On $O(n)$ Algorithms for Projection onto the Top-$k$-sum Sublevel Set
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914419193151488 |
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| author | Roth, Jake Cui, Ying |
| author_facet | Roth, Jake Cui, Ying |
| contents | The \emph{top-$k$-sum} operator computes the sum of the largest $k$ components of a given vector. The Euclidean projection onto the top-$k$-sum sublevel set serves as a crucial subroutine in iterative methods to solve composite superquantile optimization problems. In this paper, we introduce a solver that implements two finite-termination algorithms to compute this projection. Both algorithms have $O(n)$ complexity of floating point operations when applied to a sorted $n$-dimensional input vector, where the absorbed constant is \emph{independent of $k$}. This stands in contrast to an existing grid-search-inspired method that has $O(k(n-k))$ complexity, a partition-based method with $O(n+D\log D)$ complexity, where $D\leq n$ is the number of distinct elements in the input vector, and a semismooth Newon method with a finite termination property but unspecified floating point complexity. The improvement of our methods over the first method is significant when $k$ is linearly dependent on $n$, which is frequently encountered in practical superquantile optimization applications. In instances where the input vector is unsorted, an additional cost is incurred to (partially) sort the vector, whereas a full sort of the input vector seems unavoidable for the other two methods. To reduce this cost, we further derive a rigorous procedure that leverages approximate sorting to compute the projection, which is particularly useful when solving a sequence of similar projection problems. Numerical results show that our methods solve problems of scale $n=10^7$ and $k=10^4$ within $0.05$ seconds, whereas the most competitive alternative, the semismooth Newton-based method, takes about $1$ second. The existing grid-search method and Gurobi's QP solver can take from minutes to hours. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_07224 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On $O(n)$ Algorithms for Projection onto the Top-$k$-sum Sublevel Set Roth, Jake Cui, Ying Optimization and Control Numerical Analysis 90-04, 90-08, 90C06, 90C20, 90C25, 90C33 The \emph{top-$k$-sum} operator computes the sum of the largest $k$ components of a given vector. The Euclidean projection onto the top-$k$-sum sublevel set serves as a crucial subroutine in iterative methods to solve composite superquantile optimization problems. In this paper, we introduce a solver that implements two finite-termination algorithms to compute this projection. Both algorithms have $O(n)$ complexity of floating point operations when applied to a sorted $n$-dimensional input vector, where the absorbed constant is \emph{independent of $k$}. This stands in contrast to an existing grid-search-inspired method that has $O(k(n-k))$ complexity, a partition-based method with $O(n+D\log D)$ complexity, where $D\leq n$ is the number of distinct elements in the input vector, and a semismooth Newon method with a finite termination property but unspecified floating point complexity. The improvement of our methods over the first method is significant when $k$ is linearly dependent on $n$, which is frequently encountered in practical superquantile optimization applications. In instances where the input vector is unsorted, an additional cost is incurred to (partially) sort the vector, whereas a full sort of the input vector seems unavoidable for the other two methods. To reduce this cost, we further derive a rigorous procedure that leverages approximate sorting to compute the projection, which is particularly useful when solving a sequence of similar projection problems. Numerical results show that our methods solve problems of scale $n=10^7$ and $k=10^4$ within $0.05$ seconds, whereas the most competitive alternative, the semismooth Newton-based method, takes about $1$ second. The existing grid-search method and Gurobi's QP solver can take from minutes to hours. |
| title | On $O(n)$ Algorithms for Projection onto the Top-$k$-sum Sublevel Set |
| topic | Optimization and Control Numerical Analysis 90-04, 90-08, 90C06, 90C20, 90C25, 90C33 |
| url | https://arxiv.org/abs/2310.07224 |