Adaptive Online Estimation of Piecewise Polynomial Trends

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 of an online learner against a time varying comparator sequence is studied. Motivated from the theory of non-parametric regression, we introduce a new variational constraint that enforces the comparator sequence to belong to a discrete $k^{th}$ order Total Variation ball of radius $C_n$. This variational constraint models comparators that have piece-wise polynomial structure which has many relevant practical applications [Tibshirani, 2014]. By establishing connections to the theory of wavelet based non-parametric regression, we design a polynomial time algorithm that achieves the nearly optimal dynamic regret of $\tilde{O}(n^{\frac{1}{2k+3}}C_n^{\frac{2}{2k+3}})$. The proposed policy is adaptive to the unknown radius $C_n$. Further, we show that the same policy is minimax optimal for several other non-parametric families of interest.