Papers › Online Forecasting of Total-Variation-bounded Sequences
Online Forecasting of Total-Variation-bounded Sequences
Dheeraj Baby, Yu-Xiang Wang
We consider the problem of online forecasting of sequences of length n with total-variation at most Cₙ using observations contaminated by independent σ-subgaussian noise. We design an O(nlogn)-time algorithm that achieves a cumulative square error of Õ(n^(1/3)Cₙ^(2/3)σ^(4/3) + Cₙ²) with high probability.We also prove a lower bound that matches the upper bound in all parameters (up to a log(n) factor). To the best of our knowledge, this is the first \emph{polynomial-time} algorithm that achieves the optimal O(n^(1/3)) rate in forecasting total variation bounded sequences and the first algorithm that \emph{adapts to unknown} Cₙ. 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 \emph{linear forecasters} that require a suboptimal regret of Ω̃(√(n)). This implies that the use of more adaptive algorithms is necessary to obtain the optimal rate.
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