86 research outputs found
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
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
Computing Equilibria in Markets with Budget-Additive Utilities
We present the first analysis of Fisher markets with buyers that have
budget-additive utility functions. Budget-additive utilities are elementary
concave functions with numerous applications in online adword markets and
revenue optimization problems. They extend the standard case of linear
utilities and have been studied in a variety of other market models. In
contrast to the frequently studied CES utilities, they have a global satiation
point which can imply multiple market equilibria with quite different
characteristics. Our main result is an efficient combinatorial algorithm to
compute a market equilibrium with a Pareto-optimal allocation of goods. It
relies on a new descending-price approach and, as a special case, also implies
a novel combinatorial algorithm for computing a market equilibrium in linear
Fisher markets. We complement these positive results with a number of hardness
results for related computational questions. We prove that it is NP-hard to
compute a market equilibrium that maximizes social welfare, and it is PPAD-hard
to find any market equilibrium with utility functions with separate satiation
points for each buyer and each good.Comment: 21 page
Proportional Dynamics in Exchange Economies
We study the Proportional Response dynamic in exchange economies, where each
player starts with some amount of money and a good. Every day, the players
bring one unit of their good and submit bids on goods they like, each good gets
allocated in proportion to the bid amounts, and each seller collects the bids
received. Then every player updates the bids proportionally to the contribution
of each good in their utility. This dynamic models a process of learning how to
bid and has been studied in a series of papers on Fisher and production
markets, but not in exchange economies. Our main results are as follows:
- For linear utilities, the dynamic converges to market equilibrium utilities
and allocations, while the bids and prices may cycle. We give a combinatorial
characterization of limit cycles for prices and bids.
- We introduce a lazy version of the dynamic, where players may save money
for later, and show this converges in everything: utilities, allocations, and
prices.
- For CES utilities in the substitute range , the dynamic converges
for all parameters.
This answers an open question about exchange economies with linear utilities,
where tatonnement does not converge to market equilibria, and no natural
process leading to equilibria was known. We also note that proportional
response is a process where the players exchange goods throughout time (in
out-of-equilibrium states), while tatonnement only explains how exchange
happens in the limit.Comment: 25 pages, 6 figure
The virtues and vices of equilibrium and the future of financial economics
The use of equilibrium models in economics springs from the desire for
parsimonious models of economic phenomena that take human reasoning into
account. This approach has been the cornerstone of modern economic theory. We
explain why this is so, extolling the virtues of equilibrium theory; then we
present a critique and describe why this approach is inherently limited, and
why economics needs to move in new directions if it is to continue to make
progress. We stress that this shouldn't be a question of dogma, but should be
resolved empirically. There are situations where equilibrium models provide
useful predictions and there are situations where they can never provide useful
predictions. There are also many situations where the jury is still out, i.e.,
where so far they fail to provide a good description of the world, but where
proper extensions might change this. Our goal is to convince the skeptics that
equilibrium models can be useful, but also to make traditional economists more
aware of the limitations of equilibrium models. We sketch some alternative
approaches and discuss why they should play an important role in future
research in economics.Comment: 68 pages, one figur
- …