538 research outputs found
Tight Bounds for Asymptotic and Approximate Consensus
We study the performance of asymptotic and approximate consensus algorithms
under harsh environmental conditions. The asymptotic consensus problem requires
a set of agents to repeatedly set their outputs such that the outputs converge
to a common value within the convex hull of initial values. This problem, and
the related approximate consensus problem, are fundamental building blocks in
distributed systems where exact consensus among agents is not required or
possible, e.g., man-made distributed control systems, and have applications in
the analysis of natural distributed systems, such as flocking and opinion
dynamics. We prove tight lower bounds on the contraction rates of asymptotic
consensus algorithms in dynamic networks, from which we deduce bounds on the
time complexity of approximate consensus algorithms. In particular, the
obtained bounds show optimality of asymptotic and approximate consensus
algorithms presented in [Charron-Bost et al., ICALP'16] for certain dynamic
networks, including the weakest dynamic network model in which asymptotic and
approximate consensus are solvable. As a corollary we also obtain
asymptotically tight bounds for asymptotic consensus in the classical
asynchronous model with crashes.
Central to our lower bound proofs is an extended notion of valency, the set
of reachable limits of an asymptotic consensus algorithm starting from a given
configuration. We further relate topological properties of valencies to the
solvability of exact consensus, shedding some light on the relation of these
three fundamental problems in dynamic networks
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
Market Equilibrium with Transaction Costs
Identical products being sold at different prices in different locations is a
common phenomenon. Price differences might occur due to various reasons such as
shipping costs, trade restrictions and price discrimination. To model such
scenarios, we supplement the classical Fisher model of a market by introducing
{\em transaction costs}. For every buyer and every good , there is a
transaction cost of \cij; if the price of good is , then the cost to
the buyer {\em per unit} of is p_j + \cij. This allows the same good
to be sold at different (effective) prices to different buyers.
We provide a combinatorial algorithm that computes -approximate
equilibrium prices and allocations in
operations -
where is the number goods, is the number of buyers and is the sum
of the budgets of all the buyers
Asynchronous Proportional Response Dynamics in Markets with Adversarial Scheduling
We study Proportional Response Dynamics (PRD) in linear Fisher markets where
participants act asynchronously. We model this scenario as a sequential process
in which in every step, an adversary selects a subset of the players that will
update their bids, subject to liveness constraints. We show that if every
bidder individually uses the PRD update rule whenever they are included in the
group of bidders selected by the adversary, then (in the generic case) the
entire dynamic converges to a competitive equilibrium of the market. Our proof
technique uncovers further properties of linear Fisher markets, such as the
uniqueness of the equilibrium for generic parameters and the convergence of
associated best-response dynamics and no-swap regret dynamics under certain
conditions
Parallelization of implicit finite difference schemes in computational fluid dynamics
Implicit finite difference schemes are often the preferred numerical schemes in computational fluid dynamics, requiring less stringent stability bounds than the explicit schemes. Each iteration in an implicit scheme involves global data dependencies in the form of second and higher order recurrences. Efficient parallel implementations of such iterative methods are considerably more difficult and non-intuitive. The parallelization of the implicit schemes that are used for solving the Euler and the thin layer Navier-Stokes equations and that require inversions of large linear systems in the form of block tri-diagonal and/or block penta-diagonal matrices is discussed. Three-dimensional cases are emphasized and schemes that minimize the total execution time are presented. Partitioning and scheduling schemes for alleviating the effects of the global data dependencies are described. An analysis of the communication and the computation aspects of these methods is presented. The effect of the boundary conditions on the parallel schemes is also discussed
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
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