Online Forecasting of Total-Variation-bounded Sequences

TL;DR

This paper introduces a polynomial-time online forecasting algorithm for sequences with bounded total variation, achieving optimal error rates and adapting to unknown smoothness parameters, with theoretical guarantees and practical efficiency.

Contribution

It presents the first polynomial-time algorithm that attains the optimal $O(n^{1/3})$ forecasting rate for total variation bounded sequences and adapts to unknown bounds.

Findings

Achieves a cumulative square error of $ ilde{O}(n^{1/3}C_n^{2/3}\sigma^{4/3} + C_n^2)$ with high probability.

Proves a matching lower bound up to a $ ext{log}(n)$ factor.

Demonstrates the necessity of adaptive algorithms over fixed-parameter methods in online forecasting.

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 $O(n\log n)$-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.