[Paper] An Elementary Proof of the Near Optimality of LogSumExp Smoothing

Source: arXiv - 2512.10825v1

Overview

The paper tackles a surprisingly practical problem: how to replace the non‑smooth “max” operation with a smooth approximation that works well in high‑dimensional settings. The classic LogSumExp (LSE) trick is widely used in machine learning, optimization, and graphics, but the authors show that it is essentially the best you can do—up to a modest constant factor—when you care about how far the smooth surrogate can overshoot the true max. They also present exact, tighter smoothings for low‑dimensional cases, proving that LSE isn’t always perfectly optimal.

Key Contributions

• Elementary lower‑bound construction: Proves that any smooth over‑approximation of the max function must incur an error of at least ≈ 0.8145 · ln d, where d is the dimension.
• Near‑optimality of LogSumExp: Shows that the standard LSE smoothing, which deviates by at most ln d, is optimal up to a constant factor (≈ 1.23).
• Exact optimal smoothings for small d: Provides closed‑form smooth functions that achieve the lower bound exactly in low dimensions (e.g., d = 2, 3).
• Simple, constructive proof technique: Avoids heavy functional‑analysis machinery, making the result accessible to a broader audience.

Methodology

1. Problem formalization: The authors consider overestimating smoothings f of the max function, i.e., f(x) ≥ max_i x_i for all x ∈ ℝ^d, with the additional requirement that f is differentiable (smooth) and Lipschitz‑continuous in the ∞‑norm.
2. Geometric construction: They build a family of points in ℝ^d that force any smooth over‑approximation to “pay” a certain amount of error. By carefully arranging these points on a simplex and exploiting convexity, they derive a universal lower bound on the possible error.
3. Entropy‑based comparison: The classic LogSumExp function, derived from the Gibbs/Boltzmann entropy, is analyzed to show its error is exactly ln d.
4. Low‑dimensional optimization: For small d, the authors solve a small convex program analytically to find the smooth function that meets the lower bound, demonstrating that the bound is tight.

The proof relies only on elementary calculus and convex geometry—no advanced functional analysis—making it easy to follow for developers familiar with basic optimization concepts.

Results & Findings

• Interpretation: In high dimensions the LogSumExp’s overshoot is at most about 23 % larger than the theoretical optimum, which is negligible for most practical purposes.
• Exact optimality in low dimensions: For d ≤ 4 the constructed smoothings match the lower bound exactly, confirming that the bound is tight.

Practical Implications

• Machine‑learning loss functions: Many loss functions (e.g., softmax cross‑entropy, max‑margin classifiers) rely on LSE as a smooth surrogate for max. This work reassures practitioners that they are already using a near‑optimal approximation, so there’s little to gain by searching for a “better” smooth max in high‑dimensional models.
• Differentiable programming & auto‑diff: When you need a smooth max for gradient‑based optimization (e.g., in reinforcement learning, differentiable rendering, or neural architecture search), LSE remains the go‑to choice with provable guarantees on error magnitude.
• Numerical stability: The paper’s low‑dimensional exact smoothings can be implemented when d is tiny (e.g., selecting the best of a handful of actions). They may yield slightly tighter bounds and marginally better numerical conditioning.
• Hardware‑accelerated kernels: Knowing that LSE is essentially optimal lets library developers focus on optimizing its implementation (e.g., log‑sum‑exp tricks, fused multiply‑add pipelines) rather than inventing alternative smooth max kernels.

Limitations & Future Work

• Only over‑estimating smoothings: The analysis assumes the surrogate never undershoots the true max. In some applications (e.g., regularized objectives) an under‑approximation could be acceptable, but the current bounds don’t cover that case.
• Infinity‑norm Lipschitz condition: The lower bound is derived under an ∞‑norm smoothness requirement. Different norm constraints (e.g., Euclidean) might lead to different optimal constants.
• Scalability of exact smoothings: The exact optimal constructions are only tractable for very low dimensions; extending them to moderate d would likely require numerical optimization rather than closed forms.
• Empirical validation: The paper is theoretical; a natural next step is benchmarking the exact low‑dimensional smoothings against LSE in real‑world tasks (e.g., small‑action reinforcement learning) to quantify any practical gains.

Overall, the work gives developers a solid theoretical foundation for continuing to rely on LogSumExp while also pointing to niche scenarios where tighter smoothings could be beneficial.

Authors

• Thabo Samakhoana
• Benjamin Grimmer

Paper Information

• arXiv ID: 2512.10825v1
• Categories: math.ST, cs.LG, math.OC
• Published: December 11, 2025
• PDF: Download PDF