6,823 research outputs found
Sample Complexity of Sample Average Approximation for Conditional Stochastic Optimization
In this paper, we study a class of stochastic optimization problems, referred
to as the \emph{Conditional Stochastic Optimization} (CSO), in the form of
\min_{x \in \mathcal{X}}
\EE_{\xi}f_\xi\Big({\EE_{\eta|\xi}[g_\eta(x,\xi)]}\Big), which finds a wide
spectrum of applications including portfolio selection, reinforcement learning,
robust learning, causal inference and so on. Assuming availability of samples
from the distribution \PP(\xi) and samples from the conditional distribution
\PP(\eta|\xi), we establish the sample complexity of the sample average
approximation (SAA) for CSO, under a variety of structural assumptions, such as
Lipschitz continuity, smoothness, and error bound conditions. We show that the
total sample complexity improves from \cO(d/\eps^4) to \cO(d/\eps^3) when
assuming smoothness of the outer function, and further to \cO(1/\eps^2) when
the empirical function satisfies the quadratic growth condition. We also
establish the sample complexity of a modified SAA, when and are
independent. Several numerical experiments further support our theoretical
findings.
Keywords: stochastic optimization, sample average approximation, large
deviations theoryComment: Typo corrected. Reference added. Revision comments handle
Distributed Learning for Stochastic Generalized Nash Equilibrium Problems
This work examines a stochastic formulation of the generalized Nash
equilibrium problem (GNEP) where agents are subject to randomness in the
environment of unknown statistical distribution. We focus on fully-distributed
online learning by agents and employ penalized individual cost functions to
deal with coupled constraints. Three stochastic gradient strategies are
developed with constant step-sizes. We allow the agents to use heterogeneous
step-sizes and show that the penalty solution is able to approach the Nash
equilibrium in a stable manner within , for small step-size
value and sufficiently large penalty parameters. The operation
of the algorithm is illustrated by considering the network Cournot competition
problem
Eco-reliable path finding in time-variant and stochastic networks
This paper addresses a route guidance problem for finding the most eco-reliable path in time-variant and stochastic networks such that travelers can arrive at the destination with the maximum on-time probability while meeting vehicle emission standards imposed by government regulators. To characterize the dynamics and randomness of transportation networks, the link travel times and emissions are assumed to be time-variant random variables correlated over the entire network. A 0–1 integer mathematical programming model is formulated to minimize the probability of late arrival by simultaneously considering the least expected emission constraint. Using the Lagrangian relaxation approach, the primal model is relaxed into a dualized model which is further decomposed into two simple sub-problems. A sub-gradient method is developed to reduce gaps between upper and lower bounds. Three sets of numerical experiments are tested to demonstrate the efficiency and performance of our proposed model and algorithm
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