13 research outputs found
Variants of SGD for Lipschitz Continuous Loss Functions in Low-Precision Environments
Motivated by neural network training in low-bit floating and fixed-point
environments, this work studies the convergence of variants of SGD with
computational error. Considering a general stochastic Lipschitz continuous loss
function, a novel convergence result to a Clarke stationary point is presented
assuming that only an approximation of its stochastic gradient can be computed
as well as error in computing the SGD step itself. Different variants of SGD
are then tested empirically in a variety of low-precision arithmetic
environments, with improved test set accuracy achieved compared to SGD for two
image recognition tasks
Chance Constrained Optimization for Targeted Internet Advertising
We introduce a chance constrained optimization model for the fulfillment of
guaranteed display Internet advertising campaigns. The proposed formulation for
the allocation of display inventory takes into account the uncertainty of the
supply of Internet viewers. We discuss and present theoretical and
computational features of the model via Monte Carlo sampling and convex
approximations. Theoretical upper and lower bounds are presented along with a
numerical substantiation
Risk management under Omega measure
We prove that the Omega measure, which considers all moments when assessing
portfolio performance, is equivalent to the widely used Sharpe ratio under
jointly elliptic distributions of returns. Portfolio optimization of the Sharpe
ratio is then explored, with an active-set algorithm presented for markets
prohibiting short sales. When asymmetric returns are considered we show that
the Omega measure and Sharpe ratio lead to different optimal portfolios
Perturbed Iterate SGD for Lipschitz Continuous Loss Functions
This paper presents an extension of stochastic gradient descent for the
minimization of Lipschitz continuous loss functions. Using the Clarke
-subdifferential, we prove non-asymptotic convergence bounds to an
approximate stationary point in expectation. Our results hold under the
assumption that the stochastic loss function is a Carath\'eodory function which
is almost everywhere Lipschitz continuous in the decision variables. To the
best of our knowledge this is the first non-asymptotic convergence analysis
under these minimal assumptions. Our motivation is for use in non-convex
non-smooth stochastic optimization problems, which are frequently encountered
in applications such as machine learning. We present numerical results from
training a feedforward neural network, comparing our algorithm to stochastic
gradient descent
Applications of Chance Constrained Optimization in Operations Management
In this thesis we explore three applications of chance constrained optimization in operations management. We first investigate the effect of consumer demand estimation error on new product production planning. An inventory model is proposed, whereby demand is influenced by price and advertising. The effect of parameter misspecification of the demand model is empirically examined in relation to profit and service level feasibility, and conservative approaches to estimating their effect on consumer demand is determined. We next consider optimization in Internet advertising by introducing a chance constrained model for the fulfillment of guaranteed display Internet advertising campaigns. Lower and upper bounds using Monte Carlo sampling and convex approximations are presented, as well as a branching heuristic for sample approximation lower bounds and an iterative algorithm for improved convex approximation upper bounds. The final application is in risk management for parimutuel horse racing wagering. We develop a methodology to limit potential losing streaks with high probability to the given time horizon of a gambler. A proof of concept was conducted using one season of historical race data, where losing streaks were effectively contained within different time periods for superfecta betting