Compute Less to Get More: Using ORC to Improve Sparse Filtering

Sparse Filtering is a popular feature learning algorithm for image classification pipelines. In this paper, we connect the performance of Sparse Filtering with spectral properties of the corresponding feature matrices. This connection provides new insights into Sparse Filtering; in particular, it suggests early stopping of Sparse Filtering. We therefore introduce the Optimal Roundness Criterion (ORC), a novel stopping criterion for Sparse Filtering. We show that this stopping criterion is related with pre-processing procedures such as Statistical Whitening and demonstrate that it can make image classification with Sparse Filtering considerably faster and more accurate

Sequential Design for Optimal Stopping Problems

We propose a new approach to solve optimal stopping problems via simulation. Working within the backward dynamic programming/Snell envelope framework, we augment the methodology of Longstaff-Schwartz that focuses on approximating the stopping strategy. Namely, we introduce adaptive generation of the stochastic grids anchoring the simulated sample paths of the underlying state process. This allows for active learning of the classifiers partitioning the state space into the continuation and stopping regions. To this end, we examine sequential design schemes that adaptively place new design points close to the stopping boundaries. We then discuss dynamic regression algorithms that can implement such recursive estimation and local refinement of the classifiers. The new algorithm is illustrated with a variety of numerical experiments, showing that an order of magnitude savings in terms of design size can be achieved. We also compare with existing benchmarks in the context of pricing multi-dimensional Bermudan options.Comment: 24 page

Q-Learning and Enhanced Policy Iteration in Discounted Dynamic Programming

We consider the classical nite-state discounted Markovian decision problem, and we introduce a new policy iteration-like algorithm for fi nding the optimal Q-factors. Instead of policy evaluation by solving a linear system of equations, our algorithm requires (possibly inexact) solution of a nonlinear system of equations, involving estimates of state costs as well as Q-factors. This is Bellman's equation for an optimal stopping problem that can be solved with simple Q-learning iterations, in the case where a lookup table representation is used; it can also be solved with the Q-learning algorithm of Tsitsiklis and Van Roy [TsV99], in the case where feature-based Q-factor approximations are used. In exact/lookup table representation form, our algorithm admits asynchronous and stochastic iterative implementations, in the spirit of asynchronous/modi ed policy iteration, with lower overhead and more reliable convergence advantages over existing Q-learning schemes. Furthermore, for large-scale problems, where linear basis function approximations and simulation-based temporal di erence implementations are used, our algorithm resolves e ffectively the inherent difficulties of existing schemes due to inadequate exploration

Q-Learning and Enhanced Policy Iteration in Discounted Dynamic Programming (Revised)

The revised technical report C-2010-10We consider the classical finite-state discounted Markovian decision problem, and we introduce a new policy iteration-like algorithm for finding the optimal Q-factors. Instead of policy evaluation by solving a linear system of equations, our algorithm requires (possibly inexact) solution of a nonlinear system of equations, involving estimates of state costs as well as Q-factors. This is Bellman's equation for an optimal stopping problem that can be solved with simple Q-learning iterations, in the case where a lookup table representation is used; it can also be solved with the Q-learning algorithm of Tsitsiklis and Van Roy [TsV99], in the case where feature-based Q-factor approximations are used. In exact/lookup table representation form, our algorithm admits asynchronous and stochastic iterative implementations, in the spirit of asynchronous/modified policy iteration, with lower overhead and/or more reliable convergence advantages over existing Q-learning schemes. Furthermore, for large-scale problems, where linear basis function approximations and simulation-based temporal difference implementations are used, our algorithm resolves effectively the inherent difficulties of existing schemes due to inadequate exploration

Semi-parametric forecasts of the implied volatility surface using regression trees

We present a new semi-parametric model for the prediction of implied volatility surfaces that can be estimated using machine learning algorithms. Given a reasonable starting model, a boosting algorithm based on regression trees sequentially minimizes generalized residuals computed as differences between observed and estimated implied volatilities. To overcome the poor predictive power of existing models, we include a grid in the region of interest, and implement a cross-validation strategy to find an optimal stopping value for the boosting procedure. Back testing the out-of-sample performance on a large data set of implied volatilities from S&P 500 options, we provide empirical evidence of the strong predictive power of our mode