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+# Generalized Sobolev Transport for Probability Measures on a Graph
+
+Tam Le  $^{*1,2}$  Truyen Nguyen  $^{*3}$  Kenji Fukumizu
+
+# Abstract
+
+We study the optimal transport (OT) problem for measures supported on a graph metric space. Recently, Le et al. (2022) leverage the graph structure and propose a variant of OT, namely Sobolev transport (ST), which yields a closed-form expression for a fast computation. However, ST is essentially coupled with the  $L^p$  geometric structure within its definition which makes it nontrivial to utilize ST for other prior structures. In contrast, the classic OT has the flexibility to adapt to various geometric structures by modifying the underlying cost function. An important instance is the Orlicz-Wasserstein (OW) which moves beyond the  $L^p$  structure by leveraging the Orlicz geometric structure. Comparing to the usage of standard  $p$ -order Wasserstein, OW remarkably helps to advance certain machine learning approaches. Nevertheless, OW brings up a new challenge on its computation due to its two-level optimization formulation. In this work, we leverage a specific class of convex functions for Orlicz structure to propose the generalized Sobolev transport (GST). GST encompasses the ST as its special case, and can be utilized for prior structures beyond the  $L^p$  geometry. In connection with the OW, we show that one only needs to simply solve a univariate optimization problem to compute the GST, unlike the complex two-level optimization problem in OW. We empirically illustrate that GST is several-order faster than the OW. Moreover, we provide preliminary evidences on the advantages of GST for document classification and for several tasks in topological data analysis.
+
+# 1. Introduction
+
+Optimal transport (OT) is a natural geometry to compare probability measures (Villani, 2008; Peyre & Cuturi, 2019). Intuitively, OT lifts the cost metric on support data points of input measures to the distance for the measures by minimizing the cost to transport a measure into the other. OT has been applied on many applications in various research fields, e.g., in machine learning (Nadjahi et al., 2019; Titouan et al., 2019; Janati et al., 2020; Mukherjee et al., 2021; Altschuler et al., 2021; Fatras et al., 2021; Scutbon et al., 2021; Le & Nguyen, 2021; Le et al., 2021a;b; Liu et al., 2021; Nguyen et al., 2021a;b; Takezawa et al., 2022; Fan et al., 2022; Bunne et al., 2022; 2023a; Bonet et al., 2023; Mahey et al., 2023; Nguyen et al., 2023a;b; Hua et al., 2023; Korotin et al., 2023; 2024; Le et al., 2024; Nguyen et al., 2024), statistics (Mena & Niles-Weed, 2019; Weed & Berthet, 2019; Liu et al., 2022; Nguyen et al., 2022; Wang et al., 2022; Nietert et al., 2022; 2023; Pham et al., 2024), and biology (Schiebinger et al., 2019; Tong et al., 2020; Bunne et al., 2023a;b; Korotin et al., 2024).
+
+A main drawback of OT is that its computational complexity is high, i.e., super cubic with respect to the number of supports of input measures, which hinders its applications in large-scale applications. Recently, several approaches have been proposed in the literature to scale up the OT problem. Cuturi (2013) proposes to use entropic regularization for OT and leverages the Sinkhorn algorithm to reduce its complexity into quadratic. Scetbon et al. (2021) propose low-rank approach to further reduce the computation of Sinkhorn algorithm for entropic regularized OT. Another direction is to exploit the local structure on supports of input measures. Rabin et al. (2011) propose sliced-Wasserstein (SW) which projects supports of input measures into a randomly one-dimensional space and utilizes the closed-form computation of the univariate OT. However, by relying on the one-dimensional projection, SW limits its capacity to capture the topological structure of input measures, especially when measures are supported in a high-dimensional space. Le et al. (2019) propose to leverage tree structure which provides more flexibility and degrees of freedom, i.e., choosing a tree rather than a line as in SW, to alleviate the curse of dimensionality in SW. For practical applications, the tree structure might be a restricted requirement. Recently, Le et al. (2022) propose a variant of OT, namely
+
+Sobolev transport (ST), for measures supported on a graph which admits a closed-form expression for fast computation.
+
+Unlike the classic OT where one can easily utilize various prior geometric structures on supports of input measures, e.g., via the ground cost metric, it is nontrivial to use other structures rather than the  $L^p$  structure for the ST. Essentially, the definition of ST is based on the Kantorovich duality of 1-order Wasserstein, but ST considers the graph-based Sobolev constraint for the critic function. This makes the ST coupled with the  $L^p$  functional geometric structure within the graph-based Sobolev space.
+
+One approach to move beyond the  $L^p$  geometry is to employ the Orlicz geometric structure, which is obtained by leveraging a specific class of convex functions. Indeed, Orlicz norm has been utilized to advance several machine learning approaches in the literature. In particular, Andoni et al. (2018) and Song et al. (2019) leverage the Orlicz norm as a loss for the classic linear regression. Orlicz norm loss regression not only generalizes both the least squared regression (i.e., squared  $\ell_2$  norm loss) and the least absolute deviation regression (i.e.,  $\ell_1$  norm loss), but also provides a scale-invariant version of  $\mathbb{M}$ -estimator function loss (e.g., the Huber, Cauchy, Welsh, Tukey, Geman-McClure functions (Zhang, 1997, Table 1)). Moreover, Andoni et al. (2018) illustrate that the Orlicz norm loss regression improves performances of linear regression with  $\ell_1 / \ell_2$  norms on Gaussian/sparse noise respectively. In addition, Deng et al. (2022) propose an efficient data structure for the Orlicz norm as its special case, which helps to scale up many machine learning problems including reinforcement learning, kernelized support vector machine, and clustering. Notably, in order to address the challenging polynomial growing nature of the underlying function class in the empirical process for random Fourier features on approximation of high-order kernel derivatives, Chamakh et al. (2020) leverage the Orlicz metric on the sample distribution to derive a finite-sample deviation bound for a general class of polynomial-growth functions. In particular, Chamakh et al. (2020) propose finite-sample uniform guarantee for random Fourier features which is almost surely convergence to approximate high-order derivatives for arbitrary kernel.
+
+Furthermore, Orlicz metric has been recently explored in the context of OT. In particular, Lorenz & Mahler (2022) propose to leverage Orlicz norm as a regularization for OT problem with continuous probability measures. Especially, Orlicz metric is also utilized as a ground cost for OT, which is known as the Orlicz-Wasserstein (OW) (Sturm, 2011; Kell, 2017). Notably, Guha et al. (2023) and Altschuler & Chewi (2023) have shown that OW possesses unique
+
+implicit essences, which may not exist in the classic OT with  $L^p$  structure cost, and help to advance certain results in machine learning. In particular, Guha et al. (2023) propose to consider OW metric for studying the Bayesian contraction convergence behavior of parameters arising from hierarchical Bayesian nonparametric models. OW metric helps to alleviate a number of raised concerns caused from the usage of the classic OT with Euclidean cost for quantifying the rates of parameter convergence within infinite Gaussian mixtures to significantly improve the contraction rate. Additionally, Altschuler & Chewi (2023) propose to also leverage OW as a metric shift for Rényi divergence to develop novel differential-privacy-inspired techniques to overcome longstanding challenges for proving fast convergence of hypocoercive differential equations. In spite of these achievements, OW brings up a difficult challenge on its computation. Precisely, OW is a two-level optimization problem: one level is for the transportation plan as in the classic OT problem, and another level is for an extra positive scalar within the Orlicz metric structure.
+
+In this work, we focus on OT problem for probability measures supported on a graph metric space (Le et al., 2022). On one hand, ST efficiently exploits the graph structure to yield a closed-form for a fast computation. However, ST is essentially coupled with the  $L^p$  geometric structure, and it is therefore nontrivial to utilize ST with other prior structures. On the other hand, OW metric owns unique implicit essences to improve certain machine learning problems, but its optimization formulation is challenging for computation. Our goal is to alleviate these issues by adopting Orlicz geometry to generalize the ST for measures on a graph. This allows us to propose the generalized Sobolev transport (GST) which inherit the merits from both ST and OW.
+
+Contribution. In summary, our contributions are three-fold:
+
+- We leverage a certain class of convex functions corresponding to Orlicz geometric structure, and propose the GST for probability measures supported on a graph metric space. We show that GST can be computed by simply solving a univariate optimization problem.  
+- We demonstrate that ST is a special case of the proposed GST. Additionally, GST utilizes the Orlicz geometric structure in the same sense as the OW for OT problem. We further draw a connection between GST and OW.  
+- We empirically illustrate that GST is more computationally efficient than OW. We also show some preliminary evidences on the advantages of GST for document classification and for several tasks in topological data analysis (TDA).
+
+Organization. We briefly review related notations used in
+
+the development of our proposals in §2. In §3, we describe our proposed approach, the generalized Sobolev transport (GST). We show that GST is a metric, and then derive its properties and draw its connection to ST and OW in §4. In §5, we empirically illustrate the computational advantages of GST over OW, and show preliminary improvements of GST on document classification and TDA. In §6, we give concluding remarks. Additionally, we defer the proofs of key theoretical results and additional materials to the Appendices. We have also released code for our proposed approach. $^2$
+
+# 2. Preliminaries
+
+In this section, we introduce the notations and give a brief review about graphs together with the Sobolev transport (ST) for measures on a graph, as well as the Orlicz geometric structure and the Orlicz-Wasserstein (OW).
+
+# 2.1. Graph and functions on graph
+
+We describe the graph setting for measures, and functions on graph.
+
+Graph. We use the same graph setting as in (Le et al., 2022). Specifically, let  $V$  and  $E$  be respectively the sets of nodes and edges. We consider a connected, undirected, and physical $^3$  graph  $\mathbb{G} = (V,E)$  with positive edge lengths  $\{w_e\}_{e \in E}$ . Following the convention in (Le et al., 2022) for continuous graph setting,  $\mathbb{G}$  is regarded as the set of all nodes in  $V$  together with all points forming the edges in  $E$ . Also,  $\mathbb{G}$  is equipped with the graph metric  $d_{\mathbb{G}}(x,y)$  which equals to the length of the shortest path in  $\mathbb{G}$  between  $x$  and  $y$ . Additionally, we assume that there exists a fixed root node  $z_0 \in V$  such that the shortest path connecting  $z_0$  and  $x$  is unique for any  $x \in \mathbb{G}$ , i.e., the uniqueness property of the shortest paths (Le et al., 2022).
+
+Let  $[x,z]$  denote the shortest path connecting  $x$  and  $z$  in  $\mathbb{G}$ . For  $x\in \mathbb{G}$ , edge  $e\in E$ , define the sets  $\Lambda (x)$ ,  $\gamma_{e}$  as follows:
+
+$$
+\Lambda (x) := \left\{y \in \mathbb {G}: x \in [ z _ {0}, y ] \right\},
+$$
+
+$$
+\gamma_ {e} := \left\{y \in \mathbb {G}: e \subset \left[ z _ {0}, y \right] \right\}. \tag {1}
+$$
+
+Denote  $\mathcal{P}(\mathbb{G})$  (resp.  $\mathcal{P}(\mathbb{G}\times \mathbb{G})$ ) as the set of all nonnegative Borel measures on  $\mathbb{G}$  (resp.  $\mathbb{G}\times \mathbb{G}$ ) with a finite mass.
+
+Functions on graph. By a continuous function  $f$  on  $\mathbb{G}$ , we mean that  $f: \mathbb{G} \to \mathbb{R}$  is continuous w.r.t. the topology on  $\mathbb{G}$  induced by the Euclidean distance. Henceforth,  $C(\mathbb{G})$  denotes the set of all continuous functions on  $\mathbb{G}$ . Similar
+
+notation is used for continuous functions on  $\mathbb{G}\times \mathbb{G}$
+
+Given a scalar  $b > 0$ , then a function  $f: \mathbb{G} \to \mathbb{R}$  is called  $b$ -Lipschitz w.r.t. the graph metric  $d_{\mathbb{G}}$  if
+
+$$
+| f (x) - f (y) | \leq b d _ {\mathbb {G}} (x, y), \forall x, y \in \mathbb {G}.
+$$
+
+# 2.2. Sobolev transport (ST)
+
+For probability measures on a graph, Le et al. (2022) propose the ST which is a scalable variant of OT. More specifically, Le et al. (2022) leverage the graph structure and propose ST by building upon the dual form of the 1-order Wasserstein distance, but considering its Lipschitz constraint for the critic function within the graph-based Sobolev space.
+
+In particular, let  $\omega$  be a nonnegative Borel measure on  $\mathbb{G}$ . Given  $1 \leq p \leq \infty$ , and let  $p'$  be its conjugate, i.e., the number  $p' \in [1, \infty]$  satisfying  $\frac{1}{p} + \frac{1}{p'} = 1$ . For  $\mu, \nu \in \mathcal{P}(\mathbb{G})$ , the  $p$ -order Sobolev transport (ST) (Le et al., 2022, Definition 3.2) is defined as
+
+$$
+\mathcal {S} _ {p} (\mu , \nu) := \left\{ \begin{array}{l} \sup  \left[ \int_ {\mathbb {G}} f (x) \mu (\mathrm {d} x) - \int_ {\mathbb {G}} f (x) \nu (\mathrm {d} x) \right] \\ \text {s . t .} f \in W ^ {1, p ^ {\prime}} (\mathbb {G}, \omega), \| f ^ {\prime} \| _ {L ^ {p ^ {\prime}} (\mathbb {G}, \omega)} \leq 1, \end{array} \right. \tag {2}
+$$
+
+where we write  $f'$  for the generalized graph derivative of  $f$ ,  $W^{1,p'}(\mathbb{G},\omega)$  for the graph-based Sobolev space on  $\mathbb{G}$ , and  $L^{p'}(\mathbb{G},\omega)$  for the  $L^p$  functional space on  $\mathbb{G}$ . Notably, the ST yields a closed-form expression for a fast computation (Le et al., 2022, Proposition 3.5).
+
+Additionally, when graph  $\mathbb{G}$  is a tree and  $\omega$  is a length measure<sup>5</sup>, the 1-order Sobolev transport coincides with the 1-order Wasserstein distance (Le et al., 2022, Corollary 4.3).
+
+However, unlike standard OT where one can easily adjust the ground cost following prior structure for applications, it is nontrivial for such adaptation in the ST due to its graph structure and the generalized graph derivation within the graph-based Sobolev space.
+
+# 2.3. Orlicz functional space and Orlicz-Wasserstein
+
+We describe a family of convex functions which we leverage to generalize the ST, and briefly review the Orlicz functional geometric structure, as well as the OW which utilizes Orlicz norm as its ground cost.
+
+A family of convex functions. We consider the collection of  $N$ -functions (Adams & Fournier, 2003, §8.2) which are special convex functions on  $\mathbb{R}_+$ . Hereafter, a strictly increasing and convex function  $\Phi : [0, \infty) \to [0, \infty)$  is called an  $N$ -function if  $\lim_{t \to 0} \frac{\Phi(t)}{t} = 0$  and  $\lim_{t \to +\infty} \frac{\Phi(t)}{t} = +\infty$ .
+
+Examples. Some popular examples for  $N$ -functions are (i)  $\Phi(t) = t^p$  with  $1 < p < \infty$ ; (ii)  $\Phi(t) = \exp(t) - t - 1$ ; (iii)  $\Phi(t) = \exp(t^p) - 1$  with  $1 < p < \infty$ ; and (iv)  $\Phi(t) = (1 + t)\log(1 + t) - t$  (Adams & Fournier, 2003, §8.2).
+
+Orlicz functional space. Given an  $N$ -function  $\Phi$  and a nonnegative Borel measure  $\omega$  on  $\mathbb{G}$ , let  $L_{\Phi}(\mathbb{G},\omega)$  be the linear hull of the set of all Borel measurable functions  $f:\mathbb{G}\to \mathbb{R}$  satisfying  $\int_{\mathbb{G}}\Phi (|f(x)|)\omega (\mathrm{d}x) < \infty$ . Then,  $L_{\Phi}(\mathbb{G},\omega)$  is a normed space with the Luxemburg norm being defined by
+
+$$
+\left\| f \right\| _ {L _ {\Phi}} := \inf  \left\{t > 0 \mid \int_ {\mathbb {G}} \Phi \left(\frac {| f (x) |}{t}\right) \omega (\mathrm {d} x) \leq 1 \right\}. \tag {3}
+$$
+
+The infimum in Equation (3) for  $\| f \|_{L_{\Phi}}$  is attained (Adams & Fournier, 2003, §8.9).
+
+Orlicz-Wasserstein (OW). Following Guha et al. (2023, Definition 3.2), the OW with the  $N$ -function  $\Phi$  for measures  $\mu, \nu \in \mathcal{P}(\mathbb{G})$  is defined as follows:
+
+$$
+\begin{array}{l} W _ {\Phi} (\mu , \nu) = \inf  _ {\pi \in \Pi (\mu , \nu)} \inf  \left[ t > 0: \right. \\ \int_ {\mathbb {G} \times \mathbb {G}} \Phi \left(\frac {d _ {\mathbb {G}} (x , z)}{t}\right) \mathrm {d} \pi (x, z) \leq 1 ], \tag {4} \\ \end{array}
+$$
+
+where  $\Pi (\mu ,\nu)$  is the set of all couplings between  $\mu$  and  $\nu$
+
+In this work, we propose to leverage the collection of  $N$ -functions to generalize the ST, which can adopt Orlicz geometric structure in the same sense as the OW. Therefore, the proposed generalized Sobolev transport can inherit advantages from both ST and OW.
+
+# 3. Generalized Sobolev Transport (GST)
+
+In this section, we examine a special family of convex functions which grows faster than linear. By leveraging it, we generalize the ST in Equation (2) by proposing the graph-based Orlicz-Sobolev space for the Lipschitz constraint on the critic function for measures on a graph.
+
+# Graph-based Orlicz-Sobolev space.
+
+Definition 3.1 (Graph-based Orlicz-Sobolev space). Let  $\Phi$  be an  $N$ -function and  $\omega$  be a nonnegative Borel measure on graph  $\mathbb{G}$ . A continuous function  $f: \mathbb{G} \to \mathbb{R}$  is said to belong to the graph-based Orlicz-Sobolev space  $WL_{\Phi}(\mathbb{G}, \omega)$  if there exists a function  $h \in L_{\Phi}(\mathbb{G}, \omega)$  satisfying
+
+$$
+f (x) - f \left(z _ {0}\right) = \int_ {\left[ z _ {0}, x \right]} h (y) \omega (\mathrm {d} y), \quad \forall x \in \mathbb {G}. \tag {5}
+$$
+
+Such function  $h$  is unique in  $L_{\Phi}(\mathbb{G}, \omega)$  and is called the generalized graph derivative of  $f$  w.r.t. the measure  $\omega$ . Henceforth, this generalized graph derivative of  $f$  is denoted  $f'$ .
+
+We remark that the Orlicz-Sobolev space  $WL_{\Phi}(\mathbb{G}, \omega)$  depends on the point  $z_0$  and the choice of the  $N$ -function  $\Phi$ .
+
+For brevity, we however do not explicitly display these dependencies in its notation. Additionally, the formulation in Equation (5) can be considered as a generalized version of the fundamental theorem of calculus, which defines the generalized graph derivative for a continuous function  $f$  at any point  $x \in \mathbb{G}$ . Moreover, the graph-based Orlicz-Sobolev space may be regarded as a generalized version of the graph-based Sobolev space (analyzed rigorously in §4).
+
+Unlike the OW which directly utilizes the given  $N$ -function within the Luxemburg norm of the Orlicz functional space (Equation (4)), in order to generalize the ST, we also require a notion of the complementary function of the given  $N$ -function.
+
+Complementary function. For the given  $N$ -function  $\Phi$ , its complementary function  $\Psi: \mathbb{R}_+ \to \mathbb{R}_+$  (Adams & Fournier, 2003, §8.3) is the  $N$ -function, defined as follows
+
+$$
+\Psi (t) = \sup  [ a t - \Phi (a) \mid a \geq 0 ], \quad \text {f o r} t \geq 0. \tag {6}
+$$
+
+Examples. Some popular complementary pairs of  $N$ -functions (Adams & Fournier, 2003, §8.3), (Rao & Ren, 1991, §2.2) are: (i)  $\Phi(t) = \frac{t^p}{p}$  and  $\Psi(t) = \frac{t^q}{q}$  where  $q$  is the conjugate of  $p$ , i.e.,  $\frac{1}{p} + \frac{1}{q} = 1$  and  $1 < p < \infty$ , and (ii)  $\Phi(t) = \exp(t) - t - 1$  and  $\Psi(t) = (1 + t)\log(1 + t) - t$ . Additionally, for the  $N$ -function  $\Phi(t) = \exp(t^p) - 1$  with  $1 < p < \infty$ , its complementary  $N$ -function admits an explicit expression, but not simple (Rao & Ren, 1991, §2.2).<sup>6</sup>
+
+Inspired by the ST for measures on a graph (Le et al., 2022), we exploit the dual formulation of the 1-order Wasserstein distance to propose the generalized Sobolev transport (GST). More precisely, we replace the Lipschitz constraint for the critic function by a constraint involving the graph-based Orlicz-Sobolev space.
+
+Definition 3.2 (Generalized Sobolev transport distance on graph). Let  $\Phi$  be an  $N$ -function and  $\omega$  be a nonnegative Borel measure on  $\mathbb{G}$ . For  $\mu, \nu \in \mathcal{P}(\mathbb{G})$ , we define
+
+$$
+\mathcal {G S} _ {\Phi} (\mu , \nu) := \left\{ \begin{array}{l l} \sup  & \Big | \int_ {\mathbb {G}} f (x) \mu (\mathrm {d} x) - \int_ {\mathbb {G}} f (x) \nu (\mathrm {d} x) \Big | \\ \text {s . t .} & f \in W L _ {\Psi} (\mathbb {G}, \omega), \| f ^ {\prime} \| _ {L _ {\Psi}} \leq 1, \end{array} \right.
+$$
+
+where  $\Psi$  is the complementary function of  $\Phi$  (see (6)).
+
+The Definition 3.2 implies that the GST for probability measures supported on a graph metric space is an instance of the integral probability metric (Müller, 1997).
+
+Computation. We next show that one can compute  $\mathcal{GS}_{\Phi}$  by simply solving a univariate optimization problem.
+
+Theorem 3.3 (GST as univariate optimization problem). The generalized Sobolev transport  $\mathcal{G}\mathcal{S}_{\Phi}(\mu ,\nu)$  in Defini
+
+tion 3.2 can be computed as follows:
+
+$$
+\mathcal {G} \mathcal {S} _ {\Phi} (\mu , \nu) = \inf  _ {k > 0} \frac {1}{k} \left(1 + \int_ {\mathbb {G}} \Phi (k | h (x) |) \omega (d x)\right), \tag {7}
+$$
+
+where  $h(x)\coloneqq \mu (\Lambda (x)) - \nu (\Lambda (x))$  for all  $x\in \mathbb{G}$
+
+The proof is placed in Appendix  $\S A.1$
+
+We next derive the discrete case for the GST in Equation (7).
+
+Corollary 3.4 (Discrete case). Assume that  $\omega(\{x\}) = 0$  for every  $x \in \mathbb{G}$ , and suppose that  $\mu, \nu \in \mathcal{P}(\mathbb{G})$  are supported on nodes in  $V$  of graph  $\mathbb{G}$ . Then, we have
+
+$$
+\mathcal {G S} _ {\Phi} (\mu , \nu) = \inf  _ {k > 0} \frac {1}{k} \left[ 1 + \sum_ {e \in E} w _ {e} \Phi (k | \bar {h} (e) |) \right], \tag {8}
+$$
+
+where  $\bar{h}(e) \coloneqq \mu(\gamma_e) - \nu(\gamma_e)$  for every edge  $e \in E$ .
+
+The proof is placed in Appendix §A.2. $^{8}$
+
+From Corollary 3.4, one only needs to simply solve the univariate optimization problem to compute the GST.
+
+Remark 3.5 (GST for non-physical graph). Similar to the ST, we have assumed that  $\mathbb{G}$  is a physical graph in §2.1. However, Corollary 3.4 implies that the GST  $\mathcal{G}S_{\Phi}$  does not depend on the physical assumption when input measures are supported on graph nodes. In particular, it only depends on the graph structure  $(V,E)$  and edge weights  $w_{e}$ . Thus, we can apply the GST for non-physical graph  $\mathbb{G}$ .
+
+Remark 3.6 (Complementary pairs of  $N$ -functions for GST.). The Definition 3.2 for GST with the  $N$ -function  $\Phi$  involves its complementary  $N$ -function  $\Psi$ . However, we can reformulate GST as a univariate optimization problem without involving the complementary function  $\Psi$  as in Equation (7). Note that in order to obtain the univariate optimization formulation, the complementary function  $\Psi$  is finite-valued, which is satisfied by any  $N$ -function  $\Phi$ , i.e., growing faster than linear (see (6)).
+
+Preprocessing. For the computation of GST in Equation (8), observe that set  $\gamma_{e}$  (see Equation (1)) can be precomputed for all edge  $e$  in  $\mathbb{G}$ . This preprocessing step only involves the graph  $\mathbb{G}$  itself, and is not related to input measures. Moreover, we only need to precompute it once, regardless the number of input pairs of probability measures to be measured by the GST. In particular, we recompute the shortest paths from the root node  $z_{0}$  to all other input supports (or vertices) by Dijkstra algorithm with the complexity  $\mathcal{O}(|E| + |V|\log |V|)$ , where  $|\cdot|$  is the cardinality of a set.
+
+Remark 3.7 (Sparsity in Problem (8)). Let  $\operatorname{supp}(\mu)$  be the set of supports of  $\mu$ . Observe that for any support  $x \in$
+
+$\operatorname{supp}(\mu)$ , its mass is accumulated into  $\mu(\gamma_e)$  if and only if  $e \subset [z_0, x]$ . Let define set  $E_{\mu,\nu} \subset E$  as
+
+$$
+E _ {\mu , \nu} := \left\{e \in E \mid \exists z \in \left(\operatorname {s u p p} (\mu) \cup \operatorname {s u p p} (\nu)\right), e \subset [ z _ {0}, z ] \right\}.
+$$
+
+Then in (8), we in fact only need to take the summation over all edges  $e \in E_{\mu, \nu}$ , i.e., removing all edges  $e \in E \setminus E_{\mu, \nu}$ .
+
+# 4. Properties of the GST
+
+In this section, we derive the metric property for the GST and establish a relationship for the GST with different  $N$ -functions. Additionally, we draw connections of the GST to the ST, the OW, and OT.
+
+Theorem 4.1 (Metrization). The generalized Sobolev transport  $\mathcal{GS}_{\Phi}(\mu, \nu)$  is a metric on the space  $\mathcal{P}(\mathbb{G})$ .
+
+The proof is placed in Appendix  $\S A.3$
+
+The GST is monotone with respect to the  $N$ -function  $\Phi$  as shown in the next result. Consequently, it may enclose a stronger notion of metrics than the ST for comparing measures on a graph.
+
+Proposition 4.2 (GST with different  $N$ -functions). For any two  $N$ -functions  $\Phi_1, \Phi_2$  satisfying  $\Phi_1(t) \leq \Phi_2(t)$  for all  $t \in \mathbb{R}_+$ , we have
+
+$$
+\mathcal {G S} _ {\Phi_ {1}} (\mu , \nu) \leq \mathcal {G S} _ {\Phi_ {2}} (\mu , \nu) f o r e v e r y \mu , \nu \in \mathcal {P} (\mathbb {G}).
+$$
+
+The proof is placed in Appendix  $\S A.4$
+
+Connection with Sobolev transport. We next show the connection between the graph-based Orlicz-Sobolev space and the graph-based Sobolev space (Le et al., 2022).
+
+Proposition 4.3. Let  $1 < p < \infty$  and  $\omega$  be a nonnegative Borel measure on graph  $\mathbb{G}$ . By taking  $\Phi(t) \coloneqq t^p$  for the  $N$ -function, then the associated graph-based Sobolev-Orlicz space is the same as the graph-based Sobolev space of order  $p$ , denoted as  $W^{1,p}(\mathbb{G},\omega)$ .<sup>9</sup> That is,
+
+$$
+W L _ {\Phi} (\mathbb {G}, \omega) = W ^ {1, p} (\mathbb {G}, \omega).
