14,459 research outputs found
Optimal algorithms for smooth and strongly convex distributed optimization in networks
In this paper, we determine the optimal convergence rates for strongly convex and smooth distributed optimization in two settings: centralized and decentralized communications over a network. For centralized (i.e. master/slave) algorithms, we show that distributing Nesterov's accelerated gradient descent is optimal and achieves a precision in time , where is the condition number of the (global) function to optimize, is the diameter of the network, and (resp. ) is the time needed to communicate values between two neighbors (resp. perform local computations). For decentralized algorithms based on gossip, we provide the first optimal algorithm, called the multi-step dual accelerated (MSDA) method, that achieves a precision in time , where is the condition number of the local functions and is the (normalized) eigengap of the gossip matrix used for communication between nodes. We then verify the efficiency of MSDA against state-of-the-art methods for two problems: least-squares regression and classification by logistic regression
Distributed ADMM over directed networks
Distributed optimization over a network of agents is ubiquitous with
applications in areas including power system control, robotics and statistical
learning. In many settings, the communication network is directed, i.e., the
communication links between agents are unidirectional. While several variations
of gradient-descent-based primal methods have been proposed for distributed
optimization over directed networks, an extension of dual-ascent methods to
directed networks remains a less-explored area. In this paper, we propose a
distributed version of the Alternating Direction Method of Multipliers (ADMM)
for directed networks. ADMM is a dual-ascent method that is known to perform
well in practice. We show that if the objective function is smooth and strongly
convex, our distributed ADMM algorithm achieves a geometric rate of convergence
to the optimal point. Through numerical examples, we observe that the
performance of our algorithm is comparable with some state-of-the-art
distributed optimization algorithms over directed graphs. Additionally, our
algorithm is observed to be robust to changes in its parameters
Multi-consensus Decentralized Accelerated Gradient Descent
This paper considers the decentralized optimization problem, which has
applications in large scale machine learning, sensor networks, and control
theory. We propose a novel algorithm that can achieve near optimal
communication complexity, matching the known lower bound up to a logarithmic
factor of the condition number of the problem. Our theoretical results give
affirmative answers to the open problem on whether there exists an algorithm
that can achieve a communication complexity (nearly) matching the lower bound
depending on the global condition number instead of the local one. Moreover,
the proposed algorithm achieves the optimal computation complexity matching the
lower bound up to universal constants. Furthermore, to achieve a linear
convergence rate, our algorithm \emph{doesn't} require the individual functions
to be (strongly) convex. Our method relies on a novel combination of known
techniques including Nesterov's accelerated gradient descent, multi-consensus
and gradient-tracking. The analysis is new, and may be applied to other related
problems. Empirical studies demonstrate the effectiveness of our method for
machine learning applications
- …