SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proposition 6 (Wasserstein dependency matrix from KR-rearrangement). Let  $(\mathcal{Z}^d, \Sigma, \mu)$  be a probability space with  $\mu = T_{\sharp}\nu^{d}$  for the KR-rearrangement  $T \colon \mathcal{Z}^d \to \mathcal{Z}^d$  and a reference measure  $\nu^d$  on  $\mathcal{Z}^d$ . Let each  $\mathcal{Z}$  be equipped with a metric  $\rho$  and have finite diameter  $\|\rho\| < \infty$ . Let  $f \colon \mathcal{Z}^d \to \mathbb{R}$  be a Lipschitz function with respect to the product metric  $\rho^d$ . Let  $\mathcal{B} \subseteq \mathcal{Z}^d$  denote a set of bad inputs and define the corresponding set  $\mathcal{A} = T^{-1}(\mathcal{B}) \subseteq \mathcal{Z}^d$ . Let  $\widehat{T}$  be the unique KR-rearrangement that satisfies  $\widehat{T}_{\sharp}\nu^{d} = \nu^{d}|\mathcal{A}^{c}$  and denote  $\widetilde{T} = T \circ \widehat{T}$ . Suppose there exist constants  $L_{ij}$  such that for all  $v, z \in \mathcal{B}^{c}$  with  $v^{[d] \setminus \{i\}} = z^{[d] \setminus \{i\}}$  it holds
$$
\mathbb {E} _ {\tau \sim \nu^ {(i, d ]}} \left[ \rho (\widetilde {T} ^ {(i, d ]} (\widehat {v} ^ {[ i ]}, \tau) _ {j}, \widetilde {T} ^ {(i, d ]} (\widehat {z} ^ {[ i ]}, \tau) _ {j}) \right] \leq L _ {i j} \rho (v _ {i}, z _ {i}) \tag {16}
$$
where  $\widehat{v}^{[i]}$  and  $\widehat{z}^{[i]}$  are uniquely defined through  $\widetilde{T}^{[i]}(\widehat{v}^{[i]}) = v^{[i]}$  and  $\widetilde{T}^{[i]}(\widehat{z}^{[i]}) = z^{[i]}$ . Then  $\Gamma = \frac{\|\rho\|}{d} D$  is a Wasserstein dependency matrix for  $\mu |\mathcal{B}^c$  with
$$
D _ {i j} = \left\{ \begin{array}{l l} 0 & \text {i f} i > j, \\ 1 & \text {i f} i = j, \\ L _ {i j} & \text {i f} i <   j. \end{array} \right. \tag {17}
$$
We remark that  $\Gamma$  indeed distills the dependency structure of  $\mu$ . To illustrate this, let  $\mu^{\{i\}}$  be independent from  $\mu^{\{j\}}$  for some  $i \in [d]$ ,  $j \in (i,d]$ . Then conditioning on a different value of  $\mu^{\{i\}}$  does not change the distribution  $\mu^{\{j\}}$ . Thus,
$$
\widetilde {T} ^ {(i, d ]} \left(\bar {x} ^ {[ i ]}, \tau\right) _ {j} = \widetilde {T} ^ {(i, d ]} \left(\bar {z} ^ {[ i ]}, \tau\right) _ {j}, \quad \forall \tau \in \mathcal {Z} ^ {(i, d ]}, \quad \bar {x} ^ {[ d ] \backslash \{i \}} = \bar {z} ^ {[ d ] \backslash \{i \}}, \quad j \in (i, d ] \tag {18}
$$
and the choice  $L_{ij} = 0$  satisfies (16). It directly follows that  $\Gamma_{ij} = 0$ .
The following theorem states the main result of the present work.