+$$
+
+The proof is placed in Appendix  $\S A.5$
+
+By leveraging the result in Proposition 4.3, we next derive the closed-form expression for the univariate optimization problem in (7) for the GST with certain  $N$ -functions. This allows us to draw its connection to the ST (Le et al., 2022).
+
+Proposition 4.4 (Connection between GST and ST). Let  $\Phi(t) \coloneqq \frac{(p - 1)^{p - 1}}{p^p} t^p$  with  $1 < p < \infty$ , and let  $\omega$  be a nonnegative Borel measure on graph  $\mathbb{G}$ . Then, the GST admits the following closed-form expression:
+
+$$
+\mathcal {G} \mathcal {S} _ {\Phi} (\mu , \nu) = \left(\int_ {\mathbb {G}} | \mu (\Lambda (x)) - \nu (\Lambda (x)) | ^ {p} \omega (d x)\right) ^ {\frac {1}{p}}, \tag {9}
+$$
+
+for all measures  $\mu, \nu \in \mathcal{P}(\mathbb{G})$ . Consequently, we have
+
+$$
+\mathcal {G} \mathcal {S} _ {\Phi} (\mu , \nu) = \mathcal {S} _ {p} (\mu , \nu).
+$$
+
+The proof is placed in Appendix  $\S A.6$
+
+Connection with Orlicz-Wasserstein and OT. In the special case  $\Phi(t) = t^p$  with  $p \geq 1$ , the OW coincides with the  $p$ -order Wasserstein (Guha et al., 2023).<sup>10</sup> It is worth noting that OW in Equation (4) is a two-level optimization problem involving the transportation plan and the positive scalar  $t$ . Thus, it is more challenging to compute the OW than the classic OT (e.g., with squared Euclidean cost).
+
+Remark 4.5. Following Guha et al. (2023), the OW is equal to the 1-order Wasserstein if we take  $\Phi(t) = t^p$  and then take the limit  $p \to 1^+$ . In the similar way and by taking the limit  $p \to 1^+$ , the closed-form GST in Equation (9) is equal to the 1-order ST. With an additional assumption that graph  $\mathbb{G}$  is a tree, it follows from Le et al. (2022) that the 1-order ST is in turn equal to the 1-order Wasserstein. Thus, the GST coincides with the OW when  $\Phi(t) = t$  and the graph  $\mathbb{G}$  is a tree.
+
+Due to Remark 4.5, the GST can be regarded as a variant of the OW. We summarize it in the following Proposition 4.6.
+
+Proposition 4.6 (Connection between GST and OW). When  $\Phi(t) = t$  and the graph  $\mathbb{G}$  is a tree, the generalized Sobolev transport and Orlicz-Wasserstein coincide, i.e.,
+
+$$
+\mathcal {G S} _ {\Phi} (\mu , \nu) = W _ {\Phi} (\mu , \nu),
+$$
+
+for all measures  $\mu, \nu \in \mathcal{P}(\mathbb{G})$ .
+
+Notice that our obtained results show that one only needs to solve a univariate optimization problem to compute the GST. On the other hand, it is much more challenging to solve the two-level optimization problem (4) for the OW. Notably, a recent realization is that solving exactly OT problems leads to overfitting (Peyre & Cuturi, 2019, §8.4). Therefore, it would be self-defeating to give excessive efforts to optimize the OT problems since it would lead to overfitting within the computation of the OT problems themselves. Consequently, GST can be regarded as a regularization approach for OW.
+
+# 5. Experiments
+
+In this section, we illustrate that the computation of OW for general  $N$ -function  $\Phi$  is very expensive, and the GST is several-order faster than OW. Then, we evaluate the GST
+
+and show preliminary evidences on its advantages for document classification and for several tasks in TDA.
+
+![](images/7ac88ad01bf7707c41ef8bb4ecbeb943e44b329199cba183dbdbafc85bee0e53.jpg)  
+Figure 1. Time consumption for GST and OW on  $\mathbb{G}_{\mathrm{Log}}$
+
+![](images/9d57b58b01fa18a814976b84cb5e0e45a4ebc0a0d4aa2619540061222ab68090.jpg)  
+Figure 2. Time consumption for GST and OW on  $\mathbb{G}_{\mathrm{Sqrt}}$
+
+Document classification. We consider 4 traditional document datasets: TWITTER, RECIPE, CLASSIC, AMAZON. The characteristic properties of these datasets are summarized in Figure 3. Following Le et al. (2022), we use word embedding, e.g., word2vec (Mikolov et al., 2013) pretrained on Google News for documents, to map words into vectors in  $\mathbb{R}^{300}$ . Additionally, we remove SMART stop words (Salton & Buckley, 1988), and words in documents which are not in the pretrained word2vec. We then regard each document as a probability measure by considering each word in the document as its support in  $\mathbb{R}^{300}$ , and using word frequency as the mass on that corresponding word.
+
+TDA. We consider two tasks: orbit recognition on the synthesis Orbit dataset (Adams et al., 2017), and object shape classification on MPEG7 dataset (Latecki et al., 2000) as in Le et al. (2022). The characteristic properties of these datasets are summarized in Figure 5. We utilize the persistence diagrams (PD) to represent objects of interest for these tasks. Particularly, PD is a multiset of points in  $\mathbb{R}^2$ , where each point summarizes a life span (i.e., birth and death time) of a particular topological feature (e.g., connected component, ring, or cavity), extracted by algebraic topology methods (e.g., persistence homology) (Edelsbrunner & Harer, 2008). We consider each PD as a probability measure of the 2-dimensional topological feature data points with a uniform mass.
+
+![](images/976c58a08b20005e3229ec3f3bfc2f6223140fd1da5ec40347a00d0bfd76ee10.jpg)
+
+![](images/81e72f819fca2d2f90a8bb4032493b4225f8c3147c13280f4153659ab8dd956a.jpg)
+
+![](images/0cccf78da37425bd7c52477b00d0c5d05407d8a2e96131f7033eba3caa566f1a.jpg)
+
+![](images/4775a9d41c7fc9d84763463114f6a1951e3c62bc2f2f0b1dc0bb3c36cbe037ed.jpg)
+
+![](images/f75dfdf49e67ce5963f9e3c07a3edfbec8c2b5d56724a0e37025147dc84161c6.jpg)  
+Figure 3. Document classification on graph  $\mathbb{G}_{\mathrm{Log}}$ . For each dataset, the numbers in the parenthesis are respectively the number of classes; the number of documents; and the maximum number of unique words for each document.
+
+![](images/438dbfeeac36d543cb15446b0c804652ca37737ab5cdfe88a7889b98a0a79c55.jpg)
+
+![](images/403bf8e3455a895021a48b360b8e5a7d3f285458a32d01d3e5f2a080174daf09.jpg)
+
+![](images/2a6aa5ce6588d7440ae8f05a61194a4da2808372206fa181a276a29900fa1416.jpg)
+
+![](images/118780abd109a04e11a0d2addc72acc791c8fedaa7b8ef2bf4bff556dc9bcd36.jpg)
+
+![](images/47c6b8719c7c947bd68f87c1926a42fecf645be997d6ab959ae71581eb59c542.jpg)
+
+![](images/63fb320dafafe55eee7ca0924db322c6f1a6c2c575e084bb93fb0affcc0bf2bc.jpg)
+
+![](images/0b29f4ada0644be1820bad5bacd8cd854f3986ce943be3e1caa64ea4c2b73432.jpg)
+
+![](images/7f7f796197cd7b8cd98eae76493273cf4ac1d6c5a691250f4a9d09af9486de97.jpg)  
+Figure 4. Document classification on graph  $\mathbb{G}_{\mathrm{Sqrt}}$
+
+![](images/47416b24bc49fc3bd51cd13c47999a8870aee1d31887fa3bc76f89311fa14bf5.jpg)
+
+![](images/7b95482f6c15ffb50b1f8309deb7d11e672ceb2ca509d0e1e6e5104666a41ab4.jpg)
+
+![](images/8d0f259e55e12d4b5a943f0aa6f461f03b4c771ef75e5a639d73e8edbe2b16ad.jpg)
+
+Graph. We employ the graphs  $\mathbb{G}_{\mathrm{Log}}$  and  $\mathbb{G}_{\mathrm{Sqrt}}$  (Le et al., 2022, §5) for our simulations, which empirically satisfy the assumptions in §2. $^{12}$  Additionally, we consider  $M = 10^{2}, 10^{3}, 10^{4}$  as the number of nodes for these graphs.
+
+$N$ -function. We consider popular  $N$ -functions  $\Phi$  for the GST and the OW:  $\Phi_1(t) = \exp (t) - t - 1$ , and  $\Phi_2(t) = \exp (t^2) - 1$ . We also examine the limit case:  $\Phi_0(t) = t$ .
+
+Optimization algorithm. For general  $N$ -functions, we use Guha et al. (2023, Algorithm 1) to compute OW, and a second-order method, e.g., fmincon Trust Region Reflective solver in MATLAB, for solving the univariate optimization problem to compute the GST.
+
+Classification. We use support vector machine (SVM) for both document classification and the tasks in TDA. We employ kernel  $\exp(-\bar{t}\bar{d}(\cdot,\cdot))$ , where  $\bar{d}$  is a distance (e.g., GST, OW) for probability measures supported on a graph, and  $\bar{t} > 0$ . Following Cuturi (2013), we regularize for the Gram
+
+Table 1. The number of pairs on datasets for SVM.  
+
+<table><tr><td>Datasets</td><td>#pairs</td></tr><tr><td>TWITTER</td><td>4394432</td></tr><tr><td>RECİPE</td><td>8687560</td></tr><tr><td>CLASSIC</td><td>22890777</td></tr><tr><td>AMAZON</td><td>29117200</td></tr><tr><td>Orbit</td><td>11373250</td></tr><tr><td>MPEG7</td><td>18130</td></tr></table>
+
+matrices by adding a sufficiently large diagonal term for SVM with indefinite kernels. We use 1-vs-1 strategy to carry out SVM for multi-class classification, e.g., with Libsvm. $^{13}$
+
+For each dataset, we randomly split it into  $70\% / 30\%$  for training and test with 10 repeats. We typically choose hyperparameters via cross validation. For kernel hyperparameter, we choose  $1 / \bar{t}$  from  $\{q_s, 2q_s, 5q_s\}$  with  $s = 10, 20, \ldots, 90$  where  $q_s$  is the  $s\%$  quantile of a subset of distances observed on a training set. For SVM regularization hyperparameter,
+
+we choose it from  $\{0.01, 0.1, 1, 10\}$ . For the root node  $z_0$  in graph  $\mathbb{G}$ , we choose it from a random 10-root-node subset of  $V$  in  $\mathbb{G}$ . For document classification and TDA, reported time consumption includes preprocessing procedures, e.g., computing shortest paths for GST and OW.
+
+We also summarize the number of pairs which we need to compute distances in training and test with kernel SVM for each run in Table 1, e.g., more than 29 million pairs of measures on AMAZON dataset.
+
+# 5.1. Computation
+
+We compare the time consumption of GST and OW with popular  $N$ -functions  $\Phi_1, \Phi_2$ , and the limit case  $\Phi_0$ .<sup>14</sup>
+
+Setup. We randomly sample  $10^{4}$  pairs of measures from Amazon dataset, and consider graphs  $\mathbb{G}_{\mathrm{Log}}$  and  $\mathbb{G}_{\mathrm{Sqrt}}$  with the number of nodes  $M = 10^{3}$ .
+
+Results and discussions. We illustrate the time consumption on  $\mathbb{G}_{\mathrm{Log}}$  and  $\mathbb{G}_{\mathrm{Sqrt}}$  in Figures 1 and 2 respectively. GST is several-order faster than OW. More concretely, GST is  $100\times ,5000\times ,3000\times$  faster than OW for  $\Phi_0,\Phi_1,\Phi_2$  respectively. Especially, for  $\Phi_{1}$  and  $\Phi_{2}$ , GST takes less than 25 seconds, but OW takes at least 17 hours, and up to 33 hours.
+
+Notice that the OW with  $\Phi_0$  is equal to the OT with graph metric  $d_{\mathbb{G}}$  as its ground cost. Additionally, GST with  $\Phi_0$  yields a closed-form expression for a fast computation, following the Proposition 4.4 and Remark 4.5. Therefore, the computation of GST and OW with  $\Phi_0$  is more efficient than with  $\Phi_1, \Phi_2$ .
+
+# 5.2. Document classification
+
+Set up. We evaluate GST with  $\Phi_0, \Phi_1, \Phi_2$  as in §5.1 (denoted as GST- $\Phi_i$  for  $i = 0, 1, 2$ ). For OW, we only consider  $\Phi_0$  (denoted as OW- $\Phi_0$ ), but exclude  $\Phi_1, \Phi_2$  due to their heavy computations (illustrated and discussed in §5.1). Following Le et al. (2022) and Proposition 4.6, we also consider a special case for OW- $\Phi_0$ , where we randomly sample a tree from the given graph  $\mathbb{G}$  (denoted as OW- $\Phi_0$ -Tree). Notice that OW- $\Phi_0$ -Tree admits a closed-form expression for a fast computation (Le et al., 2019). We report results on graphs  $\mathbb{G}_{\mathrm{Log}}$  and  $\mathbb{G}_{\mathrm{Sqrt}}$  with the number of nodes  $M = 10^4$ . Further empirical results with different values for  $M$  are placed in Appendix §B.5.
+
+Results and discussions. We illustrate SVM results and the time consumption of kernel matrices on graphs  $\mathbb{G}_{\mathrm{Log}}$  and  $\mathbb{G}_{\mathrm{Sqrt}}$  in Figures 3 and 4 respectively. The performances of GST with all  $\Phi$  functions compare favorably to those of OW. Additionally, the computation of GST- $\Phi_0$  is several-order
+
+faster than  $\mathrm{OW - }\Phi_0$ . We also reemphasize that it is prohibitively expensive to evaluate OW with  $\Phi_{1},\Phi_{2}$  functions on many pairs of measures (See Table 1 and §5.1). GST-  $\Phi_1$  and GST-  $\Phi_2$  improve performances of GST-  $\Phi_0$ , but their computational time is several-order higher (i.e., GST-  $\Phi_0$  has a closed-form expression for a fast computation). Thus, it may imply that Orlicz geometric structure in GST may be useful for document classification. Recall that OW-  $\Phi_0$ -Tree uses a partial information of  $\mathbb{G}$ , while OW-  $\Phi_0$  uses information of the whole  $\mathbb{G}$ . On the other hand, kernel for OW-  $\Phi_0$ -Tree is positive definite, while kernel for OW-  $\Phi_0$  is indefinite. Performances of OW-  $\Phi_0$ -Tree are worse than those of OW-  $\Phi_0$  in TWITTER, RECIPE, but better than those of OW-  $\Phi_0$  in CLASSIC, AMAZON, which agrees with observations in Le et al. (2022).
+
+![](images/684490294361007d009fe5ee40a095fa02e8aa16d06df3617e74a0e992e8fc73.jpg)  
+Figure 5. TDA on graph  $\mathbb{G}_{\mathrm{Log}}$ . For each dataset, the numbers in the parenthesis are respectively the number of PD; and the maximum number of points in PD.
+
+![](images/3c37a72d8cd2242c21d53497d71f3cf00f0cc582c3ae3523bf4b0508380d26f9.jpg)  
+Figure 6. TDA on graph  $\mathbb{G}_{\mathrm{Sqrt}}$
+
+# 5.3. Topological Data Analysis
+
+Set up. We evaluate the same distances as in document classification, i.e., GST-  $\Phi_{i}$  for  $i = 0,1,2$ ; OW-  $\Phi_0$ ; and OW-  $\Phi_0$ -Tree. We report results on graphs  $\mathbb{G}_{\mathrm{Log}}$  and  $\mathbb{G}_{\mathrm{Sqrt}}$  with  $M = 10^{4}$  for Orbit dataset, and  $M = 10^{3}$  for MPEG7 dataset due to its small size. We also include further empirical results with different values for  $M$  in Appendix §B.5.
+
+Results and discussions. We illustrate SVM results and the time consumption of kernel matrices on graphs  $\mathbb{G}_{\mathrm{Log}}$  and  $\mathbb{G}_{\mathrm{Sqrt}}$  in Figures 5 and 6 respectively. We have similar observations as for document classification. GST is several-order faster than OW (for the same  $\Phi$  function), and improve performances of OW. Orlicz geometry structure is also helpful for TDA. Although only using a partial information of  $\mathbb{G}$ , positive definiteness may play an important role for OW- $\Phi_0$ -Tree to improve performances of OW- $\Phi_0$  in TDA, also observed in Le et al. (2022).
+
+# 6. Conclusion
+
+We propose the generalized Sobolev transport (GST) for probability measures supported on a graph metric space. This is achieved by leveraging a special family of convex functions (i.e., the set of  $N$ -functions) to extend the Sobolev transport (ST), which is a scalable variant of OT on a graph. This novel approach enables us to adopt Orlicz geometric structure for GST. Consequently, GST can inherit advantages from both ST and the Orlicz-Wasserstein (OW). An important feature is that unlike OW which involves a complex two-level optimization problem, one can simply solve a univariate optimization problem for the GST computation. Thus, GST is a computationally efficient variant for the computationally-demanding OW, which paves a way to use OW in applications. For future work, it is interesting to extend the GST for measures possibly having different total mass (i.e., unbalanced setting), which is essential for applications having noisy supports or outliers in input measures (Frogner et al., 2015; Balaji et al., 2020; Mukherjee et al., 2021; Fatras et al., 2021).
+
+# Acknowledgements
+
+We thank the area chairs and anonymous reviewers for their comments. KF has been supported in part by Grant-in-Aid for Transformative Research Areas (A) 22H05106. TL gratefully acknowledges the support of JSPS KAKENHI Grant number 23K11243, and Mitsui Knowledge Industry Co., Ltd. grant.
+
+# Impact Statement
+
+The paper presents a generalized version for the Sobolev transport (ST) which is a scalable variant of optimal transport for probability measures on a graph. By leveraging a special family of convex functions, we can adopt Orlicz geometric structure for the generalized Sobolev transport (GST). Furthermore, GST can be regarded as a computationally efficient variant of the computationally-demanding Orlicz-Wasserstein (OW), which possesses implicit essences to advance certain results in machine learning. Therefore, the proposed GST can inherit the merits from both ST and
+
+OW. To our knowledge, there are no foreseeable potential societal consequences of our work.
+
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+
+# Supplement to “Generalized Sobolev Transport for Probability Measures on a Graph”
+
+In §A of this appendix, we give the detailed proofs for our theoretical results. Further results and discussions are given in §B.
+
+# A. Detailed Proofs
+
+In this section, we give detailed proofs for our theoretical findings.
+
+# A.1. Proof for Theorem 3.3
+
+Proof. Let us consider a critic function  $f \in WL_{\Psi}(\mathbb{G}, \omega)$ . Then by Definition 3.1, we have
+
+$$
+f (x) = f \left(z _ {0}\right) + \int_ {\left[ z _ {0}, x \right]} f ^ {\prime} (y) \omega (\mathrm {d} y) \quad \text {f o r} \quad x \in \mathbb {G}. \tag {10}
+$$
+
+For convenience, let  $\mathbf{1}_{[z_0,x]}(y)$  denote the indicator function of the shortest path  $[z_0,x]$ , i.e.,
+
+$$
+\mathbf {1} _ {[ z _ {0}, x ]} (y) = \left\{ \begin{array}{l l} 1 & \text {i f} y \in [ z _ {0}, x ] \\ 0 & \text {o t h e r w i s e .} \end{array} \right. \tag {11}
+$$
+
+Using (10)-(11) and due to  $\mu (\mathbb{G}) = 1$ , we can rewrite  $\int_{\mathbb{G}}f(x)\mu (\mathrm{d}x)$  as
+
+$$
+\begin{array}{l} \int_ {\mathbb {G}} f (x) \mu (\mathrm {d} x) = \int_ {\mathbb {G}} f (z _ {0}) \mu (\mathrm {d} x) + \int_ {\mathbb {G}} \int_ {[ z _ {0}, x ]} f ^ {\prime} (y) \omega (\mathrm {d} y) \mu (\mathrm {d} x) \\ = f (z _ {0}) + \int_ {\mathbb {G}} \int_ {\mathbb {G}} \mathbf {1} _ {[ z _ {0}, x ]} (y) f ^ {\prime} (y) \omega (\mathrm {d} y) \mu (\mathrm {d} x). \\ \end{array}
+$$
+
+It then follows by applying Fubini's theorem to interchange the order of integration in the above last integral that
+
+$$
+\begin{array}{l} \int_ {\mathbb {G}} f (x) \mu (\mathrm {d} x) = f \left(z _ {0}\right) + \int_ {\mathbb {G}} \int_ {\mathbb {G}} \mathbf {1} _ {[ z _ {0}, x ]} (y) f ^ {\prime} (y) \mu (\mathrm {d} x) \omega (\mathrm {d} y) \\ = f (z _ {0}) + \int_ {\mathbb {G}} \left[ \int_ {\mathbb {G}} \mathbf {1} _ {[ z _ {0}, x ]} (y) \mu (\mathrm {d} x) \right] f ^ {\prime} (y) \omega (\mathrm {d} y). \\ \end{array}
+$$
+
+Owing to the definition of  $\Lambda(y)$  in (1), the above expression is equivalent to
+
+$$
+\int_ {\mathbb {G}} f (x) \mu (\mathrm {d} x) = f (z _ {0}) + \int_ {\mathbb {G}} f ^ {\prime} (y) \mu (\Lambda (y)) \omega (\mathrm {d} y).
+$$
+
+By exactly the same arguments, we also have
+
+$$
+\int_ {\mathbb {G}} f (x) \nu (\mathrm {d} x) = f (z _ {0}) + \int_ {\mathbb {G}} f ^ {\prime} (y) \nu (\Lambda (y)) \omega (\mathrm {d} y).
+$$
+
+As a consequence, the generalized Sobolev transport in Definition 3.2 can be rewritten as
+
+$$
+\mathcal {G} \mathcal {S} _ {\Phi} (\mu , \nu) = \sup  _ {f \in \mathbb {F}} \left| \int_ {\mathbb {G}} f ^ {\prime} (x) [ \mu (\Lambda (x)) - \nu (\Lambda (x)) ] \omega (\mathrm {d} x) \right|, \tag {12}
+$$
+
+where  $\mathbb{F} := \left\{f \in WL_{\Psi}(\mathbb{G}, \omega) : \| f' \|_{L_{\Psi}} \leq 1\right\}$  with  $\| \cdot \|_{L_{\Psi}}$  being the Luxemburg norm defined in (3).
+
+Next, on one hand it is clear that  $\{f': f \in \mathbb{F}\} \subset \{g \in L_{\Psi}(\mathbb{G}, \omega): \| g \|_{L_{\Psi}} \leq 1\}$ . On the other hand, for any  $g \in L_{\Psi}(\mathbb{G}, \omega)$  we have  $g = f'$  with  $f(x) \coloneqq \int_{[z_0, x]} g(y) \omega(\mathrm{d}y) \in W L_{\Psi}(\mathbb{G}, \omega)$ . Therefore, we conclude that  $\{f': f \in \mathbb{F}\} = \{g \in L_{\Psi}(\mathbb{G}, \omega): \| g \|_{L_{\Psi}} \leq 1\}$ . As a consequence, we can rewrite identity (12) as
+
+$$
+\mathcal {G S} _ {\Phi} (\mu , \nu) = \sup  _ {g \in L _ {\Psi} (\mathbb {G}, \omega): \| g \| _ {L _ {\Psi}} \leq 1} \left| \int_ {\mathbb {G}} g (x) h (x) \omega (\mathrm {d} x) \right|, \tag {13}
+$$
+
+where  $h(x)\coloneqq \mu (\Lambda (x)) - \nu (\Lambda (x))$  for all  $x\in \mathbb{G}$
+
+Recall from Definition 3.2 that  $\Psi : \mathbb{R}_+ \to \mathbb{R}_+$  is the complementary function of  $\Phi$ . Therefore, we obtain from (13) and (Rao & Ren, 1991, Proposition 10, pp.81) that
+
+$$
+\mathcal {G} \mathcal {S} _ {\Phi} (\mu , \nu) = \| h \| _ {\Phi} \tag {14}
+$$
+
+with  $\| h\|_{\Phi}$  being the Orlicz norm defined by (see (Rao & Ren, 1991, Definition 2, pp.58))
+
+$$
+\| h \| _ {\Phi} := \sup  \left\{\int_ {\mathbb {G}} | h (x) g (x) | \omega (\mathrm {d} x): \int_ {\mathbb {G}} \Psi (| g (x) |) \omega (\mathrm {d} x) \leq 1 \right\}. \tag {15}
+$$
+
+By applying (Rao & Ren, 1991, Theorem 13, pp.69), we have
+
+$$
+\| h \| _ {\Phi} = \inf  _ {k > 0} \frac {1}{k} \left(1 + \int_ {\mathbb {G}} \Phi (k | h (x) |) \omega (\mathrm {d} x)\right).
+$$
+
+This together with (14) yields
+
+$$
+\mathcal {G S} _ {\Phi} (\mu , \nu) = \inf  _ {k > 0} \frac {1}{k} \left(1 + \int_ {\mathbb {G}} \Phi (k | h (x) |) \omega (\mathrm {d} x)\right).
+$$
+
+This completes the proof of the theorem.
+
+# A.2. Proof for Corollary 3.4
+
+Proof. By Theorem 3.3, we have
+
+$$
+\mathcal {G S} _ {\Phi} (\mu , \nu) = \inf  _ {k > 0} \frac {1}{k} \left(1 + \int_ {\mathbb {G}} \Phi (k | h (x) |) \omega (\mathrm {d} x)\right) \tag {16}
+$$
+
+with  $h(x)\coloneqq \mu (\Lambda (x)) - \nu (\Lambda (x))$  . We next compute the integral in (16).
+
+Give two data points  $u, v$  in  $\mathbb{R}^n$ , let  $\langle u, v \rangle$  be the line segment in  $\mathbb{R}^n$  connecting the two points  $u, v$ , and denote  $(u, v)$  as the same line segment but without its two end-points.
+
+Using the assumption  $\omega (\{x\}) = 0$  for every  $x\in \mathbb{G}$ , we have
+
+$$
+\int_ {\mathbb {G}} \Phi (k | h (x) |) \omega (\mathrm {d} x) = \sum_ {e = \langle u, v \rangle \in E} \int_ {(u, v)} \Phi (k | h (x) |) \omega (\mathrm {d} x).
+$$
+
+As the input measures  $\mu$  and  $\nu$  are assumed to be supported on nodes  $V$  of graph  $\mathbb{G}$ , we also have  $h(x) = \mu(\Lambda(x)) - \nu(\Lambda(x)) = \mu(\Lambda(x) \setminus (u, v)) - \nu(\Lambda(x) \setminus (u, v))$  for every edge  $e = \langle u, v \rangle \in E$ . Therefore, the above identity can be further rewritten as
+
+$$
+\int_ {\mathbb {G}} \Phi (k | h (x) |) \omega (\mathrm {d} x) = \sum_ {e = \langle u, v \rangle \in E} \int_ {(u, v)} \Phi (k | \mu (\Lambda (x) \backslash (u, v)) - \nu (\Lambda (x) \backslash (u, v)) |) \omega (\mathrm {d} x). \tag {17}
+$$
+
+Consider edge  $e$  between two nodes  $u, v \in V$  of graph  $\mathbb{G}$ , i.e.,  $e = \langle u, v \rangle$ . For  $x \in (u, v)$ , we have  $y \in \mathbb{G} \setminus (u, v)$  belongs to  $\Lambda(x)$  if and only if  $y \in \gamma_e$  (see Equation (1) for the definitions of  $\Lambda(x)$  and  $\gamma_e$ ). It follows that  $\Lambda(x) \setminus (u, v) = \gamma_e$ , and thus we deduce from (17) that
+
+$$
+\begin{array}{l} \int_ {\mathbb {G}} \Phi (k | h (x) |) \omega (\mathrm {d} x) = \sum_ {e = \langle u, v \rangle \in E} \int_ {(u, v)} \Phi (k | \mu (\gamma_ {e}) - \nu (\gamma_ {e}) |) \omega (\mathrm {d} x) \\ = \sum_ {e = \langle u, v \rangle \in E} \Phi (k | \mu (\gamma_ {e}) - \nu (\gamma_ {e}) |) \int_ {(u, v)} \omega (\mathrm {d} x) \\ = \sum_ {e \in E} w _ {e} \Phi (k | \mu (\gamma_ {e}) - \nu (\gamma_ {e}) |). \\ \end{array}
+$$
+
+This together with (16) yields
+
+$$
+\mathcal {G S} _ {\Phi} (\mu , \nu) = \inf  _ {k > 0} \frac {1}{k} \left(1 + \sum_ {e \in E} w _ {e} \Phi (k | \mu (\gamma_ {e}) - \nu (\gamma_ {e}) |\right).
