Fast LapSum: Exact Differentiable Top-$k$ at Million Scale

Selecting the top-$k$ elements is a fundamental operation for inducing sparsity in large-scale models and optimization problems, enabling robust expert activation, token routing or attention pruning. However, hard top-$k$ is non-differentiable, while existing differentiable alternatives become increasingly expensive as the number of coordinates grows. We introduce Fast LapSum, a scalable solver for the LapSum soft top-$k$ formulation that preserves an exact selection mass of $k$, while supporting end-to-end differentiation. In Fast LapSum, we reduce sorting cost using probabilistic bracketing, which restricts sorting to a narrow band of scores around the threshold using a binomial order-statistic from kernel-noised samples. A certification pass upgrades the probabilistic localization to a verified one at the cost of one additional linear pass, with a full-sort fallback that covers the worst-case scenario. Our certified GPU implementation processes $10^6$, $10^7$, and $10^8$ scores in median times of $0.92$, $1.39$, and $7.24$ ms, respectively, making exact-budget soft top-$k$ practical within million-scale optimization loops. We demonstrate this capability in two applications: megapixel sparse adversarial examples with a small fraction of initial image pixels, where Fast LapSum achieves an order-of-magnitude speedup over state-of-the-art methods, and 3D Gaussian splatting. In the latter, we use Fast LapSum to reduce the number of Gaussians produced by Adaptive Density Control to a substantially smaller number while retaining nearly the same rendering quality and massively reducing computation. These results demonstrate that exact-budget differentiable top-$k$ can be incorporated into practical million-scale optimization pipelines.