Bounds on the Minimax Rate for Estimating a Prior over a VC Class from Independent Learning Tasks

By the standard PAC analysis [9, 3], for any $\gamma>0$, with probability greater than $1-\gamma$, a sample of $k=O((d/\gamma)\log(1/\gamma))$ random points will partition $\mathbb{C}$ into regions of width less than $\gamma$ (under $L_{1}(\mathcal{D})$). For brevity, we omit the $t$ subscripts and superscripts on quantities such as $\mathcal{Z}^{t}_{k}(\theta)$ throughout the following analysis, since the claims hold for any arbitrary value of $t$.

For any $\theta\in\Theta$, let $\pi_{\theta}^{\prime}$ denote a (conditional on $X_{1},\ldots,X_{k}$) distribution defined as follows. Let $f_{\theta}^{\prime}$ denote the (conditional on $X_{1},\ldots,X_{k}$) density function of $\pi_{\theta}^{\prime}$ with respect to $\pi_{0}$, and for any $g\in\mathbb{C}$, let $f_{\theta}^{\prime}(g)=\frac{\pi_{\theta}(\{h\in\mathbb{C}:\forall i\leq k,h(X_{i})=g(X_{i})\})}{\pi_{0}(\{h\in\mathbb{C}:\forall i\leq k,h(X_{i})=g(X_{i})\})}$ (or $0$ if $\pi_{0}(\{h\in\mathbb{C}:\forall i\leq k,h(X_{i})=g(X_{i})\})=0$). In other words, $\pi_{\theta}^{\prime}$ has the same probability mass as $\pi_{\theta}$ for each of the equivalence classes induced by $X_{1},\ldots,X_{k}$, but conditioned on the equivalence class, simply has a constant-density distribution over that equivalence class. Note that every $h\in\mathbb{C}$ has $f_{\theta}^{\prime}(h)$ between the smallest and largest values of $f_{\theta}(g)$ among $g\in\mathbb{C}$ with $\forall i\leq k,g(X_{i})=h(X_{i})$; therefore, by the smoothness condition, on the event (of probability greater than $1-\gamma$) that each of these regions has diameter less than $\gamma$, we have $\forall h\in\mathbb{C},|f_{\theta}(h)-f_{\theta}^{\prime}(h)|<L\gamma^{\alpha}$. On this event, for any $\theta,\theta^{\prime}\in\Theta$,

$$\|\pi_{\theta}-\pi_{\theta^{\prime}}\|=(1/2)\int|f_{\theta}-f_{\theta^{\prime}}|{\rm d}\pi_{0}<L\gamma^{\alpha}+(1/2)\int|f_{\theta}^{\prime}-f_{\theta^{\prime}}^{\prime}|{\rm d}\pi_{0}.$$

Furthermore, since the regions that define $f_{\theta}^{\prime}$ and $f_{\theta^{\prime}}^{\prime}$ are the same (namely, the partition induced by $X_{1},\ldots,X_{k}$), we have

	$\displaystyle(1/2)\int|f_{\theta}^{\prime}-f_{\theta^{\prime}}^{\prime}|{\rm d}\pi_{0}$	$\displaystyle=(1/2)\!\!\!\!\!\!\!\!\!\sum_{y_{1},\ldots,y_{k}\in\{-1,+1\}}\!\!\!|\pi_{\theta}(\{h\in\mathbb{C}:\forall i\leq k,h(X_{i})=y_{i}\})$
		$\displaystyle\phantom{aaaaaaaaaaaaaaaaa}-\pi_{\theta^{\prime}}(\{h\in\mathbb{C}:\forall i\leq k,h(X_{i})=y_{i}\})|$
		$\displaystyle=\|\mathbb{P}_{\mathbb{Y}_{k}(\theta)|\mathbb{X}_{k}}-\mathbb{P}_{\mathbb{Y}_{k}(\theta^{\prime})|\mathbb{X}_{k}}\|.$

Thus, we have that with probability at least $1-\gamma$,

$$\|\pi_{\theta}-\pi_{\theta^{\prime}}\|<L\gamma^{\alpha}+\|\mathbb{P}_{\mathbb{Y}_{k}(\theta)|\mathbb{X}_{k}}-\mathbb{P}_{\mathbb{Y}_{k}(\theta^{\prime})|\mathbb{X}_{k}}\|.$$