Theorem 7 (PAC-Bayesian risk certificate for structured prediction). Fix  $\delta \in (0, \exp(-e^{-1}))$ , let  $\mu$  be a data distribution on  $\mathcal{Z}^d$  with  $T_{\sharp}\nu^d = \mu$  the Knothe-Rosenblatt rearrangement for a reference measure  $\nu^d$  on  $\mathcal{Z}^d$  and fix a measurable set  $\mathcal{B} \subseteq \mathcal{Z}^d$  of bad inputs with  $\mu(\mathcal{B}) \leq \xi$ . Fix a PAC-Bayes prior  $\pi$  on a hypothesis class  $\mathcal{H}$  of functions  $\phi: \mathcal{X}^d \to \mathcal{Y}^d$  and a loss function  $\ell$  which assumes values in  $[0,1]$ . Define the oscillation vector  $\widetilde{\delta}$  by
$$
\widetilde {\delta} _ {i} = \sup  _ {h \in \mathcal {H}} \left. \delta_ {i} \left(L (h, \cdot) \right| _ {\mathcal {B} ^ {c}}\right), \quad i \in [ d ] \tag {19}
$$
where  $L(h,\cdot)\big|_{\mathcal{B}^c}$  denotes the restriction of  $L(h,\cdot)$  to  $\mathcal{Z}^d\setminus \mathcal{B}$ . Suppose all oscillations  $\widetilde{\delta}_i$  are finite, suppose the condition (16) is satisfied and denote by  $D$  the matrix with entries (17). Then, with probability at least  $1 - \delta$  over realizations of a training set  $\mathcal{D}_m = (Z^{(k)})_{k = 1}^m$  drawn from  $(\mu |\mathcal{B}^c)^m$  it holds for all PAC-Bayes posteriors  $\zeta$  on  $\mathcal{H}$  that
$$
\mathcal {R} (\zeta) \leq \mathcal {R} _ {m} (\zeta , \mathcal {D} _ {m}) + 2 \frac {\| \rho \|}{d} \| D \widetilde {\delta} \| _ {2} \sqrt {\frac {\log \frac {1}{\delta} + \mathrm {K L} [ \zeta : \pi ]}{2 m}} + \xi . \tag {20}
$$
Note that the generalization gap on the right hand side of (20) decays with  $d$ . This accounts for generalization from a single example. In fact, if only  $m = 1$  structured example is available, but  $d \gg 1$ , Theorem 7 still certifies risk. This effect can however be negated by the norm  $\|D\widetilde{\delta}\|$ . If structured data contain strong global dependence, then  $\|D\widetilde{\delta}\|$  will not be bounded independently of  $d$  and thus, in the worst case of  $\|D\widetilde{\delta}\| \in \mathcal{O}(d)$  the assertion is no stronger than PAC-Bayesian bounds for unstructured data. The same point was observed in [38].
The measure of the bad set under the data distribution  $\mu$  is assumed to be bounded by  $\xi$ . This is to account for a small number of data which contain strong dependence. In order to prevent these bad data from dominating  $D$ , thereby negating the decay of the bound in  $d$  as described above, it is preferable to exclude them from the sample, reduce the sample size  $m$  and pay the penalty  $\xi$  in (20).
To prove Theorem 7, we broadly follow the argument put forward in [38]. This augments typical PAC-Bayesian constructions in the literature by the inclusion of a set of bad inputs. We first reconcile the data being conditioned on  $\mathcal{B}^c$  with risk certification for unconditioned data, leading to the addition of  $\xi$  on the right hand side of (20). The model complexity term  $\mathrm{KL}[\zeta : \pi]$  is due to Donsker and Varadhand's variational formula. Subsequently, the moment generating function bound of Theorem 4 is instantiated through the Wasserstein dependency matrix constructed in Proposition 6. Markov's inequality then gives a pointwise risk bound for fixed value of a free parameter. In order to optimize this parameter, the bound is made uniform on a discrete set of values through a union bound. A full proof is presented in Appendix A.
In Theorem 7, we combine the PAC-Bayesian construction of [38] with the more general concentration of measure theory of [35]. Crucially, concentration of measure results used in [38] are predicated on the assumption of data generated by a Markov random field [57, 34]. Our work is more flexible in two major ways.
(1) Our assumption on the data-generating distribution is likely more representative of real-world data as measure transport models have repeatedly been shown to yield convincing data generators.
(2) Markov random fields are difficult to handle computationally because inference in general Markov random fields is NP-hard [57] such that one is forced to learn based on approximate inference procedures [31, 22, 55].
