53 research outputs found
Performance Limits of Stochastic Sub-Gradient Learning, Part II: Multi-Agent Case
The analysis in Part I revealed interesting properties for subgradient
learning algorithms in the context of stochastic optimization when gradient
noise is present. These algorithms are used when the risk functions are
non-smooth and involve non-differentiable components. They have been long
recognized as being slow converging methods. However, it was revealed in Part I
that the rate of convergence becomes linear for stochastic optimization
problems, with the error iterate converging at an exponential rate
to within an neighborhood of the optimizer, for some and small step-size . The conclusion was established under weaker
assumptions than the prior literature and, moreover, several important problems
(such as LASSO, SVM, and Total Variation) were shown to satisfy these weaker
assumptions automatically (but not the previously used conditions from the
literature). These results revealed that sub-gradient learning methods have
more favorable behavior than originally thought when used to enable continuous
adaptation and learning. The results of Part I were exclusive to single-agent
adaptation. The purpose of the current Part II is to examine the implications
of these discoveries when a collection of networked agents employs subgradient
learning as their cooperative mechanism. The analysis will show that, despite
the coupled dynamics that arises in a networked scenario, the agents are still
able to attain linear convergence in the stochastic case; they are also able to
reach agreement within of the optimizer
Supervised Learning Under Distributed Features
This work studies the problem of learning under both large datasets and
large-dimensional feature space scenarios. The feature information is assumed
to be spread across agents in a network, where each agent observes some of the
features. Through local cooperation, the agents are supposed to interact with
each other to solve an inference problem and converge towards the global
minimizer of an empirical risk. We study this problem exclusively in the primal
domain, and propose new and effective distributed solutions with guaranteed
convergence to the minimizer with linear rate under strong convexity. This is
achieved by combining a dynamic diffusion construction, a pipeline strategy,
and variance-reduced techniques. Simulation results illustrate the conclusions
Social Learning over Weakly-Connected Graphs
In this paper, we study diffusion social learning over weakly-connected
graphs. We show that the asymmetric flow of information hinders the learning
abilities of certain agents regardless of their local observations. Under some
circumstances that we clarify in this work, a scenario of total influence (or
"mind-control") arises where a set of influential agents ends up shaping the
beliefs of non-influential agents. We derive useful closed-form expressions
that characterize this influence, and which can be used to motivate design
problems to control it. We provide simulation examples to illustrate the
results.Comment: To appear in 2017 in the IEEE Transactions on Signal and Information
Processing over Network
Variance-Reduced Stochastic Learning by Networked Agents under Random Reshuffling
A new amortized variance-reduced gradient (AVRG) algorithm was developed in
\cite{ying2017convergence}, which has constant storage requirement in
comparison to SAGA and balanced gradient computations in comparison to SVRG.
One key advantage of the AVRG strategy is its amenability to decentralized
implementations. In this work, we show how AVRG can be extended to the network
case where multiple learning agents are assumed to be connected by a graph
topology. In this scenario, each agent observes data that is spatially
distributed and all agents are only allowed to communicate with direct
neighbors. Moreover, the amount of data observed by the individual agents may
differ drastically. For such situations, the balanced gradient computation
property of AVRG becomes a real advantage in reducing idle time caused by
unbalanced local data storage requirements, which is characteristic of other
reduced-variance gradient algorithms. The resulting diffusion-AVRG algorithm is
shown to have linear convergence to the exact solution, and is much more memory
efficient than other alternative algorithms. In addition, we propose a
mini-batch strategy to balance the communication and computation efficiency for
diffusion-AVRG. When a proper batch size is employed, it is observed in
simulations that diffusion-AVRG is more computationally efficient than exact
diffusion or EXTRA while maintaining almost the same communication efficiency.Comment: 23 pages, 12 figures, submitted for publicatio
Stochastic Learning under Random Reshuffling with Constant Step-sizes
In empirical risk optimization, it has been observed that stochastic gradient
implementations that rely on random reshuffling of the data achieve better
performance than implementations that rely on sampling the data uniformly.
Recent works have pursued justifications for this behavior by examining the
convergence rate of the learning process under diminishing step-sizes. This
work focuses on the constant step-size case and strongly convex loss function.
In this case, convergence is guaranteed to a small neighborhood of the
optimizer albeit at a linear rate. The analysis establishes analytically that
random reshuffling outperforms uniform sampling by showing explicitly that
iterates approach a smaller neighborhood of size around the
minimizer rather than . Furthermore, we derive an analytical expression
for the steady-state mean-square-error performance of the algorithm, which
helps clarify in greater detail the differences between sampling with and
without replacement. We also explain the periodic behavior that is observed in
random reshuffling implementations
Dynamic Average Diffusion with randomized Coordinate Updates
This work derives and analyzes an online learning strategy for tracking the
average of time-varying distributed signals by relying on randomized
coordinate-descent updates. During each iteration, each agent selects or
observes a random entry of the observation vector, and different agents may
select different entries of their observations before engaging in a
consultation step. Careful coordination of the interactions among agents is
necessary to avoid bias and ensure convergence. We provide a convergence
analysis for the proposed methods, and illustrate the results by means of
simulations
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