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• W2948334328 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 $sigma$-subgaussian noise. We design an $O(nlog n)$-time algorithm that achieves a cumulative square error of $tilde{O}(n^{1/3}C_n^{2/3}sigma^{4/3} + C_n^2)$ with high probability.We also prove a lower bound that matches the upper bound in all parameters (up to a $log(n)$ 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} $C_n$. 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 emph{linear forecasters} that require a suboptimal regret of $tilde{Omega}(sqrt{n})$. This implies that the use of more adaptive algorithms is necessary to obtain the optimal rate." @default.
• W2948334328 created "2019-06-14" @default.
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• W2948334328 date "2019-06-07" @default.
• W2948334328 modified "2023-10-18" @default.
• W2948334328 title "Online Forecasting of Total-Variation-bounded Sequences" @default.
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• W2948334328 doi "https://doi.org/10.48550/arxiv.1906.03364" @default.
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