Following analogous to the inductive argument of [12], suppose $I\subseteq\{1,\ldots,k\}$, fix $\bar{x}_{I}\in\mathcal{X}^{|I|}$ and $\bar{y}_{I}\in\{-1,+1\}^{|I|}$. Then the $\tilde{y}_{I}\in\{-1,+1\}^{|I|}$ for which $\|\bar{y}_{I}-\tilde{y}_{I}\|_{1}$ is minimal, subject to the constraint that no $h\in\mathbb{C}$ has $h(\bar{x}_{I})=\tilde{y}_{I}$, has $(1/2)\|\bar{y}_{I}-\tilde{y}_{I}\|_{1}\leq d+1$; also, for any $i\in I$ with $\bar{y}_{i}\neq\tilde{y}_{i}$, letting $\bar{y}^{\prime}_{j}=\bar{y}_{j}$ for $j\in I\setminus\{i\}$ and $\bar{y}^{\prime}_{i}=\tilde{y}_{i}$, we have

$$\mathbb{P}_{\mathbb{Y}_{I}(\theta)|\mathbb{X}_{I}}(\bar{y}_{I}|\bar{x}_{I})=\mathbb{P}_{\mathbb{Y}_{I\setminus\{i\}}(\theta)|\mathbb{X}_{I\setminus\{i\}}}(\bar{y}_{I\setminus\{i\}}|\bar{x}_{I\setminus\{i\}})-\mathbb{P}_{\mathbb{Y}_{I}(\theta)|\mathbb{X}_{I}}(\bar{y}^{\prime}_{I}|\bar{x}_{I}),$$

and similarly for $\theta^{\prime}$, so that

	$\displaystyle|\mathbb{P}_{\mathbb{Y}_{I}(\theta)|\mathbb{X}_{I}}(\bar{y}_{I}|\bar{x}_{I})-\mathbb{P}_{\mathbb{Y}_{I}(\theta^{\prime})|\mathbb{X}_{I}}(\bar{y}_{I}|\bar{x}_{I})|$
	$\displaystyle\leq|\mathbb{P}_{\mathbb{Y}_{I\setminus\{i\}}(\theta)|\mathbb{X}_{I\setminus\{i\}}}(\bar{y}_{I\setminus\{i\}}|\bar{x}_{I\setminus\{i\}})-\mathbb{P}_{\mathbb{Y}_{I\setminus\{i\}}(\theta^{\prime})|\mathbb{X}_{I\setminus\{i\}}}(\bar{y}_{I\setminus\{i\}}|\bar{x}_{I\setminus\{i\}})|$
	$\displaystyle\phantom{aaaa}+|\mathbb{P}_{\mathbb{Y}_{I}(\theta)|\mathbb{X}_{I}}(\bar{y}^{\prime}_{I}|\bar{x}_{I})-\mathbb{P}_{\mathbb{Y}_{I}(\theta^{\prime})|\mathbb{X}_{I}}(\bar{y}^{\prime}_{I}|\bar{x}_{I})|.$

Now consider that these two terms inductively define a binary tree. Every time the tree branches left once, it arrives at a difference of probabilities for a set $I$ of one less element than that of its parent. Every time the tree branches right once, it arrives at a difference of probabilities for a $\bar{y}_{I}$ one closer to an unrealized $\tilde{y}_{I}$ than that of its parent. Say we stop branching the tree upon reaching a set $I$ and a $\bar{y}_{I}$ such that either $\bar{y}_{I}$ is an unrealized labeling, or $|I|=d$. Thus, we can bound the original (root node) difference of probabilities by the sum of the differences of probabilities for the leaf nodes with $|I|=d$. Any path in the tree can branch left at most $k-d$ times (total) before reaching a set $I$ with only $d$ elements, and can branch right at most $d+1$ times in a row before reaching a $\bar{y}_{I}$ such that both probabilities are zero, so that the difference is zero. So the depth of any leaf node with $|I|=d$ is at most $(k-d)d$. Furthermore, at any level of the tree, from left to right the nodes have strictly decreasing $|I|$ values, so that the maximum width of the tree is at most $k-d$. So the total number of leaf nodes with $|I|=d$ is at most $(k-d)^{2}d$. Thus, for any $\bar{y}\in\{-1,+1\}^{k}$ and $\bar{x}\in\mathcal{X}^{k}$,

