An Elementary Proof of the Near Optimality of LogSumExp Smoothing
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912833025867776 |
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| author | Samakhoana, Thabo Grimmer, Benjamin |
| author_facet | Samakhoana, Thabo Grimmer, Benjamin |
| contents | We consider the design of smoothings of the (coordinate-wise) max function in $\mathbb{R}^d$ in the infinity norm. The LogSumExp function $f(x)=\ln(\sum^d_i\exp(x_i))$ provides a classical smoothing, differing from the max function in value by at most $\ln(d)$. We provide an elementary construction of a lower bound, establishing that every overestimating smoothing of the max function must differ by at least $\sim 0.8145\ln(d)$. Hence, LogSumExp is optimal up to small constant factors. However, in small dimensions, we provide stronger, exactly optimal smoothings attaining our lower bound, showing that the entropy-based LogSumExp approach to smoothing is not exactly optimal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10825 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Elementary Proof of the Near Optimality of LogSumExp Smoothing Samakhoana, Thabo Grimmer, Benjamin Statistics Theory Machine Learning Optimization and Control We consider the design of smoothings of the (coordinate-wise) max function in $\mathbb{R}^d$ in the infinity norm. The LogSumExp function $f(x)=\ln(\sum^d_i\exp(x_i))$ provides a classical smoothing, differing from the max function in value by at most $\ln(d)$. We provide an elementary construction of a lower bound, establishing that every overestimating smoothing of the max function must differ by at least $\sim 0.8145\ln(d)$. Hence, LogSumExp is optimal up to small constant factors. However, in small dimensions, we provide stronger, exactly optimal smoothings attaining our lower bound, showing that the entropy-based LogSumExp approach to smoothing is not exactly optimal. |
| title | An Elementary Proof of the Near Optimality of LogSumExp Smoothing |
| topic | Statistics Theory Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2512.10825 |