A Quantum Algorithm for Shapley Value Estimation

TL;DR

This work proposes quantum algorithms which can extract Shapley values within some confidence interval and demonstrates the validity of each approach under specific examples of cooperative voting games.

Abstract

In the classical context, the cooperative game theory concept of the Shapley value has been adapted for post hoc explanations of machine learning models. However, this approach does not easily translate to eXplainable Quantum ML (XQML). Finding Shapley values can be highly computationally complex. We propose quantum algorithms which can extract Shapley values within some confidence interval. Our results perform in polynomial time. We demonstrate the validity of each approach under specific examples of cooperative voting games.

Figures (4)

• Figure 1: This circuit $R$ is a controlled rotation, where $R_y(\theta) = (\cos(\theta/2), -\sin(\theta/2);\sin(\theta/2), \cos(\theta/2))$. (Note: Library used for visualizing circuits can be found here: kay_2023)
• Figure 2: Visual representation of $\beta_{n,m}$ being approximated using Riemann sums with partition $P_\ell$ over function $x^m(1-x)^{n-m}$, $t \in [0,1]$, $n=4$, $m=1$. The $k^\text{th}$ rectangle's height is $(t'_\ell(k))^m(1-~t'_\ell(k))^{n-m}$, and its width is $w_\ell(k)$.
• Figure 3: Circuit of $U_W$ for a weighted voting game. This circuit takes an input $x$ and outputs $W(S_x)$ in the utility register (Recall, $S_x$ is defined in Section \ref{['sec:algorithm']}). The auxiliary register contains the total vote count. Just before $U_W'$, the Aux register is in a basis state corresponding to the vote count of $S_x$. $U_W'$ uses the auxiliary register as an input, and outputs whether the vote count is in the correct range for player $i$ to be a deciding vote. Note that there is no $+w_i$ gate activation, as $S_x$ does not include player $i$ by definition. Results and simulation code are available in a https://github.com/iain-burge/QuantumShapleyValueAlgorithmgithubEntry.
• Figure 4: We generated 32 random weighted voting games for each condition. We generated random weights $w_j\in \mathbb{N}$ for each case such that $q\leq \sum_j w_j < 2q$. There were three primary scenarios: (1) Four players, voting threshold $q=8$; (2) Eight players, voting threshold $q=16$; and (3) 12 players, voting threshold $q=32$. We approximated every player's Shapley value for each scenario with our quantum algorithm using various $\ell$'s, assuming $\epsilon=0$. Next, we found the absolute error of our approximation by comparing each approximated Shapley value to its true value. Finally, we took the reciprocal of the mean for all Shapley value errors in each random game for each condition.

Theorems & Definitions (17)

• Definition 1: Coalitional game
• Definition 2: Payoff vector
• Definition 3: Shapley value shapley1952value
• Definition 4: monotonic game shapley1952value
• Theorem 1
• proof
• Theorem 2
• proof
• Definition 5: Beta function
• Lemma 1