29,719 research outputs found
Time-Average Stochastic Optimization with Non-convex Decision Set and its Convergence
Abstract-This paper considers time-average stochastic optimization, where a time average decision vector, an average of decision vectors chosen in every time step from a timevarying (possibly non-convex) set, minimizes a convex objective function and satisfies convex constraints. This formulation has applications in networking and operations research. In general, time-average stochastic optimization can be solved by a Lyapunov optimization technique. This paper shows that the technique exhibits a transient phase and a steady state phase. When the problem has a unique vector of Lagrange multipliers, the convergence time can be improved. By starting the time average in the steady state, the convergence times become O(1/ ) under a locally-polyhedral assumption and O(1/ 1.5 ) under a locallynon-polyhedral assumption, where denotes the proximity to the optimal objective cost
Distributed Big-Data Optimization via Block-Iterative Convexification and Averaging
In this paper, we study distributed big-data nonconvex optimization in
multi-agent networks. We consider the (constrained) minimization of the sum of
a smooth (possibly) nonconvex function, i.e., the agents' sum-utility, plus a
convex (possibly) nonsmooth regularizer. Our interest is in big-data problems
wherein there is a large number of variables to optimize. If treated by means
of standard distributed optimization algorithms, these large-scale problems may
be intractable, due to the prohibitive local computation and communication
burden at each node. We propose a novel distributed solution method whereby at
each iteration agents optimize and then communicate (in an uncoordinated
fashion) only a subset of their decision variables. To deal with non-convexity
of the cost function, the novel scheme hinges on Successive Convex
Approximation (SCA) techniques coupled with i) a tracking mechanism
instrumental to locally estimate gradient averages; and ii) a novel block-wise
consensus-based protocol to perform local block-averaging operations and
gradient tacking. Asymptotic convergence to stationary solutions of the
nonconvex problem is established. Finally, numerical results show the
effectiveness of the proposed algorithm and highlight how the block dimension
impacts on the communication overhead and practical convergence speed
Robust and Communication-Efficient Collaborative Learning
We consider a decentralized learning problem, where a set of computing nodes
aim at solving a non-convex optimization problem collaboratively. It is
well-known that decentralized optimization schemes face two major system
bottlenecks: stragglers' delay and communication overhead. In this paper, we
tackle these bottlenecks by proposing a novel decentralized and gradient-based
optimization algorithm named as QuanTimed-DSGD. Our algorithm stands on two
main ideas: (i) we impose a deadline on the local gradient computations of each
node at each iteration of the algorithm, and (ii) the nodes exchange quantized
versions of their local models. The first idea robustifies to straggling nodes
and the second alleviates communication efficiency. The key technical
contribution of our work is to prove that with non-vanishing noises for
quantization and stochastic gradients, the proposed method exactly converges to
the global optimal for convex loss functions, and finds a first-order
stationary point in non-convex scenarios. Our numerical evaluations of the
QuanTimed-DSGD on training benchmark datasets, MNIST and CIFAR-10, demonstrate
speedups of up to 3x in run-time, compared to state-of-the-art decentralized
optimization methods
Data-driven satisficing measure and ranking
We propose an computational framework for real-time risk assessment and
prioritizing for random outcomes without prior information on probability
distributions. The basic model is built based on satisficing measure (SM) which
yields a single index for risk comparison. Since SM is a dual representation
for a family of risk measures, we consider problems constrained by general
convex risk measures and specifically by Conditional value-at-risk. Starting
from offline optimization, we apply sample average approximation technique and
argue the convergence rate and validation of optimal solutions. In online
stochastic optimization case, we develop primal-dual stochastic approximation
algorithms respectively for general risk constrained problems, and derive their
regret bounds. For both offline and online cases, we illustrate the
relationship between risk ranking accuracy with sample size (or iterations).Comment: 26 Pages, 6 Figure
Distributed Big-Data Optimization via Block Communications
We study distributed multi-agent large-scale optimization problems, wherein
the cost function is composed of a smooth possibly nonconvex sum-utility plus a
DC (Difference-of-Convex) regularizer. We consider the scenario where the
dimension of the optimization variables is so large that optimizing and/or
transmitting the entire set of variables could cause unaffordable computation
and communication overhead. To address this issue, we propose the first
distributed algorithm whereby agents optimize and communicate only a portion of
their local variables. The scheme hinges on successive convex approximation
(SCA) to handle the nonconvexity of the objective function, coupled with a
novel block-signal tracking scheme, aiming at locally estimating the average of
the agents' gradients. Asymptotic convergence to stationary solutions of the
nonconvex problem is established. Numerical results on a sparse regression
problem show the effectiveness of the proposed algorithm and the impact of the
block size on its practical convergence speed and communication cost
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