Sparse Contextual CDF Regression
Abstract: Estimating cumulative distribution functions (CDFs) of context dependent random variables is a central statistical task underpinning numerous applications in machine learning and economics. In this work, we extend a recent line of theoretical inquiry into this domain by analyzing the problem of \emph{sparse contextual CDF regression}, wherein data points are sampled from a convex combination of $s$ context dependent CDFs chosen from a set of $d$ basis functions. We show that adaptations of several canonical regression methods serve as tractable estimators in this functional sparse regression setting under standard assumptions on the conditioning of the basis functions. In particular, given $n$ data samples, we prove estimation error upper bounds of $\tilde{O}(\sqrt{s/n})$ for functional versions of the lasso and Dantzig selector estimators, and $\tilde{O}(\sqrt{s}/\sqrt[4]{n})$ for a functional version of the elastic net estimator. Our results match the corresponding error bounds for finite dimensional regression and improve upon CDF ridge regression which has $\tilde{O}(\sqrt{d/n})$ sample complexity. Finally, we obtain a matching information-theoretic lower bound which establishes the minimax optimality of the lasso and Dantzig selector estimators up to logarithmic factors.
Submission Length: Regular submission (no more than 12 pages of main content)
Assigned Action Editor: ~Lihong_Li1
Submission Number: 2336
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