Non-stationary Online Regression

Online forecasting under a changing environment has been a problem of increasing importance in many real-world applications. In this paper, we consider the meta-algorithm presented in \citet{zhang2017dynamic} combined with different subroutines. We show that an expected cumulative error of order $\tilde{O}(n^{1/3} C_n^{2/3})$ can be obtained for non-stationary online linear regression where the total variation of parameter sequence is bounded by $C_n$. Our paper extends the result of online forecasting of one-dimensional time-series as proposed in \cite{baby2019online} to general $d$-dimensional non-stationary linear regression. We improve the rate $O(\sqrt{n C_n})$ obtained by Zhang et al. 2017 and Besbes et al. 2015. We further extend our analysis to non-stationary online kernel regression. Similar to the non-stationary online regression case, we use the meta-procedure of Zhang et al. 2017 combined with Kernel-AWV (Jezequel et al. 2020) to achieve an expected cumulative controlled by the effective dimension of the RKHS and the total variation of the sequence. To the best of our knowledge, this work is the first extension of non-stationary online regression to non-stationary kernel regression. Lastly, we evaluate our method empirically with several existing benchmarks and also compare it with the theoretical bound obtained in this paper.