{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/online-forecasting-of-total-variation-bounded","title":"Online Forecasting of Total-Variation-bounded Sequences","arxiv_id":"1906.03364","date":"2019-06-08","proceeding":"NeurIPS 2019 12","authors":["Dheeraj Baby","Yu-Xiang Wang"],"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.","url_abs":"https://arxiv.org/abs/1906.03364v2","url_pdf":"https://arxiv.org/pdf/1906.03364v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"online-forecasting-of-total-variation-bounded","repo_url":"https://github.com/yuxiangw/tv_online","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"stochastic-optimization","task_name":"Stochastic Optimization"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1906.03364","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}