{"ID":2830355,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.10825","arxiv_id":"2512.10825","title":"An Elementary Proof of the Near Optimality of LogSumExp Smoothing","abstract":"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.","short_abstract":"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, es...","url_abs":"https://arxiv.org/abs/2512.10825","url_pdf":"https://arxiv.org/pdf/2512.10825v2","authors":"[\"Thabo Samakhoana\",\"Benjamin Grimmer\"]","published":"2025-12-11T17:17:48Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"cs.LG\",\"math.OC\"]","methods":"[]","has_code":false}