent. Since the standard
technique of "ergodic averaging" offers no tangible benefits beyond convex
programming, we focus directly on the algorithm's last generated sample (its
"last iterate"), and we show that it converges with probabiility if the
underlying problem is coherent. We further consider a localized version of
variational coherence which ensures local convergence of stochastic mirror
descent (SMD) with high probability. These results contribute to the landscape
of non-convex stochastic optimization by showing that (quasi-)convexity is not
essential for convergence to a global minimum: rather, variational coherence, a
much weaker requirement, suffices. Finally, building on the above, we reveal an
interesting insight regarding the convergence speed of SMD: in problems with
sharp minima (such as generic linear programs or concave minimization
problems), SMD reaches a minimum point in a finite number of steps (a.s.), even
in the presence of persistent gradient noise. This result is to be contrasted
with existing black-box convergence rate estimates that are only asymptotic.Comment: 30 pages, 5 figure
Theory and Applications of Simulated Annealing for Nonlinear Constrained Optimization
A general mixed-integer nonlinear programming problem (MINLP) is formulated as follows: where z = (x, y) T ∈ Z; x ∈ Rv and y ∈ D w are, respectively, bounded continuous and discrete variables; f(z) is a lower-bounded objective function; g(z) = (g1(z),…, gr(z)) T is a vector of r inequality constraint functions; 2 and h(z) = (h1(z),…,hm(z)) T is a vector of m equality constrain
Entropy-driven dynamics and robust learning procedures in games
In this paper, we introduce a new class of game dynamics made of a pay-off replicator-like term modulated by an entropy barrier which keeps players away from the boundary of the strategy space. We show that these {\it entropy-driven} dynamics are equivalent to players computing a score as their on-going exponentially discounted cumulative payoff and then using a quantal choice model on the scores to pick an action. This dual perspective on {\it entropy-driven} dynamics helps us to extend the folk theorem on convergence to quantal response equilibria to this case, for potential games. It also provides the main ingredients to design a discrete time effective learning algorithm that is fully distributed and only requires partial information to converge to QRE. This convergence is resilient to stochastic perturbations and observation errors and does not require any synchronization between the players