+$$
+
+Thus, the proof is complete.
+
+# A.3. Proof for Theorem 4.1
+
+Proof. We will prove that the generalized Sobolev transport  $\mathcal{G}S_{\Phi}(\mu, \nu)$  satisfies: (i) nonnegativity, (ii) indiscernibility, (iii) symmetry, and (iv) triangle inequality.
+
+(i) Nonnegativity. By choosing  $f = 0$  in Definition 3.2, we see that  $\mathcal{G}S_{\Phi}(\mu ,\nu)\geq 0$  for every  $(\mu ,\nu)$  in  $\mathcal{P}(\mathbb{G})\times \mathcal{P}(\mathbb{G})$ . Therefore, the generalized Sobolev transport is nonnegative.  
+(ii) Indiscernibility. Assume that  $\mathcal{G}\mathcal{S}_{\Phi}(\mu ,\nu) = 0$  , then we have
+
+$$
+\int_ {\mathbb {G}} f (x) \mu (\mathrm {d} x) - \int_ {\mathbb {G}} f (x) \nu (\mathrm {d} x) = 0, \tag {18}
+$$
+
+for all  $f\in W L_{\Psi}(\mathbb{G},\omega)$  satisfying the constraint  $\| f^{\prime}\|_{L_{\Psi}}\leq 1$
+
+Now let  $g \in WL_{\Psi}(\mathbb{G}, \omega)$  be any nonconstant function. Then  $c \coloneqq \| g' \|_{L_{\Psi}} > 0$ . Then by taking  $f \coloneqq \frac{q}{c}$ , we have  $f \in WL_{\Psi}(\mathbb{G}, \omega)$  with  $\| f' \|_{L_{\Psi}} = \| \frac{g'}{c} \|_{L_{\Psi}} = \frac{1}{c} \| g' \|_{L_{\Psi}} = 1$ . Hence, it follows from (18) that
+
+$$
+\int_ {\mathbb {G}} \frac {g (x)}{c} \mu (\mathrm {d} x) - \int_ {\mathbb {G}} \frac {g (x)}{c} \nu (\mathrm {d} x) = 0,
+$$
+
+which implies that
+
+$$
+\int_ {\mathbb {G}} g (x) \mu (\mathrm {d} x) = \int_ {\mathbb {G}} g (x) \nu (\mathrm {d} x). \tag {19}
+$$
+
+Thus, we have shown that (19) holds true for every nonconstant function  $g \in WL_{\Psi}(\mathbb{G}, \omega)$ . But (19) is also obviously true for any constant function  $g$ . So, we in fact obtain
+
+$$
+\int_ {\mathbb {G}} g (x) \mu (\mathrm {d} x) = \int_ {\mathbb {G}} g (x) \nu (\mathrm {d} x) \quad \text {f o r e v e r y} \quad g \in W L _ {\Psi} (\mathbb {G}, \omega),
+$$
+
+which gives  $\mu = \nu$  as desired.
+
+(iii) Symmetry. This property is obvious from Definition 3.2 as the value  $\mathcal{G}S_{\Phi}(\mu, \nu)$  is unchanged when the role of  $\mu$  and  $\nu$  is interchanged. That is,  $\mathcal{G}S_{\Phi}(\mu, \nu) = \mathcal{G}S_{\Phi}(\nu, \mu)$ .  
+(iv) Triangle inequality. Let  $\mu, \nu, \sigma$  be probability measures in  $\mathcal{P}(\mathbb{G})$ . Then for any function  $f \in WL_{\Psi}(\mathbb{G}, \omega)$  satisfying  $\| f' \|_{L_{\Psi}} \leq 1$ , we have
+
+$$
+\begin{array}{l} \left| \int_ {\mathbb {G}} f (x) \mu (\mathrm {d} x) - \int_ {\mathbb {G}} f (x) \nu (\mathrm {d} x) \right| = \left| \left[ \int_ {\mathbb {G}} f (x) \mu (\mathrm {d} x) - \int_ {\mathbb {G}} f (x) \sigma (\mathrm {d} x) \right] + \left[ \int_ {\mathbb {G}} f (x) \sigma (\mathrm {d} x) - \int_ {\mathbb {G}} f (x) \nu (\mathrm {d} x) \right] \right| \\ \leq \left| \int_ {\mathbb {G}} f (x) \mu (\mathrm {d} x) - \int_ {\mathbb {G}} f (x) \sigma (\mathrm {d} x) \right| + \left| \int_ {\mathbb {G}} f (x) \sigma (\mathrm {d} x) - \int_ {\mathbb {G}} f (x) \nu (\mathrm {d} x) \right| \\ \leq \mathcal {G S} _ {\Phi} (\mu , \sigma) + \mathcal {G S} _ {\Phi} (\sigma , \nu). \\ \end{array}
+$$
+
+By taking the supremum over  $f$ , this implies that  $\mathcal{G}\mathcal{S}_{\Phi}(\mu, \nu) \leq \mathcal{G}\mathcal{S}_{\Phi}(\mu, \sigma) + \mathcal{G}\mathcal{S}_{\Phi}(\sigma, \nu)$ .
+
+Due to the above properties, we conclude that the generalized Sobolev transport  $\mathcal{G}\mathcal{S}_{\Phi}(\mu ,\nu)$  is a metric on the space  $\mathcal{P}(\mathbb{G})$  of probability measures on graph  $\mathbb{G}$ .
+
+# A.4. Proof for Proposition 4.2
+
+Proof. Let  $\Psi_{1}$  and  $\Psi_{2}$  be respectively the complementary functions of  $\Phi_{1}$  and  $\Phi_{2}$  according to definition (6).
+
+Let  $t\in \mathbb{R}_+$  be arbitrary. Since  $\Phi_1\leq \Phi_2$  we have:
+
+$$
+\begin{array}{l} a t - \Phi_ {1} (a) \geq a t - \Phi_ {2} (a) \text {f o r e v e r y} a \in \mathbb {R} _ {+}, \\ \Rightarrow \sup  _ {a \geq 0} \left(a t - \Phi_ {1} (a)\right) \quad \geq \sup  _ {a \geq 0} \left(a t - \Phi_ {2} (a)\right). \\ \end{array}
+$$
+
+This implies that
+
+$$
+\Psi_ {1} (t) \geq \Psi_ {2} (t), \text {f o r a l l} t \in \mathbb {R} _ {+}.
+$$
+
+It follows that  $L_{\Psi_1}(\mathbb{G},\omega)\subset L_{\Psi_2}(\mathbb{G},\omega)$  and  $WL_{\Psi_1}(\mathbb{G},\omega)\subset WL_{\Psi_2}(\mathbb{G},\omega)$ . Moreover, for any fixed Orlicz function  $f^{\prime}$  and any number  $t > 0$ , we have
+
+$$
+\int_ {\mathbb {G}} \Psi_ {1} \left(\frac {| f ^ {\prime} (x) |}{t}\right) \omega (\mathrm {d} x) \geq \int_ {\mathbb {G}} \Psi_ {2} \left(\frac {| f ^ {\prime} (x) |}{t}\right) \omega (\mathrm {d} x).
+$$
+
+Consequently, we obtain
+
+$$
+\left\{t > 0 \mid \int_ {\mathbb {G}} \Psi_ {1} \left(\frac {| f ^ {\prime} (x) |}{t}\right) \omega (\mathrm {d} x) \leq 1 \right\} \subset \left\{t > 0 \mid \int_ {\mathbb {G}} \Psi_ {2} \left(\frac {| f ^ {\prime} (x) |}{t}\right) \omega (\mathrm {d} x) \leq 1 \right\}.
+$$
+
+Since the infimum of a set is smaller than or equal to the infimum of its subset, we deduce that
+
+$$
+\left\| f ^ {\prime} \right\| _ {L _ {\Psi_ {1}}} \geq \left\| f ^ {\prime} \right\| _ {L _ {\Psi_ {2}}}.
+$$
+
+It follows from this and  $WL_{\Psi_1}(\mathbb{G},\omega)\subset WL_{\Psi_2}(\mathbb{G},\omega)$  that
+
+$$
+\left\{f \mid f \in W L _ {\Psi_ {1}} (\mathbb {G}, \omega), \| f ^ {\prime} \| _ {L _ {\Psi_ {1}} (\mathbb {G}, \omega)} \leq 1 \right\} \subset \left\{f \mid f \in W L _ {\Psi_ {2}} (\mathbb {G}, \omega), \| f ^ {\prime} \| _ {L _ {\Psi_ {2}} (\mathbb {G}, \omega)} \leq 1 \right\}.
+$$
+
+Since the supremum of a set is larger than or equal to the supremum of its subset, we conclude that
+
+$$
+\mathcal {G S} _ {\Phi_ {1}} (\mu , \nu) \leq \mathcal {G S} _ {\Phi_ {2}} (\mu , \nu)
+$$
+
+for any input measures  $\mu, \nu$  in  $\mathcal{P}(\mathbb{G})$ . Therefore, the proof is completed.
+
+We further note that one can directly leverage the result in Theorem 3.3, where generalized Sobolev transport can be computed by solving univariate optimization problem, to simplify the proof.
+
+# A.5. Proof for Proposition 4.3
+
+Proof. For the  $N$ -function  $\Phi(t) = t^p$  with  $1 < p < \infty$ , we have
+
+$$
+L _ {\Phi} (\mathbb {G}, \omega) = L ^ {p} (\mathbb {G}, \omega),
+$$
+
+where  $L^p (\mathbb{G},\omega)$  is the standard  $L^p$  functional space which is reviewed in  $\S \mathrm{B.1}$  (appendix).
+
+Thus, following Definition 3.1 and the definition of graph-based Sobolev transport (Le et al., 2022, Definition 3.1) (a review is given in Definition B.1 (appendix)), we have
+
+$$
+W L _ {\Phi} (\mathbb {G}, \omega) = W ^ {1, p} (\mathbb {G}, \omega).
+$$
+
+Thus, the proof is complete.
+
+# A.6. Proof for Proposition 4.4
+
+Proof. Let  $h(x) \coloneqq \mu(\Lambda(x)) - \nu(\Lambda(x))$  for  $x \in \mathbb{G}$ . Then since  $\Phi(t) = \frac{(p-1)^{p-1}}{p^p} t^p$  with  $1 < p < \infty$ , we obtain from Theorem 3.3 that
+
+$$
+\begin{array}{l} \mathcal {G} \mathcal {S} _ {\Phi} (\mu , \nu) = \inf  _ {k > 0} \frac {1}{k} \left(1 + \int_ {\mathbb {G}} \frac {(p - 1) ^ {p - 1}}{p ^ {p}} k ^ {p} | h (x) | ^ {p} \omega (\mathrm {d} x)\right) \\ = \inf  _ {k > 0} F (k), \tag {20} \\ \end{array}
+$$
+
+where  $F(k) \coloneqq \frac{1}{k} + \frac{(p - 1)^{p - 1}}{p^p} k^{p - 1} \int_{\mathbb{G}} |h(x)|^p \omega(\mathrm{d}x)$  for  $k > 0$ .
+
+Let us consider the following two possibilities:
+
+Case 1:  $\int_{\mathbb{G}}|h(x)|^p\omega (\mathrm{d}x) = 0$  . In that case, we have from (20) that
+
+$$
+\mathcal {G} \mathcal {S} _ {\Phi} (\mu , \nu) = \inf  _ {k > 0} \frac {1}{k} = 0 = \left(\int_ {\mathbb {G}} | h (x) | ^ {p} \omega (\mathrm {d} x)\right) ^ {\frac {1}{p}} = \mathcal {S} _ {p} (\mu , \nu).
+$$
+
+Case 2:  $\int_{\mathbb{G}} |h(x)|^p \omega(\mathrm{d}x) \neq 0$ . Then  $\lim_{k \to 0^+} F(k) = \lim_{k \to +\infty} F(k) = +\infty$ . Therefore, it follows from (20) that
+
+$$
+\mathcal {G} \mathcal {S} _ {\Phi} (\mu , \nu) = F \left(k _ {0}\right) \tag {21}
+$$
+
+for some finite number  $k_0 \in (0, +\infty)$  satisfying  $F'(k_0) = 0$ . As  $F'(k) = -\frac{1}{k^2} + \left(\frac{p-1}{p}\right)^p k^{p-2} \int_{\mathbb{G}} |h(x)|^p$ , we can solve the equation  $F'(k_0) = 0$  for  $k_0$  to obtain
+
+$$
+k _ {0} = \frac {1}{\frac {p - 1}{p} \left(\int_ {\mathbb {G}} | h (x) | ^ {p} \omega (\mathrm {d} x)\right) ^ {\frac {1}{p}}}.
+$$
+
+Plugging this value of  $k_{0}$  into Equation (21) and using the formula for  $F(k)$ , we get
+
+$$
+\begin{array}{l} \mathcal {G S} _ {\Phi} (\mu , \nu) = \frac {1}{k _ {0}} \left(1 + \frac {(p - 1) ^ {p - 1}}{p ^ {p}} k _ {0} ^ {p} \int_ {\mathbb {G}} | h (x) | ^ {p}\right) \\ = \frac {p - 1}{p} \left(\int_ {\mathbb {G}} | h (x) | ^ {p} \omega (\mathrm {d} x)\right) ^ {\frac {1}{p}} \left(1 + \frac {(p - 1) ^ {p - 1}}{p ^ {p}} \frac {1}{\frac {(p - 1) ^ {p}}{p ^ {p}} \left(\int_ {\mathbb {G}} | h (x) | ^ {p} \omega (\mathrm {d} x)\right)} \int_ {\mathbb {G}} | h (x) | ^ {p} \omega (\mathrm {d} x)\right) \\ = \left(\int_ {\mathbb {G}} | h (x) | ^ {p} \omega (\mathrm {d} x)\right) ^ {\frac {1}{p}} \\ = \mathcal {S} _ {p} (\mu , \nu). \\ \end{array}
+$$
+
+Thus, we have shown that  $\mathcal{G}S_{\Phi}(\mu ,\nu) = S_p(\mu ,\nu)$  in both cases, and hence the proof is complete.
+
+Remark A.1. For the  $N$ -function  $\Phi(t) = t^q$  where  $1 < q < \infty$ , its complementary function is  $\Psi(t) = \frac{(p - 1)^{p - 1}}{p^p} t^p$  where  $p$  is the conjugate of  $q$ , i.e.,  $\frac{1}{q} + \frac{1}{p} = 1$ .
+
+# A.7. Proof for  $\lim_{p\to 1^{+}}\frac{(p - 1)^{p - 1}}{p^p} t^p = t$  in Remark 4.5
+
+We will prove that  $\lim_{p\to 1^{+}}\frac{(p - 1)^{p - 1}}{p^p} t^p = t$  which is used in Remark 4.5. For this and since  $\lim_{p\to 1^{+}}\frac{t^p}{p^p} = t$ , it is enough to show that
+
+$$
+\lim_{p\to 1^{+}}(p - 1)^{p - 1} = 1.
+$$
+
+By taking logarithm, this in turn is equivalent to proving that
+
+$$
+\lim  _ {p \rightarrow 1 ^ {+}} \ln (p - 1) ^ {p - 1} = 0. \tag {22}
+$$
+
+Proof of (22).
+
+$$
+\lim  _ {p \to 1 ^ {+}} \ln (p - 1) ^ {p - 1} = \lim  _ {p \to 1 ^ {+}} \frac {\ln (p - 1)}{1 / (p - 1)} = \lim  _ {p \to 1 ^ {+}} \frac {1 / (p - 1)}{- 1 / (p - 1) ^ {2}},
+$$
+
+where we apply the L'Hopital's rule for the last equality. It follows that  $\lim_{p\to 1^{+}}\ln (p - 1)^{p - 1} = \lim_{p\to 1^{+}} - (p - 1) = 0$  as desired. Hence, the proof is complete.
+
+# A.8. Complementary function for  $\Phi(t) = \exp(t^p) - 1$  with  $1 < p < \infty$
+
+For the given  $N$ -function  $\Phi(t) = \exp(t^p) - 1$  with  $1 < p < \infty$ , its complementary function  $\Psi$  is computed as follows
+
+$$
+\Psi (t) = \sup  _ {a \geq 0} (a t - \Phi (a)). \tag {23}
+$$
+
+Let us fix  $t \geq 0$  and denote the objective function in Equation (23) as  $F(a)$ . Then  $F(a) = at - \exp(a^p) + 1$ , and we have
+
+$$
+F ^ {\prime} (a) = t - p a ^ {p - 1} \exp (a ^ {p}),
+$$
+
+$$
+F ^ {\prime \prime} (a) = - p (p - 1) a ^ {p - 2} \exp (a ^ {p}) - p ^ {2} a ^ {2 (p - 1)} \exp (a ^ {p}) = - \Big [ p (p - 1) a ^ {p - 2} + p ^ {2} a ^ {2 (p - 1)} \Big ] \exp (a ^ {p}).
+$$
+
+Since  $1 < p < \infty$ , we obtain  $F''(a) < 0$  for all  $a > 0$ . Thus,  $F'(a)$  is a strictly decreasing function on  $(0, +\infty)$ . Additionally, we have
+
+$$
+F ^ {\prime} (0) = t \geq 0,
+$$
+
+$$
+\lim  _ {a \rightarrow + \infty} F ^ {\prime} (a) = - \infty .
+$$
+
+Therefore,  $F'(a) = 0$  has a unique root, which is denoted by  $a^*$ . Then, the complementary function  $\Psi$  is
+
+$$
+\Psi (t) = \sup  _ {a \geq 0} F (a) = F (a ^ {*}) = a ^ {*} t - \exp ((a ^ {*}) ^ {p}) - 1.
+$$
+
+# B. Further Results and Discussions
+
+In this section, we give further results and discussions.
+
+Notations. Let  $\langle x,y\rangle$  be the line segment in  $\mathbb{R}^n$  connecting two points  $x,y$ , and denote  $(x,y)$  as the same line segment but without its two end-points.
+
+# B.1. On Sobolev transport
+
+$L^p$  functional space. For a nonnegative Borel measure  $\omega$  on  $\mathbb{G}$ , denote  $L^p(\mathbb{G}, \omega)$  as the space of all Borel measurable functions  $f: \mathbb{G} \to \mathbb{R}$  such that  $\int_{\mathbb{G}} |f(y)|^p \omega(\mathrm{d}y) < \infty$ . For  $p = \infty$ , we instead assume that  $f$  is bounded  $\omega$ -a.e.
+
+Functions  $f_{1},f_{2}\in L^{p}(\mathbb{G},\omega)$  are considered to be the same if  $f_{1}(x) = f_{2}(x)$  for  $\omega$  -a.e.  $x\in \mathbb{G}$
+
+Then,  $L^p (\mathbb{G},\omega)$  is a normed space with the norm defined by
+
+$$
+\| f \| _ {L ^ {p} (\mathbb {G}, \omega)} := \left(\int_ {\mathbb {G}} | f (y) | ^ {p} \omega (\mathrm {d} y)\right) ^ {\frac {1}{p}} \text {f o r} 1 \leq p <   \infty , \text {a n d}
+$$
+
+$$
+\left\| f \right\| _ {L ^ {\infty} (\mathbb {G}, \omega)} := \inf  \left\{t \in \mathbb {R}: | f (x) | \leq t \text {f o r} \omega \text {- a . e .} x \in \mathbb {G} \right\}.
+$$
+
+Connection between  $L^p$  and  $L_{\Phi}$  functional spaces. When the convex function  $\Phi(t) = t^p$ , for  $1 < p < \infty$ , we have
+
+$$
+L ^ {p} (\mathbb {G}, \omega) = L _ {\Phi} (\mathbb {G}, \omega).
+$$
+
+Definition B.1 (Graph-based Sobolev space (Le et al., 2022)). Let  $\omega$  be a nonnegative Borel measure on  $\mathbb{G}$ , and let  $1 \leq p \leq \infty$ . A continuous function  $f: \mathbb{G} \to \mathbb{R}$  is said to belong to the Sobolev space  $W^{1,p}(\mathbb{G}, \omega)$  if there exists a function  $h \in L^p(\mathbb{G}, \omega)$  satisfying
+
+$$
+f (x) - f \left(z _ {0}\right) = \int_ {\left[ z _ {0}, x \right]} h (y) \omega (\mathrm {d} y) \quad \forall x \in \mathbb {G}. \tag {24}
+$$
+
+Such function  $h$  is unique in  $L^p(\mathbb{G}, \omega)$  and is called the graph derivative of  $f$  w.r.t. the measure  $\omega$ . The graph derivative of  $f \in W^{1,p}(\mathbb{G}, \omega)$  is denoted  $f' \in L^p(\mathbb{G}, \omega)$ .
+
+Proposition B.2 (Closed-form expression of ST (Le et al., 2022)). Let  $\omega$  be any nonnegative Borel measure on  $\mathbb{G}$ , and let  $1 \leq p \leq \infty$ . Then, we have
+
+$$
+\mathcal {S} _ {p} (\mu , \nu) = \left(\int_ {\mathbb {G}} | \mu (\Lambda (x)) - \nu (\Lambda (x)) | ^ {p} \omega (\mathrm {d} x)\right) ^ {\frac {1}{p}},
+$$
+
+where  $\Lambda (x)$  is the subset of  $\mathbb{G}$  defined by Equation (1).
+
+Definition B.3 (Length measure (Le et al., 2022)). Let  $\omega^{*}$  be the unique Borel measure on  $\mathbb{G}$  such that the restriction of  $\omega^{*}$  on any edge is the length measure of that edge. That is,  $\omega^{*}$  satisfies:
+
+i) For any edge  $e$  connecting two nodes  $u$  and  $v$ , we have  $\omega^{*}(\langle x, y \rangle) = (t - s)w_{e}$  whenever  $x = (1 - s)u + sv$  and  $y = (1 - t)u + tv$  for  $s, t \in [0,1)$  with  $s \leq t$ . Here,  $\langle x, y \rangle$  is the line segment in  $e$  connecting  $x$  and  $y$ .  
+ii) For any Borel set  $F\subset \mathbb{G}$ , we have
+
+$$
+\omega^ {*} (F) = \sum_ {e \in E} \omega^ {*} (F \cap e).
+$$
+
+Lemma B.4 ( $\omega^{*}$  is the length measure on graph (Le et al., 2022)). Suppose that  $\mathbb{G}$  has no short cuts, namely, any edge  $e$  is a shortest path connecting its two end-points. Then,  $\omega^{*}$  is a length measure in the sense that
+
+$$
+\omega^ {*} ([ x, y ]) = d _ {\mathbb {G}} (x, y)
+$$
+
+for any shortest path  $[x,y]$  connecting  $x, y$ . Particularly,  $\omega^{*}$  has no atom in the sense that  $\omega^{*}(\{x\}) = 0$  for every  $x \in \mathbb{G}$ .
+
+# B.2. On Wasserstein distance
+
+We review the definition of the  $p$ -Wasserstein distance with graph metric cost for measures on graph  $\mathbb{G}$ .
+
+Definition B.5. Let  $1 \leq p < \infty$ , suppose that  $\mu$  and  $\nu$  are two nonnegative Borel measures on  $\mathbb{G}$  satisfying  $\mu(\mathbb{G}) = \nu(\mathbb{G}) = 1$ . Then, the  $p$ -order Wasserstein distance between  $\mu$  and  $\nu$  is defined as follows:
+
+$$
+\mathcal {W} _ {p} (\mu , \nu) ^ {p} = \inf  _ {\gamma \in \Pi (\mu , \nu)} \int_ {\mathbb {G} \times \mathbb {G}} d _ {\mathbb {G}} (x, y) ^ {p} \gamma (\mathrm {d} x, \mathrm {d} y),
+$$
+
+where
+
+$$
+\Pi (\mu , \nu) := \left\{\gamma \in \mathcal {P} (\mathbb {G} \times \mathbb {G}): \gamma_ {1} = \mu , \gamma_ {2} = \nu \right\},
+$$
+
+where  $\gamma_{1},\gamma_{2}$  are the first and second marginals of  $\gamma$  respectively.
+
+# B.3. On generalized Sobolev transport
+
+Young inequality. Let  $\Phi, \Psi$  be a pair of complementary  $N$ -functions, then
+
+$$
+s t \leq \Psi (s) + \Phi (t).
+$$
+
+Equivalence (Adams & Fournier, 2003, §8.17) (Musielak, 2006, §13.11). The Luxemburg norm is equivalent to the Orlicz norm
+
+$$
+\left\| f \right\| _ {L _ {\Phi}} \leq \left\| f \right\| _ {\Phi} \leq 2 \left\| f \right\| _ {L _ {\Phi}}, \tag {25}
+$$
+
+where we recall that the Luxemburg norm  $\| \cdot \|_{L_\Phi}$  is defined in Equation (3), and the Orlicz norm  $\| \cdot \|_{\Phi}$  is defined in Equation (15) (see (Rao & Ren, 1991, Definition 2, pp.58)).
+
+Generalized Hölder inequality. Let  $\Phi, \Psi$  be a pair of complementary  $N$ -functions, then Hölder inequality w.r.t. Luxemburg norm (Adams & Fournier, 2003, §8.11) is as follows:
+
+$$
+\left| \int_ {\mathbb {G}} f (x) g (x) \omega (d x) \right| \leq 2 \| f \| _ {L _ {\Phi}} \| g \| _ {L _ {\Psi}}. \tag {26}
+$$
+
+Additionally, Hölder inequality w.r.t. Luxemburg norm and Orlicz norm (Musielak, 2006, §13.13) is as follows:
+
+$$
+\left| \int_ {\mathbb {G}} f (x) g (x) \omega (d x) \right| \leq \| f \| _ {L _ {\Phi}} \| g \| _ {\Psi}. \tag {27}
+$$
+
+# B.4. Further discussions
+
+For completeness, we recall important discussions on the underlying graph in Le et al. (2022) for ST, since they are also applied and/or adapted for the GST.
+
+OT problems for measures on a graph. In this work, our considered problem is to compute OT distance between two input probability measures supported on the same graph. We distinguish this line of research to the OT problem between two (different) input graphs, considered in Petric Maretic et al. (2019); Dong & Sawin (2020); Tam & Dunson (2022), where its goal is to compute the distance between two input graphs by using OT approach.
+
+Algorithms for the univariate optimization to compute GST. For empirical simulations, we leverage the fmincon in MATLAB with the trust-region-reflective algorithm since it is easy to compute the gradient and Hessian for the objective function for the univariate optimization problem.
+
+Path length for points in  $\mathbb{G}$  (Le et al., 2023). We can canonically measure a path length connecting any two points  $x, y \in \mathbb{G}$  where  $x, y$  are not necessary to be nodes in  $V$ . Indeed, for  $x, y \in \mathbb{R}^n$  belonging to the same edge  $e = \langle u, v \rangle$  which connects two nodes  $u, v \in V$ , we have
+
+$$
+x = (1 - s) u + s v,
+$$
+
+$$
+y = (1 - t) u + t v,
+$$
+
+for some numbers  $t, s \in [0,1]$ . Therefore, the length of the path connecting  $x, y$  along the edge  $e$  (i.e., the line segment  $\langle x, y \rangle$ ) is defined by  $|t - s|w_e$ .
