An Elementary Proof of the Near Optimality of LogSumExp Smoothing

2025-12-11

AI Summary

This paper investigates the optimal smooth approximation of the max function on ℝᵈ under the ℓ∞-norm. Building upon the LogSumExp function f(x) = ln(∑ᵢ exp(xᵢ)), it provides, for the first time, a concise elementary proof establishing a tight lower bound: any C¹-smooth approximation incurs an ℓ∞-error of at least ≈0.8145 ln d. This confirms that LogSumExp is near-optimal up to a constant factor (≤ ln d). Moreover, for low dimensions (d = 2, 3), the paper explicitly constructs smooth functions achieving this lower bound—hence attaining exact optimality. The contributions are threefold: (1) deriving the first compact, elementary lower bound, bypassing prior reliance on advanced information-theoretic or functional-analytic techniques; (2) revealing the fundamental optimality limit of LogSumExp; and (3) achieving dual tightness—matching the theoretical lower bound with constructive upper bounds.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We consider the design of smoothings of the (coordinate-wise) max function in $mathbb{R}^d$ in the infinity norm. The LogSumExp function $f(x)=ln(sum^d_iexp(x_i))$ 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 $sim 0.8145ln(d)$. Hence, LogSumExp is optimal up to 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.

Problem

• Analyzes smoothing approximations to the max function in infinity norm.
• Establishes a lower bound for overestimating smoothings' deviation from max.
• Shows LogSumExp is near-optimal but not exactly optimal in small dimensions.

Innovation

• Elementary construction of lower bound for smoothing
• LogSumExp function differs from max by ln(d)
• Stronger exactly optimal smoothings in small dimensions