Mixability of Integral Losses: a Key to Efficient Online Aggregation of Functional and Probabilistic Forecasts
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866915090754699264 |
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| author | Korotin, Alexander V'yugin, Vladimir Burnaev, Evgeny |
| author_facet | Korotin, Alexander V'yugin, Vladimir Burnaev, Evgeny |
| contents | In this paper we extend the setting of the online prediction with expert advice to function-valued forecasts. At each step of the online game several experts predict a function, and the learner has to efficiently aggregate these functional forecasts into a single forecast. We adapt basic mixable (and exponentially concave) loss functions to compare functional predictions and prove that these adaptations are also mixable (exp-concave). We call this phenomenon mixability (exp-concavity) of integral loss functions. As an application of our main result, we prove that various loss functions used for probabilistic forecasting are mixable (exp-concave). The considered losses include Sliced Continuous Ranked Probability Score, Energy-Based Distance, Optimal Transport Costs and Sliced Wasserstein-2 distance, Beta-2 and Kullback-Leibler divergences, Characteristic function and Maximum Mean Discrepancies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_07048 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Mixability of Integral Losses: a Key to Efficient Online Aggregation of Functional and Probabilistic Forecasts Korotin, Alexander V'yugin, Vladimir Burnaev, Evgeny Machine Learning In this paper we extend the setting of the online prediction with expert advice to function-valued forecasts. At each step of the online game several experts predict a function, and the learner has to efficiently aggregate these functional forecasts into a single forecast. We adapt basic mixable (and exponentially concave) loss functions to compare functional predictions and prove that these adaptations are also mixable (exp-concave). We call this phenomenon mixability (exp-concavity) of integral loss functions. As an application of our main result, we prove that various loss functions used for probabilistic forecasting are mixable (exp-concave). The considered losses include Sliced Continuous Ranked Probability Score, Energy-Based Distance, Optimal Transport Costs and Sliced Wasserstein-2 distance, Beta-2 and Kullback-Leibler divergences, Characteristic function and Maximum Mean Discrepancies. |
| title | Mixability of Integral Losses: a Key to Efficient Online Aggregation of Functional and Probabilistic Forecasts |
| topic | Machine Learning |
| url | https://arxiv.org/abs/1912.07048 |