Sketching stochastic valuation functions

TL;DR

This paper introduces a scalable method for creating discretized sketches of stochastic valuation functions that approximate true values within a constant factor, applicable to various practical valuation models.

Contribution

It develops a technique to discretize item value distributions with small support that yields constant-factor approximations for monotone subadditive or submodular valuations, scalable for large problems.

Findings

Discretized distributions with support size O(k log k) provide constant-factor approximations.

The method is efficiently computable independently for each item.

Applicable to common valuation functions like max-based and CES functions.

Abstract

We consider the problem of sketching set valuation functions, defined as the expectation of a valuation function applied to independent random item values. For valuation functions that are monotone and either subadditive or submodular, and that satisfy a weak homogeneity condition (or other structural conditions), we show that there exist discretized versions of the item value distributions -- each with support size $O(k \log k)$ -- that yield a sketch valuation function providing a constant-factor approximation to the true valuation for any subset of items of size at most $k$. These discretized distributions can be computed efficiently for each item independently, making the approach highly scalable. Our results apply broadly to valuation functions commonly encountered in practice, including team performance based on the best member (e.g., maximum functions), constant elasticity of substitution (CES) production functions with diminishing returns in economics, and others. Sketch valuation functions are especially useful in optimization problems such as best set selection and welfare maximization, where exact value computations are costly or intricate. They enable efficient approximate evaluation of value oracle queries while preserving provable approximation guarantees for the original stochastic optimization problem.