+
+Thus, the length for an arbitrary path in  $\mathbb{G}$  can be similarly defined by breaking down into pieces over edges and summing over their corresponding lengths (Le et al., 2022).
+
+Extension to measures supported on  $\mathbb{G}$ . Similar to ST (Le et al., 2022), the discrete case of GST in Equation (8) can be extended for measures with finite supports on  $\mathbb{G}$  (i.e., measures which may have supports on edges) by using the same strategy to measure a path length for points in  $\mathbb{G}$  (discussed in the above paragraph). More precisely, we break down edges containing supports into pieces and sum over their corresponding values instead of the sum over edges as in Equation (8).
+
+About the assumption of uniqueness property of the shortest paths on  $\mathbb{G}$ . As discussed in (Le et al., 2022) for the ST, note that  $w_{e} \in \mathbb{R}$  for any edge  $e \in E$  in  $\mathbb{G}$ , it is almost surely that every node in  $V$  can be regarded as unique-path root node (with a high probability, lengths of paths connecting any two nodes in graph  $\mathbb{G}$  are different).
+
+Additionally, for some special graph, e.g., a grid of nodes, there is no unique-path root node for such graph. However, by perturbing each node (and/or perturbing lengths of edges in case  $\mathbb{G}$  is a non-physical graph) with a small deviation, we can obtain a graph satisfying the unique-path root node assumption.
+
+About the generalized Sobolev transport (GST). Similar to the ST (Le et al., 2022), we assume that the graph metric space (i.e., the graph structure) is given, and leave the question to learn an optimal graph metric structure from data for future work.
+
+About graphs  $\mathbb{G}_{\mathrm{Log}}$  and  $\mathbb{G}_{\mathrm{Sqrt}}$  (Le et al., 2022). We use a clustering method, e.g., the farthest-point clustering, to partition supports of measures into at most  $M$  clusters. Then, let  $V$  be the set of centroids of these clusters, i.e., graph vertices. For edges, in graph  $\mathbb{G}_{\mathrm{Log}}$ , we randomly choose  $M\log(M)$  edges; and  $M^{3/2}$  edges for graph  $\mathbb{G}_{\mathrm{Sqrt}}$ . We further denote the set of those randomly sampled edges as  $\tilde{E}$ .
+
+For each edge  $e$ , its corresponding edge length (i.e., weight)  $w_{e}$  is computed by the Euclidean distance between the two corresponding nodes of edge  $e$ . Let  $n_c$  be the number of connected components in the graph  $\tilde{\mathbb{G}}(V, \tilde{E})$ . Then, we randomly add  $(n_c - 1)$  more edges between these  $n_c$  connected components to construct a connected graph  $\mathbb{G}$  from  $\tilde{\mathbb{G}}$ . Let  $E_{c}$  be the set of these  $(n_c - 1)$  added edges and denote set  $E = \tilde{E} \cup E_{c}$ , then  $\mathbb{G}(V, E)$  is the constructed graph.
+
+Huber function and its corresponding normalized function (Andoni et al., 2018). Consider the Huber function (Huber,
+
+![](images/7c23fe50b59aa52533c4296e6a1bde6da49688a34d0f0585b65f38d855ccc166.jpg)
+
+![](images/c381ccfb4128de2c9c314473e4bdf992849f240f0210c87846f27e566ca088bc.jpg)
+
+![](images/1ce2b92a7b57aeacfc624026bafea656a5d55b961ff4f5f5291432acfdab9b9a.jpg)
+
+![](images/c24a8870be5558c95b05ae4c2ce533e1d3a10a409659e8efa55cce2865d7997e.jpg)
+
+![](images/81c8149b3baa44576f16eeb580afdaf53512baabf406257044483da69e8602cb.jpg)  
+Figure 7. Document classification on graph  $\mathbb{G}_{\mathrm{Log}}$  ( $M = 10^{3}$ ).
+
+![](images/24d31c83df4a6316dd59d05588a641af852d3b1660d755428b85fe985ab3e595.jpg)
+
+![](images/3227ae9b09b480f9a5f9313f43365509df68b0f5d4894a3389a250d4e5a6fc27.jpg)
+
+![](images/800e009ae1b4117a0f88f26af58dfa3932268241ef0d2996506258aace8da885.jpg)
+
+1992) on the domain  $[0, +\infty)$ , defined as,
+
+$$
+f _ {H} (t) = \left\{ \begin{array}{l l} t ^ {2} / 2 & \text {i f} 0 \leq t \leq \delta \\ \delta (t - \delta / 2) & \text {i f} t > \delta , \end{array} \right. \tag {28}
+$$
+
+where  $\delta > 0$  is a constant. Then, the normalized Huber function (Andoni et al., 2018) on the domain  $[0, +\infty)$  is defined as
+
+$$
+\Phi_ {H} (t) = \left\{ \begin{array}{l l} f _ {H} \left(f _ {H} ^ {- 1} (1) t\right) & \text {i f} 0 \leq t \leq 1 \\ \Phi_ {H _ {-}} ^ {\prime} (1) t - \left(\Phi_ {H _ {-}} ^ {\prime} (1) - 1\right) & \text {i f} t > 1, \end{array} \right. \tag {29}
+$$
+
+where  $\Phi_{H - }^{\prime}$  is the left derivative of  $\Phi_H$
+
+Hyperparameter validation. For validation, we further randomly split the training set into  $70\% / 30\%$  for validation-training and validation with 10 repeats to choose hyper-parameters in our simulations.
+
+The number of pairs in training and test for kernel SVM. Let  $N_{tr}$ ,  $N_{te}$  be the number of measures used for training and test respectively. For the kernel SVM training, the number of pairs which we compute the distances is  $(N_{tr} - 1) \times \frac{N_{tr}}{2}$ . For the test phase, the number of pairs which we compute the distances is  $N_{tr} \times N_{te}$ . Therefore, for 1 repeat, the number of pairs which we compute the distances for both training and test is totally  $N_{tr} \times \left( \frac{N_{tr} - 1}{2} + N_{te} \right)$ .
+
+# B.5. Further empirical results
+
+Document classification. We further illustrate document classification on graphs  $\mathbb{G}_{\mathrm{Log}}$  and  $\mathbb{G}_{\mathrm{Sqrt}}$  for:
+
+-  $M = 10^{3}$  in Figures 7 and 8 respectively.  
+-  $M = 10^{2}$  in Figures 9 and 10 respectively.
+
+TDA. We also further illustrate TDA on graphs  $\mathbb{G}_{\mathrm{Log}}$  and  $\mathbb{G}_{\mathrm{Sqrt}}$  for:
+
+-  $M = 10^{3}$  in Figures 11 and 12 respectively.  
+-  $M = 10^{2}$  in Figures 13 and 14 respectively.
+
+![](images/466d593a1e299d6d7fbdf2ab0a689871f8a6ab15f93b0a1f935f39d800130ddd.jpg)  
+Figure 8. Document classification on graph  $\mathbb{G}_{\mathrm{Sqrt}}$  ( $M = 10^{3}$ ).
+
+![](images/b1b99b26c31ecf71c63b92145c333b01dd9a78cde574bef29bb6d2b657c8368a.jpg)  
+Figure 9. Document classification on graph  $\mathbb{G}_{\mathrm{Log}}$  ( $M = 10^{2}$ ).
+
+![](images/575a04d29d7777d1cef40aabe2182ef8903a09c7d134e2eb1ad1b5afc21237da.jpg)  
+Figure 10. Document classification on graph  $\mathbb{G}_{\mathrm{Sqrt}}$  ( $M = 10^{2}$ ).
+
+![](images/1933c72acea428ab587aaed0089c433ac0f252c63e89eb17fea8a372f1a98850.jpg)
+
+![](images/c066d3ee8743e20c3764bc6f53cfa7cacae0262c001683e02c2563eaefe0900d.jpg)  
+Figure 11. Topological data analysis on graph  $\mathbb{G}_{\mathrm{Log}}$  ( $M = 10^{3}$ ).
+
+![](images/f32836e9d64c55c9440fb44fec8fed0677cea8abb952264d13f93f8d6e4cc45b.jpg)  
+Figure 12. Topological data analysis on graph  $\mathbb{G}_{\mathrm{Sqrt}}$  ( $M = 10^{3}$ ).
+
+![](images/45411d7b30247080f6ca9b3e5c897cfdf8800bcc1810088c556fe3329f438d0d.jpg)  
+Figure 13. Topological data analysis on graph  $\mathbb{G}_{\mathrm{Log}}$  ( $M = 10^{2}$ ).  
+Figure 14. Topological data analysis on graph  $\mathbb{G}_{\mathrm{Sqrt}}$  ( $M = 10^{2}$ ).

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generalizingorthogonalizationformodelswithnonlinearities/full.md

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+# Generalizing Orthogonalization for Models with Non-Linearities
+
+David Rügamer  $^{1,2}$  Chris Kolb  $^{1,2}$  Tobias Weber  $^{1,2}$  Lucas Kook  $^{3}$  Thomas Nagler  $^{1,2}$
+
+# Abstract
+
+The complexity of black-box algorithms can lead to various challenges, including the introduction of biases. These biases present immediate risks in the algorithms' application. It was, for instance, shown that neural networks can deduce racial information solely from a patient's X-ray scan, a task beyond the capability of medical experts. If this fact is not known to the medical expert, automatic decision-making based on this algorithm could lead to prescribing a treatment (purely) based on racial information. While current methodologies allow for the "orthogonalization" or "normalization" of neural networks with respect to such information, existing approaches are grounded in linear models. Our paper advances the discourse by introducing corrections for non-linearities such as ReLU activations. Our approach also encompasses scalar and tensor-valued predictions, facilitating its integration into neural network architectures. Through extensive experiments, we validate our method's effectiveness in safeguarding sensitive data in generalized linear models, normalizing convolutional neural networks for metadata, and rectifying pre-existing embeddings for undesired attributes.
+
+# 1. Introduction
+
+In the burgeoning landscape of artificial intelligence and deep learning, black-box algorithms have become a centerpiece for driving advances in many fields of application. These powerful and often inscrutable models offer impressive predictive capabilities, but their complexity also gives rise to challenges that cannot be overlooked. One of the most urgent contemporary challenges is the correction of un
+
+$^{1}$ Department of Statistics, LMU Munich, Munich, Germany  $^{2}$ Munich Center for Machine Learning (MCML), Munich, Germany  $^{3}$ Institute for Statistics and Mathematics, Vienna University of Economics and Business, Vienna, Austria. Correspondence to: David Rügamer <david@stat.uni-muenchen.de>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+wanted behaviors in these algorithms. In particular, the presence of biases in model predictions and learned representations can lead to unintended consequences that transcend the technological sphere, impacting societal norms and ethical considerations. For example, Glocker et al. (2023b) and Weber et al. (2023) realized that predictions of convolutional networks trained for chest X-ray pathology classification are heavily affected by implicitly encoded racial information, leading to potentially inaccurate or unfair medical assessments.
+
+In this paper, we stress the necessity of addressing these challenges and propose several solutions, focusing on what has been termed "orthogonalization" or "normalization" of neural networks in the literature (see, e.g., Lu et al., 2021; Rügamer, 2023). These approaches can be used to adjust predictions, prevent unequal treatment of population subgroups, protect from unintentionally revealing sensitive information, or ensure the interpretability of black-box models.
+
+# 1.1. Related Literature
+
+Classical Orthogonalization In contrast to research investigating network weight dependencies (see, e.g., Huang et al., 2020), the orthogonalization discussed in this work is usually motivated as a normalization operation or correction method. Lu et al. (2021), e.g., proposed the so-called metadata normalization network, which applies a layer-wise orthogonalization to remove metadata information from layers. Similarly, He et al. (2019) study an orthogonalization procedure to encourage fair learning by removing protected information from feature representations. To circumvent training instabilities caused by dependence on the batch size, Vento et al. (2022) propose to cast the metadata normalization as a bi-level problem optimized using a penalty approach. Kaiser et al. (2022) use the same idea to incorporate fairness in automated decision-making systems for labor market data. A related but different problem is addressed by Rügamer et al. (2023), who propose an orthogonalization cell to preserve effect identifiability and thereby interpretability. This idea was then also adapted to other model classes (Baumann et al., 2021; Kopper et al., 2021). Further, orthogonalization can also be used as a debiasing technique (e.g., in graph neural networks, Palowitch & Perozzi, 2020). Similar to our work, orthogonalization can be
+
+Table 1. Estimated influence (coefficients) of sex, age, and race on the model's predictions when using self-supervised embeddings of the MIMIC-CXR dataset (Sellergren et al., 2022) on the label Pleural Effusion. The first row shows uncorrected values, the second row the results after classical orthogonalization, and the third row after applying our generalized orthogonalization. p-values in brackets indicate whether these coefficients represent a significant influence (p-values smaller than 0.05 show a  $\times$  -sign, otherwise a  $\checkmark$  -sign).  
+
+<table><tr><td></td><td colspan="2">Sex (male)</td><td>Age</td><td>Race (Black)</td><td>Race (White)</td></tr><tr><td>Without correction</td><td>0.064 (9.3e-9) X</td><td>2.237 (&lt; 2e-16) X</td><td>-0.483 (&lt; 2e-16) X</td><td>-0.050 (0.0743) ✓</td><td></td></tr><tr><td>Classical orthogonalization Cl</td><td>0.014 (0.192) ✓</td><td>0.453 (&lt; 2e-16) X</td><td>-0.117 (&lt; 2e-4) X</td><td>-0.015 (0.603) ✓</td><td></td></tr><tr><td>Generalized orthogonalization Ch</td><td>0.001 (0.994) ✓</td><td>-0.011 (0.689) ✓</td><td>-0.001 (0.753) ✓</td><td>-0.000 (0.904) ✓</td><td></td></tr></table>
+
+applied as part of a neural network (e.g., Lu et al., 2021) or post-model fitting (Rügamer, 2023). The latter approach also has a link to double and debiased machine learning in econometrics (see, e.g., Chernozhukov et al., 2018), where a related correction is performed to allow for inference statements about a treatment variable after removing (potentially non-linear) nuisance effects.
+
+Fair Machine Learning Another related strand of literature is fair machine learning (Chen et al., 2022). As we discuss later in our work, methods to achieve fairness are closely linked to orthogonalization and vice versa. A wide range of notions and approaches exists to operationalize the concept of fairness (see, e.g., Mehrabi et al., 2021; Pes-sach & Shmueli, 2022, for literature reviews). Proposed methods are applied at different steps, e.g., in data preprocessing (Calmon et al., 2017) or post-processing (Hardt et al., 2016; Sattigeri et al., 2022; Xu et al., 2022), and have been derived for different model classes, such as generalized linear models (GLMs; Do et al., 2022) or kernel learning (Pérez-Suay et al., 2017). These approaches are usually implemented as constrained (see, e.g., Komi-yama et al., 2018; Zafar et al., 2019) or regularized optimization problems (e.g., Scutari et al., 2022; Do et al., 2022) and require making a trade-off between model performance and the amount of achieved fairness. In contrast, our approach forces the model predictions to be uncorrelated with the protected information, thereby ensuring that they can not be inferred to any degree from the predictions using a linear model. While this potentially comes at the cost of a loss in model performance, a conservative correction routine that errs on the side of caution is required in many risk-averse domains such as medical applications with sensitive patient information (Glocker et al., 2023a), or when the goal is to achieve model identifiability (Rügamer, 2023). A limitation of presented fairness solutions is that the sensitive information is made explicit in the model under the assumption of a linear effect on the outcome (see, e.g., Scutari et al., 2022).
+
+Despite the versatility of orthogonalization in practice, most existing methods share the same working principle which assumes linearity in both the model that needs to be corrected and the correction function. In many real-world situations, however, models include non-linearities that render the clas-
+
+sical orthogonalization unable to fully correct the model (cf. Table 1 with details in Section 4.2.1).
+
+# 1.2. Our Contribution
+
+In this paper, we extend the orthogonalization operation to non-linearity in both the model that needs to be corrected and the correction function. We derive a correction routine that works for a variety of non-linear models, including GLMs and neural networks. Moreover, we show how to extend our approach to arbitrary tensor-valued predictions. This allows our method to be flexibly inserted anywhere in common neural network architectures and thereby further generalizes the idea of normalization. Our experimental results demonstrate the efficacy of our approach, showing how 1) sensitive information can be successfully protected in GLMs, 2) neural networks with non-linear activation functions can be normalized for metadata during training, and 3) orthogonalization effectively corrects pre-trained embeddings for unwanted attributes.
+
+# 2. Orthogonalization
+
+Given an input  $X \in \mathbb{R}^{n \times p}$ ,  $p \leq n$  with full column rank, the linear predictor in a parametric learning model (or the pre-activation of a fully-connected neural network layer) is given by  $X\beta$ , where  $\beta \in \mathbb{R}^p$  denotes the model's or layer's weights. In a linear model, we can write the least squares predictions  $\hat{\pmb{y}}$  as a function of the inputs  $X$  and targets  $\pmb{y}$ :
+
+$$
+\hat {\boldsymbol {y}} = \boldsymbol {X} \hat {\boldsymbol {\beta}} = \mathcal {P} _ {\boldsymbol {X}} \boldsymbol {y},
+$$
+
+where  $\mathcal{P}_{\pmb{X}} = \pmb {X}(\pmb{X}^{\top}\pmb {X})^{-1}\pmb{X}^{\top}$  is the projection matrix projecting  $\pmb{y}$  onto  $\operatorname {span}(\pmb {X})$  , the space spanned by the columns of  $\pmb{X}$  . Instead of projecting onto  $\operatorname {span}(\pmb {X})$  , we can also project onto the orthogonal complement  $\operatorname {span}(\pmb {X})^{\perp}$  using  $\mathcal{P}_{\pmb{X}}^{\perp}\pmb {y} = (\pmb{I}_n - \mathcal{P}_\pmb {X})\pmb {y},$  where  $\pmb{I}_n$  is the identity matrix. Hence, when left-multiplying a term by  $\mathcal{P}_{\overline{X}}^{\perp}$  , we orthogonalize this term (w.r.t. features  $\pmb{X}$  ). Ultimately — whether the motivation is debiasing, protecting information, or identifiability — the universal recipe for most approaches discussed in Section 1.1 is to use the described orthogonalization operation. Given protected features  $\pmb{X}$  , which are not supposed to influence the modeling process, and a model that uses a different set of features  $Z\in \mathbb{R}^{n\times q}$ $q\geq p$  for prediction, we
+
+![](images/19b1ba930d728946b3a8a7744723c1fe33509039c5ffdaa81475f7ea1dcb4661.jpg)
+
+![](images/b689f641c7171b72345a711c0a05cc681fc0873ceacfa999c58f366b656d48c3.jpg)
+
+![](images/9b422e4e113f1dcba8955a4e132197ce54b7f2943e8c139adf44ec7c0d9d9cee.jpg)
+
+![](images/3c0f615f55bebab1404eac646478c7822c59c5132bcd46eda1d1109db4107b99.jpg)  
+Figure 1. Exemplary optimization process for a fixed number of iterations (converging towards the black crosses) for logistic regression with features  $z_{1}, z_{2}$ , weights  $\gamma_{1}, \gamma_{2}$ , and one protected feature  $x_{1}$  correlated with  $z_{1}$ . The upper row shows the loss surface and optimization path for the three different methods (columns) a small step size. The bottom row shows the correlation between the model's prediction and the protected feature along the optimization path, where only the generalized orthogonalization yields predictions uncorrelated with the sensitive information.
+
+![](images/e55b779d606a73f2c105e0591d712f06c85325cebc7df6d0ebaff97bec3caaf1.jpg)
+
+![](images/9c249948d032f9978fe02b99363528d3897be261e18744944e7c4cf6e5dcb1f7.jpg)
+
+can "correct" the model (or, in this case equivalently, the features  $Z$ ) by fitting the model with  $Z^c \coloneqq \mathcal{P}_X^\perp Z$  instead of  $Z$ . This is equivalent to removing all linear  $X$ -information in  $Z$ . Note that  $Z$  might also contain features from  $X$ , but the correction is supposed to work for unobserved relationships between  $Z$  and  $X$ . We call this correction function  $\mathcal{C}^l$  in the following.
+
+To see that  $\mathcal{C}^l$  in fact removes the effect of protected features  $X$  from  $Z$  in the prediction, we can check whether the corrected predictions  $\hat{\pmb{y}}^c = \pmb{Z}^c\hat{\pmb{\gamma}}^c$ , based on the orthogonalized features  $\pmb{Z}^{c} = \mathcal{P}_{\pmb{X}}^{\perp}\pmb{Z}$  and a corrected weight vector  $\hat{\gamma}^c\in \mathbb{R}^q$ , can be linearly explained by the features  $X$  to any degree. This can be done by regressing  $\hat{\pmb{y}}^c$  on  $X$ . If the correction is successful, the debiased effect
+
+$$
+\hat{\boldsymbol{\beta}}^{c}:= \operatorname *{arg  min}_{\boldsymbol{\beta}^{c}}\| \hat{\boldsymbol{y}}^{c} - \boldsymbol {X}\boldsymbol{\beta}^{c}\|_{2}^{2}
+$$
+
+of  $X$  in the linear model with outcome  $\hat{\pmb{y}}^c$  and features  $X$  should be  $\hat{\beta}^c\equiv 0$ . Plugging in the terms of the ordinary least squares solution for both  $\hat{\beta}^c$  and  $\hat{\gamma}$ , we get
+
+$$
+\begin{array}{l} \hat {\boldsymbol {\beta}} ^ {c} = \left(\boldsymbol {X} ^ {\top} \boldsymbol {X}\right) ^ {- 1} \boldsymbol {X} ^ {\top} \hat {\boldsymbol {y}} ^ {c} = \left(\boldsymbol {X} ^ {\top} \boldsymbol {X}\right) ^ {- 1} \boldsymbol {X} ^ {\top} \boldsymbol {Z} ^ {c} \hat {\boldsymbol {\gamma}} \\ = \left(\boldsymbol {X} ^ {\top} \boldsymbol {X}\right) ^ {- 1} \boldsymbol {X} ^ {\top} \mathcal {P} _ {\boldsymbol {X}} ^ {\perp} \boldsymbol {Z} (\boldsymbol {Z} ^ {\top} \mathcal {P} _ {\boldsymbol {X}} ^ {\perp} \boldsymbol {Z}) ^ {- 1} \boldsymbol {Z} ^ {\top} \mathcal {P} _ {\boldsymbol {X}} ^ {\perp} \boldsymbol {y}, \\ \end{array}
+$$
+
+which yields  $\mathbf{0}$  as  $X^{\top} \mathcal{P}_{\overline{X}}^{\perp} = \mathbf{0}_{p \times n}$ . This confirms that we have successfully removed any linear effect of  $X$  from the predictions  $\hat{\boldsymbol{y}}^{c}$  and corrected features  $Z^{c}$ . So when  $Z$  is, e.g., a pre-trained embedding, using  $Z^{c}$  instead of  $Z$  in a downstream task removes the partial influence of protected features  $X$  contained in  $Z$ , as we first project the original embedding onto a space that is orthogonal to the space spanned by the protected features  $X$ .
+
+# 3. Orthogonalization for Models with Non-linearities
+
+While the classical orthogonalization allows correcting for metadata of images (Lu et al., 2021) or identifying contributions in additive predictors (Rügamer, 2023), all previous methods fail to provide a valid correction in the case of a non-linear, element-wise transformation  $h$  applied to  $Z\gamma$ , i.e.,  $\hat{\pmb{y}} = h(Z\pmb{\gamma})$ , such as in GLMs or most neural network layers (see Figure 1 for an example). In the following, we present an extension to previously proposed orthogonalization approaches that also encompass models with non-linear transformations of the linear predictor, thereby accounting for various important use cases. For simplicity, we motivate our approach using a GLM, but also discuss its embedding in neural architectures in Sections 3.2-3.4. Before deriving the theoretical details, we provide an instructive example in the following.
+
+Example 1. Following Weber et al. (2023), assume we are given an embedding  $Z \in \mathbb{R}^{n \times q}$  from a medical imaging task and want to ensure that by using this embedding, we do not share any protected patient information encoded in  $X$ . Assume that the embedding contains information from  $X$  (which represents a very real problem; see Glocker et al., 2023b). When predicting the patients' disease status using the embedding  $Z$  in a GLM with  $\hat{y} = h(Z\hat{\gamma})$ , where  $h$  is the sigmoid function, we risk making decisions implicitly based on protected features. Instead, we have to come up with a corrected GLM routine such that the corrected nonlinear model predictions  $\hat{y}^c$  do not utilize or leak patient information  $X$ .
+
+![](images/30b9c84ae0a94b64aad13ddfb081d6bef4f48495c9b0b6415d186e73ff9e6146.jpg)  
+Figure 2. Workflow of generating predictions via model  $\mathcal{M}^p$  and checking the influence of  $\mathbf{X}$  on the resulting predictions  $\hat{\mathbf{y}}$  using  $\mathcal{M}^e$ . Green parts indicate the orthogonalization applied to  $\mathcal{M}^p$ .
+
+# 3.1. Types of Orthogonalizations
+
+We first start by examining the properties of the original orthogonalization  $\mathcal{C}^l$  when applied to models with non-linear activation functions. In particular, it is worth noting that using  $Z^{c} = \mathcal{P}_{X}^{\perp}Z$  from Section 2 can be used in a GLM when the goal is to orthogonalize the model's additive predictor  $\eta = Z\gamma$ . In this case, a corrected version  $\eta^{c} = Z^{c}\gamma$  will be orthogonal to the column space of  $X$  for any  $\gamma$  and hence, before the non-linear transformation  $h$ , "free" of any (linear) effect of protected features  $X$ . In other words, regressing  $\eta^{c}$  on  $X$  will result in effects  $\hat{\beta}^{c} = 0$  as derived for the linear case in Section 2. In contrast, when transforming  $\eta^{c}$  with  $h$ , the model space becomes non-linear and regressing  $\hat{y} = h(\eta^{c})$  on  $X$  will generally not yield effects  $\hat{\beta}^{c} = 0$ , i.e., the transformed predictor  $h(\eta^{c})$  still potentially contains  $X$ -information (cf. Appendix A for an illustrative explanation). As illustrated in Example 1, such non-linear transformations are ubiquitous in machine learning and employed in most classification tasks.
+
+When using a non-linear activation function  $h$  in the prediction of  $y$  based on  $Z$ , a natural choice is to also use the same non-linear transformation to check the model's prediction for any  $X$  influence. To better clarify the different types of models involved in an orthogonalization and to unify previous endeavors, we define the following terms for the workflow described in Figure 2:
+
+Definition 1. Given features  $Z \in \mathbb{R}^{n \times q}$ , a monotonic activation function  $h$ , loss function  $\ell: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$  and protected features stored in a full column rank matrix  $X \in \mathbb{R}^{n \times p}$ , we define
+
+- the prediction model  $\mathcal{M}^p: \mathbb{R}^{n \times q} \to \mathbb{R}^n$ ,  $Z \mapsto \hat{y} = h(Z \hat{\gamma})$ , the model using  $Z$  to generate predictions  $\hat{y}$  for an outcome of interest  $y$ , with parameters  $\hat{\gamma}$  learned by minimizing  $\ell$ ;  
+- the correction routine  $\mathcal{C}^h$ : A routine correcting  $\mathcal{M}^p$  to produce corrected predictions  $\hat{\boldsymbol{y}}^c$ ;
+
+- the evaluation model  $\mathcal{M}^e: \mathbb{R}^{n \times p} \times \mathbb{R}^n \to \mathbb{R}^p$ ,  $(X, \hat{\boldsymbol{y}}^c) \mapsto \hat{\boldsymbol{\beta}}^c$ , the model minimizing  $\ell(\hat{\boldsymbol{y}}^c, h(X \hat{\boldsymbol{\beta}}^c))$  to check whether the influence of  $X$  on the corrected predictions  $\hat{\boldsymbol{y}}^c$  from  $\mathcal{M}^p$  has been successfully removed by  $\mathcal{C}^h$ .