Our work also allows for more general metrics  $\rho$  as opposed to the singular choice of Hamming norm required in [38]. Additionally, the key results of [38] are constructed to ensure all data drawn from the unconditioned distribution  $\mu$  are in the good set. This reduces the probability of correctness  $1 - \delta$  by  $m\xi$ . Instead, we assume data drawn from  $\mu|\mathcal{B}^c$ , effectively reducing the number of available samples by a factor of  $1 - \xi$ , but keeping the probability of correctness high. This allows the set of bad inputs to be used more effectively as a computational tool in Section 5.
Comparing the dependency of (20) on  $d$  with the respective result in [38], it first appears as though our bound decays with a faster rate ( $d$  instead of  $\sqrt{d}$ ). However, this will not typically be the case in practice because  $\| D\widetilde{\delta}\|_2$  grows with rate  $\sqrt{d}$  in most situations. To see this, consider the case of local
dependency in the sense that
$$
L _ {i j} = \left\{ \begin{array}{l l} 1, & \text {i f} j \in \mathcal {N} _ {i}, \\ 0, & \text {e l s e} \end{array} \right. \tag {21}
$$
for local neighborhoods  $\mathcal{N}_i\subseteq [d]$  which contain a fixed number of  $c$  elements and let  $\widetilde{\delta}_i = \alpha$  for all  $i\in [d]$  and some constant value  $\alpha >0$ . Then
$$
\| D \widetilde {\delta} \| _ {2} = \sqrt {\sum_ {i \in [ d ]} (\alpha | \mathcal {N} _ {i} |) ^ {2}} = c \alpha \sqrt {d}. \tag {22}
$$
Clearly, if dependence is localized and the oscillations  $\widetilde{\delta}$  do not decay in  $d$ , then  $\|D\widetilde{\delta}\|_2$  contains a factor that grows with rate  $\sqrt{d}$ , leading to the same asymptotic rate of (20) already observed in [38].
Note that [38] additionally allows for a set of bad hypotheses  $\mathcal{B}_{\mathcal{F}} \subseteq \mathcal{H}$  which do not conform to stability assumptions. In our construction, this means restricting the bound (19) to oscillations on the set of good hypotheses. We omit this extension for clarity of exposition, but do not expect it to necessitate major changes to the presented proofs. The same applies to the derandomization strategy proposed by [38] which is based on hypothesis stability.
Further, [38] considers a large number of applicable orderings for random variables by introducing a filtration of their index set. This notion is not easily compatible with our assumption of KR-rearrangement, because triangularity of transport depends on the order of variables.
5 Bounding the Bad Set
With regard to numerical risk certificates, a key technical aspect of Section 4 concerns the quantities  $L_{ij}$  in (16). Here, we propose a way to use the set of bad inputs as a computational tool to this end. Suppose we assign arbitrary fixed values to  $L_{ij}$  and subsequently define  $\mathcal{B} \subseteq \mathcal{Z}^d$  as the set of inputs on which the condition (16) fails. Then we have fulfilled the prerequisites of Proposition 6 by construction and are left with bounding  $\mu(\mathcal{B})$ . Note that
$$
\mu (\mathcal {B}) = \mathbb {P} _ {Z \sim \mu} (Z \in \mathcal {B}) = \mathbb {E} _ {Z \sim \mu} [ \mathbf {1} \{Z \in \mathcal {B} \} ] \tag {23}
$$
and the indicator function 1 assumes values in the bounded set  $\{0,1\}$ . Therefore, Hoeffding's inequality gives the following.
Proposition 8 (Upper bound on the bad set). Let  $\mu$  be a data distribution and let  $\widetilde{\mathcal{D}}_n\sim \mu^n$  be a sample of size  $n$ . Fix an error probability  $\epsilon \in (0,1)$ . Then
$$
\mu (\mathcal {B}) \leq \frac {1}{n} \sum_ {Z \in \widetilde {\mathcal {D}} _ {n}} \mathbf {1} \{Z \in \mathcal {B} \} + \sqrt {\frac {1}{2 n} \log \frac {2}{\epsilon}} \tag {24}
$$
with probability at least  $1 - \epsilon$  over the sample.
