Smooth Approximations of the Rounding Function

Stanislav Semenov

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arXiv · 2504.19026

Smooth Approximations of the Rounding Function

Stanislav Semenov

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Abstract

We propose novel smooth approximations to the classical rounding function, suitable for differentiable optimization and machine learning applications. Our constructions are based on two approaches: (1) localized sigmoid window functions centered at each integer, and (2) normalized weighted sums of sigmoid derivatives representing local densities. The first method approximates the step-like behavior of rounding through differences of shifted sigmoids, while the second method achieves smooth interpolation between integers via density-based weighting. Both methods converge pointwise to the classical rounding function as the sharpness parameter k tends to infinity, and allow controlled trade-offs between smoothness and approximation accuracy. We demonstrate that by restricting the summation to a small set of nearest integers, the computational cost remains low without sacrificing precision. These constructions provide fully differentiable alternatives to hard rounding, which are valuable in contexts where gradient-based methods are essential.

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Stanislav Semenov. 2025-04-26. Smooth Approximations of the Rounding Function. https://arxiv.org/abs/2504.19026

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