Online Forecasting of Total-Variation-bounded Sequences

Code

• github.com/yuxiangw/tv_online
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Abstract

We consider the problem of online forecasting of sequences of length n with total-variation at most C_n using observations contaminated by independent -subgaussian noise. We design an O(n n)-time algorithm that achieves a cumulative square error of O(n^1/3C_n^2/3^4/3 + C_n^2) with high probability.We also prove a lower bound that matches the upper bound in all parameters (up to a (n) factor). To the best of our knowledge, this is the first polynomial-time algorithm that achieves the optimal O(n^1/3) rate in forecasting total variation bounded sequences and the first algorithm that adapts to unknown C_n. Our proof techniques leverage the special localized structure of Haar wavelet basis and the adaptivity to unknown smoothness parameters in the classical wavelet smoothing [Donoho et al., 1998]. We also compare our model to the rich literature of dynamic regret minimization and nonstationary stochastic optimization, where our problem can be treated as a special case. We show that the workhorse in those settings --- online gradient descent and its variants with a fixed restarting schedule --- are instances of a class of linear forecasters that require a suboptimal regret of (n). This implies that the use of more adaptive algorithms is necessary to obtain the optimal rate.

Tasks

Reproductions