Checking the condition  $Z \in \mathcal{B}$  requires evaluating (16) which comes down to finding a Lipschitz constant for a one-dimensional function. If this is computationally feasible for the given data model, then the concentration argument (24) can be used to bound  $\mu(\mathcal{B})$  with high probability. Because (24) decays only in the number of structured examples  $\mathcal{O}(\sqrt{n})$ , it can not be used to show generalization from a single structured example. However, PAC-Bayesian risk certificates are typically dominated by the KL complexity term in (20) which decays with the size of structured examples as well. Thus, Proposition 8 should still be useful in practice.
Note that Proposition 8 makes a pointwise statement about a fixed value of  $L_{ij}$  which has limited utility for learning  $L_{ij}$  from data. To remedy this problem, we can first define a discrete set  $\mathcal{L} = (L^{(k)})_{k\in [l]}$  of candidate matrices and select error probabilities  $\epsilon_{k}$  for each of the events
$$
\mu \left(\mathcal {B} \left(L ^ {(k)}\right)\right) \geq \frac {1}{n} \sum_ {Z \in \widetilde {\mathcal {D}} _ {n}} \mathbf {1} \{Z \in \mathcal {B} \left(L ^ {(k)}\right) \} + \sqrt {\frac {1}{2 n} \log \frac {2}{\epsilon_ {k}}} \tag {25}
$$
such that  $\sum_{k\in [l]}\epsilon_k = \epsilon$ . Then, by a union bound with probability at least  $1 - \epsilon$  over the sample none of the events (25) occurs. We have thus constructed a uniform bound over the set of candidate matrices which allows us to select the one which minimizes the generalization gap in (20).
In order to make this strategy most effective, domain knowledge on the application at hand should be applied when constructing candidate matrices and assigning error probabilities. For instance, the limited empirical findings of [14] on ImageNet [17] indicate that the majority of natural images contain mostly local signal. In image segmentation, this is conducive to concentration, because it can lead to many small values in an optimal Wasserstein dependency matrix. In particular, if dependency decays with distance in the image domain, one should select configurations  $L^{(k)}$  in which  $L_{ij}^{(k)}$  is small if  $i$  is distant from  $j$  in the image domain and allowed to assume larger values if  $i$  is close to  $j$  in the image domain.
We give an intuitive interpretation of the relationship between Proposition 8 and Theorem 7 as follows. Suppose the majority of samples from a structured data distribution contain mostly local signal. The locality of signal in samples indicates weak global dependence of random variables which in turn manifests in small entries of a Wasserstein dependency matrix. However, a small number of bad data may contain only weak local signal. For instance, an image in which every pixel has the same value does not give more information to a learner if it is doubled in size. Even worse, a small (but not null) set of bad data will dominate the Wasserstein dependency matrix and prevent generalization that scales with  $d$ . Proposition 8 thus estimates an upper bound on the likelihood of bad data under  $\mu$  which are then excluded from the concentration argument underlying the bound (20). This relationship is further illustrated in a numerical toy example in Appendix C.
We assume that structured data are drawn from a distribution  $\mu$  which arises from a reference measure through KR-rearrangement. This is motivated by the fact that a large class of data distributions can be represented in this way and that some assumption on the data-generating distribution is required in order to make statements on conditional distributions required to quantify dependence. However, it is an open question how closely a measure transport distribution learned from data approximates the unknown distribution from which the data are drawn. A first step towards this goal is made in the recent work [4], but a non-asymptotic theory of generalization for generative models is left for future work. Accordingly, we do not present any empirical results on real-world data.
We describe how dependency in the data distribution which is relevant for generalization of discriminative models can be written as properties of the KR-rearrangement through a Wasserstein dependency matrix. Section 5 further outlines how the construction of bad data can be used as a computational tool to this end. We do not cover how to compute the involved Lipschitz constants  $L_{ij}$  in (16). In practice, this will require nontrivial numerical machinery because a deterministic bound on the expected value under the reference distribution appears needed. One possible approach is to employ quasi-Monte-Carlo methods [19, 18] which come with deterministic error bounds and have recently been applied to PAC-Bayesian certification of ordinary (non-structured) classification risk [7]. This entails a fine-grained stability analysis of the involved integrands and is likely to be computationally expensive because deterministic error bounds tend to be pessimistic compared to their stochastic counterparts.
