37,485 research outputs found
Online Modified Greedy Algorithm for Storage Control under Uncertainty
This paper studies the general problem of operating energy storage under
uncertainty. Two fundamental sources of uncertainty are considered, namely the
uncertainty in the unexpected fluctuation of the net demand process and the
uncertainty in the locational marginal prices. We propose a very simple
algorithm termed Online Modified Greedy (OMG) algorithm for this problem. A
stylized analysis for the algorithm is performed, which shows that comparing to
the optimal cost of the corresponding stochastic control problem, the
sub-optimality of OMG is bounded and approaches zero in various scenarios. This
suggests that, albeit simple, OMG is guaranteed to have good performance in
some cases; and in other cases, OMG together with the sub-optimality bound can
be used to provide a lower bound for the optimal cost. Such a lower bound can
be valuable in evaluating other heuristic algorithms. For the latter cases, a
semidefinite program is derived to minimize the sub-optimality bound of OMG.
Numerical experiments are conducted to verify our theoretical analysis and to
demonstrate the use of the algorithm.Comment: 14 page version of a paper submitted to IEEE trans on Power System
Constrained Deep Networks: Lagrangian Optimization via Log-Barrier Extensions
This study investigates the optimization aspects of imposing hard inequality
constraints on the outputs of CNNs. In the context of deep networks,
constraints are commonly handled with penalties for their simplicity, and
despite their well-known limitations. Lagrangian-dual optimization has been
largely avoided, except for a few recent works, mainly due to the computational
complexity and stability/convergence issues caused by alternating explicit dual
updates/projections and stochastic optimization. Several studies showed that,
surprisingly for deep CNNs, the theoretical and practical advantages of
Lagrangian optimization over penalties do not materialize in practice. We
propose log-barrier extensions, which approximate Lagrangian optimization of
constrained-CNN problems with a sequence of unconstrained losses. Unlike
standard interior-point and log-barrier methods, our formulation does not need
an initial feasible solution. Furthermore, we provide a new technical result,
which shows that the proposed extensions yield an upper bound on the duality
gap. This generalizes the duality-gap result of standard log-barriers, yielding
sub-optimality certificates for feasible solutions. While sub-optimality is not
guaranteed for non-convex problems, our result shows that log-barrier
extensions are a principled way to approximate Lagrangian optimization for
constrained CNNs via implicit dual variables. We report comprehensive weakly
supervised segmentation experiments, with various constraints, showing that our
formulation outperforms substantially the existing constrained-CNN methods,
both in terms of accuracy, constraint satisfaction and training stability, more
so when dealing with a large number of constraints
Optimality of the Maximum Likelihood estimator in Astrometry
The problem of astrometry is revisited from the perspective of analyzing the
attainability of well-known performance limits (the Cramer-Rao bound) for the
estimation of the relative position of light-emitting (usually point-like)
sources on a CCD-like detector using commonly adopted estimators such as the
weighted least squares and the maximum likelihood. Novel technical results are
presented to determine the performance of an estimator that corresponds to the
solution of an optimization problem in the context of astrometry. Using these
results we are able to place stringent bounds on the bias and the variance of
the estimators in close form as a function of the data. We confirm these
results through comparisons to numerical simulations under a broad range of
realistic observing conditions. The maximum likelihood and the weighted least
square estimators are analyzed. We confirm the sub-optimality of the weighted
least squares scheme from medium to high signal-to-noise found in an earlier
study for the (unweighted) least squares method. We find that the maximum
likelihood estimator achieves optimal performance limits across a wide range of
relevant observational conditions. Furthermore, from our results, we provide
concrete insights for adopting an adaptive weighted least square estimator that
can be regarded as a computationally efficient alternative to the optimal
maximum likelihood solution. We provide, for the first time, close-form
analytical expressions that bound the bias and the variance of the weighted
least square and maximum likelihood implicit estimators for astrometry using a
Poisson-driven detector. These expressions can be used to formally assess the
precision attainable by these estimators in comparison with the minimum
variance bound.Comment: 24 pages, 7 figures, 2 tables, 3 appendices. Accepted by Astronomy &
Astrophysic
Lagrange Coded Computing: Optimal Design for Resiliency, Security and Privacy
We consider a scenario involving computations over a massive dataset stored
distributedly across multiple workers, which is at the core of distributed
learning algorithms. We propose Lagrange Coded Computing (LCC), a new framework
to simultaneously provide (1) resiliency against stragglers that may prolong
computations; (2) security against Byzantine (or malicious) workers that
deliberately modify the computation for their benefit; and (3)
(information-theoretic) privacy of the dataset amidst possible collusion of
workers. LCC, which leverages the well-known Lagrange polynomial to create
computation redundancy in a novel coded form across workers, can be applied to
any computation scenario in which the function of interest is an arbitrary
multivariate polynomial of the input dataset, hence covering many computations
of interest in machine learning. LCC significantly generalizes prior works to
go beyond linear computations. It also enables secure and private computing in
distributed settings, improving the computation and communication efficiency of
the state-of-the-art. Furthermore, we prove the optimality of LCC by showing
that it achieves the optimal tradeoff between resiliency, security, and
privacy, i.e., in terms of tolerating the maximum number of stragglers and
adversaries, and providing data privacy against the maximum number of colluding
workers. Finally, we show via experiments on Amazon EC2 that LCC speeds up the
conventional uncoded implementation of distributed least-squares linear
regression by up to , and also achieves a
- speedup over the state-of-the-art straggler
mitigation strategies
Inexact Convex Relaxations for AC Optimal Power Flow: Towards AC Feasibility
Convex relaxations of AC optimal power flow (AC-OPF) problems have attracted
significant interest as in several instances they provably yield the global
optimum to the original non-convex problem. If, however, the relaxation is
inexact, the obtained solution is not AC-feasible. The quality of the obtained
solution is essential for several practical applications of AC-OPF, but
detailed analyses are lacking in existing literature. This paper aims to cover
this gap. We provide an in-depth investigation of the solution characteristics
when convex relaxations are inexact, we assess the most promising AC
feasibility recovery methods for large-scale systems, and we propose two new
metrics that lead to a better understanding of the quality of the identified
solutions. We perform a comprehensive assessment on 96 different test cases,
ranging from 14 to 3120 buses, and we show the following: (i) Despite an
optimality gap of less than 1%, several test cases still exhibit substantial
distances to both AC feasibility and local optimality and the newly proposed
metrics characterize these deviations. (ii) Penalization methods fail to
recover an AC-feasible solution in 15 out of 45 cases, and using the proposed
metrics, we show that most failed test instances exhibit substantial distances
to both AC-feasibility and local optimality. For failed test instances with
small distances, we show how our proposed metrics inform a fine-tuning of
penalty weights to obtain AC-feasible solutions. (iii) The computational
benefits of warm-starting non-convex solvers have significant variation, but a
computational speedup exists in over 75% of the cases
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