Sketching stochastic valuation functions

TL;DR

This work shows that for monotone subadditive or submodular valuation functions satisfying a weak homogeneity condition, or certain other conditions, there exist discretized distributions of item values with support sizes that yield a sketch valuation function which is a constant-factor approximation for any value query for a set of items of cardinality less than or equal to $k$.

Abstract

We consider the problem of sketching a set valuation function, which is defined as the expectation of a valuation function of independent random item values. We show that for monotone subadditive or submodular valuation functions satisfying a weak homogeneity condition, or certain other conditions, there exist discretized distributions of item values with $O(k\log(k))$ support sizes that yield a sketch valuation function which is a constant-factor approximation, for any value query for a set of items of cardinality less than or equal to $k$. The discretized distributions can be efficiently computed by an algorithm for each item's value distribution separately. Our results hold under conditions that accommodate a wide range of valuation functions arising in applications, such as the value of a team corresponding to the best performance of a team member, constant elasticity of substitution production functions exhibiting diminishing returns used in economics and consumer theory, and others. Sketch valuation functions are particularly valuable for finding approximate solutions to optimization problems such as best set selection and welfare maximization. They enable computationally efficient evaluation of approximate value oracle queries and provide an approximation guarantee for the underlying optimization problem.

Figures (12)

• Figure 1: Sketching by discretizing distributions: each item's value distribution is discretized, independently of other items, such that the resulting discrete distributions have finite supports and provide a satisfactory approximation guarantee when used to evaluate the corresponding sketch set valuation function.
• Figure 2: Algorithm \ref{['alg:disc']}: (left) input distribution $P$ and (right) output distribution $Q$.
• Figure 3: Approximation factor $(1+c)e^{2c/(1-c)}$ versus $c$.
• Figure 4: The approximation ratio for various objective functions and item value distributions: (left) exponential distributions and (right) Pareto distributions.
• Figure 5: Results showing the effect of different values of $\epsilon$: (top) exponential distribution and (bottom) Pareto distribution.

Theorems & Definitions (21)

• Lemma 2.1: Lemma 3 AN16
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• Lemma 3.2
• Definition 3.1
• Theorem 3.3
• Corollary 3.1
• Definition 3.2