79,763 research outputs found
Stochastic Subgradient Algorithms for Strongly Convex Optimization over Distributed Networks
We study diffusion and consensus based optimization of a sum of unknown
convex objective functions over distributed networks. The only access to these
functions is through stochastic gradient oracles, each of which is only
available at a different node, and a limited number of gradient oracle calls is
allowed at each node. In this framework, we introduce a convex optimization
algorithm based on the stochastic gradient descent (SGD) updates. Particularly,
we use a carefully designed time-dependent weighted averaging of the SGD
iterates, which yields a convergence rate of
after gradient updates for each node on
a network of nodes. We then show that after gradient oracle calls, the
average SGD iterate achieves a mean square deviation (MSD) of
. This rate of convergence is optimal as it
matches the performance lower bound up to constant terms. Similar to the SGD
algorithm, the computational complexity of the proposed algorithm also scales
linearly with the dimensionality of the data. Furthermore, the communication
load of the proposed method is the same as the communication load of the SGD
algorithm. Thus, the proposed algorithm is highly efficient in terms of
complexity and communication load. We illustrate the merits of the algorithm
with respect to the state-of-art methods over benchmark real life data sets and
widely studied network topologies
Residual Weighted Learning for Estimating Individualized Treatment Rules
Personalized medicine has received increasing attention among statisticians,
computer scientists, and clinical practitioners. A major component of
personalized medicine is the estimation of individualized treatment rules
(ITRs). Recently, Zhao et al. (2012) proposed outcome weighted learning (OWL)
to construct ITRs that directly optimize the clinical outcome. Although OWL
opens the door to introducing machine learning techniques to optimal treatment
regimes, it still has some problems in performance. In this article, we propose
a general framework, called Residual Weighted Learning (RWL), to improve finite
sample performance. Unlike OWL which weights misclassification errors by
clinical outcomes, RWL weights these errors by residuals of the outcome from a
regression fit on clinical covariates excluding treatment assignment. We
utilize the smoothed ramp loss function in RWL, and provide a difference of
convex (d.c.) algorithm to solve the corresponding non-convex optimization
problem. By estimating residuals with linear models or generalized linear
models, RWL can effectively deal with different types of outcomes, such as
continuous, binary and count outcomes. We also propose variable selection
methods for linear and nonlinear rules, respectively, to further improve the
performance. We show that the resulting estimator of the treatment rule is
consistent. We further obtain a rate of convergence for the difference between
the expected outcome using the estimated ITR and that of the optimal treatment
rule. The performance of the proposed RWL methods is illustrated in simulation
studies and in an analysis of cystic fibrosis clinical trial data.Comment: 48 pages, 3 figure
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