Query strategies for priced information, revisited

TL;DR

This paper analyzes a universal query strategy for determining a function's value with minimal cost, relating its performance to the function's influence, and connects improvements to longstanding open problems in learning decision trees.

Contribution

It introduces a simple, universal query strategy applicable to all functions and relates its performance to the influence measure, providing new guarantees and insights.

Findings

Expected cost is proportional to optimal cost times influence divided by epsilon squared.

Strategy errs on at most an O(epsilon) fraction of inputs.

Improving parameters relates to open problems in learning decision trees.

Abstract

We consider the problem of designing query strategies for priced information, introduced by Charikar et al. In this problem the algorithm designer is given a function $f : \{0,1\}^n \to \{-1,1\}$ and a price associated with each of the $n$ coordinates. The goal is to design a query strategy for determining $f$'s value on unknown inputs for minimum cost. Prior works on this problem have focused on specific classes of functions. We analyze a simple and natural strategy that applies to all functions $f$, and show that its performance relative to the optimal strategy can be expressed in terms of a basic complexity measure of $f$, its influence. For $\varepsilon \in (0,\frac1{2})$, writing $\mathsf{opt}$ to denote the expected cost of the optimal strategy that errs on at most an $\varepsilon$-fraction of inputs, our strategy has expected cost $\mathsf{opt} \cdot \mathrm{Inf}(f)/\varepsilon^2$ and also errs on at most an $O(\varepsilon)$-fraction of inputs. This connection yields new guarantees that complement existing ones for a number of function classes that have been studied in this context, as well as new guarantees for new classes. Finally, we show that improving on the parameters that we achieve will require making progress on the longstanding open problem of properly learning decision trees.