325 research outputs found
Tighter Bounds on the Inefficiency Ratio of Stable Equilibria in Load Balancing Games
In this paper we study the inefficiency ratio of stable equilibria in load
balancing games introduced by Asadpour and Saberi [3]. We prove tighter lower
and upper bounds of 7/6 and 4/3, respectively. This improves over the best
known bounds in problem (19/18 and 3/2, respectively). Equivalently, the
results apply to the question of how well the optimum for the -norm can
approximate the -norm (makespan) in identical machines scheduling
Exact Recovery for a Family of Community-Detection Generative Models
Generative models for networks with communities have been studied extensively
for being a fertile ground to establish information-theoretic and computational
thresholds. In this paper we propose a new toy model for planted generative
models called planted Random Energy Model (REM), inspired by Derrida's REM. For
this model we provide the asymptotic behaviour of the probability of error for
the maximum likelihood estimator and hence the exact recovery threshold. As an
application, we further consider the 2 non-equally sized community Weighted
Stochastic Block Model (2-WSBM) on -uniform hypergraphs, that is equivalent
to the P-REM on both sides of the spectrum, for high and low edge cardinality
. We provide upper and lower bounds for the exact recoverability for any
, mapping these problems to the aforementioned P-REM. To the best of our
knowledge these are the first consistency results for the 2-WSBM on graphs and
on hypergraphs with non-equally sized community
Convergence to Equilibrium of Logit Dynamics for Strategic Games
We present the first general bounds on the mixing time of the Markov chain
associated to the logit dynamics for wide classes of strategic games. The logit
dynamics with inverse noise beta describes the behavior of a complex system
whose individual components act selfishly and keep responding according to some
partial ("noisy") knowledge of the system, where the capacity of the agent to
know the system and compute her best move is measured by the inverse of the
parameter beta.
In particular, we prove nearly tight bounds for potential games and games
with dominant strategies. Our results show that, for potential games, the
mixing time is upper and lower bounded by an exponential in the inverse of the
noise and in the maximum potential difference. Instead, for games with dominant
strategies, the mixing time cannot grow arbitrarily with the inverse of the
noise.
Finally, we refine our analysis for a subclass of potential games called
graphical coordination games, a class of games that have been previously
studied in Physics and, more recently, in Computer Science in the context of
diffusion of new technologies. We give evidence that the mixing time of the
logit dynamics for these games strongly depends on the structure of the
underlying graph. We prove that the mixing time of the logit dynamics for these
games can be upper bounded by a function that is exponential in the cutwidth of
the underlying graph and in the inverse of noise. Moreover, we consider two
specific and popular network topologies, the clique and the ring. For games
played on a clique we prove an almost matching lower bound on the mixing time
of the logit dynamics that is exponential in the inverse of the noise and in
the maximum potential difference, while for games played on a ring we prove
that the time of convergence of the logit dynamics to its stationary
distribution is significantly shorter
Solving Zero-Sum Games through Alternating Projections
In this work, we establish near-linear and strong convergence for a natural
first-order iterative algorithm that simulates Von Neumann's Alternating
Projections method in zero-sum games. First, we provide a precise analysis of
Optimistic Gradient Descent/Ascent (OGDA) -- an optimistic variant of Gradient
Descent/Ascent \cite{DBLP:journals/corr/abs-1711-00141} -- for
\emph{unconstrained} bilinear games, extending and strengthening prior results
along several directions. Our characterization is based on a closed-form
solution we derive for the dynamics, while our results also reveal several
surprising properties. Indeed, our main algorithmic contribution is founded on
a geometric feature of OGDA we discovered; namely, the limit points of the
dynamics are the orthogonal projection of the initial state to the space of
attractors.
Motivated by this property, we show that the equilibria for a natural class
of \emph{constrained} bilinear games are the intersection of the unconstrained
stationary points with the corresponding probability simplexes. Thus, we employ
OGDA to implement an Alternating Projections procedure, converging to an
-approximate Nash equilibrium in
iterations. Although our
algorithm closely resembles the no-regret projected OGDA dynamics, it surpasses
the optimal no-regret convergence rate of
\cite{DASKALAKIS2015327}, while it also supplements the recent work in pursuing
last-iterate guarantees in saddle-point problems
\cite{daskalakis2018lastiterate,mertikopoulos2018optimistic}. Finally, we
illustrate an -- in principle -- trivial reduction from any game to the assumed
class of instances, without altering the space of equilibria
Truthful Mechanisms for Delivery with Agents
We study the game-theoretic task of selecting mobile agents to deliver multiple items on a network. An instance is given by packages (physical objects) which have to be transported between specified source-target pairs in an undirected graph, and mobile heterogeneous agents, each being able to transport one package at a time. Following a recent model [Baertschi et al. 2017], each agent i has a different rate of energy consumption per unit distance traveled, i.e., its weight. We are interested in optimizing or approximating the total energy consumption over all selected agents.
Unlike previous research, we assume the weights to be private values known only to the respective agents. We present three different mechanisms which select, route and pay the agents in a truthful way that guarantees voluntary participation of the agents, while approximating the optimum energy consumption by a constant factor. To this end, we analyze a previous structural result and an approximation algorithm given in [Baertschi et al. 2017]. Finally, we show that for some instances in the case of a single package, the sum of the payments can be bounded in terms of the optimum
A Robust Framework for Analyzing Gradient-Based Dynamics in Bilinear Games
In this work, we establish a frequency-domain framework for analyzing
gradient-based algorithms in linear minimax optimization problems;
specifically, our approach is based on the Z-transform, a powerful tool applied
in Control Theory and Signal Processing in order to characterize linear
discrete-time systems. We employ our framework to obtain the first tight
analysis of stability of Optimistic Gradient Descent/Ascent (OGDA), a natural
variant of Gradient Descent/Ascent that was shown to exhibit last-iterate
convergence in bilinear games by Daskalakis et al.
\cite{DBLP:journals/corr/abs-1711-00141}. Importantly, our analysis is
considerably simpler and more concise than the existing ones.
Moreover, building on the intuition of OGDA, we consider a general family of
gradient-based algorithms that augment the memory of the optimization through
multiple historical steps. We reduce the convergence -- to a saddle-point -- of
the dynamics in bilinear games to the stability of a polynomial, for which
efficient algorithmic schemes are well-established. As an immediate corollary,
we obtain a broad class of algorithms -- that contains OGDA as a special case
-- with a last-iterate convergence guarantee to the space of Nash equilibria of
the game
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