We have presented a PAC-Bayesian risk certificate for structured prediction. Compared to earlier work, we make the assumption of data generated by KR-rearrangement of a reference measure. This approach is able to represent all atom-free data distributions and yields an explicit quantification of dependence via Wasserstein dependency matrices. It also indicates a way of leveraging powerful generative models to compute risk certificates for downstream discriminative tasks.
This work is funded by the Deutsche Forschungsgemeinschaft (DFG), grant SCHN 457/17-1, within the priority programme SPP 2298: "Theoretical Foundations of Deep Learning". This work is funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany's Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster).
[46] G. Papamakarios, E. Nalisnick, D.J. Rezende, S. Mohamed, and B. Lakshminarayanan. Normalizing Flows for Probabilistic Modeling and Inference. J. Machine Learning Research, 22(57):1-64, 2021.
[47] Maria Perez-Ortiz, Omar Rivasplata, Benjamin Guedj, Matthew Gleeson, Jingyu Zhang, John Shawe-Taylor, Miroslaw Bober, and Josef Kittler. Learning pac-bayes priors for probabilistic neural networks. arXiv preprint arXiv:2109.10304, 2021.
[48] Maria Pérez-Ortiz, Omar Rivasplata, John Shawe-Taylor, and Csaba Szepesvári. Tighter Risk Certificates for Neural Networks. Journal of Machine Learning Research, 22(227):1-40, 2021.
[49] Murray Rosenblatt. Remarks on a multivariate transformation. The annals of mathematical statistics, 23(3):470-472, 1952.
[50] L. Ruthotto and E. Haber. An Introduction to Deep Generative Modeling. GAMM Mitt., 44(2):24 pages, 2021.
[51] John Shawe-Taylor and Robert C Williamson. A pac analysis of a bayesian estimator. In Proceedings of the tenth annual conference on Computational learning theory, pages 2-9, 1997.
[52] Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole. Score-Based Generative Modeling Through Stochastic Differential Equations. In ICLR, 2021.
[53] Esteban G Tabak and Cristina V Turner. A family of nonparametric density estimation algorithms. Communications on Pure and Applied Mathematics, 66(2):145-164, 2013.
[54] Esteban G Tabak and Eric Vanden-Eijnden. Density estimation by dual ascent of the log-likelihood. Communications in Mathematical Sciences, 8(1):217–233, 2010.
[55] Vera Trajkovska, Paul Swoboda, Freddie Åström, and Stefania Petra. Graphical model parameter learning by inverse linear programming. In François Lauze, Yiqiu Dong, and Anders Bjorholm Dahl, editors, Scale Space and Variational Methods in Computer Vision, pages 323–334, Cham, 2017. Springer International Publishing.
[56] Brian L. Trippe and Richard E. Turner. Conditional Density Estimation with Bayesian Normalising Flows, February 2018. arXiv:1802.04908 [stat].
[57] M.J. Wainwright and M.I. Jordan. Graphical Models, Exponential Families, and Variational Inference. Found. Trends Mach. Learn., 1(1-2):1-305, 2008.
A Full Proofs of Presented Results
In this appendix, we present full proofs of all results which are not already complete in the main text.
