67,126 research outputs found
Global Continuous Optimization with Error Bound and Fast Convergence
This paper considers global optimization with a black-box unknown objective
function that can be non-convex and non-differentiable. Such a difficult
optimization problem arises in many real-world applications, such as parameter
tuning in machine learning, engineering design problem, and planning with a
complex physics simulator. This paper proposes a new global optimization
algorithm, called Locally Oriented Global Optimization (LOGO), to aim for both
fast convergence in practice and finite-time error bound in theory. The
advantage and usage of the new algorithm are illustrated via theoretical
analysis and an experiment conducted with 11 benchmark test functions. Further,
we modify the LOGO algorithm to specifically solve a planning problem via
policy search with continuous state/action space and long time horizon while
maintaining its finite-time error bound. We apply the proposed planning method
to accident management of a nuclear power plant. The result of the application
study demonstrates the practical utility of our method
Convergence Analysis of Mixed Timescale Cross-Layer Stochastic Optimization
This paper considers a cross-layer optimization problem driven by
multi-timescale stochastic exogenous processes in wireless communication
networks. Due to the hierarchical information structure in a wireless network,
a mixed timescale stochastic iterative algorithm is proposed to track the
time-varying optimal solution of the cross-layer optimization problem, where
the variables are partitioned into short-term controls updated in a faster
timescale, and long-term controls updated in a slower timescale. We focus on
establishing a convergence analysis framework for such multi-timescale
algorithms, which is difficult due to the timescale separation of the algorithm
and the time-varying nature of the exogenous processes. To cope with this
challenge, we model the algorithm dynamics using stochastic differential
equations (SDEs) and show that the study of the algorithm convergence is
equivalent to the study of the stochastic stability of a virtual stochastic
dynamic system (VSDS). Leveraging the techniques of Lyapunov stability, we
derive a sufficient condition for the algorithm stability and a tracking error
bound in terms of the parameters of the multi-timescale exogenous processes.
Based on these results, an adaptive compensation algorithm is proposed to
enhance the tracking performance. Finally, we illustrate the framework by an
application example in wireless heterogeneous network
On linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems
In this paper we propose a distributed dual gradient algorithm for minimizing
linearly constrained separable convex problems and analyze its rate of
convergence. In particular, we prove that under the assumption of strong
convexity and Lipshitz continuity of the gradient of the primal objective
function we have a global error bound type property for the dual problem. Using
this error bound property we devise a fully distributed dual gradient scheme,
i.e. a gradient scheme based on a weighted step size, for which we derive
global linear rate of convergence for both dual and primal suboptimality and
for primal feasibility violation. Many real applications, e.g. distributed
model predictive control, network utility maximization or optimal power flow,
can be posed as linearly constrained separable convex problems for which dual
gradient type methods from literature have sublinear convergence rate. In the
present paper we prove for the first time that in fact we can achieve linear
convergence rate for such algorithms when they are used for solving these
applications. Numerical simulations are also provided to confirm our theory.Comment: 14 pages, 4 figures, submitted to Automatica Journal, February 2014.
arXiv admin note: substantial text overlap with arXiv:1401.4398. We revised
the paper, adding more simulations and checking for typo
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
Barrier Frank-Wolfe for Marginal Inference
We introduce a globally-convergent algorithm for optimizing the
tree-reweighted (TRW) variational objective over the marginal polytope. The
algorithm is based on the conditional gradient method (Frank-Wolfe) and moves
pseudomarginals within the marginal polytope through repeated maximum a
posteriori (MAP) calls. This modular structure enables us to leverage black-box
MAP solvers (both exact and approximate) for variational inference, and obtains
more accurate results than tree-reweighted algorithms that optimize over the
local consistency relaxation. Theoretically, we bound the sub-optimality for
the proposed algorithm despite the TRW objective having unbounded gradients at
the boundary of the marginal polytope. Empirically, we demonstrate the
increased quality of results found by tightening the relaxation over the
marginal polytope as well as the spanning tree polytope on synthetic and
real-world instances.Comment: 25 pages, 12 figures, To appear in Neural Information Processing
Systems (NIPS) 2015, Corrected reference and cleaned up bibliograph
Inexact Block Coordinate Descent Algorithms for Nonsmooth Nonconvex Optimization
In this paper, we propose an inexact block coordinate descent algorithm for
large-scale nonsmooth nonconvex optimization problems. At each iteration, a
particular block variable is selected and updated by inexactly solving the
original optimization problem with respect to that block variable. More
precisely, a local approximation of the original optimization problem is
solved. The proposed algorithm has several attractive features, namely, i) high
flexibility, as the approximation function only needs to be strictly convex and
it does not have to be a global upper bound of the original function; ii) fast
convergence, as the approximation function can be designed to exploit the
problem structure at hand and the stepsize is calculated by the line search;
iii) low complexity, as the approximation subproblems are much easier to solve
and the line search scheme is carried out over a properly constructed
differentiable function; iv) guaranteed convergence of a subsequence to a
stationary point, even when the objective function does not have a Lipschitz
continuous gradient. Interestingly, when the approximation subproblem is solved
by a descent algorithm, convergence of a subsequence to a stationary point is
still guaranteed even if the approximation subproblem is solved inexactly by
terminating the descent algorithm after a finite number of iterations. These
features make the proposed algorithm suitable for large-scale problems where
the dimension exceeds the memory and/or the processing capability of the
existing hardware. These features are also illustrated by several applications
in signal processing and machine learning, for instance, network anomaly
detection and phase retrieval
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