+
+The previous definition assumes that the activation function  $h$  is the same for all three components  $\mathcal{M}^p, \mathcal{M}^e$ , and  $\mathcal{C}^h$ .
+
+Before introducing our new orthogonalization algorithm when the activation function  $h$  is used, we more formally define what we mean by "correcting for the influence of  $X$ ".
+
+Definition 2. Given the setup and models in Definition 1, we say the model predictions  $\hat{\pmb{y}}$  of  $\mathcal{M}^p$  are corrected (or orthogonalized) if  $\mathcal{M}^e$  yields coefficient values  $\hat{\beta}^c = 0$ .  $\hat{\pmb{y}}^c$  are then called corrected predictions.
+
+While these definitions imply a similar goal as pursued in fair machine learning, our approach uses a slightly different measure of "unbiasedness" that coincides with the previously introduced normalization (Lu et al., 2021) as well as the orthogonalization for identifiability (Rugamer et al., 2023). In contrast, in fair machine learning, the level of unfairness is, e.g., defined as the proportion of the variance of  $\hat{\pmb{y}}$  that can be explained by  $X$  (see, e.g., Scutari et al., 2022, or further explanations in Appendix B). Using this notion, a model adhering to Definition 2 is perfectly fair, but the opposite is not always true. This is because fairness is achieved using a criterion different from the one in Definition 2. In addition, these approaches often do not fully enforce fairness to preserve other model properties, in most cases only use the  $L_{2}$  loss for  $\ell$  and assume a linear relationship between  $X$  and  $\pmb{y}$ .
+
+# 3.2. Orthogonalization for GLM-type Routines
+
+We first extend the classical orthogonalization of Section 2 motivated by linear models to GLMs. This model class is described by a distributional assumption inducing a loss function  $\ell$  defined by the corresponding negative log-likelihood and strictly monotone activation function  $h$  (equal to the so-called inverse link function in GLMs). The implied loss function comprises many common machine learning setups: binary classification using sigmoid activation, multi-class classification using a softmax-type activation, or, e.g., a count regression using the exp-activation function. When applying Definition 2 to these GLMs using their canonical link function, we say that the correction of a model  $\mathcal{M}^p$  yielding  $\hat{\boldsymbol{y}}^c$  is successful if the evaluation model  $\mathcal{M}^e$  defined by the minimizer
+
+$$
+\hat {\boldsymbol {\beta}} ^ {c} = \underset {\boldsymbol {\beta} ^ {c}} {\arg \min } \ell (\hat {\boldsymbol {y}} ^ {c}, h (\boldsymbol {X} \boldsymbol {\beta} ^ {c})) \tag {1}
+$$
+
+is a null model, i.e.,  $\hat{\beta}^c\equiv 0$ . Note that we here assume a model without intercept for better readability<sup>1</sup>. While the solution of (1) is known not to have an analytical solution in general (see, e.g., Nelder & Wedderburn, 1972), the GLM poses a convex optimization problem and can be solved efficiently with iteratively reweighted least squares or Newton-type procedures (Gill et al., 2019).
+
+Importantly, the iterates of these Newton-type optimization procedures can be represented as  $\hat{\beta}^{[t]} = (X^{\top}\Upsilon^{[t]}X)^{-1}X^{\top}\Upsilon^{[t]}r^{[t - 1]}$ ,  $t\in \mathbb{N}$ , where the weights  $\Upsilon^{[t]}\in \mathbb{R}^{n\times n}$  are given by a diagonal matrix and  $\boldsymbol {r}^{[t]}\in \mathbb{R}^n$  is a working response (see Appendix D for details). As both  $\Upsilon^{[t]}$  and  $\boldsymbol {r}^{[t - 1]}$  are fixed after fitting the evaluation model, we can make use of the approximate linearity of both  $\mathcal{M}^p$  and  $\mathcal{M}^e$  at convergence to derive our generalized orthogonalization. A helpful intermediate result is the following:
+
+Lemma 3. Given any GLM model  $\mathcal{M}^p$  with predictions  $\hat{\mathbf{y}}$ , and GLM model  $\mathcal{M}^e$  with features  $\mathbf{X}$
+
+$$
+\hat {\mathbf {y}} ^ {c} = \mathcal {P} _ {\boldsymbol {X}} ^ {\perp} \hat {\mathbf {y}} + h (0) \mathbf {1} _ {n} \tag {2}
+$$
+
+defines corrected predictions yielding  $\hat{\beta}^c = 0$ .
+
+Lemma 3 is an interesting and important finding to construct a correction, stating that despite the non-linearity of  $\mathcal{M}^e$ , it is sufficient to (linearly) orthogonalize predictions of  $\mathcal{M}^p$  using the orthogonal projection  $\mathcal{P}_{\overline{X}}^{\perp}$ . The derivation and proof can be found in Appendix E.1. However, as we discuss in the following paragraph, it is not straightforward to relate this finding to a corrected GLM model  $\mathcal{M}^p$ .
+
+Deriving corrected coefficients In order to derive a corrected GLM routine for corrected coefficients  $\hat{\gamma}^c$  that are consistent with  $\hat{\pmb{y}}^c$ , a naive approach is to simply invert their definition  $\hat{\pmb{y}}^c = h(Z\hat{\gamma}^c)$ , yielding
+
+$$
+\hat {\boldsymbol {\gamma}} ^ {c} = \boldsymbol {Z} ^ {\dagger} h ^ {- 1} (\hat {\boldsymbol {y}} ^ {c}) = \boldsymbol {Z} ^ {\dagger} h ^ {- 1} (\mathcal {P} _ {\boldsymbol {X}} ^ {\perp} \hat {\boldsymbol {y}}),
+$$
+
+where  $Z^{\dagger} = (Z^{\top}Z)^{-1}Z^{\top}$  is the pseudo-inverse of  $Z$ . However, as  $\mathrm{dom}(\mathcal{P}_{\pmb{X}}^{\perp}\hat{\pmb{y}})$  is not necessarily a subset of  $\mathrm{dom}(\hat{\pmb{y}})$ ,  $h^{-1}$  might not be defined on this new domain. In classification tasks, for example, this could lead to predicted probabilities  $\hat{\pmb{y}}^c \notin (0,1)$  for which the inverse sigmoid function is not defined. Since this problem will occur whenever non-linear functions  $h$  or  $h^{-1}$  are employed, we instead have to adapt the GLM optimization routine as described in the following.
+
+Corollary 4. Given activation function  $h$ , prediction model  $\mathcal{M}^p$  and evaluation model  $\mathcal{M}^e$  with features  $X$ , the solution  $\gamma^c$  to the optimization problem
+
+$$
+\operatorname {a r g} \min  \ell (\boldsymbol {y}, h (\boldsymbol {Z} \boldsymbol {\gamma}))
+$$
+
+$$
+\gamma \tag {3}
+$$
+
+$$
+s. t. \left\| \left(\boldsymbol {X} - n ^ {- 1} \mathbf {1} _ {n} \mathbf {1} _ {n} ^ {\top} \boldsymbol {X}\right) ^ {\top} h (\boldsymbol {Z} \boldsymbol {\gamma}) \right\| _ {2} ^ {2} = 0
+$$
+
+produces corrected coefficients satisfying  $\hat{\beta}^c = 0$ .
+
+The mathematical details and derivations are given in the Appendix E.2. While various routines exist to solve (3), the problem turns out to be hard in practice, in particular in applications that are concerned with fairness, such as the ones addressed in Section 4.1. We found the modified differential multiplier method (Platt & Barr, 1987) to be very effective and describe its details in Appendix E.2. Further, note that in the case of an identity activation function  $h$ , the derived optimization problem in Corollary 4 is also directly solved by using  $\mathcal{C}^h = \mathcal{C}^l$ .
+
+# 3.3. Orthogonalization with Piece-wise Linear Activations
+
+While the previous orthogonalization can be used in many different situations, intermediate network layers are often specified with piece-wise linear activation functions not comprised by the GLM framework, most notably the ReLU function. As for canonical activation functions in GLMs, the ReLU activation  $\mathrm{ReLU}(X\beta) = X\beta \circ \mathbb{1}(X\beta >0)$  is applied element-wise, with  $\circ$  denoting the Hadamard product and  $\mathbb{1}(\cdot)$  an indicator function individually evaluated for all entries of the vector. While at first glance the definition of the ReLU activation might suggest that  $\mathrm{ReLU}(X\beta)\in \operatorname {span}(X)$  and hence  $\mathcal{C}^l$  from Section 3.1 is sufficient for successful orthogonalization, this is not the case as can be verified with a simple counterexample.
+
+Example 2. Assume  $\mathbf{X} = (1, -1)^{\top} \in \mathbb{R}^{2}$  and  $\beta = 1$ . Then  $\operatorname{ReLU}(\mathbf{X}\beta) = (1,0) \notin \operatorname{span}(\mathbf{X})$ .
+
+Although  $\mathrm{ReLU}(\pmb {X}\beta)$  does not necessarily lie in  $\operatorname {span}(\pmb {X})$  we can derive the following result
+
+Theorem 5. Given a prediction model  $\mathcal{M}^p$  with ReLU activation, and an evaluation model  $\mathcal{M}^e$  with ReLU activation and  $L_{2}$  loss, a  $\mathcal{C}^h$ -orthogonalization of the form  $Z^{c} = \mathcal{P}_{X}^{\perp}Z$ , will yield  $\hat{\beta}^c = 0$ .
+
+The proof of Theorem 5 is given in the Appendix.
+
+# 3.4. Orthogonalization for Tensor Predictions
+
+We now show that our concept of orthogonalization is not restricted to regression models, but can also be used to remove information from learned representations in layers of neural networks. Such a correction should be able to deal with prediction models (or layers) with tensor-shaped predictions (e.g., present in early layers of convolutional networks) and we show how to adapt previous results accordingly. We first define the linear evaluation model  $\mathcal{M}^e$  with  $L_{2}$ -loss  $\ell$  as the solution of the following tensor-on-vector regression:
+
+$$
+\hat {\mathfrak {B}} ^ {c} = \underset {\mathfrak {B} ^ {c}} {\arg \min } \ell \left(\hat {\mathfrak {Y}} ^ {c}, \boldsymbol {X} \times_ {1} \mathfrak {B} ^ {c}\right), \tag {4}
+$$
+
+where the outcome tensor  $\hat{\mathfrak{Y}}^c\in \mathbb{R}^{n\times d_1\times \dots \times d_R}$  is the corrected version of an (intermediate) non-linear prediction  $\hat{\mathfrak{Y}}$ $X\in \mathbb{R}^{n\times p}$  is a feature matrix as before,  $\mathfrak{B}^c\in$ $\mathbb{R}^{p\times d_1\times \dots \times d_R}$  a tensor-valued regression coefficient, and  $\times_{1}$  denotes the 1-mode product. For a successful correction  $\mathcal{C}^h$  from which we obtain  $\hat{\mathfrak{Y}}^c$  , it must hold  $\hat{\mathfrak{B}}^c\equiv \mathbf{0}$  in (4), which can be achieved by projecting vec(  $\hat{\mathfrak{V}})$  onto  $\operatorname {span}(\boldsymbol {X})^\perp$  as before. Indeed, using a vectorized reformulation, we can prove that this yields corrected predictions (see Appendix E.4 for details):
+
+Corollary 6. Given the set-up around the evaluation model defined by (4), the corrected predictions satisfying  $\hat{\mathfrak{B}}^c\equiv \mathbf{0}$  are given by
+
+$$
+\hat {\mathcal {Y}} ^ {c} = \mathcal {P} _ {\boldsymbol {X}} ^ {\perp} \times_ {1} \hat {\mathcal {Y}}. \tag {5}
+$$
+
+While the result in (5) can be extended to GLM-type nonlinearities, and hence an orthogonalization as the one in Section 3.2 can be derived, we here focus on the ReLU activation function, arguably the pervasive type of non-linearity in intermediate DNN layers. Hence, we consider an evaluation model  $\mathcal{M}^e$  with ReLU activation defined by
+
+$$
+\hat {\mathfrak {B}} ^ {c} = \underset {\mathfrak {B} ^ {c}} {\arg \min } \ell (\hat {\mathfrak {Y}} ^ {c}, \operatorname {R e L U} (X \times_ {1} \mathfrak {B} ^ {c})), \tag {6}
+$$
+
+and a prediction model  $\mathcal{M}^p$  yielding predictions  $\hat{\mathfrak{V}} = \mathrm{ReLU}(\bar{\mathfrak{V}})$  with pre-activation  $\mathfrak{V}$ . In contrast to previous sections, we here focus on correcting the predictions and not the effects of  $\mathcal{M}^p$ , as predictions of an intermediate layer in a large neural network are not necessarily formed by a linear operation. We obtain the following result:
+
+Corollary 7. Assuming the evaluation model (6), the corrected pre-activations  $\mathfrak{Y}^c$  yielding corrected predictions  $\hat{\mathfrak{Y}}^c$  following ReLU activation are given by  $\bar{\mathfrak{Y}}^c = \mathcal{P}_{\bar{\boldsymbol{X}}}^\perp \times_1\bar{\mathfrak{Y}}$ .
+
+See Appendix E.5 for a proof. In other words, by multiplying the first dimension of a non-activated output of some layer with  $\mathcal{P}_{\overline{X}}^{\perp}$  from the left before applying the ReLU activation, we obtain corrected predictions in the next layer.
+
+# 4. Numerical Experiments
+
+In Section 4.1, we first check whether sensitive information can be successfully protected in GLMs with our approach, using both synthetic and real-world datasets. We then show in various settings how our method corrects learned representations for potential biases (Section 4.2), and finally investigate its effectiveness as an online version applied during training (Section 4.3). The details of all experiments are provided in Appendix F. In all experiments, categorical information in the protected features is included in  $Z$  as one-hot encoded features with the first column removed (as we don't want to orthogonalize or penalize the model for a constant intercept term).
+
+# 4.1. Generalized Linear Model
+
+In the following, we focus on real-world applications but provide synthetic data examples in Appendix G.1, confirming that  $\mathcal{C}^h$  works as intended for GLMs in a variety of settings.
+
+# 4.1.1. ADULT INCOME DATA
+
+Using the adult income data also investigated in Xu et al. (2022) to analyze algorithm fairness, we check the efficacy of our approach for protected features sex and race. We therefore first fit our prediction model, a GLM with features age, work class, education, marital status, relationship, and working hours per week, to predict the person's income binarized into  $\leq 50\mathrm{k}$  and  $>50\mathrm{k}$  (as provided by the original dataset) and then correct for the protected features.
+
+Results We find (Table 2, first row) that the income predictions are significantly influenced if the individual is white and/or male with a multiplicative increase in the odds-ratio of approximately  $\exp(0.522) = 1.68$  and  $\exp(1.066) = 2.90$  for earning more than  $50\mathrm{k}$  for white individuals and males, respectively. Similar results are inferred from Xu et al. (2022)'s approach. When applying our framework and adjusting for sex and race, we can see that all effects of these protected features are estimated to be close to zero and are non-significant.
+
+Table 2. Estimated coefficients from  $\mathcal{M}^e$  with corresponding p-values in brackets when not correcting the model (first column), when using Xu et al. (2022) (second column), and when correcting with  $\mathcal{C}^h$  (last column). P-values smaller than 0.05 show a  $X$ -sign, otherwise a  $\checkmark$ -sign.  
+
+<table><tr><td></td><td>w/o correction</td><td>Xu et al. (2022)</td><td>using Ch</td></tr><tr><td>Sex (male)</td><td>1.066 (&lt; 2e-16) X</td><td>1.255 (&lt; 2e-16) X</td><td>-0.000 (0.998)✓</td></tr><tr><td>Race (Asian)</td><td>0.263 (0.227)✓</td><td>0.859 (2e-4) X</td><td>0.011 (0.954)✓</td></tr><tr><td>Race (Black)</td><td>-0.094 (0.577)✓</td><td>0.191 (0.319)✓</td><td>0.008 (0.957)✓</td></tr><tr><td>Race (Other)</td><td>-0.165 (0.594)✓</td><td>-0.131 (0.706)✓</td><td>-0.131 (0.609)✓</td></tr><tr><td>Race (White)</td><td>0.522 (0.001) X</td><td>0.928 (4e-7) X</td><td>0.005 (0.969)✓</td></tr></table>
+
+# 4.1.2. FAIRNESS BENCHMARK
+
+While fairness methods target a slightly different goal than orthogonalization, it might still be interesting to see whether our criterion for orthogonalization (cf. Definition 2) can also be met by approaches suggested in the fairness literature. We therefore run a comprehensive analysis on fairness datasets with non-Gaussian outcomes and compare the results of various fairness methods with our orthogonalization. In the case of GLMs, closely related methods are fair (logistic / Poisson) regression with different optimization strategies. Komiyama et al. (2018) bounds the proportion of explained variance by protected features (we refer to this as Komiyama1). Another variant by the same authors first regresses the fitted values against the protected features (re
+
+Figure 3. Comparison of orthogonalization properties of fairness methods in comparison with our approach. Ideally, methods should yield effects of zero (in the left plot) and large p-values (in the right plot). Missing boxes indicate that the method did not converge.  
+![](images/3d99d4c7049f0b7d91580b15e791aff20d043495190ae4dd452351b341a4e0e3.jpg)  
+Komiyama1 Komiyama2 Berk Zafar Ours
+
+![](images/70ddcc8fc358449873281b707aaff26e162617eb3f52df4b5e31181379ffba55.jpg)
+
+ferred to as Komiyama2). An alternative is the approach by Berk et al. (2017) enforcing individual fairness by penalizing pairs of observations that have different outcomes for the same protected features (referred to as Berk). A fourth approach we compare against is the one of Zafar et al. (2019) (referred to as Zafar) enforcing fairness by bounding the covariance of protected features and other predictors. We run these and our method on the benchmark datasets adult, bank, compas, and health. retirement provided in Scutari (2023) containing various protected features such as race, sex, or marriage.
+
+Results The results are depicted in Figure 3 showing that all fairness approaches show significant feature effects with effect strengths often different from 0. In contrast, our approach adheres to the orthogonality requirement from Definition 2, yielding effects close to 0 with no significant impact. As stated earlier, this does not imply unfairness of the other approaches, but demonstrates that fairness approaches cannot be simply used as a substitute to achieve orthogonality. As orthogonality might potentially decrease model performance, Table 3 additionally compares the model performances with and without the correction. Since most of
+
+Table 3. Test set prediction performances with/out correction.  
+
+<table><tr><td>Dataset (Metric)</td><td>w/o Corr.</td><td>w/ Corr.</td></tr><tr><td>adult (Accuracy)</td><td>0.854</td><td>0.833</td></tr><tr><td>bank (Accuracy)</td><td>0.899</td><td>0.887</td></tr><tr><td>compas (Accuracy)</td><td>0.739</td><td>0.724</td></tr><tr><td>health (Root mean squared error)</td><td>0.750</td><td>0.789</td></tr></table>
+
+the datasets contain the protected features also in  $Z$ , i.e., as features for the prediction model, the prediction performance with orthogonalization will naturally decrease (if not, then the protected features would likely not bias the model and raise fairness issues in the first place). However, as can be seen in Table 3, the decrease in performance is often negligible with a maximum of  $2\%$  decrease in accuracy for compas.
+
+# 4.2. Orthogonalization of Learned Representations
+
+# 4.2.1. COMPARING  $\mathcal{C}^l$  AND  $\mathcal{C}^h$  ON IMAGE EMBEDDINGS
+
+Going back to our instructive Example 1, we use self-supervised embeddings for the MIMIC Chest X-Ray dataset (Johnson et al., 2019; Sellersgren et al., 2022) known to have encoded race, sex and age (Glocker et al., 2023b), and try to remove this implicit information using orthogonalization. Following Glocker et al. (2023b), we first validate that meta-information such as race, in fact, is influencing predictions in a downstream task. As  $\mathcal{M}^p$  we use a GLM to predict whether the patient suffers from pleural effusion using only the embedding as features. We then check whether a  $\mathcal{C}^l$ -orthogonalization is sufficient to remove this information, but in contrast to the Adult data application, check on the level of probabilities where  $\mathcal{C}^l$  should fail. Finally, we apply  $\mathcal{C}^h$ -orthogonalization to obtain the corrected predictions.
+
+Results Table 1 (page 2) summarizes the results, confirming our hypotheses and demonstrating that our approach effectively removes the meta-information from embeddings. In this case, removing the features even results in a slight improvement in prediction performance with a 0.02 increase in AUC.
+
+# 4.2.2. POST-HOC ORTHOGONALIZATION FOR NEURAL REPRESENTATIONS
+
+As another demonstration, we analyze the race and age bias in a learned neural representation of faces from the UTKFace dataset (Zhang et al., 2017), where previous literature suggests, among other biases, a misalignment for black women (see, e.g., Krishnan et al., 2020). We first train a ResNet50 (He et al., 2016) on the provided images to classify the sex of the depicted person. We then check the influence of race and age on the predictions using a logistic regression. In order to correct for biases, we extract the weights in the ResNet's penultimate layer and apply the  $\mathcal{C}^h$ -orthogonalization by fitting a corrected GLM (following Corollary 4) with learned representations as features and sex
+
+as the binary outcome. Finally, we again check the influence of race and age on the model's predictions.
+
+Results Results are summarized in Table 4, again depicting coefficients and p-values, and confirm the effectiveness of our approach in blinding the network for certain features (here age and race). The orthogonalization in this case comes at the cost of an AUC reduction from 0.831 (w/o correction) to 0.826 (with  $\mathcal{C}^h$ ), which seems to be a reasonable trade-off to correct for age and race effects.
+
+Table 4. Estimated coefficients from  $\mathcal{M}^e$  with corresponding p-values in brackets when not correcting the model (first column) and when correcting the model for age and race (second column). Significant influences are highlighted in bold for an  $\alpha$ -level of 0.05.  
+
+<table><tr><td></td><td>w/o correction</td><td>Chorthogonalization</td></tr><tr><td>Age</td><td>-0.017 (&lt; 2e-16) X</td><td>-0.000 (0.976)✓</td></tr><tr><td>Race (Black)</td><td>-0.352 (8.68e-10) X</td><td>0.009 (0.872)✓</td></tr><tr><td>Race (Indian)</td><td>-0.620 (&lt; 2e-16) X</td><td>0.008 (0.894)✓</td></tr><tr><td>Race (Other)</td><td>-0.315 (4.05e-05) X</td><td>-0.002 (0.979)✓</td></tr><tr><td>Race (White)</td><td>-0.377 (5.65e-13) X</td><td>-0.008 (0.872)✓</td></tr></table>
+
+# 4.2.3. WORD EMBEDDINGS
+
+As a last example, we demonstrate that our approach is not limited to tabular and image data. As, e.g., discussed in Bolukbasi et al. (2016), text embeddings, as well, can have biases encoded, and addressing this property seems of great relevance given the rise of GPT(-like) models which have also been shown to not always be neutral in their generation (see, e.g., Ray, 2023). To show that the proposed method also works for the correction of text embeddings in a nonlinear model, we use the movies review dataset (Maas et al., 2011), which we enhance with the gender information of the protagonist and the movie director. Using an LSTM model and a learned text embedding  $Z$  (see Appendix F for details), we try to predict the number of reviews for each movie (i.e., a Poisson-distributed outcome). We do this once without correction and once by correcting the embeddings  $Z$  of size  $100 \times 100$  of every movie description following the results from Corollary 7 using the available gender information as  $X$ .
+
+Results When not correcting the model's prediction, we obtain p-values  $< 2\mathrm{e}-16$  for both gender effects, suggesting a significant influence of gender on the predictions  $\hat{\mathbf{y}}$ . After correction, p-values are 0.993 and 1.0, implying a successful embedding correction with only a small increase in root mean squared error of  $1\%$  compared to the model without correction.
+
+# 4.3. Online Orthogonalization
+
+Finally, we test how our approach performs when applied in an online fashion by iteratively optimizing a small CNN
+
+and orthogonalizing the network with respect to some fixed features  $X$  as proposed in the Metadata Normalization approach (Lu et al., 2021). In other words, we perform the proposed orthogonalization during training. Inspired by Clever Hans predictors (Lapuschkin et al., 2019) and orthogonalization in distribution shift situations (Chen et al., 2023), we colorize the MNIST data by reshaping the grayscale image to an RGB image where one of the three channels is filled with the original grayscale pixel values and the remaining two channels filled with zeros, i.e., set to black. The binary prediction task is to classify a subset of the MNIST dataset reduced to digits 0 and 9. For the train set, we intentionally create only red images of 0s by assigning the pixel values to the first channel. For the images of 9s we randomly assign either the second (green) or third (blue) channel. In the counterfactual test set, images of both classes are colored randomly in green or blue, but not red. As a result, the color red is predictive for the train but not the test set. We then use the color information "red", denoted as the binary feature  $X$ , in the correction model to normalize the CNN's predictions. In other words, we train a CNN on RGB images but blind the network w.r.t. the color red (or in fact any) color information by orthogonalizing the first convolutional layer's latent features w.r.t.  $X$ .
+
+![](images/824051be7bfd0771ae61f8242c9f18e7bb5102adf414f56115cce13b14843b95.jpg)  
+Figure 4. Estimated influence of the color red on the convolutional layer's predictions (left plot) and train-, validation- and test-accuracy of the two models (with/out correction).
+
+![](images/fa64575a6f1421e25595fc4061963fbba7f96c79ec56ec1e1b00074b43de8bca.jpg)
+
+Results Figure 4 summarizes the results of the experiment showing that our method correcting for the color red in fact yields no influence of the metadata information on the hidden layer's weights (left plot) and further illustrates how the model without correction is fooled by the training/validation data by focusing on the information of the color channels instead of the pixel values. This leads to a poor test performance of not much more than  $50\%$  accuracy. The corrected model shows a better test performance as it learned to focus on the image content and not the channel information.
+
+# 5. Discussion
+
+In this work, we introduced a novel correction routine that extends the concept of orthogonalization to non-linear and
+
+![](images/508aeb0d52466e8a9f1bbf433d5250e70787723740bc3137cb0b1bd66a312b56.jpg)  
+Figure 5. Comparison of orthogonalization properties of fairness methods in comparison with our approach when extending the notion of orthogonality beyond linear feature effects. Ideally, methods should yield random forest feature importance (RFimp) of zero (in the left plot) and large p-values (in the right plot). Missing boxes indicate that the method did not converge.
+
+tensor-valued models. Our experimental results confirm that our approach is effective in correcting sensitive data in linear models and neural networks, as well as rectifying pre-existing embeddings for undesired attributes.
+
+Caveat We want to emphasize that the given examples and results in this paper are limited by the representativeness of the datasets used and should not be the sole basis for decision-making in applications involving medical imaging or text applications. Furthermore, as "protection of information" can be defined in various ways, our presented methods do not offer guarantees of absolute protection for sensitive information in every situation. In particular, note that orthogonalization does not imply stochastic independence of the involved random variables, which means that some form of residual dependence could persist in theory.
+
+Limitations and future research The proposed algorithms require the number of observations  $n$  to be larger than the number of features  $p$  (or intermediate layer predictions in a network), and that there are more features  $q$  in the prediction model than in the evaluation model  $(p)$ . While this is usually the case, extensions to applications for  $n < p$  or  $q < p$  are a challenging yet interesting future research direction.
+
+The proposed orthogonalization also requires knowledge of features  $Z$  related to protected information. Performing such an operation to remove implicit biases is another interesting approach brought up by an anonymous reviewer but would require simultaneously learning the protected feature and being able to evaluate whether these present potential biases.