Proof of Theorem 4 For any  $i\in [d]$  and  $z\in \mathcal{Z}^d$  define
$$
M ^ {(i)} = \mathbb {E} _ {Z \sim \mu} [ f (Z) | \mathcal {B} ^ {c}, Z ^ {[ i ]} = z ^ {[ i ]} ] - \mathbb {E} _ {Z \sim \mu} [ f (Z) | \mathcal {B} ^ {c}, Z ^ {[ i - 1 ]} = z ^ {[ i - 1 ]} ] \tag {26}
$$
with the edge case
$$
M ^ {(1)} = \mathbb {E} _ {Z \sim \mu} [ f (Z) | \mathcal {B} ^ {c}, Z _ {1} = z _ {1} ] - \mathbb {E} _ {Z \sim \mu} [ f (Z) | \mathcal {B} ^ {c} ]. \tag {27}
$$
Due to  $\mathbb{E}_{Z\sim \mu}[f(Z)|\mathcal{B}^c,Z = z] = f(z)$  for  $z\in \mathcal{B}^c$  we have
$$
f - \mathbb {E} _ {Z \sim \mu} [ f (Z) | \mathcal {B} ^ {c} ] = \sum_ {i = 1} ^ {d} M ^ {(i)}. \tag {28}
$$
Since the conditions  $Z^{[i]} = z^{[i]}$  generate a nested sequence of  $\sigma$ -algebras, the quantities  $K^{(i+1)}f(z) = \mathbb{E}_{\mu}[f(Z)|\mathcal{B}^c,Z^{[i]} = z^{[i]}]$  are a Doob martingale and (26) is a martingale difference sequence. In order to bound the moment generating function of  $f$ , we will bound every  $M^{(i)}$  from above and below and apply the Azuma-Hoeffding theorem 10. We have
$$
\begin{array}{l} M ^ {(i)} = \mathbb {E} _ {\mu} [ f (Z) | \mathcal {B} ^ {c}, Z ^ {[ i ]} = z ^ {[ i ]} ] - \mathbb {E} _ {\mu} [ f (Z) | \mathcal {B} ^ {c}, Z ^ {[ i - 1 ]} = z ^ {[ i - 1 ]} ] (29a) \\ = \mathbb {E} _ {\mu} [ f (Z) | \mathcal {B} ^ {c}, Z ^ {[ i ]} = z ^ {[ i ]} ] - \mathbb {E} _ {\mu} [ \mathbb {E} _ {\mu} [ f (Z) | \mathcal {B} ^ {c}, Z ^ {[ i - 1 ]} = z ^ {[ i - 1 ]}, Z _ {i} ] | \mathcal {B} ^ {c}, Z ^ {[ i - 1 ]} = z ^ {[ i - 1 ]} ] (29b) \\ = \int f (z ^ {[ i ]} w ^ {(i, d ]}) \mu (d w ^ {(i, d ]} | z ^ {[ i ]}, \mathcal {B} ^ {c}) \\ - \int \left(\int f \left(z ^ {[ i ]} u ^ {(i, d ]}\right) \mu \left(d u ^ {(i, d ]} \mid z ^ {[ i - 1 ]}, w _ {i}, \mathcal {B} ^ {c}\right)\right) \mu \left(d w ^ {[ i, d ]} \mid z ^ {[ i - 1 ]}, \mathcal {B} ^ {c}\right) (29c) \\ \end{array}
$$
by the tower property of conditional expectations. Because  $\mu (dw^{[i,d]}|z^{[i - 1]},\mathcal{B}^c)$  is a probability measure, it holds
$$
\int f \left(z ^ {[ i ]} w ^ {(i, d ]}\right) \mu \left(d w ^ {(i, d ]} \mid z ^ {[ i ]}, \mathcal {B} ^ {c}\right) = \int \left(\int f \left(z ^ {[ i - 1 ]} z _ {i} u ^ {(i, d ]}\right) \mu \left(d u ^ {(i, d ]} \mid z ^ {[ i ]}, \mathcal {B} ^ {c}\right)\right) \mu \left(d w ^ {[ i, d ]} \mid z ^ {[ i - 1 ]}, \mathcal {B} ^ {c}\right) \tag {30}
$$
and we find
$$
\begin{array}{l} M ^ {(i)} = \int \mu (d w ^ {[ i, d ]} | z ^ {[ i - 1 ]}, \mathcal {B} ^ {c}) \Big (\int f (z ^ {[ i - 1 ]} z _ {i} u ^ {(i, d ]}) \mu (d u ^ {(i, d ]} | z ^ {[ i ]}, \mathcal {B} ^ {c}) \\ \left. - \int f \left(z ^ {[ i ]} u ^ {(i, d ]}\right) \mu \left(d u ^ {(i, d ]} \mid z ^ {[ i - 1 ]}, w _ {i}, \mathcal {B} ^ {c}\right)\right) \tag {31} \\ \end{array}
