1,017 research outputs found
Theoretical and Practical Advances on Smoothing for Extensive-Form Games
Sparse iterative methods, in particular first-order methods, are known to be
among the most effective in solving large-scale two-player zero-sum
extensive-form games. The convergence rates of these methods depend heavily on
the properties of the distance-generating function that they are based on. We
investigate the acceleration of first-order methods for solving extensive-form
games through better design of the dilated entropy function---a class of
distance-generating functions related to the domains associated with the
extensive-form games. By introducing a new weighting scheme for the dilated
entropy function, we develop the first distance-generating function for the
strategy spaces of sequential games that has no dependence on the branching
factor of the player. This result improves the convergence rate of several
first-order methods by a factor of , where is the branching
factor of the player, and is the depth of the game tree.
Thus far, counterfactual regret minimization methods have been faster in
practice, and more popular, than first-order methods despite their
theoretically inferior convergence rates. Using our new weighting scheme and
practical tuning we show that, for the first time, the excessive gap technique
can be made faster than the fastest counterfactual regret minimization
algorithm, CFR+, in practice
Computing large market equilibria using abstractions
Computing market equilibria is an important practical problem for market
design (e.g. fair division, item allocation). However, computing equilibria
requires large amounts of information (e.g. all valuations for all buyers for
all items) and compute power. We consider ameliorating these issues by applying
a method used for solving complex games: constructing a coarsened abstraction
of a given market, solving for the equilibrium in the abstraction, and lifting
the prices and allocations back to the original market. We show how to bound
important quantities such as regret, envy, Nash social welfare, Pareto
optimality, and maximin share when the abstracted prices and allocations are
used in place of the real equilibrium. We then study two abstraction methods of
interest for practitioners: 1) filling in unknown valuations using techniques
from matrix completion, 2) reducing the problem size by aggregating groups of
buyers/items into smaller numbers of representative buyers/items and solving
for equilibrium in this coarsened market. We find that in real data
allocations/prices that are relatively close to equilibria can be computed from
even very coarse abstractions
Linear Coupling: An Ultimate Unification of Gradient and Mirror Descent
First-order methods play a central role in large-scale machine learning. Even
though many variations exist, each suited to a particular problem, almost all
such methods fundamentally rely on two types of algorithmic steps: gradient
descent, which yields primal progress, and mirror descent, which yields dual
progress.
We observe that the performances of gradient and mirror descent are
complementary, so that faster algorithms can be designed by LINEARLY COUPLING
the two. We show how to reconstruct Nesterov's accelerated gradient methods
using linear coupling, which gives a cleaner interpretation than Nesterov's
original proofs. We also discuss the power of linear coupling by extending it
to many other settings that Nesterov's methods cannot apply to.Comment: A new section added; polished writin
A duality-based approach for distributed min-max optimization with application to demand side management
In this paper we consider a distributed optimization scenario in which a set
of processors aims at minimizing the maximum of a collection of "separable
convex functions" subject to local constraints. This set-up is motivated by
peak-demand minimization problems in smart grids. Here, the goal is to minimize
the peak value over a finite horizon with: (i) the demand at each time instant
being the sum of contributions from different devices, and (ii) the local
states at different time instants being coupled through local dynamics. The
min-max structure and the double coupling (through the devices and over the
time horizon) makes this problem challenging in a distributed set-up (e.g.,
well-known distributed dual decomposition approaches cannot be applied). We
propose a distributed algorithm based on the combination of duality methods and
properties from min-max optimization. Specifically, we derive a series of
equivalent problems by introducing ad-hoc slack variables and by going back and
forth from primal and dual formulations. On the resulting problem we apply a
dual subgradient method, which turns out to be a distributed algorithm. We
prove the correctness of the proposed algorithm and show its effectiveness via
numerical computations.Comment: arXiv admin note: substantial text overlap with arXiv:1611.0916
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