More PAC-Bayes bounds: From bounded losses, to losses with general tail behaviors, to anytime validity
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| Format: | Preprint |
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2023
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| _version_ | 1866916273494949888 |
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| author | Rodríguez-Gálvez, Borja Thobaben, Ragnar Skoglund, Mikael |
| author_facet | Rodríguez-Gálvez, Borja Thobaben, Ragnar Skoglund, Mikael |
| contents | In this paper, we present new high-probability PAC-Bayes bounds for different types of losses. Firstly, for losses with a bounded range, we recover a strengthened version of Catoni's bound that holds uniformly for all parameter values. This leads to new fast-rate and mixed-rate bounds that are interpretable and tighter than previous bounds in the literature. In particular, the fast-rate bound is equivalent to the Seeger--Langford bound. Secondly, for losses with more general tail behaviors, we introduce two new parameter-free bounds: a PAC-Bayes Chernoff analogue when the loss' cumulative generating function is bounded, and a bound when the loss' second moment is bounded. These two bounds are obtained using a new technique based on a discretization of the space of possible events for the ``in probability'' parameter optimization problem. This technique is both simpler and more general than previous approaches optimizing over a grid on the parameters' space. Finally, using a simple technique that is applicable to any existing bound, we extend all previous results to anytime-valid bounds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_12214 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | More PAC-Bayes bounds: From bounded losses, to losses with general tail behaviors, to anytime validity Rodríguez-Gálvez, Borja Thobaben, Ragnar Skoglund, Mikael Machine Learning In this paper, we present new high-probability PAC-Bayes bounds for different types of losses. Firstly, for losses with a bounded range, we recover a strengthened version of Catoni's bound that holds uniformly for all parameter values. This leads to new fast-rate and mixed-rate bounds that are interpretable and tighter than previous bounds in the literature. In particular, the fast-rate bound is equivalent to the Seeger--Langford bound. Secondly, for losses with more general tail behaviors, we introduce two new parameter-free bounds: a PAC-Bayes Chernoff analogue when the loss' cumulative generating function is bounded, and a bound when the loss' second moment is bounded. These two bounds are obtained using a new technique based on a discretization of the space of possible events for the ``in probability'' parameter optimization problem. This technique is both simpler and more general than previous approaches optimizing over a grid on the parameters' space. Finally, using a simple technique that is applicable to any existing bound, we extend all previous results to anytime-valid bounds. |
| title | More PAC-Bayes bounds: From bounded losses, to losses with general tail behaviors, to anytime validity |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2306.12214 |