# Online Forecasting of Total-Variation-bounded Sequences

arXiv: https://arxiv.org/abs/1906.03364 · HTML: https://mlsanity.com/paper/1906.03364/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2019.

## Built on by (most cited first, 30 of 31)

- [Adaptivity and Non-stationarity: Problem-dependent Dynamic Regret for Online Convex Optimization](https://mlsanity.com/paper/2112.14368/) (2021)
- [Optimal Dynamic Regret in Exp-Concave Online Learning](https://mlsanity.com/paper/2104.11824/) (2021)
- [Non-stationary Online Learning with Memory and Non-stochastic Control](https://mlsanity.com/paper/2102.03758/) (2021)
- [Optimal Dynamic Regret in Proper Online Learning with Strongly Convex Losses and Beyond](https://mlsanity.com/paper/2201.08905/) (2022)
- [Adapting to Online Label Shift with Provable Guarantees](https://mlsanity.com/paper/2207.02121/) (2022)
- [Online Bilevel Optimization: Regret Analysis of Online Alternating Gradient Methods](https://mlsanity.com/paper/2207.02829/) (2022)
- [Adapting to Continuous Covariate Shift via Online Density Ratio Estimation](https://mlsanity.com/paper/2302.02552/) (2023)
- [Dynamic Regret of Online Markov Decision Processes](https://mlsanity.com/paper/2208.12483/) (2022)
- [Non-stationary Projection-free Online Learning with Dynamic and Adaptive Regret Guarantees](https://mlsanity.com/paper/2305.11726/) (2023)
- [Unconstrained Dynamic Regret via Sparse Coding](https://mlsanity.com/paper/2301.13349/) (2023)
- [Online Label Shift: Optimal Dynamic Regret meets Practical Algorithms](https://mlsanity.com/paper/2305.19570/) (2023)
- [Deep Learning meets Nonparametric Regression: Are Weight-Decayed DNNs Locally Adaptive?](https://mlsanity.com/paper/2204.09664/) (2022)
- [An Optimal Reduction of TV-Denoising to Adaptive Online Learning](https://mlsanity.com/paper/2101.09438/) (2021)
- [An Equivalence Between Static and Dynamic Regret Minimization](https://mlsanity.com/paper/2406.01577/) (2024)
- [Improved Dynamic Regret for Online Frank-Wolfe](https://mlsanity.com/paper/2302.05620/) (2023)
- [Second Order Path Variationals in Non-Stationary Online Learning](https://mlsanity.com/paper/2205.01921/) (2022)
- [Optimal Dynamic Regret in LQR Control](https://mlsanity.com/paper/2206.09257/) (2022)
- [Nonparametric Density Estimation under Distribution Drift](https://mlsanity.com/paper/2302.02460/) (2023)
- [Optimal Online Generalized Linear Regression with Stochastic Noise and Its Application to Heteroscedastic Bandits](https://mlsanity.com/paper/2202.13603/) (2022)
- [Dynamic Regret Reduces to Kernelized Static Regret](https://mlsanity.com/paper/2507.05478/) (2025)
- [Revisiting Weighted Strategy for Non-stationary Parametric Bandits and MDPs](https://mlsanity.com/paper/2601.01069/) (2026)
- [Non-stationary Delayed Online Convex Optimization: From Full-information to Bandit Setting](https://mlsanity.com/paper/2305.12131/) (2023)
- [Sparsity-Based Interpolation of External, Internal and Swap Regret](https://mlsanity.com/paper/2502.04543/) (2025)
- [Adaptive Estimation and Learning under Temporal Distribution Shift](https://mlsanity.com/paper/2505.15803/) (2025)
- [Spatially Adaptive Online Prediction of Piecewise Regular Functions](https://mlsanity.com/paper/2203.16587/) (2022)
- [From Switching to Dynamic Regret: A Simple Reduction via Unbiased Random Sequences](https://mlsanity.com/paper/2609.20968/) (2026)
- [Online (Non-)Convex Learning via Tempered Optimism](https://mlsanity.com/paper/2301.07530/) (2023)
- [Universal Online Optimization in Dynamic Environments via Uniclass Prediction](https://mlsanity.com/paper/2302.06066/) (2023)
- [Revisiting Weighted Strategy for Non-stationary Parametric Bandits](https://mlsanity.com/paper/2303.02691/) (2023)
- [The Cost of Learning Under Multiple Change Points](https://mlsanity.com/paper/2602.11406/) (2026)

## Abstract (arXiv)

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(nlog n)-time algorithm that achieves a cumulative square error of tildeO(n1/3Cn2/3σ4/3 + 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 polynomial-time algorithm that achieves the optimal O(n1/3) rate in forecasting total variation bounded sequences and the first algorithm that 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 linear forecasters that require a suboptimal regret of tildeOmega(√n). This implies that the use of more adaptive algorithms is necessary to obtain the optimal rate.