SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $\mathbf{Z} \in \{0,1\}^{2^d \times d}$  is a row-wise binary encoding of all subsets of  $T \subseteq D$ ,  $\mathbf{W} \in \mathbb{R}^{2^d \times 2^d}$  is a diagonal matrix with the subset weights  $\mu$  and  $\mathbf{y}$  consists of the evaluations of  $\nu_0(T)$  for each subset. To include the optimization constraint, the (otherwise undefined) weights  $\mu(d), \mu(0)$  of  $D$  and  $\emptyset$  are set to a high positive constant. Solving (5) still requires  $2^d$  model evaluations and thus KSH [26] approximates  $I^{\mathrm{SV}}$  by considering (5) as an expectation
$$
\sum_ {T \in \mathcal {T} _ {1}} \mu (t) \left(\nu_ {0} (T) - \sum_ {i \in T} \beta_ {i}\right) ^ {2} \propto \mathbb {E} _ {T \sim p (T)} \left[ \left(\nu_ {0} (T) - \sum_ {i \in T} \beta_ {i}\right) ^ {2} \right]
$$
with  $p(T) \propto \mu(t)$ . Note that the optimization problem is invariant in terms of scaling. This expectation is the approximated similarly to SHAP-IQ, by computing high and low subset sizes explicitly and using Monte Carlo integration for the center sizes. The KSH estimator is difficult to analyze and it is only known that it is asymptotically unbiased [46].
KSH constructs a collection of subsets by determining a sampling order  $k_0$ , such that subsets with  $k_0 \leq t \leq d - k_0$  are sampled from  $p(T) \propto \mu(t)$  and for  $t < k_0$  or  $t > d - k_0$  all possible subsets are used. The value of  $k_0$  is thereby found by successively comparing the expected number of subsets with the total number of subsets of that size. As the number of subsets  $\binom{d}{t}$  for fixed  $d$  is a symmetric
log-concave sequence of positive terms, it has a maximum at the middle term(s)  $\lfloor \frac{d}{2} \rfloor$ ,  $\lceil \frac{d}{2} \rceil$  and grows monotonically and symmetrically as  $\binom{d}{t} = \binom{d}{d-t}$  towards this maximum from both sides. Thus, the implementation starts the comparison at  $k_0 = 0$  and iteratively increases the  $k_0$  candidate.