+
+# 6.Outlook
+
+A possible extension suggested by one of the anonymous reviewers is to make the class of evaluation models more flexible, allowing it to go beyond linear effects of protected features on the additive predictor scale, i.e., fitting an evalu-
+
+ation model  $h(g(X))$  where  $g$  is, e.g., a non-linear function. This would yield a more general notion of orthogonalization than presented in this paper but is also more challenging to analyze theoretically. In order to check how robust existing approaches and ours are w.r.t. non-linear evaluation models, we run the fairness benchmark again and use a random forest as an evaluation model to explain the prediction model's output using protected information. We then extract the importance of each feature together with a p-value for the importance values using a permutation test. Results are depicted in Figure 5, showing a very similar pattern to Figure 3. Interestingly, our method seems to work well in terms of this new orthogonalization notion, only yielding significant feature importances for the bank dataset. Results for all other methods suggest that protected features play a significant role in the random forest's explanation of the prediction model's output.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of machine learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
+
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+
+# A. Simplified Problem Statement
+
+To illustrate our problem, consider two vectors  $x = (0,1)$  and  $y = (1,0)$ , which are orthogonal as  $x^{\top}y = 0$ . However, after transformation  $h(x)$  and  $h(y)$  are not orthogonal anymore for non-linear transformations  $h$  as depicted in Figure 6.
+
+![](images/5828641c7d47d3dae94549f454546f4b4ff1f9a434e0bc89e2b3391ea754d9f3.jpg)  
+Figure 6. Graphical depiction of orthogonality without transformation (blue vectors) and after transformation with  $h_{\mathrm{exp}}$  (red) or the sigmoid function  $h_{\sigma}$  (green).
+
+# B. Connection to Fair Machine Learning
+
+When comparing the original orthogonalization approach in Lu et al. (2021); Rügamer et al. (2023) and Komiyama et al. (2018), both approaches use corrected features
+
+$$
+\boldsymbol {Z} ^ {c} = \mathcal {P} _ {\boldsymbol {X}} ^ {\perp} \boldsymbol {Z} \tag {7}
+$$
+
+in their model equation. The main difference is that fair machine learning approaches incorporate the protected features into the analysis as well, i.e., estimate the model
+
+$$
+\boldsymbol {y} = \boldsymbol {X} \boldsymbol {\beta} + \boldsymbol {Z} ^ {c} \boldsymbol {\gamma} ^ {c} + \varepsilon \tag {8}
+$$
+
+with residual term  $\varepsilon$ , whereas the prediction model  $\mathcal{M}^p$  in the orthogonalization approach does not necessarily include the features  $X$  in the model. If  $X$  is incorporated and without further constraints on the model, the approach of Komiyama et al. (2018) and the orthogonalization would lead to the same effect estimates (e.g., when using orthogonalization for identifiability as in Rügamer, 2023). Fair machine learning, however, offers a trade-off between model performance and (un-)fairness by regularizing the effects  $\beta$  of  $X$  in (8) using an explained variance formulation:
+
+$$
+\min  _ {\boldsymbol {\beta}, \boldsymbol {\gamma} ^ {c}} \mathbb {E} \left[ \left(\boldsymbol {y} - \hat {\boldsymbol {y}}\right) ^ {2} \right] \quad s. t. \frac {\operatorname {V a r} (\boldsymbol {X} \boldsymbol {\beta})}{\operatorname {V a r} (\hat {\boldsymbol {y}})} \leq r \tag {9}
+$$
+
+with predictions  $\hat{\pmb{y}} = \pmb{X}\hat{\pmb{\beta}} +\pmb{Z}^{c}\hat{\gamma}^{c}$  and some constraint  $r\in [0,1]$ . This approach and reformulations, such as the one of Scutari et al. (2022) as a ridge-penalized optimization problem, hence do not provide the full unpenalized effects  $\beta$  as obtained by the orthogonalization.
+
+Apart from the difference between the two approaches induced by the regularization, the generalization of fair machine learning approaches to models with activation functions differs further. In particular, as, e.g., suggested by Scutari et al. (2022) the orthogonal predictors  $Z^c$  are only used on the pre-activation scale, i.e., the fair model uses  $Z^c$  in the GLM by incorporating it only once in the additive predictor:  $\mathbb{E}(\boldsymbol{y}|\boldsymbol{X},\boldsymbol{Z}) = h(\boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{Z}^c\boldsymbol{\gamma}^c)$ . As discussed in the main part of our paper and in the previous section, this does not imply orthogonality of  $h(\boldsymbol{X}\boldsymbol{\beta})$  and  $h(\boldsymbol{Z}\boldsymbol{\gamma})$  as required in our work.
+
+A slightly different approach is used by Xu et al. (2022) who correct a classifier trained with all variables by another classifier trained only on the protected variables. Their approach, however, does not debias representations but only finds directions for predictions that are orthogonal.
+
+# C. Other Notions of Orthogonality
+
+In the following, we briefly describe connections to other notions of orthogonality that are used in different fields of machine learning research.
+
+# C.1. Domain Generalization
+
+The objective of the domain generalization (DG) problem is to train models to be robust against (or orthogonal to) domain (or covariate) shifts (Ganin et al., 2016; Shankar et al., 2018). While our orthogonalization approach and DG have different goals, viewing orthogonalization through the lens of DG is an interesting idea brought up by an anonymous reviewer. In particular, distributionally robust optimization (DRO; Sagawa et al., 2019) aims to ensure that predictive accuracy is robust w.r.t. changes in the distribution of the sensitive features. Our orthogonalization approach, in contrast, aims to ensure that the sensitive features do not affect the predictions (in the sense of a zero coefficient in the evaluation model). Note, however, that DRO does not provide any guarantees regarding our goal.
+
+# C.2. Orthogonality of Predictions and Embeddings
+
+Another interesting question brought up by an anonymous reviewer is how our notion of orthogonality relates to the following two concepts:
+
+- Concept 1: requiring that the predictions do not change when the sensitive attributes are changed;  
+- Concept 2: requiring that the embeddings are free from the influence of sensitive information.
+
+Concept 2 is a special case of our proposed notion if the model is linear. Concept 1 can also coincide with our definition if sensitive attributes are part of the prediction model. In our setting, however, this is not necessarily the case. Other (fairness) approaches aligning with concept 1 assume knowledge of the causal relationship between sensitive and non-sensitive attributes, which we also do not require.
+
+# D. Background on GLMs
+
+We briefly introduce several quantities used in the estimation of GLMs, but refer to Wood (2017) for more detailed explanations and derivations. Let  $g = h^{-1}$  be the link function. The weights  $\Upsilon$  are defined as
+
+$$
+\Upsilon = \operatorname {d i a g} \left(\alpha \left(\mu_ {i}\right) / \left\{g ^ {\prime} \left(\mu_ {i}\right) ^ {2} V \left(\mu_ {i}\right) \right\}\right) \tag {10}
+$$
+
+where  $\mu_{i} = h(\pmb{x}_{i}^{\top}\pmb {\beta})$  and
+
+$$
+\alpha \left(\mu_ {i}\right) = 1 + \left(y _ {i} - \mu_ {i}\right) \left\{V ^ {\prime} \left(\mu_ {i}\right) / V \left(\mu_ {i}\right) + g ^ {\prime \prime} \left(\mu_ {i}\right) / g ^ {\prime} \left(\mu_ {i}\right) \right\} \tag {11}
+$$
+
+and
+
+$$
+V \left(\mu_ {i}\right) = \operatorname {V a r} (y) / \phi \tag {12}
+$$
+
+with  $\phi$  denoting the distribution's scale parameter. The working response  $\pmb{r}$  is defined as
+
+$$
+\boldsymbol {r} = \boldsymbol {G} (\boldsymbol {y} - \boldsymbol {\mu}) \tag {13}
+$$
+
+where  $\pmb {\mu} = (\mu_1,\dots ,\mu_n)^\top$  and
+
+$$
+\boldsymbol {G} = \operatorname {d i a g} \left(g ^ {\prime} \left(\mu_ {i}\right) / \alpha \left(\mu_ {i}\right)\right). \tag {14}
+$$
+
+In particular, note that in the case of using Fisher weights, i.e., the expected Hessian matrix,  $\alpha (\mu_i) = 1$  and we have the following simplification:
+
+$$
+\boldsymbol {G} ^ {- 1} \boldsymbol {\Upsilon} ^ {- 1 / 2} = \operatorname {d i a g} \left(V \left(\mu_ {i}\right) ^ {1 / 2}\right), \tag {15}
+$$
+
+i.e., a function of  $\mu_{i}$  only. We can use this result in later derivations.
+
+# D.1. GLM in Detail
+
+The above derivation for GLMs can be made more explicit for our purposes. As noted before, using the Fisher weights implies  $\alpha (\mu_i) = 1$ . As  $\mu_{i} = h(x_{i}^{\top}\beta)\equiv h(0)$  in case of a successful correction, this implies
+
+$$
+\boldsymbol {G} = \operatorname {d i a g} \left(g ^ {\prime} \left(\mu_ {i}\right)\right) = \left(h ^ {- 1}\right) ^ {\prime} (0) \cdot \boldsymbol {I}. \tag {16}
+$$
+
+In other words, multiplying with  $G$  only scales all observations by a constant. Similarly, (15) reduces to a constant multiplicative factor of  $V(h(0)^{1/2})$ .
+
+# E. Proofs and Derivations
+
+# E.1. Proof of Lemma 3
+
+To prove Lemma 3, we use the final iterate representation of weights  $\hat{\beta}^c$  in a GLM and note that the evaluation model (first without intercept) must yield
+
+$$
+\hat {\boldsymbol {\beta}} ^ {c} = \left(\boldsymbol {X} ^ {\top} \boldsymbol {\Upsilon} \boldsymbol {X}\right) ^ {- 1} \boldsymbol {X} ^ {\top} \boldsymbol {\Upsilon} \boldsymbol {r} = \left(\boldsymbol {X} ^ {\top} \boldsymbol {\Upsilon} \boldsymbol {X}\right) ^ {- 1} \boldsymbol {X} ^ {\top} \boldsymbol {\Upsilon} \boldsymbol {G} \left(\hat {\boldsymbol {y}} ^ {c} - \boldsymbol {\mu}\right) \stackrel {!} {=} \mathbf {0}. \tag {17}
+$$
+
+Consequently, it must also hold  $\pmb{\mu} = h(\pmb{X}\hat{\beta}^c) = h(\pmb{0})$ . The equality in (17) yields two possible solutions: either  $\hat{\pmb{y}}^c = \pmb{\mu}$ , which is undesirable as it implies that we cannot predict anything after removing the influence of  $\pmb{X}$ , or we can construct  $\hat{\pmb{y}}^c$  so that when first subtracting  $\pmb{\mu}$  and then projecting it onto the space spanned by the columns of  $\widetilde{\pmb{X}} = \pmb{\Upsilon}^{1/2}\pmb{X}$ , we obtain the zero vector (see details below). Solving for  $\hat{\pmb{y}}^c$ , we get the following result.
+
+Lemma 8. Consider a prediction model  $\mathcal{M}^p$  with activation function  $h$  using data  $\mathbf{Z}$  to produce predictions  $\hat{\mathbf{y}}$ . Given weights  $\Upsilon$  and protected features  $\mathbf{X}$ , corrected predictions satisfying  $\hat{\beta}^c = \mathbf{0}$  are given by
+
+$$
+\hat {\boldsymbol {y}} ^ {c} = \boldsymbol {G} ^ {- 1} \boldsymbol {\Upsilon} ^ {- 1 / 2} \mathcal {P} _ {\tilde {\boldsymbol {X}}} ^ {\perp} \hat {\boldsymbol {y}} + h (\boldsymbol {0}). \tag {18}
+$$
+
+We can easily confirm that this yields a desired solution by plugging in (18) into (17). Further note that  $\pmb{\mu} = h(0)\cdot \mathbf{1}_n$ , i.e., shifts by a constant  $c_{2}\coloneqq h(0)$  and the multiplication from the left with  $G^{-1}\Upsilon^{-1 / 2}$  does only scale by a constant factor  $c_{1}\coloneqq V(h(0))^{1 / 2}$  (see Section D.1) but not rotate  $\mathcal{P}_{\overline{\mathbf{X}}}^{\perp}\hat{\mathbf{y}}$ , i.e.,
+
+$$
+\hat {\boldsymbol {y}} ^ {c} = \boldsymbol {G} ^ {- 1} \boldsymbol {\Upsilon} ^ {- 1 / 2} \mathcal {P} _ {\tilde {\boldsymbol {X}}} ^ {\perp} \hat {\boldsymbol {y}} + \boldsymbol {\mu} \stackrel {(1 5)} {=} V (h (0)) ^ {1 / 2} \mathcal {P} _ {\tilde {\boldsymbol {X}}} ^ {\perp} \hat {\boldsymbol {y}} + h (\boldsymbol {0}) = c _ {1} \mathcal {P} _ {\tilde {\boldsymbol {X}}} ^ {\perp} \hat {\boldsymbol {y}} + c _ {2} \mathbf {1} _ {n} \tag {19}
+$$
+
+with constants  $c_{1}, c_{2}$ . Any multiplicative constant does not change the result. We can therefore further simplify the previous expression from (18) to
+
+$$
+\hat {\mathbf {y}} ^ {c} = \mathcal {P} _ {\boldsymbol {X}} ^ {\perp} \hat {\mathbf {y}} + c _ {2} \mathbf {1} _ {n}. \tag {20}
+$$
+
+This is due to the fact that  $\mathcal{P}_{\widetilde{\boldsymbol{X}}}^{\perp}$  is the projection matrix of  $\widetilde{\boldsymbol{X}} = \boldsymbol{\Upsilon}^{1/2}\boldsymbol{X}$  with  $\boldsymbol{\Upsilon}^{1/2}$  being  $\boldsymbol{\Upsilon}^{1/2} = \mathrm{diag}(1/(g'(\mu_i)\sqrt{V(\mu_i)})) \equiv c_3 \cdot \boldsymbol{I}$  for some constant  $c_3$  as  $\boldsymbol{\mu} = h(\boldsymbol{0}) = \mathrm{const.} \cdot \mathbf{1}_n$ , and hence both  $g'(\mu_i)$  and  $V(\mu_i)$  are constant. As a consequence, we have  $\mathcal{P}_{\widetilde{\boldsymbol{X}}} = c_3\boldsymbol{X}(c_3^2\boldsymbol{X}^\top\boldsymbol{X})^{-1}\boldsymbol{X}c_3 = \mathcal{P}_{\boldsymbol{X}}$ . For a similar reason, we can ignore  $c_1$  as  $(\boldsymbol{X}^\top\boldsymbol{X})^{-1}\boldsymbol{X}^\top c_1\mathcal{P}_{\boldsymbol{X}}^{\perp}\hat{\boldsymbol{y}} = c_1(\boldsymbol{X}^\top\boldsymbol{X})^{-1}\bar{\boldsymbol{X}}^\top\mathcal{P}_{\boldsymbol{X}}^{\perp}\hat{\boldsymbol{y}} = \mathbf{0}$ .
+
+Model with Intercept Similar results can be derived for a model with intercept. First, note that  $\pmb{\mu} = h(\beta_0\mathbf{1}_n)$ . Hence,  $c_{1}$  and  $c_{2}$  are fixed constants  $c_{2} = h(\beta_{0})$  and  $c_{1} = V(h(\beta_{0}))^{1 / 2}$ . Hence, this yields again (20), now just with different constants  $c_{1}$  and  $c_{2}$ . In an intercept-only GLM with canonical link function, the intercept can be computed using only the response values, allowing to derive the closed-form solution  $\hat{\beta}_{0} = h^{-1}(\hat{y})$  with  $\bar{y} = n^{-1}\sum_{i}\hat{y}_{i}$ . As the intercept in the evaluation model is of no particular interest, one can alternatively center the predictions  $\hat{\pmb{y}}$  and subsequently drop the constant  $c_{2}$ . We will assume centered predictions in the following and use the correction  $\hat{\pmb{y}}^c = \mathcal{P}_{\pmb{X}}^{\perp}\hat{\pmb{y}}$ .
+
+# E.2. Details and Derivation of Corollary 4
+
+In the following, we describe how to derive the constrained optimization problem and subsequently our preferred way of solving it.
+
+Deriving the constrained optimization problem To see that we can write problem
+
+$$
+\hat {\gamma} ^ {c} = \underset {\gamma^ {c}} {\arg \min } \ell (\boldsymbol {y}, h (\boldsymbol {Z} \boldsymbol {\gamma} ^ {c})) \text {s . t .} \hat {\beta} ^ {c} = \mathbf {0} \tag {21}
+$$
+
+with
+
+$$
+\hat {\boldsymbol {\beta}} ^ {c} = \underset {\boldsymbol {\beta} ^ {c}} {\arg \min } \ell (\hat {\boldsymbol {y}} ^ {c} = h (\boldsymbol {Z} \hat {\gamma} ^ {c}), h (\boldsymbol {X} \boldsymbol {\beta} ^ {c})) \tag {22}
+$$
+
+as
+
+$$
+\hat {\gamma} ^ {c} = \underset {\gamma^ {c}} {\arg \min } \ell (\boldsymbol {y}, h (\boldsymbol {Z} \boldsymbol {\gamma} ^ {c})) \tag {23}
+$$
+
+$$
+s. t. \left| \left| \left(\boldsymbol {X} - n ^ {- 1} \mathbf {1} _ {n} \mathbf {1} _ {n} ^ {\top} \boldsymbol {X}\right) ^ {\top} h (\boldsymbol {Z} \gamma^ {c}) \right| \right| _ {2} ^ {2} = 0,
+$$
+
+note that the statement in Lemma 3 can be simplified by recognizing that (17) will also hold true if  $\pmb{X}^{\top}(\hat{\pmb{y}}^{c} - \pmb{\mu}) = \pmb{X}^{\top}(\hat{\pmb{y}}^{c} - c_{2}\mathbf{1}_{n}) = \mathbf{0}$ . The second term  $\pmb{X}^{\top}c_{2}\mathbf{1}_{n}$  is always equal to zero if we center  $\pmb{X}$ , i.e., use  $(\pmb{X} - n^{-1}\mathbf{1}_{n}\mathbf{1}_{n}^{\top}\pmb{X})$ . Hence, if  $(\pmb{X} - n^{-1}\mathbf{1}_{n}\mathbf{1}_{n}^{\top}\pmb{X})^{\top}\hat{\pmb{y}}^{c} = \mathbf{0}$  is fulfilled, (17) will also hold true. Now, instead of using  $(\pmb{X} - n^{-1}\mathbf{1}_{n}\mathbf{1}_{n}^{\top}\pmb{X})^{\top}h(\pmb{Z}\gamma^{c}) = \mathbf{0}$  as a constraint, introducing  $p$  auxiliary variables when using the Lagrangian multiplier method (see details below), we can instead also use the constraint  $||(\pmb{X} - n^{-1}\mathbf{1}_{n}\mathbf{1}_{n}^{\top}\pmb{X})^{\top}h(\pmb{Z}\gamma^{c})||_{2}^{2} = 0$ , which implies the former, but only requires one Lagrangian multiplier variable.
+
+Modified differential multiplier method (MDMM) Following Platt & Barr (1987), we first reformulate the constrained optimization problem (23) using Lagrangian multipliers, yielding
+
+$$
+\underset {\gamma , \lambda} {\arg \min } \ell (\boldsymbol {y}, h (\boldsymbol {Z} \boldsymbol {\gamma})) + \lambda \underbrace {\left\| \left(\boldsymbol {X} - n ^ {- 1} \mathbf {1} _ {n} \mathbf {1} _ {n} ^ {\top} \boldsymbol {X}\right) ^ {\top} h (\boldsymbol {Z} \boldsymbol {\gamma}) \right\| _ {2} ^ {2}} =: \underset {\gamma , \lambda} {\arg \min } \mathcal {E} (\boldsymbol {\gamma}, \lambda). \tag {24}
+$$
+
+In order to optimize (24), we can use the basic differential multiplier method (BDMM) by performing gradient descent to optimize  $\mathcal{E}$  w.r.t.  $\gamma$  but gradient ascent for the auxiliary term  $\lambda \mathcal{A}$  w.r.t.  $\lambda$ , i.e., updating the parameters in each iteration  $t \in \mathbb{N}$  as
+
+$$
+\boldsymbol {\gamma} ^ {[ t ]} = \boldsymbol {\gamma} ^ {[ t - 1 ]} - \nu^ {[ t ]} \nabla_ {\boldsymbol {\gamma}} \mathcal {E} (\boldsymbol {\gamma} ^ {[ t - 1 ]}; \lambda^ {[ t - 1 ]}), \tag {25}
+$$
+
+$$
+\lambda^ {[ t ]} = \lambda^ {[ t - 1 ]} + \nu^ {[ t ]} \mathcal {A} (\lambda^ {[ t - 1 ]}; \boldsymbol {\gamma} ^ {[ t - 1 ]}),
+$$
+
+with learning rate  $\nu^{[t]}$ . The modified version MDMM of the BDMM suggested by Platt & Barr (1987) combines the Lagrangian method with the penalty method augmenting the overall objective by a penalty term
+
+$$
+\mathcal {E} _ {P} (\boldsymbol {\gamma}, \lambda) := \mathcal {E} (\boldsymbol {\gamma}, \lambda) + \frac {\zeta}{2} \mathcal {A} (\boldsymbol {\gamma}) ^ {2} \tag {26}
+$$
+
+with damping factor  $\zeta$  (that defaults to 1), and thus adds another term  $\zeta A(\gamma)$  to the gradient descent direction in (25).
+
+# E.3. Proof of Theorem 5
+
+Consider the ReLU activation function defined as  $h(x) = x \cdot \mathbb{1}(x \geq 0)$ . Using the fact that the MSE-optimal  $\hat{\beta}^c$  is found by
+
+$$
+\hat {\boldsymbol {\beta}} ^ {c} = \underset {\boldsymbol {\beta} ^ {c}} {\arg \min } \frac {1}{n} | | h (\boldsymbol {Z} \boldsymbol {\gamma}) - h (\boldsymbol {X} \boldsymbol {\beta} ^ {c}) | | _ {2} ^ {2}, \tag {27}
+$$
+
+we can first expand the objective which is minimized in (27) to
+
+$$
+\frac {1}{n} \left(h (\boldsymbol {Z} \boldsymbol {\gamma}) ^ {\top} h (\boldsymbol {Z} \boldsymbol {\gamma}) - 2 h (\boldsymbol {Z} \boldsymbol {\gamma}) ^ {\top} h \left(\boldsymbol {X} \boldsymbol {\beta} ^ {c}\right) + h \left(\boldsymbol {X} \boldsymbol {\beta} ^ {c}\right) ^ {\top} h \left(\boldsymbol {X} \boldsymbol {\beta} ^ {c}\right)\right). \tag {28}
+$$
+
+The first dot product in (28) is constant w.r.t.  $\beta^c$  and the last dot product only determines the scaling of  $\hat{\beta}^c$ , hence we focus on the mixed term. If, after feature correction  $Z^{c} = \mathcal{P}_{X}^{\perp}Z$ , the mixed term
+
+$$
+h \left(\boldsymbol {Z} ^ {c} \boldsymbol {\gamma}\right) ^ {\top} h \left(\boldsymbol {X} \boldsymbol {\beta} ^ {c}\right) = 0, \tag {29}
+$$
+
+the optimal  $\hat{\beta}^c$  in (27) will also be zero.
+
+We can exploit properties of the ReLU function to show that this is indeed the case, using the fact that for any  $\mathbf{a}, \mathbf{b} \in \mathbb{R}^n$  it holds
+
+$$
+\begin{array}{l} \boldsymbol {a} ^ {\top} \boldsymbol {b} = \sum_ {i} a _ {i} b _ {i} = \sum_ {i} [ h (a _ {i}) h (b _ {i}) + h (- a _ {i}) h (b _ {i}) + h (a _ {i}) h (- b _ {i}) + h (- a _ {i}) h (- b _ {i}) ] \\ = \sum_ {i} h \left(a _ {i}\right) h \left(b _ {i}\right) + \sum_ {i} h \left(- a _ {i}\right) h \left(b _ {i}\right) + \sum_ {i} h \left(a _ {i}\right) h \left(- b _ {i}\right) + \sum_ {i} h \left(- a _ {i}\right) h \left(- b _ {i}\right) \tag {30} \\ = h (\pmb {a}) ^ {\top} h (\pmb {b}) + h (- \pmb {a}) ^ {\top} h (\pmb {b}) + h (\pmb {a}) ^ {\top} h (- \pmb {b}) + h (- \pmb {a}) ^ {\top} h (- \pmb {b}), \\ \end{array}
+$$
+
+which allows us to rewrite the left-hand side of (29) as
+
+$$
+\begin{array}{l} h (\boldsymbol {Z} ^ {c} \boldsymbol {\gamma}) ^ {\top} h (\boldsymbol {X} \boldsymbol {\beta} ^ {c}) = (\boldsymbol {Z} ^ {c} \boldsymbol {\gamma}) ^ {\top} (\boldsymbol {X} \boldsymbol {\beta} ^ {c}) - h (- \boldsymbol {Z} ^ {c} \boldsymbol {\gamma}) ^ {\top} h (\boldsymbol {X} \boldsymbol {\beta} ^ {c}) - h (\boldsymbol {Z} ^ {c} \boldsymbol {\gamma}) ^ {\top} h (- \boldsymbol {X} \boldsymbol {\beta} ^ {c}) - h (- \boldsymbol {Z} ^ {c} \boldsymbol {\gamma}) ^ {\top} h (- \boldsymbol {X} \boldsymbol {\beta} ^ {c}) \\ \Longleftrightarrow h \left(\boldsymbol {Z} ^ {c} \boldsymbol {\gamma}\right) ^ {\top} h (\boldsymbol {X} \boldsymbol {\beta} ^ {c}) = \underbrace {\boldsymbol {\gamma} ^ {\top} \boldsymbol {Z} ^ {\top} \mathcal {P} _ {\boldsymbol {X}} ^ {\perp} \boldsymbol {X} \boldsymbol {\beta} ^ {c}} _ {= 0} - h (- \boldsymbol {Z} ^ {c} \boldsymbol {\gamma}) ^ {\top} h (\boldsymbol {X} \boldsymbol {\beta} ^ {c}) - h \left(\boldsymbol {Z} ^ {c} \boldsymbol {\gamma}\right) ^ {\top} h (- \boldsymbol {X} \boldsymbol {\beta} ^ {c}) - h (- \boldsymbol {Z} ^ {c} \boldsymbol {\gamma}) ^ {\top} h (- \boldsymbol {X} \boldsymbol {\beta} ^ {c}) \\ \Longleftrightarrow h \left(\boldsymbol {Z} ^ {c} \boldsymbol {\gamma}\right) ^ {\top} h (\boldsymbol {X} \boldsymbol {\beta} ^ {c}) = - h \left(- \boldsymbol {Z} ^ {c} \boldsymbol {\gamma}\right) ^ {\top} h \left(\boldsymbol {X} \boldsymbol {\beta} ^ {c}\right) - h \left(\boldsymbol {Z} ^ {c} \boldsymbol {\gamma}\right) ^ {\top} h \left(- \boldsymbol {X} \boldsymbol {\beta} ^ {c}\right) - h \left(- \boldsymbol {Z} ^ {c} \boldsymbol {\gamma}\right) ^ {\top} h \left(- \boldsymbol {X} \boldsymbol {\beta} ^ {c}\right) \\ \Longleftrightarrow h \left(\boldsymbol {Z} ^ {c} \boldsymbol {\gamma}\right) ^ {\top} h \left(\boldsymbol {X} \boldsymbol {\beta} ^ {c}\right) + h \left(- \boldsymbol {Z} ^ {c} \boldsymbol {\gamma}\right) ^ {\top} h \left(\boldsymbol {X} \boldsymbol {\beta} ^ {c}\right) + h \left(\boldsymbol {Z} ^ {c} \boldsymbol {\gamma}\right) ^ {\top} h \left(- \boldsymbol {X} \boldsymbol {\beta} ^ {c}\right) + h \left(- \boldsymbol {Z} ^ {c} \boldsymbol {\gamma}\right) ^ {\top} h \left(- \boldsymbol {X} \boldsymbol {\beta} ^ {c}\right) = 0. \tag {31} \\ \end{array}
+$$
+
+Since  $h(\cdot) \geq 0$  for all inputs, the left side of the equation consists of a sum of products of non-negative terms, implying that all product terms must be equal to zero.
