37 research outputs found
Distributionally Robust Games with Risk-averse Players
We present a new model of incomplete information games without private
information in which the players use a distributionally robust optimization
approach to cope with the payoff uncertainty. With some specific restrictions,
we show that our "Distributionally Robust Game" constitutes a true
generalization of three popular finite games. These are the Complete
Information Games, Bayesian Games and Robust Games. Subsequently, we prove that
the set of equilibria of an arbitrary distributionally robust game with
specified ambiguity set can be computed as the component-wise projection of the
solution set of a multi-linear system of equations and inequalities. For
special cases of such games we show equivalence to complete information finite
games (Nash Games) with the same number of players and same action spaces.
Thus, when our game falls within these special cases one can simply solve the
corresponding Nash Game. Finally, we demonstrate the applicability of our new
model of games and highlight its importance.Comment: 11 pages, 3 figures, Proceedings of 5th the International Conference
on Operations Research and Enterprise Systems ({ICORES} 2016), Rome, Italy,
February 23-25, 201
A New Perspective on Randomized Gossip Algorithms
In this short note we propose a new approach for the design and analysis of
randomized gossip algorithms which can be used to solve the average consensus
problem. We show how that Randomized Block Kaczmarz (RBK) method - a method for
solving linear systems - works as gossip algorithm when applied to a special
system encoding the underlying network. The famous pairwise gossip algorithm
arises as a special case. Subsequently, we reveal a hidden duality of
randomized gossip algorithms, with the dual iterative process maintaining a set
of numbers attached to the edges as opposed to nodes of the network. We prove
that RBK obtains a superlinear speedup in the size of the block, and
demonstrate this effect through experiments
Momentum and Stochastic Momentum for Stochastic Gradient, Newton, Proximal Point and Subspace Descent Methods
In this paper we study several classes of stochastic optimization algorithms
enriched with heavy ball momentum. Among the methods studied are: stochastic
gradient descent, stochastic Newton, stochastic proximal point and stochastic
dual subspace ascent. This is the first time momentum variants of several of
these methods are studied. We choose to perform our analysis in a setting in
which all of the above methods are equivalent. We prove global nonassymptotic
linear convergence rates for all methods and various measures of success,
including primal function values, primal iterates (in L2 sense), and dual
function values. We also show that the primal iterates converge at an
accelerated linear rate in the L1 sense. This is the first time a linear rate
is shown for the stochastic heavy ball method (i.e., stochastic gradient
descent method with momentum). Under somewhat weaker conditions, we establish a
sublinear convergence rate for Cesaro averages of primal iterates. Moreover, we
propose a novel concept, which we call stochastic momentum, aimed at decreasing
the cost of performing the momentum step. We prove linear convergence of
several stochastic methods with stochastic momentum, and show that in some
sparse data regimes and for sufficiently small momentum parameters, these
methods enjoy better overall complexity than methods with deterministic
momentum. Finally, we perform extensive numerical testing on artificial and
real datasets, including data coming from average consensus problems.Comment: 47 pages, 7 figures, 7 table