12,672 research outputs found
Optimal Algorithms for Non-Smooth Distributed Optimization in Networks
In this work, we consider the distributed optimization of non-smooth convex
functions using a network of computing units. We investigate this problem under
two regularity assumptions: (1) the Lipschitz continuity of the global
objective function, and (2) the Lipschitz continuity of local individual
functions. Under the local regularity assumption, we provide the first optimal
first-order decentralized algorithm called multi-step primal-dual (MSPD) and
its corresponding optimal convergence rate. A notable aspect of this result is
that, for non-smooth functions, while the dominant term of the error is in
, the structure of the communication network only impacts a
second-order term in , where is time. In other words, the error due
to limits in communication resources decreases at a fast rate even in the case
of non-strongly-convex objective functions. Under the global regularity
assumption, we provide a simple yet efficient algorithm called distributed
randomized smoothing (DRS) based on a local smoothing of the objective
function, and show that DRS is within a multiplicative factor of the
optimal convergence rate, where is the underlying dimension.Comment: 17 page
Practical Inexact Proximal Quasi-Newton Method with Global Complexity Analysis
Recently several methods were proposed for sparse optimization which make
careful use of second-order information [10, 28, 16, 3] to improve local
convergence rates. These methods construct a composite quadratic approximation
using Hessian information, optimize this approximation using a first-order
method, such as coordinate descent and employ a line search to ensure
sufficient descent. Here we propose a general framework, which includes
slightly modified versions of existing algorithms and also a new algorithm,
which uses limited memory BFGS Hessian approximations, and provide a novel
global convergence rate analysis, which covers methods that solve subproblems
via coordinate descent
Adaptive Regret Minimization in Bounded-Memory Games
Online learning algorithms that minimize regret provide strong guarantees in
situations that involve repeatedly making decisions in an uncertain
environment, e.g. a driver deciding what route to drive to work every day.
While regret minimization has been extensively studied in repeated games, we
study regret minimization for a richer class of games called bounded memory
games. In each round of a two-player bounded memory-m game, both players
simultaneously play an action, observe an outcome and receive a reward. The
reward may depend on the last m outcomes as well as the actions of the players
in the current round. The standard notion of regret for repeated games is no
longer suitable because actions and rewards can depend on the history of play.
To account for this generality, we introduce the notion of k-adaptive regret,
which compares the reward obtained by playing actions prescribed by the
algorithm against a hypothetical k-adaptive adversary with the reward obtained
by the best expert in hindsight against the same adversary. Roughly, a
hypothetical k-adaptive adversary adapts her strategy to the defender's actions
exactly as the real adversary would within each window of k rounds. Our
definition is parametrized by a set of experts, which can include both fixed
and adaptive defender strategies.
We investigate the inherent complexity of and design algorithms for adaptive
regret minimization in bounded memory games of perfect and imperfect
information. We prove a hardness result showing that, with imperfect
information, any k-adaptive regret minimizing algorithm (with fixed strategies
as experts) must be inefficient unless NP=RP even when playing against an
oblivious adversary. In contrast, for bounded memory games of perfect and
imperfect information we present approximate 0-adaptive regret minimization
algorithms against an oblivious adversary running in time n^{O(1)}.Comment: Full Version. GameSec 2013 (Invited Paper
An Asynchronous Parallel Randomized Kaczmarz Algorithm
We describe an asynchronous parallel variant of the randomized Kaczmarz (RK)
algorithm for solving the linear system . The analysis shows linear
convergence and indicates that nearly linear speedup can be expected if the
number of processors is bounded by a multiple of the number of rows in
Proactive Caching for Energy-Efficiency in Wireless Networks: A Markov Decision Process Approach
Content caching in wireless networks provides a substantial opportunity to
trade off low cost memory storage with energy consumption, yet finding the
optimal causal policy with low computational complexity remains a challenge.
This paper models the Joint Pushing and Caching (JPC) problem as a Markov
Decision Process (MDP) and provides a solution to determine the optimal
randomized policy. A novel approach to decouple the influence from buffer
occupancy and user requests is proposed to turn the high-dimensional
optimization problem into three low-dimensional ones. Furthermore, a
non-iterative algorithm to solve one of the sub-problems is presented,
exploiting a structural property we found as \textit{generalized monotonicity},
and hence significantly reduces the computational complexity. The result
attains close performance in comparison with theoretical bounds from
non-practical policies, while benefiting from higher time efficiency than the
unadapted MDP solution.Comment: 6 pages, 6 figures, submitted to IEEE International Conference on
Communications 201
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