mlsanityData to 14 Oct 2026

An Elementary Proof of the Near Optimality of LogSumExp Smoothing

arXiv 2512.10825 ↗First posted 11 Dec 2025math.ST cs.LG math.OCcited by 0

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Abstract (arXiv)

We consider the design of smoothings of the (coordinate-wise) max function in ℝd in the infinity norm. The LogSumExp function f(x)=ln(sumdiexp(xi)) 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 ~ 0.8145ln(d). Hence, LogSumExp is optimal up to small constant factors. However, we provide strictly stronger smoothings showing the entropy-based LogSumExp approach is not exactly optimal. In small dimensions, we propose exactly optimal smoothings, attaining our lower bound.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 9 of its 17 references that are in the map. First posted on arXiv 11 Dec 2025.

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References · 9

  1. 2017Attention Is All You Needcited by 60,361
  2. 2019Online Learning: A Modern Introduction Using Convex Optimizationcited by 417
  3. 2016From Softmax to Sparsemax: A Sparse Model of Attention and Multi-Label Classificationcited by 341
  4. 2013On Lower Complexity Bounds for Large-Scale Smooth Convex Optimizationcited by 24
  5. 2024The Price of Adaptivity in Stochastic Convex Optimizationcited by 11
  6. 2020Optimal Approximation -- Smoothness Tradeoffs for Soft-Max Functionscited by 10
  7. 2021Radial Duality Part II: Applications and Algorithmscited by 4
  8. 2015Accelerated First-Order Methods for Hyperbolic Programmingcited by 3
  9. 2025The Optimal Smoothings of Sublinear Functions and Convex Conescited by 1

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