	$\displaystyle|\mathbb{P}_{\mathbb{Y}_{k}(\theta)|\mathbb{X}_{k}}(\bar{y}|\bar{x})-\mathbb{P}_{\mathbb{Y}_{k}(\theta^{\prime})|\mathbb{X}_{k}}(\bar{y}|\bar{x})|$
	$\displaystyle\leq(k-d)^{2}d\cdot\max_{\bar{y}^{d}\in\{-1,+1\}^{d}}\max_{D\in\{1,\ldots,k\}^{d}}|\mathbb{P}_{\mathbb{Y}_{d}(\theta)|\mathbb{X}_{d}}(\bar{y}^{d}|\bar{x}_{D})-\mathbb{P}_{\mathbb{Y}_{d}(\theta^{\prime})|\mathbb{X}_{d}}(\bar{y}^{d}|\bar{x}_{D})|.$

Since

$$\|\mathbb{P}_{\mathbb{Y}_{k}(\theta)|\mathbb{X}_{k}}-\mathbb{P}_{\mathbb{Y}_{k}(\theta^{\prime})|\mathbb{X}_{k}}\|=(1/2)\sum_{\bar{y}^{k}\in\{-1,+1\}^{k}}|\mathbb{P}_{\mathbb{Y}_{k}(\theta)|\mathbb{X}_{k}}(\bar{y}^{k})-\mathbb{P}_{\mathbb{Y}_{k}(\theta^{\prime})|\mathbb{X}_{k}}(\bar{y}^{k})|,$$

and by Sauer’s Lemma this is at most

$$(ek)^{d}\max_{\bar{y}^{k}\in\{-1,+1\}^{k}}|\mathbb{P}_{\mathbb{Y}_{k}(\theta)|\mathbb{X}_{k}}(\bar{y}^{k})-\mathbb{P}_{\mathbb{Y}_{k}(\theta^{\prime})|\mathbb{X}_{k}}(\bar{y}^{k})|,$$

we have that

	$\displaystyle\|\mathbb{P}_{\mathbb{Y}_{k}(\theta)|\mathbb{X}_{k}}-\mathbb{P}_{\mathbb{Y}_{k}(\theta^{\prime})|\mathbb{X}_{k}}\|$
	$\displaystyle\leq(ek)^{d}k^{2}d\max_{\bar{y}^{d}\in\{-1,+1\}^{d}}\max_{D\in\{1,\ldots,k\}^{d}}|\mathbb{P}_{\mathbb{Y}_{d}(\theta)|\mathbb{X}_{D}}(\bar{y}^{d})-\mathbb{P}_{\mathbb{Y}_{d}(\theta^{\prime})|\mathbb{X}_{D}}(\bar{y}^{d})|.$

Thus, we have that

	$\displaystyle\|\pi_{\theta}-\pi_{\theta^{\prime}}\|=\mathbb{E}\|\pi_{\theta}-\pi_{\theta^{\prime}}\|$
	$\displaystyle<\gamma\!+\!L\gamma^{\alpha}\!+\!(ek)^{d}k^{2}d\mathbb{E}\bigg{[}\max_{\bar{y}^{d}\in\{-1,+1\}^{d}}\max_{D\in\{1,\ldots,k\}^{d}}\mathbb{P}_{\mathbb{Y}_{d}(\theta)|\mathbb{X}_{D}}(\bar{y}^{d})-\mathbb{P}_{\mathbb{Y}_{d}(\theta^{\prime})|\mathbb{X}_{D}}(\bar{y}^{d})|\bigg{]}.$