$$
Now bound  $A^{(i)}\leq M^{(i)}\leq B^{(i)}$  almost surely with
$$
\begin{array}{l} A ^ {(i)} = \int \mu (d w ^ {[ i, d ]} | z ^ {[ i - 1 ]}, \mathcal {B} ^ {c}) \inf  _ {z _ {i} \in \mathcal {B} _ {i} ^ {c} (z ^ {[ i - 1 ]})} \left(\int f (z ^ {[ i - 1 ]} z _ {i} u ^ {(i, d ]}) \mu (d u ^ {(i, d ]} | z ^ {[ i ]}, \mathcal {B} ^ {c}) \right. \\ \left. - \int f \left(z ^ {[ i ]} u ^ {(i, d ]}\right) \mu \left(d u ^ {(i, d ]} \mid z ^ {[ i - 1 ]}, w _ {i}, \mathcal {B} ^ {c}\right)\right) \tag {32a} \\ \end{array}
$$
$$
\begin{array}{l} B ^ {(i)} = \int \mu (d w ^ {[ i, d ]} | z ^ {[ i - 1 ]}, \mathcal {B} ^ {c}) \sup  _ {z _ {i} \in \mathcal {B} _ {i} ^ {c} (z ^ {[ i - 1 ]})} \Big (\int f (z ^ {[ i - 1 ]} z _ {i} u ^ {(i, d ]}) \mu (d u ^ {(i, d ]} | z ^ {[ i ]}, \mathcal {B} ^ {c}) \\ \left. - \int f \left(z ^ {[ i ]} u ^ {(i, d ]}\right) \mu \left(d u ^ {(i, d ]} \mid z ^ {[ i - 1 ]}, w _ {i}, \mathcal {B} ^ {c}\right)\right) \tag {32b} \\ \end{array}
$$
where  $\mathcal{B}_i^c (z^{[i - 1]})$  contains all  $z_{i}\in \mathcal{Z}$  such that there exist  $z^{(i,d]}\in \mathcal{Z}^{d - i}$  with  $(z^{[i - 1]},z_i,z^{(i,d]})\in \mathcal{B}^c$ . Because every realization of a random variable conditioned on  $\mathcal{B}^c$  is in the set of good inputs, the difference  $\| B^{(i)} - A^{(i)}\|_{\infty}$  can be written as
$$
\sup  _ {v, z \in \mathcal {B} ^ {c}, v ^ {[ d ] \backslash \{i \}} = z ^ {[ d ] \backslash \{i \}}} \int f (v ^ {[ i ]} u ^ {(i, d ])} \mu (d u ^ {(i, d ]} | v ^ {[ i ]}, \mathcal {B} ^ {c}) - \int f (z ^ {[ i ]} u ^ {(i, d ])} \mu (d u ^ {(i, d ]} | z ^ {[ i ]}, \mathcal {B} ^ {c}) \tag {33}
$$
By seeing this expression in terms of oscillation of the kernel action  $K^{(i + 1)}f$ , we find
$$
\left\| B ^ {(i)} - A ^ {(i)} \right\| _ {\infty} \leq \| \rho \| \delta_ {i} \left(K ^ {(i + 1)} \widetilde {f}\right) \leq \| \rho \| \left(V ^ {(i + 1)} \delta (\widetilde {f})\right) _ {i} = (\Gamma \delta (\widetilde {f})) _ {i} \tag {34}
$$
where  $\widetilde{f}\colon \mathcal{B}^c\to \mathbb{R}$  is the restriction of  $f$  to  $\mathcal{B}^c$ . The assertion then follows from the Azuma-Hoeffding theorem [35, Theorem 4.1] which we recite as Theorem 10 to make this paper self-contained.
Proof of Proposition 6 For arbitrary  $z, z' \in \mathcal{Z}^d$  it holds
$$
\left| f (z) - f \left(z ^ {\prime}\right) \right| \leq \delta_ {j} (f) \rho \left(z _ {j}, z _ {j} ^ {\prime}\right), \quad \forall i \in [ d ] \tag {35}
$$
and thus, by summing over all indices we get
$$
\left| f (z) - f \left(z ^ {\prime}\right) \right| \leq \frac {1}{d} \sum_ {j \in [ d ]} \delta_ {j} (f) \rho \left(z _ {j}, z _ {j} ^ {\prime}\right) \tag {36}
$$