Discounted Online Convex Optimization: Uniform Regret Across a Continuous Interval
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| Format: | Preprint |
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2025
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| _version_ | 1866912394733682688 |
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| author | Yang, Wenhao Yang, Sifan Zhang, Lijun |
| author_facet | Yang, Wenhao Yang, Sifan Zhang, Lijun |
| contents | Reflecting the greater significance of recent history over the distant past in non-stationary environments, $λ$-discounted regret has been introduced in online convex optimization (OCO) to gracefully forget past data as new information arrives. When the discount factor $λ$ is given, online gradient descent with an appropriate step size achieves an $O(1/\sqrt{1-λ})$ discounted regret. However, the value of $λ$ is often not predetermined in real-world scenarios. This gives rise to a significant open question: is it possible to develop a discounted algorithm that adapts to an unknown discount factor. In this paper, we affirmatively answer this question by providing a novel analysis to demonstrate that smoothed OGD (SOGD) achieves a uniform $O(\sqrt{\log T/1-λ})$ discounted regret, holding for all values of $λ$ across a continuous interval simultaneously. The basic idea is to maintain multiple OGD instances to handle different discount factors, and aggregate their outputs sequentially by an online prediction algorithm named as Discounted-Normal-Predictor (DNP) (Kapralov and Panigrahy,2010). Our analysis reveals that DNP can combine the decisions of two experts, even when they operate on discounted regret with different discount factors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_19491 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Discounted Online Convex Optimization: Uniform Regret Across a Continuous Interval Yang, Wenhao Yang, Sifan Zhang, Lijun Machine Learning Reflecting the greater significance of recent history over the distant past in non-stationary environments, $λ$-discounted regret has been introduced in online convex optimization (OCO) to gracefully forget past data as new information arrives. When the discount factor $λ$ is given, online gradient descent with an appropriate step size achieves an $O(1/\sqrt{1-λ})$ discounted regret. However, the value of $λ$ is often not predetermined in real-world scenarios. This gives rise to a significant open question: is it possible to develop a discounted algorithm that adapts to an unknown discount factor. In this paper, we affirmatively answer this question by providing a novel analysis to demonstrate that smoothed OGD (SOGD) achieves a uniform $O(\sqrt{\log T/1-λ})$ discounted regret, holding for all values of $λ$ across a continuous interval simultaneously. The basic idea is to maintain multiple OGD instances to handle different discount factors, and aggregate their outputs sequentially by an online prediction algorithm named as Discounted-Normal-Predictor (DNP) (Kapralov and Panigrahy,2010). Our analysis reveals that DNP can combine the decisions of two experts, even when they operate on discounted regret with different discount factors. |
| title | Discounted Online Convex Optimization: Uniform Regret Across a Continuous Interval |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2505.19491 |