Note that

	$\displaystyle\mathbb{E}\bigg{[}\max_{\bar{y}^{d}\in\{-1,+1\}^{d}}\max_{D\in\{1,\ldots,k\}^{d}}|\mathbb{P}_{\mathbb{Y}_{d}(\theta)|\mathbb{X}_{D}}(\bar{y}^{d})-\mathbb{P}_{\mathbb{Y}_{d}(\theta^{\prime})|\mathbb{X}_{D}}(\bar{y}^{d})|\bigg{]}$
	$\displaystyle\leq\sum_{\bar{y}^{d}\in\{-1,+1\}^{d}}\sum_{D\in\{1,\ldots,k\}^{d}}\mathbb{E}\big{[}|\mathbb{P}_{\mathbb{Y}_{d}(\theta)|\mathbb{X}_{D}}(\bar{y}^{d})-\mathbb{P}_{\mathbb{Y}_{d}(\theta^{\prime})|\mathbb{X}_{D}}(\bar{y}^{d})|\big{]}$
	$\displaystyle\leq(2k)^{d}\max_{\bar{y}^{d}\in\{-1,+1\}^{d}}\max_{D\in\{1,\ldots,k\}^{d}}\mathbb{E}\big{[}|\mathbb{P}_{\mathbb{Y}_{d}(\theta)|\mathbb{X}_{D}}(\bar{y}^{d})-\mathbb{P}_{\mathbb{Y}_{d}(\theta^{\prime})|\mathbb{X}_{D}}(\bar{y}^{d})|\big{]},$

and by exchangeability, this last line equals

$$(2k)^{d}\max_{\bar{y}^{d}\in\{-1,+1\}^{d}}\mathbb{E}\left[|\mathbb{P}_{\mathbb{Y}_{d}(\theta)|\mathbb{X}_{d}}(\bar{y}^{d})-\mathbb{P}_{\mathbb{Y}_{d}(\theta^{\prime})|\mathbb{X}_{d}}(\bar{y}^{d})|\right].$$

[12] showed that $\mathbb{E}\left[|\mathbb{P}_{\mathbb{Y}_{d}(\theta)|\mathbb{X}_{d}}(\bar{y}^{d})-\mathbb{P}_{\mathbb{Y}_{d}(\theta^{\prime})|\mathbb{X}_{d}}(\bar{y}^{d})|\right]\leq 4\sqrt{\|\mathbb{P}_{\mathcal{Z}_{d}(\theta)}-\mathbb{P}_{\mathcal{Z}_{d}(\theta^{\prime})}\|}$, so that in total we have $\|\pi_{\theta}-\pi_{\theta^{\prime}}\|\!<\!(L\!+\!1)\gamma^{\alpha}\!+\!4(2ek)^{2d+2}\!\!\sqrt{\|\mathbb{P}_{\mathcal{Z}_{d}(\theta)}\!-\!\mathbb{P}_{\mathcal{Z}_{d}(\theta^{\prime})}\|}$. Plugging in the value of $k=c(d/\gamma)\log(1/\gamma)$, this is

$$(L\!+\!1)\gamma^{\alpha}+4\!\left(\!2ec\frac{d}{\gamma}\log\!\left(\frac{1}{\gamma}\right)\!\right)^{\!\!2d+2}\!\sqrt{\|\mathbb{P}_{\mathcal{Z}_{d}(\theta)}\!-\!\mathbb{P}_{\mathcal{Z}_{d}(\theta^{\prime})}\|}.$$

Thus, it suffices to bound the rate of convergence (in total variation distance) of some estimator of $\mathbb{P}_{\mathcal{Z}_{d}({\theta_{\star}})}$. If $N(\varepsilon)$ is the $\varepsilon$-covering number of $\{\mathbb{P}_{\mathcal{Z}_{d}(\theta)}:\theta\in\Theta\}$, then taking $\hat{\theta}_{T{\theta_{\star}}}$ as the minimum distance skeleton estimate of [13, 5] achieves expected total variation distance $\varepsilon$ from $\mathbb{P}_{\mathcal{Z}_{d}({\theta_{\star}})}$, for some $T=O((1/\varepsilon^{2})\log N(\varepsilon/4))$. We can partition $\mathbb{C}$ into $O((L/\varepsilon)^{d/\alpha})$ cells of diameter $O((\varepsilon/L)^{1/\alpha})$, and set a constant density value within each cell, on an $O(\varepsilon)$-grid of density values, and every prior with $(L,\alpha)$-Hölder smooth density will have density within $\varepsilon$ of some density so-constructed; there are then at most $(1/\varepsilon)^{O((L/\varepsilon)^{d/\alpha})}$ such densities, so this bounds the covering numbers of $\Pi_{\Theta}$. Furthermore, the covering number of $\Pi_{\Theta}$ upper bounds $N(\varepsilon)$ [12], so that $N(\varepsilon)\leq(1/\varepsilon)^{O((L/\varepsilon)^{d/\alpha})}$.