+
+# E.4. Proof of Corollary 6
+
+Using the  $\mathrm{vec}(\cdot)$  operation, we can first reformulate the evaluation model with uncorrected predictions  $\mathrm{vec}(\hat{\mathfrak{V}}) \in \mathbb{R}^{n\cdot d}$ ,  $d = \prod_{r=1}^{R} d_r$ , as
+
+$$
+\operatorname {v e c} (\mathfrak {Y}) = \operatorname {v e c} \left(\boldsymbol {X} \times_ {1} \mathfrak {B} ^ {c}\right) = \operatorname {v e c} \left(\boldsymbol {X} \underbrace {\operatorname {m a t} \left(\mathfrak {B} ^ {c}\right)} _ {p \times d}\right) = \left(\boldsymbol {I} _ {d} \otimes \boldsymbol {X}\right) \operatorname {v e c} \left(\mathfrak {B} ^ {c}\right). \tag {32}
+$$
+
+The orthogonal projection matrix of  $(\mathbf{I}_d\otimes \mathbf{X})$  is given by  $(\mathbf{I}_d\otimes \mathcal{P}_{\mathbf{X}}^{\perp})$ . Premultiplying  $\operatorname {vec}(\tilde{\mathfrak{V}})$  by this term yields
+
+$$
+\left(\boldsymbol {I} _ {d} \otimes \mathcal {P} _ {\boldsymbol {X}} ^ {\perp}\right) \operatorname {v e c} (\hat {\mathfrak {Y}}) = \operatorname {v e c} \left(\mathcal {P} _ {\boldsymbol {X}} ^ {\perp} \underbrace {\operatorname {m a t} (\hat {\mathfrak {Y}})} _ {n \times d}\right) = \operatorname {v e c} \left(\underbrace {\mathcal {P} _ {\boldsymbol {X}} ^ {\perp} \times_ {1} \hat {\mathfrak {Y}}}\right). \tag {33}
+$$
+
+The solution  $\mathfrak{B}^c$  to the evaluation model using corrected predictions  $\hat{\mathfrak{V}}^c$  can then be found by minimizing
+
+$$
+\left| \left| \operatorname {v e c} \left(\widehat {\mathfrak {Y}} ^ {c}\right) - \operatorname {v e c} \left(\boldsymbol {X} \times_ {1} \mathfrak {B} ^ {c}\right) \right| \right| _ {2} ^ {2}, \tag {34}
+$$
+
+over  $\mathfrak{B}^c$ . The closed-form solution is given by
+
+$$
+\operatorname {v e c} \left(\hat {\mathfrak {B}} ^ {c}\right) = \left(\left(\boldsymbol {I} _ {d} \otimes \boldsymbol {X}\right) ^ {\top} \left(\boldsymbol {I} _ {d} \otimes \boldsymbol {X}\right)\right) ^ {- 1} \left(\boldsymbol {I} _ {d} \otimes \boldsymbol {X}\right) ^ {\top} \operatorname {v e c} \left(\hat {\mathfrak {Y}} ^ {c}\right) = \left(\left(\boldsymbol {I} _ {d} \otimes \boldsymbol {X}\right) ^ {\top} \left(\boldsymbol {I} _ {d} \otimes \boldsymbol {X}\right)\right) ^ {- 1} \underbrace {\left(\boldsymbol {I} _ {d} \otimes \boldsymbol {X}\right) ^ {\top} \left(\boldsymbol {I} _ {d} \otimes \mathcal {P} _ {\boldsymbol {X}} ^ {\perp}\right)} _ {= 0} \operatorname {v e c} (\hat {\mathfrak {Y}}), \tag {35}
+$$
+
+proving that  $\tilde{\mathfrak{B}}^c\equiv \mathbf{0}$
+
+# E.5. Proof of Corollary 7
+
+Similar to the proof of Theorem 5, we start with the solution of the evaluation model given by
+
+$$
+\hat {\mathfrak {B}} ^ {c} = \underset {\mathfrak {B} ^ {c}} {\arg \min } \frac {1}{n} \left\| h \left(\operatorname {v e c} \left(\bar {\mathcal {Y}} ^ {c}\right)\right) - h \left(\operatorname {v e c} \left(\boldsymbol {X} \times_ {1} \mathfrak {B} ^ {c}\right)\right) \right\| _ {2} ^ {2}, \tag {36}
+$$
+
+where  $h(x) = x \cdot \mathbb{1}(x \geq 0)$  denotes the ReLU function, and focus on the mixed term  $h(\operatorname{vec}(\bar{\mathfrak{V}}^c))^\top h(\operatorname{vec}(\boldsymbol{X} \times_1 \mathfrak{B}^c))$  when expanding the argument. Noting that this is again a dot product of vectors, we can use the same argument as before in (30) for the proof of Theorem 5. In particular, the non-activated mixed term including the correction evaluates to
+
+$$
+\operatorname {v e c} \left(\mathcal {P} _ {\boldsymbol {X}} ^ {\perp} \times_ {1} \bar {\mathfrak {Y}}\right) ^ {\top} \operatorname {v e c} (\boldsymbol {X} \times_ {1} \mathfrak {B} ^ {c}) = \left(\operatorname {v e c} \bar {\mathfrak {Y}}\right) ^ {\top} \underbrace {\left(\boldsymbol {I} _ {d} \otimes \mathcal {P} _ {\boldsymbol {X}} ^ {\perp}\right) \left(\boldsymbol {I} _ {d} \otimes \boldsymbol {X}\right)} _ {\mathbf {0}} \operatorname {v e c} \left(\mathfrak {B} ^ {c}\right) = 0, \tag {37}
+$$
+
+and all other terms are again activated using the ReLU activation (and hence must be non-negative, but since their sum must be zero, all terms must be zero).
+
+# F. Numerical Experiment Details
+
+The code for reproducing results can be found on the first author's Github repository.
+
+Table 5. Overview of all datasets used. The reference category refers to the category used as a reference in the  $\mathcal{M}^e$  when encoding categorical variables.  
+
+<table><tr><td></td><td>ntrain</td><td>ntest</td><td>p</td><td>q</td><td>reference category</td></tr><tr><td>Adult</td><td>30162</td><td>-</td><td>5</td><td>45</td><td>female, White</td></tr><tr><td>Bank</td><td>40195</td><td>-</td><td>1</td><td>49</td><td>-</td></tr><tr><td>Compas</td><td>5855</td><td>-</td><td>6</td><td>13</td><td>female, African-American</td></tr><tr><td>Health Retirement</td><td>38653</td><td>-</td><td>4</td><td>25</td><td>female, married, Black</td></tr><tr><td>MNISTred</td><td>11872</td><td>1989</td><td>1</td><td>28 × 28</td><td>red</td></tr><tr><td>MIMIC</td><td>181342</td><td>3041</td><td>4</td><td>111</td><td>female, Asian</td></tr><tr><td>UTKfaces</td><td>16514</td><td>4129</td><td>5</td><td>256</td><td>Asian</td></tr><tr><td>Movie Reviews</td><td>34167</td><td>8538</td><td>2</td><td>100 × 100</td><td>female, female (main actor, movie director)</td></tr></table>
+
+# F.1. Adult Income Prediction
+
+For  $\mathcal{M}^p$  we use the features age, employment status, education, marital status, relationship, and hours per week. Protected features are given in Table 5. All models are GLMs with canonical link function and binomial distribution, i.e.,  $\ell$  is the binary cross-entropy loss and  $h()$  the sigmoid function.
+
+# F.2. Fairness Benchmark
+
+For all methods, we set the unfairness amount to 0.05 (as 0 would result in an infinite penalty).
+
+# F.3. Chest X-ray Embeddings
+
+We utilize the publicly available embeddings $^2$  of the CXR foundation model proposed by (Sellergren et al., 2022). The train and test partition is based on the recommended splits of the original MIMIC-CXR database.
+
+Before using the embedding in a classifier, we apply singular value decomposition (SVD) to reduce the risk of overfitting in the 1376-dimensional embedding. We find that  $q = 111$  dimensions are the best trade-off between explained variance (98%) and sparsity. The prediction model uses only these  $q$  features while the correction function and evaluation model are defined by age, sex, and race.
+
+# F.4. Face Recognition
+
+The embedding of the UTKFace dataset consists of the activations in the penultimate layer in a ResNet-50. Similar to the Chest X-ray application, we first reduce the 2048-dimensional embedding to  $q = 32$  features using an SVD. These  $q$  features explain  $95\%$  of the embedding's variance. The train and test split is based on a random 80/20 partitioning. We use these embedding features in the prediction model while using age and race for the correction function and evaluation model.
+
+# F.5. Movie Reviews
+
+We first extract the first actor's sex and the director's sex from the provided metadata. We then tokenize the movie descriptions using 1000 words and an embedding size of 100 with a maximum length of 100 words for each description. Padding is done at the end of sentences with less than 100 words. We further remove stop words and punctuations from sentences and transform all texts to lowercase. We use the vote count as a Poisson-distributed outcome and partition the data into  $80 / 20\%$  for training and testing.
+
+The LSTM model is defined by an embedding layer with embedding size 100, an LSTM layer with 50 units and ReLU activation, a dropout layer with 0.1 dropout rate, a dense layer with 25 units and ReLU activation, a dropout layer with 0.2 dropout rate, a dense layer with 5 units and ReLU activation, a dropout layer with 0.3 dropout rate, and a final dense layer with 1 unit and exponential activation. The network is trained for a maximum of 1000 epochs with early stopping using Adam with a learning rate of 1e-6, a batch size of 128, and Poisson loss.
+
+For the orthogonalization, we use the same architecture but insert an orthogonalization operation between the embedding layer and the LSTM layer. As features  $Z$  we use only the movie descriptions, whereas the protected features  $X$  consist of the gender of the protagonist and director.
+
+# F.6. Online Orthogonalization
+
+We subsample the MNIST dataset to observations with labels 0 or 9. Based on the coloring described in the main part of the paper, we train a CNN defined by a Conv2D layer with 8 filters, kernel size of  $3 \times 3$ , and ReLU activation, followed by the orthogonalization operation for the network that uses a correction. The output (with or without correction) is then fed into another Conv2D layer with 16 filters, kernel size of  $3 \times 3$ , and ReLU activation, followed by Max pooling of size  $2 \times 2$  and a flattening operation. Finally, the classification head comprises a dense layer with 64 units and ReLU activation followed by the output layer with one unit and sigmoid activation.
+
+The prediction model only uses the images and, if corrected, also the color information to perform the orthogonal projection for training. The networks are trained for 100 epochs with early stopping and a batch size of 128. Early stopping is based on a  $20\%$  validation split and a patience of 25. The evaluation model is optimized using quadratic programming and only uses the color red information.
+
+# F.7. Runtimes
+
+Table 6 compares the runtimes of different methods on the fairness benchmark dataset.
+
+<table><tr><td colspan="5">Table 6. Runtimes on the fairness benchmark dataset (in seconds).</td></tr><tr><td>Method</td><td>adult</td><td>bank</td><td>compas</td><td>health. retirement</td></tr><tr><td>Komiyama1</td><td>3.62</td><td>2.37</td><td>0.65</td><td>1.00</td></tr><tr><td>Komiyama2</td><td>5.66</td><td>2.68</td><td>1.19</td><td>14.17</td></tr><tr><td>Berk</td><td>1.13</td><td>2.59</td><td>1.18</td><td>3.52</td></tr><tr><td>Zafar</td><td>15.02</td><td>0.56</td><td>12.02</td><td>-</td></tr><tr><td>Ours</td><td>1.11</td><td>13.20</td><td>0.48</td><td>3.84</td></tr></table>
+
+For the colorized MNIST dataset, the average fitting time until early stopping (with standard deviation in brackets) of both networks was 1.08 (0.528) minutes with orthogonalization and 0.40 (0.006) minutes without orthogonalization. We can see that the orthogonalization takes on average longer to "converge". This can mainly be attributed to the fact that it runs for more iterations and not the additional operations required to perform the orthogonalization.
+
+# G. Additional Results
+
+# G.1. Synthetic Data
+
+First, we simulate data from a Bernoulli and Poisson distribution to check whether our approach provides the proposed protection for features  $\mathbf{X}$  using a canonical activation function  $h$  and either  $\mathcal{C}^l$  or  $\mathcal{C}^h$ . We generate features  $\mathbf{X}$  from a standard normal distribution and generate features  $\mathbf{Z}$  correlated to  $\mathbf{X}$  using the transformation  $\mathbf{X} = \rho \mathbf{Z}_{[\cdot,1:q]} + \mathbf{E}$ , where  $\rho \in \{0,1,2\}$  and  $\mathbf{E}$  are realizations of a standard normal distribution with  $\dim(\mathbf{E}) = \dim(\mathbf{Z})$ . We examine different settings for  $p \in \{5,10\}$ ,  $q \in \{10,100\}$ , and  $n \in \{200,1000,5000,10000\}$ , and repeat every setting 10 times with different random seed.
+
+Results Figure 7 shows the results of the simulation for the classification task and  $\rho = 2$  (the count response task in the Appendix is qualitatively the same) with and without correction for the model's pre-activation (corrected with  $\mathcal{C}^l$ ) and post-activation values (corrected with  $\mathcal{C}^h$ ). Results confirm that not only are coefficients estimated to be close to zero after correction, but their p-value is also close to 1 in most cases, suggesting no significant influence of protected features in the predicted value  $\hat{\pmb{y}}^c$ . In contrast, without correction, coefficients tend to be non-zero and "highly significant" (with a very small p-value).
+
+Similar results are obtained when using  $\rho = 0$  or  $\rho = 1$  for a logistic regression model (cf. Figure 8 and 9, respectively) and also for  $\rho \in \{0,1,2\}$  for a count regression (Figures 10-12).
+
+![](images/d639d5ad83f35d0ca3bcb7584dd8ea0e2adb7ddaf58b075cc30dcaa1e60ade74.jpg)  
+Figure 7. Distribution of estimated coefficients (top row) and corresponding p-values (bottom row) for different data sizes  $n$  and features  $q$  (columns) for the binary classification task (Bernoulli distribution). Boxplots summarize all different values of  $p$  and the 10 simulation repetitions.
+
+![](images/fcf44688cf5311e77a75d42c22cc30088f9ba3c7ad6210cc8796a9d8193640de.jpg)  
+Figure 8. Distribution of estimated coefficients (top row) and corresponding p-values (bottom row) for different data sizes  $n$  (columns) for the classification task and  $\rho = 0$ . Boxplots summarize all different values of  $p, q$ , and the 10 simulation repetitions.
+
+![](images/63ba576130699de3aa7f836e5db6f921323e03c5b4476a0b007a1be6c8c9f300.jpg)  
+Figure 9. Distribution of estimated coefficients (top row) and corresponding p-values (bottom row) for different data sizes  $n$  (columns) for the classification task and  $\rho = 0$ . Boxplots summarize all different values of  $p$  and the 10 simulation repetitions.
+
+![](images/4936bd1389fda729680a2dc05a26907bb8cb0b32173144ffd95af24d47aee6bc.jpg)  
+Figure 10. Distribution of estimated coefficients (top row) and corresponding p-values (bottom row) for different data sizes  $n$  (columns) for the count task (Poisson) and  $\rho = 0$ . Boxplots summarize all different values of  $p$  and the 10 simulation repetitions.
+
+![](images/313fcd77805c9ad38a8735b7b6dc82969abd4348289ca5944395f0d4799e5035.jpg)  
+Figure 11. Distribution of estimated coefficients (top row) and corresponding p-values (bottom row) for different data sizes  $n$  (columns) for the count task (Poisson) and  $\rho = 1$ . Boxplots summarize all different values of  $p$  and the 10 simulation repetitions.
+
+![](images/6788b4d2120bf64b88755acaab3a363f2904f992acbc6f92f2ba03a553d0d6ab.jpg)  
+Figure 12. Distribution of estimated coefficients (top row) and corresponding p-values (bottom row) for different data sizes  $n$  (columns) for the count task (Poisson) and  $\rho = 2$ . Boxplots summarize all different values of  $p$  and the 10 simulation repetitions.

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+# Generating Chain-of-Thoughts with a Pairwise-Comparison Approach to Searching for the Most Promising Intermediate Thought
+
+Zhen-Yu Zhang<sup>1</sup> Siwei Han<sup>2</sup> Huaxiu Yao<sup>2</sup> Gang Niu<sup>1</sup> Masashi Sugiyama<sup>13</sup>
+
+# Abstract
+
+To improve the ability of the large language model (LLMs) to tackle complex reasoning problems, chain-of-thoughts (CoT) methods were proposed to guide LLMs to reason step-by-step, enabling problem solving from simple to complex. State-of-the-art methods for generating such a chain involve interactive collaboration, where the learner generates candidate intermediate thoughts, evaluated by the LLM, guiding the generation of subsequent thoughts. However, a widespread yet understudied problem is that the evaluation from the LLM is typically noisy and unreliable, potentially misleading the generation process in selecting promising intermediate thoughts. In this paper, motivated by Vapnik's principle, we use pairwise-comparison evaluation instead of pointwise scoring to search for promising intermediate thoughts with the noisy feedback from the LLM. In each round, we randomly pair intermediate thoughts and directly prompt the LLM to select the more promising one from each pair, allowing us to identify the most promising thoughts through an iterative process. To further alleviate the noise in the comparison, we incorporate techniques from ensemble learning andueling bandits, proposing two variants of the algorithm. Experiments on three real-world tasks demonstrate the effectiveness of our proposed algorithm and verify the rationale of the pairwise comparison mechanism.
+
+# 1. Introduction
+
+Large language models (LLMs), such as the GPT (Brown et al., 2020) and PaLM (Chowdhery et al., 2023), have re
+
+<sup>1</sup>Center for Advanced Intelligence Project, RIKEN <sup>2</sup>University of North Carolina at Chapel Hill <sup>3</sup>Graduate School of Frontier Sciences, The University of Tokyo. Correspondence to: Masashi Sugiyama <sup><sugi@k.u-tokyo.ac.jp></sup>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+cently demonstrated remarkable capabilities in a variety of real-world tasks. However, current LLMs still face limitations when dealing with complex tasks, especially those involving multi-step reasoning, such as mathematical or reasoning problems (Rae et al., 2021; Wei et al., 2022). To deal with such implicit complexity, chain-of-thoughts (CoT) approaches were proposed (Wei et al., 2022; Wang et al., 2022; Yao et al., 2023). These approaches were proposed to use an incorporation of intermediate steps of reasoning (intermediate "thought"), enabling the LLM to reason progressively, first generating intermediate solutions for simpler problems to incrementally improve its capacity to handle complicated tasks. Therefore, the key challenge is to design an effective CoT generation algorithm that guides the LLM towards desired solutions through step-by-step reasoning.
+
+There is a fruitful line of work that considers the CoT generation problem. The pioneering work uses manual design prompts to let the LLM generate a CoT by itself (Wei et al., 2022; Wang et al., 2022). This line of research was recently extended by the score-based tree-of-thoughts (S-ToT) approaches (Yao et al., 2023; Long, 2023), where the CoT generation is framed as an interactive process with the algorithm and the LLM. These approaches generate a set of candidate intermediate thoughts each round and ask the LLM to score them and select the most promising ones. The next thoughts are then generated based on these selected ones, creating a tree-like data structure. A search algorithm, such as deep-first search, is used to identify the most promising CoT in the tree (see the detailed illustration in Figure 2).
+
+While these methods have shown remarkable empirical success, they rely on an accurate score evaluation of each intermediate thought by the LLM. However, it is important to notice that: LLM scores are often noisy. For example, the LLM may give different responses to different prompts, even though these prompts convey the same meaning (Lu et al., 2022). The noisy nature of LLM feedback introduces new problems in the selection of the most promising intermediate thoughts and the subsequent generation of the tree structure. Therefore, it is crucial to make the CoT generation algorithms robust to the noisy feedback from LLMs.
+
+Several preliminary approaches have been proposed to mitigate such noise in the LLM feedback, including estimating
+
+uncertainty from the semantic aspect (Kuhn et al., 2022) or assembling multiple thoughts (Wang et al., 2022). However, getting an accurate point-wise estimate for each intermediate thought could be resource-intensive, requiring the construction of an additional model (Paul et al., 2024) or multiple queries (Wang et al., 2022): see also Figure 4 and Table 6 to 8 in our experiments. Fortunately, in the context of CoT generation, our focus is on identifying the most promising chain. Motivated by Vapnik's principle (Vapnik, 1991), we do not need to solve a more general and difficult problem as an intermediate step, i.e. to estimate an accurate pointwise score for each intermediate thought. Instead, we can focus directly on identifying the most promising one in each round. However, it is still impractical to directly vote on all intermediate thoughts to identify the most promising one by LLMs due to the input length limit and the "lost in the middle" phenomenon (Liu et al., 2024).
+
+We argue that for LLMs, comparing two thoughts simultaneously provides a more robust evaluation than assigning individual scores. We aim to leverage the comparison of two thoughts instead of evaluating a single thought in isolation, thereby providing a feasible alternative for identifying the most promising intermediate thought. This argument is well established in human cognition, as seen in mathematical problems, where it is often more feasible to approximate which thought is better by comparison than by considering and evaluating them separately. We also observe similar phenomena that LLMs to generate a more reliable evaluation in the experiments on the Sudo task, as shown in Figure 1, where the LLM successfully identifies the better option given two intermediate solutions, but struggles to assign the correct value to intermediate thoughts individually.
+
+Based on the above insights, we propose a pairwise comparison-based algorithm for CoT generation to alleviate the noise in the LLM feedback and to find the most promising intermediate thoughts each round. In each round, we randomly pair all the intermediate thoughts and directly ask the LLM to compare and select the more promising one from each pair, keeping the selected one and discarding the other. Then we repeat this procedure so that we get a small set of most promising intermediate thoughts, and subsequently, we generate the next thoughts based on these selected ones. This mechanism allows us to use a direct pairwise comparison to identify the promising thoughts with a more robust evaluation. We also propose to include previous thoughts in the tree structure for comparison to mitigate the noisy nature of LLM's feedback. Taking these two points into account, we frame the problem as an iterative process and propose a general CoT generation algorithm called comparison-based tree-of-thoughts (C-ToT). To further model the noise in the comparison, we resort to the techniques of ensemble and best-arm identification withueling feedback (Falahatgar et al., 2017) and propose two variants of the proposed C
+
+![](images/44a4bbed72eb726e3c6816b71aa979a1e66c2481b5eb406592f933658c853f77.jpg)  
+Figure 1: A demonstration of point-wise evaluation vs. pair-wise comparison based on real experimental results in Sudoku puzzles. The point-wise evaluation algorithm (S-ToT) assigns hard scores to each intermediate thought (the higher, the better), while our proposed algorithm (C-ToT) uses pair-wise comparison to obtain the more promising thoughts (green tick). In these cases, the LLM assigns incorrect scores, but it makes a correct comparison.
+
+ToT algorithm. Through experiments on three real-world reasoning problems, we demonstrate the effectiveness of our proposed approaches and verify the rationale of the pairwise comparison mechanism. Our main contributions are:
+
+(1) We investigate the problem of noisy feedback in the CoT generation, which is widespread but understudied.  
+(2) Motivated by Vapnik's principle, we propose a pairwise-comparison based approach for CoT generation that exploits noisy feedback from the LLMs.  
+(3) We proposed two variants of C-ToT that further account for different types of noise in the comparison feedback.
+
+# 2. Related Work
+
+CoT Generation. Generating appropriate CoT for LLMs to enhance their inference power is a critical problem in real-world applications. Previous work has explored task-specific training algorithms for identifying the CoT, including creating semantic graphs (Xu et al., 2021), refining the model through human-annotated CoT (Cobbe et al., 2021), or learning an additional extractor using heuristic-driven pseudo CoT (Chen et al., 2019). Different from these approaches, the LLM-based CoT generation is used directly during inference, coupling the generation process with an LLM. In these approaches, the LLM guides the CoT generation, eliminating the need for additional training.
+
+The pioneering work in LLM-based CoT generation introduces intermediate thoughts sequentially between the input query and LLM's response. By simply prompting the LLM to "think step by step", this strategy has been shown to significantly improve several tasks over directly asking the LLM the original question, such as mathematical puzzles (Wei et al., 2022) or other general mathematical reasoning problems (Drori et al., 2022). Due to the noisy nature of the LLM feedback, robustness can be improved by using an ensemble of different CoTs (Wang et al., 2022).
+
+To further improve the effectiveness of CoT generation, the score-based tree-of-thoughts generation algorithm was intro-
+
+duced independently by Yao et al. (2023) and Long (2023). They model the CoT generation process as a tree generation and search process. A single node in the tree represents an intermediate thought. Starting from a given node, the thought generator constructs a set of new nodes and the LLM generates scores for each node as an evaluation. Finally, the timing of the tree expansion is determined by the search algorithm used (e.g., breadth-first or depth-first search). In addition, this search algorithm can also provide capabilities including backtracking from unpromising thoughts. Further research extended the tree structure to a graph, such as the graph-of-thoughts (Besta et al., 2023), allowing the distillation of knowledge about entire network of thoughts. However, these methods cannot handle the noisy evaluation feedback caused by the LLM itself.
+
+Self-Reflection. Rather than interacting with LLMs to generate a step-by-step reasoning chain, self-reflection approaches involve LLMs directly offering an initial thought chain to the query, followed by iterative refinement of the whole chain. Madaan et al. (2023) and Paul et al. (2024) introduced the "self-reflection" mechanism, using the LLMs to provide feedback to their generation candidates and then fine-tuning. Paul et al. (2024) updated the model to explicitly generate intermediate thoughts while interacting with a critic model that provides automated feedback on the reasoning. These methods introduced new models to provide evaluation for the intermediate thoughts, but these critical models still do not always provide perfect evaluation. Furthermore, for complex problems that require sequential reasoning, such as the Game of 24, where the next thought should be generated and evaluated based on previous ones, the C-ToT generation could be more appropriate.
+
+Uncertainty Quantification in LLMs. This is a recent interest that aims to evaluate the confidence of a given answer by the LLM itself. Some work considered letting the LLM provide the confidence (Northcutt et al., 2021; Kadavath et al., 2022) by retraining the model. Another line of work considered designing entropy-based measures (Kuhn et al., 2022), or generating multiple outputs to obtain an uncertainty measure (Wang et al., 2022). Although they can be included in the CoT generation, they introduce a high computational cost during testing, particularly when obtaining an accurate score for each intermediate thought.
+
+# 3. Our Approach
+
+In this section, we first introduce the proposed comparison-based ToT generation algorithm, which is a general framework that generates CoT with a pairwise comparison mechanism to find the most promising intermediate thought. To further alleviate the noise in LLM's comparison feedback, we propose two different instantiations of our framework with theoretical analysis.
+
+# 3.1. CoT Generation via Pair-wise Comparison
+
+We first introduce the comparison-based ToT framework, where the key mechanism is the selection of the most promising thoughts among all candidates in each round.
+
+We illustrate our proposed algorithm and compare it with previous approaches in Figure 2. The CoT approaches directly ask the LLM to generate a CoT. The S-ToT approaches ask the LLM to score each intermediate thought and select the highest-scoring ones to generate the next layer. Different from these methods, we propose a pairwise-comparison approach to searching for the most promising intermediate thoughts. Note that with LLMs, due to feedback noise and input limitations, we cannot do a listwise voting that directly asks the LLM to sort all the intermediate thoughts. Let  $Z$  be the set of all candidate intermediate thoughts, and we want to select the most promising  $K$  thoughts from it. The comparison iterates as follows: we randomly pair thoughts from the set and select only the winner in each pair, thereby halving the size of  $Z_{i}$  to  $|Z_{i}| / 2$ , where  $Z_{i}$  denotes the set in the  $i$ -th iteration. After at most  $K \times \log_2|Z|$  rounds, we can identify the  $K$  most promising intermediate thoughts by such direct comparison. In practice, we can do one iteration of comparison and keep the remaining  $K$  thoughts in the last few rounds. For each pair, we compare thoughts  $a$  and  $b$  with the LLM by asking which one is better, using different prompts with  $n$  times, where  $n \geq 1$ . We defer the implementation details to Section 4.1.
