While therewardcompensation mechanism isunknown,the learner can adapt his (her) decision to the past reward feedback so as to maximize the sum of rewards.

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We consider the problem of online forecasting of sequences of lengthn with total-variation at mostCn using observations contaminated by independentσsubgaussian noise. We design anO(nlogn)-time algorithm that achieves a cu-mulativesquare error of O(n1/3C2/3n σ4/3+C2n)with high probability.

The lasso and elastic net linear regression models impose a double-exponential prior distribution on the model parameters to achieve regression shrinkage and variable selection, allowing the inference of robust models from large data sets.

The Highly Adaptive Lasso (HAL) is a nonparametric regression method that achieves almost dimension-free convergence rates under minimal smoothness assumptions, but its implementation can be computationally prohibitive in high dimensions due to the large basis matrix it requires. The Highly Adaptive Ridge (HAR) has been proposed as a scalable alternative. Building on both procedures, we introduce the Principal Component based Highly Adaptive Lasso (PCHAL) and Principal Component based Highly Adaptive Ridge (PCHAR). These estimators constitute an outcome-blind dimension reduction which offer substantial gains in computational efficiency and match the empirical performances of HAL and HAR. We also uncover a striking spectral link between the leading principal components of the HAL/HAR Gram operator and a discrete sinusoidal basis, revealing an explicit Fourier-type structure underlying the PC truncation.

We consider the framework of non-stationary stochastic optimization [Besbes et al., 2015] with squared error losses and noisy gradient feedback where the dynamic regret ofanonline learner against atime varying comparator sequence isstudied.

We consider the framework of non-stationary stochastic optimization [Besbes et al., 2015] with squared error losses and noisy gradient feedback where the dynamic regret ofanonline learner against atime varying comparator sequence isstudied.