Solving $T\!=\!O(\varepsilon^{-2}(L/\varepsilon)^{d/\alpha}\log(1/\varepsilon))$ for $\varepsilon$, we have $\varepsilon\!=\!O\!\left(L\!\left(\frac{\log(TL)}{T}\right)^{\frac{\alpha}{d+2\alpha}}\right)$. So this bounds the rate of convergence for $\mathbb{E}\|\mathbb{P}_{\mathcal{Z}_{d}(\hat{\theta}_{T})}-\mathbb{P}_{\mathcal{Z}_{d}({\theta_{\star}})}\|$, for $\hat{\theta}_{T}$ the minimum distance skeleton estimate. Plugging this rate into the bound on the priors, combined with Jensen’s inequality, we have

$$\mathbb{E}\|\pi_{\hat{\theta}_{T}}-\pi_{{\theta_{\star}}}\|<(L+1)\gamma^{\alpha}+4\left(2ec\frac{d}{\gamma}\log\left(\frac{1}{\gamma}\right)\right)^{2d+2}\!\!\!\!\!\times O\left(L\left(\frac{\log(TL)}{T}\right)^{\frac{\alpha}{2d+4\alpha}}\right).$$

This holds for any $\gamma>0$, so minimizing this expression over $\gamma>0$ yields a bound on the rate. For instance, with $\gamma=\tilde{O}\left(T^{-\frac{\alpha}{2(d+2\alpha)(\alpha+2(d+1))}}\right)$, we have

$$\mathbb{E}\|\pi_{\hat{\theta}_{T}}-\pi_{{\theta_{\star}}}\|=\tilde{O}\left(LT^{-\frac{\alpha^{2}}{2(d+2\alpha)(\alpha+2(d+1))}}\right).$$

∎

(Sketch) We proceed by a reduction from the task of determining the bias of a coin from among two given possibilities. Specifically, fix any $\gamma\in(0,1/2)$, $n\in\mathbb{N}$, and let $B_{1}(p),\ldots,B_{n}(p)$ be i.i.d ${\rm Bernoulli}(p)$ random variables, for each $p\in[0,1]$; then it is known that, for any (possibly nondeterministic) decision rule $\hat{p}_{n}:\{0,1\}^{n}\to\{(1+\gamma)/2,(1-\gamma)/2\}$,

$$\frac{1}{2}\sum_{p\in\{(1+\gamma)/2,(1-\gamma)/2\}}\mathbb{P}(\hat{p}_{n}(B_{1}(p),\ldots,B_{n}(p))\neq p)\\ \geq(1/32)\cdot\exp\left\{-128\gamma^{2}n/3\right\}.$$

(1)

This easily follows from the results of [1], combined with a result of [7] bounding the KL divergence (see also [11])

