1,956 research outputs found
Tracing Equilibrium in Dynamic Markets via Distributed Adaptation
Competitive equilibrium is a central concept in economics with numerous
applications beyond markets, such as scheduling, fair allocation of goods, or
bandwidth distribution in networks. Computation of competitive equilibria has
received a significant amount of interest in algorithmic game theory, mainly
for the prominent case of Fisher markets. Natural and decentralized processes
like tatonnement and proportional response dynamics (PRD) converge quickly
towards equilibrium in large classes of Fisher markets. Almost all of the
literature assumes that the market is a static environment and that the
parameters of agents and goods do not change over time. In contrast, many large
real-world markets are subject to frequent and dynamic changes. In this paper,
we provide the first provable performance guarantees of discrete-time
tatonnement and PRD in markets that are subject to perturbation over time. We
analyze the prominent class of Fisher markets with CES utilities and quantify
the impact of changes in supplies of goods, budgets of agents, and utility
functions of agents on the convergence of tatonnement to market equilibrium.
Since the equilibrium becomes a dynamic object and will rarely be reached, our
analysis provides bounds expressing the distance to equilibrium that will be
maintained via tatonnement and PRD updates. Our results indicate that in many
cases, tatonnement and PRD follow the equilibrium rather closely and quickly
recover conditions of approximate market clearing. Our approach can be
generalized to analyzing a general class of Lyapunov dynamical systems with
changing system parameters, which might be of independent interest
Convex-Concave Min-Max Stackelberg Games
Min-max optimization problems (i.e., min-max games) have been attracting a
great deal of attention because of their applicability to a wide range of
machine learning problems. Although significant progress has been made
recently, the literature to date has focused on games with independent strategy
sets; little is known about solving games with dependent strategy sets, which
can be characterized as min-max Stackelberg games. We introduce two first-order
methods that solve a large class of convex-concave min-max Stackelberg games,
and show that our methods converge in polynomial time. Min-max Stackelberg
games were first studied by Wald, under the posthumous name of Wald's maximin
model, a variant of which is the main paradigm used in robust optimization,
which means that our methods can likewise solve many convex robust optimization
problems. We observe that the computation of competitive equilibria in Fisher
markets also comprises a min-max Stackelberg game. Further, we demonstrate the
efficacy and efficiency of our algorithms in practice by computing competitive
equilibria in Fisher markets with varying utility structures. Our experiments
suggest potential ways to extend our theoretical results, by demonstrating how
different smoothness properties can affect the convergence rate of our
algorithms.Comment: 25 pages, 4 tables, 1 figure, Forthcoming in NeurIPS 202
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
Amortized Analysis on Asynchronous Gradient Descent
Gradient descent is an important class of iterative algorithms for minimizing
convex functions. Classically, gradient descent has been a sequential and
synchronous process. Distributed and asynchronous variants of gradient descent
have been studied since the 1980s, and they have been experiencing a resurgence
due to demand from large-scale machine learning problems running on multi-core
processors.
We provide a version of asynchronous gradient descent (AGD) in which
communication between cores is minimal and for which there is little
synchronization overhead. We also propose a new timing model for its analysis.
With this model, we give the first amortized analysis of AGD on convex
functions. The amortization allows for bad updates (updates that increase the
value of the convex function); in contrast, most prior work makes the strong
assumption that every update must be significantly improving.
Typically, the step sizes used in AGD are smaller than those used in its
synchronous counterpart. We provide a method to determine the step sizes in AGD
based on the Hessian entries for the convex function. In certain circumstances,
the resulting step sizes are a constant fraction of those used in the
corresponding synchronous algorithm, enabling the overall performance of AGD to
improve linearly with the number of cores.
We give two applications of our amortized analysis.Comment: 40 page
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