To use this result, we construct a learning problem as follows. Fix some $m\in\mathbb{N}$ with $m\geq d$, let $\mathcal{X}=\{1,\ldots,m\}$, and let $\mathbb{C}$ be the space of all classifiers $h:\mathcal{X}\to\{-1,+1\}$ such that $|\{x\in\mathcal{X}:h(x)=+1\}|\leq d$. Clearly the VC dimension of $\mathbb{C}$ is $d$. Define the distribution $\mathcal{D}$ as uniform over $\mathcal{X}$. Finally, we specify a family of $(L,\alpha)$-Hölder smooth priors, parameterized by $\Theta=\{-1,+1\}^{\binom{m}{d}}$, as follows. Let $\gamma_{m}=(L/2)(1/m)^{\alpha}$. First, enumerate the $\binom{m}{d}$ distinct $d$-sized subsets of $\{1,\ldots,m\}$ as $\mathcal{X}_{1},\mathcal{X}_{2},\ldots,\mathcal{X}_{\binom{m}{d}}$. Define the reference distribution $\pi_{0}$ by the property that, for any $h\in\mathbb{C}$, letting $q=|\{x:h(x)=+1\}|$, $\pi_{0}(\{h\})=(\frac{1}{2})^{d}\binom{m-q}{d-q}/\binom{m}{d}$. For any $\mathbf{b}=(b_{1},\ldots,b_{\binom{m}{d}})\in\{-1,1\}^{\binom{m}{d}}$, define the prior $\pi_{\mathbf{b}}$ as the distribution of a random variable $h_{\mathbf{b}}$ specified by the following generative model. Let $i^{*}\sim{\rm Uniform}(\{1,\ldots,\binom{m}{d}\})$, let $C_{\mathbf{b}}(i^{*})\sim{\rm Bernoulli}((1+\gamma_{m}b_{i^{*}})/2)$; finally, $h_{\mathbf{b}}\sim{\rm Uniform}(\{h\in\mathbb{C}:\{x:h(x)=+1\}\subseteq\mathcal{X}_{i^{*}},{\rm Parity}(|\{x:h(x)=+1\}|)=C_{\mathbf{b}}(i^{*})\})$, where ${\rm Parity}(n)$ is $1$ if $n$ is odd, or $0$ if $n$ is even. We will refer to the variables in this generative model below. For any $h\in\mathbb{C}$, letting $H=\{x:h(x)=+1\}$ and $q=|H|$, we can equivalently express $\pi_{\mathbf{b}}(\{h\})=(\frac{1}{2})^{d}\binom{m}{d}^{-1}\sum_{i=1}^{\binom{m}{d}}\mathbbm{1}[H\subseteq\mathcal{X}_{i}](1+\gamma_{m}b_{i})^{{\rm Parity}(q)}(1-\gamma_{m}b_{i})^{1-{\rm Parity}(q)}$. From this explicit representation, it is clear that, letting $f_{\mathbf{b}}=\frac{{\rm d}\pi_{\mathbf{b}}}{{\rm d}\pi_{0}}$, we have $f_{\mathbf{b}}(h)\in[1-\gamma_{m},1+\gamma_{m}]$ for all $h\in\mathbb{C}$. The fact that $f_{\mathbf{b}}$ is Hölder smooth follows from this, since every distinct $h,g\in\mathbb{C}$ have $\mathcal{D}(\{x:h(x)\neq g(x)\})\geq 1/m=(2\gamma_{m}/L)^{1/\alpha}$.

Next we set up the reduction as follows. For any estimator $\hat{\pi}_{T}=\hat{\pi}_{T}(\mathcal{Z}^{1}_{d}({\theta_{\star}}),$ $\ldots,\mathcal{Z}^{T}_{d}({\theta_{\star}}))$, and each $i\in\{1,\ldots,\binom{m}{d}\}$, let $h_{i}$ be the classifier with $\{x:h_{i}(x)=+1\}=\mathcal{X}_{i}$; also, if $\hat{\pi}_{T}(\{h_{i}\})>(\frac{1}{2})^{d}/\binom{m}{d}$, let $\hat{b}_{i}=2{\rm Parity}(d)-1$, and otherwise $\hat{b}_{i}=1-2{\rm Parity}(d)$. We use these $\hat{b}_{i}$ values to estimate the original $b_{i}$ values. Specifically, let $\hat{p}_{i}=(1+\gamma_{m}\hat{b}_{i})/2$ and $p_{i}=(1+\gamma_{m}b_{i})/2$, where $\mathbf{b}={\theta_{\star}}$. Then

	$\displaystyle\|\hat{\pi}_{T}-\pi_{{\theta_{\star}}}\|$	$\displaystyle\geq(1/2)\sum_{i=1}^{\binom{m}{d}}|\hat{\pi}_{T}(\{h_{i}\})-\pi_{{\theta_{\star}}}(\{h_{i}\})|$
		$\displaystyle\geq(1/2)\sum_{i=1}^{\binom{m}{d}}\frac{\gamma_{m}}{2^{d}\binom{m}{d}}|\hat{b}_{i}-b_{i}|/2=(1/2)\sum_{i=1}^{\binom{m}{d}}\frac{1}{2^{d}\binom{m}{d}}|\hat{p}_{i}-p_{i}|.$

Thus, we have reduced from the problem of deciding the biases of these $\binom{m}{d}$ independent Bernoulli random variables. To complete the proof, it suffices to lower bound the expectation of the right side for an arbitrary estimator.

Toward this end, we in fact study an even easier problem. Specifically, consider an estimator $\hat{q}_{i}=\hat{q}_{i}(\mathcal{Z}^{1}_{d}({\theta_{\star}}),\ldots,\mathcal{Z}^{T}_{d}({\theta_{\star}}),i_{1}^{*},\ldots,i_{T}^{*})$, where $i_{t}^{*}$ is the $i^{*}$ random variable in the generative model that defines $h^{*}_{t{\theta_{\star}}}$; that is, $i_{t}^{*}\sim{\rm Uniform}(\{1,$ $\ldots,\binom{m}{d}\})$, $C_{t}\sim{\rm Bernoulli}((1+\gamma_{m}b_{i_{t}^{*}})/2)$, and $h^{*}_{t{\theta_{\star}}}\sim{\rm Uniform}(\{h\in\mathbb{C}:\{x:h(x)=+1\}\subseteq\mathcal{X}_{i_{t}^{*}},{\rm Parity}(|\{x:h(x)=+1\}|)=C_{t}\})$, where the $i_{t}^{*}$ are independent across $t$, as are the $C_{t}$ and $h^{*}_{t{\theta_{\star}}}$. Clearly the $\hat{p}_{i}$ from above can be viewed as an estimator of this type, which simply ignores the knowledge of $i_{t}^{*}$. The knowledge of these $i_{t}^{*}$ variables simplifies the analysis, since given $\{i_{t}^{*}:t\leq T\}$, the data can be partitioned into $\binom{m}{d}$ disjoint sets, $\{\{\mathcal{Z}^{t}_{d}({\theta_{\star}}):i_{t}^{*}=i\}:i=1,\ldots,\binom{m}{d}\}$, and we can use only the set $\{\mathcal{Z}^{t}_{d}({\theta_{\star}}):i_{t}^{*}=i\}$ to estimate $p_{i}$. Furthermore, we can use only the subset of these for which $\mathbb{X}_{td}=\mathcal{X}_{i}$, since otherwise we have zero information about the value of ${\rm Parity}(|\{x:h^{*}_{t{\theta_{\star}}}(x)=+1\}|)$. That is, given $i_{t}^{*}=i$, any $\mathcal{Z}^{t}_{d}({\theta_{\star}})$ is conditionally independent from every $b_{j}$ for $j\neq i$, and is even conditionally independent from $b_{i}$ when $\mathbb{X}_{td}$ is not completely contained in $\mathcal{X}_{i}$; specifically, in this case, regardless of $b_{i}$, the conditional distribution of $\mathbb{Y}_{td}({\theta_{\star}})$ given $i_{t}^{*}=i$ and given $\mathbb{X}_{td}$ is a product distribution, which deterministically assigns label $-1$ to those $Y_{tk}({\theta_{\star}})$ with $X_{tk}\notin\mathcal{X}_{i}$, and gives uniform random values to the subset of $\mathbb{Y}_{td}({\theta_{\star}})$ with their respective $X_{tk}\in\mathcal{X}_{i}$. Finally, letting $r_{t}={\rm Parity}(|\{k\leq d:Y_{tk}({\theta_{\star}})=+1\}|)$, we note that given $i_{t}^{*}=i$, $\mathbb{X}_{td}=\mathcal{X}_{i}$, and the value $r_{t}$, $b_{i}$ is conditionally independent from $\mathcal{Z}^{t}_{d}({\theta_{\star}})$. Thus, the set of values $C_{iT}({\theta_{\star}})=\{r_{t}:i_{t}^{*}=i,\mathbb{X}_{td}=\mathcal{X}_{i}\}$ is a sufficient statistic for $b_{i}$ (hence for $p_{i}$). Recall that, when $i_{t}^{*}=i$ and $\mathbb{X}_{td}=\mathcal{X}_{i}$, the value of $r_{t}$ is equal to $C_{t}$, a ${\rm Bernoulli}(p_{i})$ random variable. Thus, we neither lose nor gain anything (in terms of risk) by restricting ourselves to estimators $\hat{q}_{i}$ of the type $\hat{q}_{i}=\hat{q}_{i}(\mathcal{Z}^{1}_{d}({\theta_{\star}}),\ldots,\mathcal{Z}^{T}_{d}({\theta_{\star}}),i_{1}^{*},\ldots,i_{T}^{*})=\hat{q}_{i}^{\prime}(C_{iT}({\theta_{\star}}))$, for some $\hat{q}_{i}^{\prime}$ [8]: that is, estimators that are a function of the $N_{iT}({\theta_{\star}})=|C_{iT}({\theta_{\star}})|$ ${\rm Bernoulli}(p_{i})$ random variables, which we should note are conditionally i.i.d. given $N_{iT}({\theta_{\star}})$.