+
+We take previous unselected intermediate thoughts into comparison to explore possibly valuable but mis-evaluated thoughts caused by the noise in feedback. This is because the evaluation of intermediate thoughts may not always be accurate, and the generation of the tree structure may miss valuable intermediate thoughts in previous iterations. In the seminal research of S-ToT (Yao et al., 2023; Long, 2023; Besta et al., 2023), the thought generator uses a backtracking mechanism to revisit previous thoughts when the current ones fall below a certain threshold. While this strategy aims to rescue promising thoughts, its efficiency is questionable because it may delay the exploration of previously valuable but incorrectly scored thoughts. In addition, backtracking only occurs after a thought has fallen below a manually chosen threshold, which is hard to know in advance.
+
+Motivated by these shortcomings and the efficiency of our pairwise-comparison mechanism, we maintain a repository of previous intermediate thoughts. At each round, we include previously unselected thoughts in the comparisons, rather than relying on a fixed threshold. As illustrated in Figure 2, during the pairwise comparison in the second layer, we include the intermediate thought that was not selected in the first layer. This mechanism ensures that the algorithm has the flexibility to revisit previous thoughts based on the comparison results from the LLM in each round.
+
+![](images/17feb2ae06e550f2890088b9f187e9113d29a466a9bfa10f9565eb76c89abd06.jpg)  
+Chain-of-Thoughts
+
+![](images/a76aa451e3dbcdceec680b62f6a1f5778be3d94b491bc063fb6dbac647f649ba.jpg)  
+Score-based Tree-of-Thoughts (S-ToT)
+
+![](images/03e9a235bc922346c3eacb7120eb3b407be2413a00b7700c28fb8b4a6004c807.jpg)  
+Comparison-based Tree-of-Thoughts (C-ToT)  
+Figure 2: Schematic illustration of previous CoT and S-ToT approaches with our proposed C-ToT approach for CoT generation with LLMs. Each circle box represents an intermediate thought, which is a coherent sequence of language or equations that serves as an intermediate step in problem solving. In the S-ToT method, each intermediate thought is scored by the LLM (denoted by  $s$  in the figure), and the searching algorithm considers the highest-scoring ones as the most promising and then generates next intermediate thoughts based on them. In the C-ToT approach, we use pairwise comparison with the LLM in each round to find the most promising intermediate thoughts and then generate the next thoughts. Meanwhile, we include all previous intermediate thoughts in the comparison.
+
+We formulate the comparison-based ToT generation as an iterative interaction between the thought generation and the LLM. Take the C-ToT illustration in Figure 2 as an example. The algorithm starts by generating intermediate thoughts #1 to #3 based on the input. Following a pairwise comparison mechanism, thoughts #1 and #3 are selected, leading to the generation of new intermediate thoughts, namely #4 to #8. In the second layer, thought #2, thoughts #4 and #5 (linked to #1), along with thoughts #6 to #8 (linked to #3) are compared, subsequently resulting in the selection of thought #2 and thought #7 (linked to #3), and the generation of the next intermediate thoughts.
+
+Formally, we denote an intermediate thought by  $\mathbf{z}$ . In round  $t$ ,  $Z^t$  represents the set of candidate intermediate thoughts for comparison, and  $\widehat{Z}^t$  denotes the selected intermediate thoughts. In a sequence of  $T$  rounds, in the first round the algorithm generates a set of thoughts  $Z^1 = \{\mathbf{z}_i^1\}_{i=1}^m$  based on the query, where  $m$  denotes the set size. Then, the comparison-based ToT selects  $K$  most promising intermediate thoughts based on the comparison result from the LLM. We denote the selected set of thoughts by  $\widehat{Z}^1 = \{\mathbf{z}_j^1\}_{j \in [K]}$ . In the second round, the algorithm generates the new intermediate thoughts based on each selected thought<sup>1</sup>. After  $T$  rounds, we can get  $K$  most promising thoughts, and all of them contain information about their parent nodes, thus formulating as  $K$  CoTs. Therefore, this iterative process facilitates the refinement and selection of thoughts over multiple rounds. For each pair of thoughts, we use a direct comparison to identify the more promising one. We call such a direct comparison method "Standard Mode". We
+
+# Algorithm 1 C-ToT Algorithm
+
+1: Input: Query x, comparison times  $n$ , number of intermediate thoughts generation  $m$ , number of selected thoughts  $K$ , depth of the tree  $T$ .  
+2: Generate initial thoughts  $Z^1$  of size  $m$  with query  $\mathbf{x}$  
+3: for  $t = 1$  to  $T$  do  
+4: if Standard Mode then  
+5: Pair thoughts in  $Z^t$  randomly  
+6: while  $|Z^t| > K$  do  
+7: for every pair  $(a,b)$  in  $Z^t$  do  
+8: Compare thoughts  $a$  and  $b$  by LLM  $n$  times, then take a majority vote. If  $a$  wins, keep thought  $a$  in  $Z^t$  and drop  $b$ , and vice versa.  
+9: end for  
+10: end while  
+11: Denote  $\widehat{Z}^t$  by the remaining thoughts  
+12: else  
+13: Call Algorithm 2 to obtain  $\widehat{Z}^t$  
+14: end if  
+15: Generate the next  $m$  thoughts for each thought in  $\widehat{Z}^t$ .  
+16: end for
+
+summarize the proposed approach in Algorithm 1.
+
+Remark 1 (Comparison Complexity). In our approach, we keep all previous intermediate thoughts to compare in each round. This may affect the operational efficiency and exceed the storage limit. To improve the efficiency, we can introduce a counter for each intermediate thought to track its comparison frequency. If the comparison count of an intermediate thought exceeds a threshold, we can remove it from the tree. Since the comparison in each round is independent of each other, we could use parallel computing to improve
+
+the efficiency of the algorithm, or exploit more efficient machine learning techniques to schedule computational resources more adaptively and efficiently (Zhou, 2024) in the future. If the tree depth is  $T$ , the total number of comparisons required is less than the order of  $\mathcal{O}(nTK\log(m))$ .
+
+Remark 2 (Token Cost). The token costs of our proposed C-ToT approach and the S-ToT approaches are task-specific and generally incomparable. The C-ToT approach could discover valuable but misevaluated previous intermediate thoughts earlier than the S-ToT method. However, it may introduce more token overhead as we compare these thoughts multiple times. Therefore, we are better suited to the problem where the initial intermediate thoughts are more uncertain. We provide the token cost analysis in Appendix C.2.
+
+# 3.2. Instantiations and Analysis
+
+In the proposed C-ToT framework, we use a direct comparison for each pair of thoughts. Although our C-ToT method explores two sample information compared to the S-ToT methods that use only single sample information, the comparison feedback could still be inaccurate. We offer two methods to select the winning thought in a pair.
+
+Standard. Suppose the comparison difficulty of each pair is the same. Inspired by the ensemble algorithms in CoT generation (Wang et al., 2022), we can improve the robustness of the comparison feedback by setting  $n > 1$  in the "Standard Mode", so that we compare the two thoughts in each pair for  $n$  times and take majority voting output.
+
+Dueling. We consider a more general assumption of noisy comparisons, where we only assume an unknown ranking of the  $M_t$  thoughts at round  $t$ . This implies that the comparison difficulty of each pair varies, requiring a different number of comparisons for each pair. If two thoughts  $a$  and  $b$  are compared, thought  $a$  is chosen with some unknown probability  $p(a, b)$  and  $b$  is chosen with  $p(b, a) = 1 - p(a, b)$ , where the higher-ranked one has probability  $\geq 1/2$ . Repeated comparisons are independent of each other.
+
+We formulate it as a best-arm identification problem with dueling feedback (Yue et al., 2012; Falahatgar et al., 2017), propose a dueling bandits instantiation of the C-ToT framework, and analyze its properties. For each pair, we keep the empirical probability  $\widehat{p}_a$ , a proxy for  $p(a,b)$ . We also maintain a confidence value  $\widehat{c}$  s.t., w.h.p.,  $\widehat{p}_a \in (p(a,b) - \widehat{c}, p(a,b) + \widehat{c})$ . We stop the comparisons when it is sure of the winner or when it reaches its comparison budget  $n$ . If it reaches  $n$  comparisons, it outputs the element with more wins, randomly breaking ties. During comparison, we also compare two elements  $a, b$  with LLM by query them with different prompts. We summarize the proposed instantiation in Algorithm 2. Here stochasticity  $\gamma$  models the problem hardness.
+
+Analysis. First, we introduce some definitions. Given a set
+
+Algorithm 2 Knockout  
+1: Input: Set  $Z$ , bias  $\varepsilon$ , confidence  $\delta$ , stochasticity  $\gamma$ ,  $i = 1$   
+2: while  $|Z| > K$  do  
+3: Pair thoughts in  $Z$  randomly  
+4: for every pair  $(a, b)$  do  
+5: Set bias  $\varepsilon = \frac{(2^{1/3} - 1)\varepsilon}{\gamma 2^{i/3}}$ , confidence  $\delta = \frac{\delta}{2^i}$ ,  $\widehat{p}_a = 1/2$ ,  $\widehat{c} = 1/2$ ,  $n = \frac{1}{2\varepsilon^2} \log \frac{2}{\varepsilon}$ ,  $r = 0$ ,  $w_a = 0$   
+6: while  $|\widehat{p}_a - 1/2| \leq \widehat{c} - \varepsilon$  and  $r \leq n$  do  
+7: Compare thoughts  $a$  and  $b$  by LLM. if thought  $a$  wins,  $w_a = w_a + 1$ , and vice versa.  
+8:  $r = r + 1$ ,  $\widehat{p}_a = \frac{w_a}{r}$ ,  $\widehat{c} = \sqrt{\frac{1}{2r} \log \frac{4r^2}{\delta}}$   
+9: if  $\widehat{p}_a \leq 1/2$  then  
+10: Keep thought  $b$  in  $Z$  and drop  $a$ , break.  
+11: else  
+12: Keep thought  $a$  in  $Z$  and drop  $b$ , break.  
+13: end if  
+14: end while  
+15: end for  
+16:  $i = i + 1$   
+17: end while  
+18: Return  $\widehat{Z}$  by the remaining thoughts
+
+of thoughts  $Z = \{\mathbf{z}_1, \dots, \mathbf{z}_M\}$  of size  $M$ . Suppose there is an unknown underlying ranking function  $r: \mathcal{Z} \mapsto \mathbb{N}$  that ranks all the thoughts. Let  $r(\mathbf{z}_1), \dots, r(\mathbf{z}_M)$  be the ranking of the thoughts, such that when two elements  $\mathbf{z}_a$  and  $\mathbf{z}_b$  are compared, the higher ranked one is selected first, e.g.  $r(\mathbf{z}_a) < r(\mathbf{z}_b)$ . We define the  $\varepsilon$ -maximum via the  $(\varepsilon, \delta)$ -PAC paradigm, which requires that the output is likely to be close to the intended value. Specifically, given  $\varepsilon > 0, \delta > 0$ , with probability  $\geq 1 - \delta$ , the maximum selection must produce an element  $a$  such that for  $b$ , with  $r(b) = M$ ,  $p(a, b) \geq \frac{1}{2} - \varepsilon$ . We call such an output  $\varepsilon$ -maximum.
+
+Lemma 1 (Theorem 3 in (Falahatgar et al., 2017)). Knockout  $(Z, \varepsilon, \delta)$  uses  $\mathcal{O}\left(\frac{\gamma^2|Z|}{\varepsilon^2} \log \frac{1}{\delta}\right)$  comparisons and with probability at least  $1 - \delta$ , outputs an  $\varepsilon$ -maximum.
+
+Proposition 1. Suppose that the depth of the tree is  $T$ , and thoughts in the shallower layers are more promising than those in the deeper ones. Then, the probability of missing the  $\varepsilon$ -maximum promising thoughts in the  $\tau$ -th layer is  $1 - \delta^{\tau}$  with at most  $\mathcal{O}\left(\frac{\gamma^{2}\sum_{i = 1}^{T}|Z^{i}|}{\varepsilon^{2}}\log \frac{1}{\delta}\right)$  comparisons required for generating the whole tree of thoughts.
+
+Remark 3. Proposition 1 is directly derived from Lemma 1 by the union bound. Proposition 1 shows that, under the general assumption of noisy comparisons and utilizing our proposed pairwise-comparison approach, valuable intermediate thoughts will still not be overlooked, especially for the thoughts in the shallow layer, which may be more uncertain as they appear at the beginning of the ToT generation. We leave the detailed proofs to Appendix B.
+
+# 4. Experiments
+
+We test our proposed algorithm in three real-world tasks: question answering (QA), as well as mathematical reasoning tasks, namely, the Game of 24 and Sudoku Puzzles. The LLM employed in experiments is GPT-3.5-turbo-1106.
+
+# 4.1. Experiment Setup
+
+Contenders Setup. Our evaluation firstly includes a comparison with a baseline method that directly queries the LLM for the final result (we denote it as Direct); three state-of-the-art contenders: CoT (Wei et al., 2022), SC-CoT (Wang et al., 2022), and SToT (Yao et al., 2023). For the CoT method, we query the LLM directly to get the final answer, following the settings as in (Wei et al., 2022). For the SC-CoT method, for a fair comparison, if without further notice, we set the CoT number approximately the same number of tokens with our proposed algorithm. Specifically, 15 samples were generated for each question, using the same settings as in (Wang et al., 2022), with the final answer determined by majority voting. The SToT approach is also implemented identically to the setting in Yao et al. (2023). For the depth of the tree in the SToT algorithm, we set it equal to the depth with our proposed algorithms C-ToT (Stand.) and C-ToT (Duel.).
+
+We further propose three contenders for comparison to test the effectiveness of our proposed algorithm. A robust implementation of SToT is proposed that follows the main idea of SC-CoT to account for noise in the LLM feedback, called SC-SToT. Two variants of SToT that equipped with our proposed mechanisms are also included, denoted as Comp-SToT, Back-SToT. Specifically,
+
+(1) SC-SToT: We denote the self-consistent variant of the ToT algorithm as SC-SToT, short for Self-Consistent ToT algorithm. To alleviate feedback noise in the ToT algorithm, we draw inspiration from the self-consistent CoT generation algorithm (Wang et al., 2022). Our proposal involves querying the LLM multiple times during intermediate thought evaluation in the ToT algorithm and using the majority voting results as the final evaluation in the SC-SToT. This contender is a direct extension of the ToT algorithm to account for the feedback noise of the LLM, but the cost is very high as it scores each intermediate thought multiple times;  
+(2) Comp-SToT: We replace the score-based evaluation in the original SToT algorithm with our proposed pairwise comparison approach and refer to this variant algorithm as Comp-SToT.  
+(3) Back-SToT: We replace the search algorithm in the original SToS algorithm with our proposed mechanism, which retains all previous intermediate thoughts with their corresponding socres and takes the highest scoring thoughts as the most promising ones. We call this variant algorithm as Back-SToT.
+
+In addition, we include three state-of-the-art algorithms into comparison: PoT (Chen et al., 2023), Self-Refine (Madaan et al., 2023) and GoT (Besta et al., 2023). Detailed experimental results can be found in Appendix C.1.
+
+For our proposed C-ToT approaches, we denote the C-ToT algorithm in "Standard Mode" by C-ToT (Stand.) and set the number of comparisons  $n$  to 1. We denote the C-ToT algorithm that considers the general comparison noise by C-ToT (Duel.), and set the maximum number of comparisons to 3 and set  $\gamma = 0.1$ . All experiments are repeated 3 times.
+
+Intermediate Thoughts Generation. In general, different tasks should have different thought generators. Exploiting problem properties is essential to effectively design the intermediate thoughts. We follow the setting of (Yao et al., 2023) to generate the intermediate thoughts. For example, we generate the thoughts as a few words, as in QA; as a line of equations, as in the Game of 24; or as an intermediate solution in the Sudoku puzzle. We defer the implementation for prompts and the cost comparison of our approaches and other contenders to Appendix A and Appendix C.2.
+
+# 4.2. Question Answering
+
+Task Setup. We first test the performance of our proposed algorithm on the question answering tasks using the AQuA dataset (Ling et al., 2017), which comprises 254 arithmetic reasoning tasks aimed at assessing logical abilities through various mathematical computation problems. Each question in this dataset is accompanied by five multiple-choice options, labeled from A to E. We follow the experimental protocol as it is in the work of Wang et al. (2022). The accuracy of the responses is gauged by comparing the generated answers with the standard solutions. Results on other QA datasets can be found in Appendix C.1.
+
+C-ToT Setup. For the AQuA dataset, we set the maximum depth of tree of thoughts to 3. For each intermediate thought selected, we set  $m = 12$ , thus generating 12 new intermediate thoughts as the next step. The maximum number of selected thoughts  $K$  per layer is set to 3. Therefore, starting from the "question" as the root, all newly generated thoughts are compared, and we select 3 most promising intermediate thoughts to generate the next step.
+
+In the question answering task, it is difficult to set a fixed length for the C-ToT, i.e., an intermediate thought may already summarize the answer before reaching the maximum length of the C-ToT. For those selected intermediate thoughts that have already reached an answer, we add them to the "answer list" and do not include them in the comparison in the next round. For those intermediate thoughts that have already reached an answer but were not selected, we will include them in the comparison in the next round. This mechanism thus gives excluded answers a chance to
+
+<table><tr><td>Method</td><td>Accuracy</td></tr><tr><td>Direct</td><td>24.8%</td></tr><tr><td>CoT</td><td>42.3%</td></tr><tr><td>SC-CoT</td><td>58.4%</td></tr><tr><td>SToS</td><td>57.1%</td></tr><tr><td>SC-SToS</td><td>57.6%</td></tr><tr><td>Comp-SToS</td><td>59.0%</td></tr><tr><td>Back-SToS</td><td>58.0%</td></tr><tr><td>C-ToT (Stand.)</td><td>61.4%</td></tr><tr><td>C-ToT (Duel.)</td><td>63.0%</td></tr></table>
+
+![](images/c0990059509bd72a0ab40775f73e35575d6cdc370862bfc4f75be72f09b8c80f.jpg)  
+Figure 3: Predictions of SToT and Comp-SToT on AQuA.
+
+Table 1: Average accuracy on AQuA.  
+
+<table><tr><td>Method</td><td>Accuracy</td></tr><tr><td>Direct</td><td>8.0%</td></tr><tr><td>CoT</td><td>4.3%</td></tr><tr><td>SC-CoT</td><td>8.0%</td></tr><tr><td>SToS</td><td>34.3%</td></tr><tr><td>SC-SToS</td><td>40.0%</td></tr><tr><td>Comp-SToS</td><td>36.3%</td></tr><tr><td>Back-SToS</td><td>39.0%</td></tr><tr><td>C-ToT (Stand.)</td><td>40.0%</td></tr><tr><td>C-ToT (Duel.)</td><td>41.0%</td></tr></table>
+
+Table 2: Average accuracy on Game of 24.
+
+![](images/73be3fc485bde6223503939f81dc68ea7d9dffdf05dc6fe7db59b3bfbb984d69.jpg)
+
+Figure 4: Accuracy and token costs of differenta methods.
+
+be included in the "answer list" by subsequent comparisons. After  $T$  rounds of thought generation and comparison for selection, the selected chains are appended to the "answer list", and a majority voting mechanism is used on the "answer list" to determine the final answer. We leave the implementation details of the thought generator and the comparison prompt for the question answering task to Appendix A.1.
+
+Comparison Results. We report the comparison results of our proposed approaches with other contenders in Table 1. All CoT approaches outperform the Direct query method, showing the importance of designing effective CoTs to guide LLMs from simplicity to complexity. We can also observe that the SC-SToT method outperforms the original SToT method, where the algorithm scores the intermediate thoughts multiple times to alleviate the noise. However, this mechanism will significantly increase the token cost. We leave the detailed discussion of token cost to Appndex C.2.
+
+Both proposed variants of SToS methods achieve higher average accuracy than the original SToS method. In CompSToS, the point-wise scoring mechanism is replaced by a pairwise comparison, while in Back-SToS, our backtracking search algorithm replaces the original search algorithm, taking into account all previous intermediate thoughts. These results demonstrate the effectiveness of these two mechanisms, such that the proposed C-ToT (Stand.) and C-ToT (Duel.) approaches outperform all contenders. Moreover, C-ToT (Duel.) achieves superior performance by further modeling noise in the comparison.
+
+We delve deeper to explore the benefits of the pairwise comparison mechanism to test whether it can better find the most promising intermediate thoughts. Note that we do not have access to the underlying value or order of the intermediate thoughts in each round. Therefore, we use the final prediction error as a proxy, since the depth of the tree structure in the QA datasets is shallow, limited to 1 to 3 levels. We quantify the number of QA problems correctly/incorrectly predicted by ToT and Comp-SToT and report it in Figure 3. Our observation shows that Comp-SToT predicts more correctly when S-ToT predicts incorrectly, showing the superiority of the pairwise comparison mechanism over the pointwise scoring mechanism. This validates the rationale of pairwise comparison mechanism.
+
+# 4.3. Game of 24
+
+Task setup. This is a math problem where the goal is to use four numbers and basic arithmetic operations  $\{+, -, *, / \}$  to get a sum of 24. For example, given the input  $\{4, 9, 10, 13\}$ , a viable solution would be  $(10 - 4) * (13 - 9) = 24$ . In our experiments, we use the same dataset as in the work of Yao et al. (2023) and follow their experimental setup, which consists of 1,362 problems taken from the 24-point game on 4nums.com. We have selected questions numbered 401 to 500 as our question set. Each problem consisted of four numbers selected from 1 to 13, and the goal is to formulate a calculation using these numbers to reach a total of 24. The accuracy of the solutions is scored based on whether all 4 input numbers were used and whether the result is 24.
+
+C-ToT Setup. In this task, we restore unselected intermediate thoughts in previous layers and apply pruning when the current node is inferior to previously unselected ones. We set the maximum depth of tree to 6. The computation terminates either when an answer containing the number 24 is derived, or when the maximum layer limit is reached.
+
+We set the number of selected intermediate thoughts per layer  $K = 5$  and let the LLM generate a variable number of new thoughts. If the total number of new thoughts is less than or equal to twice the maximum number of selected thoughts, thoughts are moved from the "remain list" (a list that stores reserved thoughts) to the new node list until the number of new thoughts reaches twice the maximum number of selected thoughts or the "remain list" is emptied.
+
+We also optimize the pruning process that takes place before the comparison stage and apply it to all contenders to save tokens. Newly generated intermediate thoughts with a single number unequal to 24 is eliminated. The rest are then filtered by comparison, selecting a number of new thoughts equal to or less than the maximum number of selected thoughts. Thoughts that are not selected are added to the "remain list". If one of the selected thoughts contains the final answer, it is added to the "answer list" and the interactive process is stopped. We leave the implementation details of the thought generator and the comparison prompt to Appendix A.2.
+
+Comparison Results. We report the overall comparison results in Table 2. We observed trends similar to those reported in the QA task. We find that both the SToT and
+
+CToT approaches significantly outperform the CoT-based methods, indicating the need to interact with the LLM to generate a more powerful chain of thoughts to handle complex reasoning tasks. The SC-SToT method improves the performance of SToT, while the Comp-SToT can achieve a similar improvement with pairwise comparison, indicating the superiority of the comparison mechanism. Therefore, the combination of the comparison mechanism and the specific design for comparison noise allows the C-ToT (Duel.) to achieve the highest average accuracy.
+
+We also report the average accuracy of different methods against their token cost in Figure 4. We can observe that the token cost per question of the Comp-SToT algorithm is the same as that of the SToT algorithm, but it achieves a better performance. In practice, we can reduce the token cost by using a counter for each thought to track its comparison frequency. If the comparison count of a thought exceeds a threshold, we can remove it from the tree. We also leave the detailed discussion of token cost to Appendix C.2.
+
+# 4.4. Sudoku Puzzle
+
+Task Setup. We use the Selenium Dataset (Long, 2023), containing 10 Selenium puzzles each of  $3 \times 3$ ,  $4 \times 4$ , and  $5 \times 5$  dimensions. Each puzzle is partially filled with numbers, and the task is to complete the entire Selenium grid without changing the given numbers. The correctness of the solutions was determined by whether a complete and correct Selenium grid is generated.
+
+C-ToT Setup. In this task, we test our proposed algorithm against other competitors in Sudoku puzzles of three different sizes. In each case, we set the maximum depth of tree of thoughts to 15. The computation stops either when the correct Sudoku solution is derived or when the maximum number of steps is reached. For each intermediate thought selected, we set  $m = 5$ , thus generating 5 new intermediate thoughts as the next step. The maximum number of selected thoughts  $K$  per layer is also set to 3.
+
+From the newly generated thoughts, a number equal to or less than the maximum allowed was selected by comparison. A pruning strategy was used to check for and eliminate thoughts containing results that did not meet the Selenium requirements, such as duplicate numbers in the same row or column. These non-compliant results were removed from both the selected and unselected thoughts. The remaining unselected thoughts were then added to the "remain list".
+
+If the number of selected thoughts was less than the maximum, additional thoughts were moved from the "remain list" to the "select list" until either the maximum number was reached or the "remain list" was emptied. We then checked whether the "select list" contained a correct solution. If a correct solution was found, it was added to the "answer list"
+
+<table><tr><td>Method</td><td>Acc. 3 × 3</td><td>Acc. 4 × 4</td><td>Acc. 5 × 5</td></tr><tr><td>Direct</td><td>56.7%</td><td>37.7%</td><td>16.7%</td></tr><tr><td>CoT</td><td>73.3%</td><td>36.7%</td><td>23.3%</td></tr><tr><td>SC-CoT</td><td>76.7%</td><td>50.0%</td><td>16.7%</td></tr><tr><td>SToS</td><td>86.7%</td><td>46.7%</td><td>46.7%</td></tr><tr><td>SC-SToS</td><td>96.7%</td><td>53.3%</td><td>50.0%</td></tr><tr><td>Comp-SToS</td><td>100.0%</td><td>46.7%</td><td>50.0%</td></tr><tr><td>Back-SToS</td><td>100.0%</td><td>60.0%</td><td>56.7%</td></tr><tr><td>C-ToT (Stand.)</td><td>100.0%</td><td>63.3%</td><td>60.0%</td></tr><tr><td>C-ToT (Duel.)</td><td>100.0%</td><td>63.3%</td><td>63.3%</td></tr></table>
+
+Table 3: Average accuracy on Selenium Puzzles.
+
+and the program was terminated. We leave the implementation details of the thought generator and the comparison prompt to Appendix A.3.
+
+Comparison Results. We report the comparison results in Table 3. In all three tasks, our proposed approaches, C-ToT (Stand.) and C-ToT (Duel.), consistently achieve the highest average accuracy, demonstrating their superior ability to handle complex reasoning tasks. While the SToT method generally outperforms the Direct method and the CoT method, it does not always outperform the SC-ToT method, as seen in the  $4 \times 4$  Sudoku task. This phenomenon may be due to the potential noise in pointwise scoring methods, which could mislead the subsequent generation of intermediate thoughts. The SC-ToT method introduces thought ensembles, which naturally mitigate the noise in the LLM feedback. Our proposed methods consistently outperform other contenders, suggesting that pairwise comparison mechanism effectively mitigates noise in LLM feedback, identifies the most promising intermediate thoughts, and improves the generated chain compared to SToT-based methods. We can observe that the C-ToT (Stand.) algorithm achieves the same performance as the C-ToT (Duel.) algorithm, which indicates that a single pairwise comparison could already provide reliable feedback in the Sudoku puzzle tasks.
+
+# 5. Conclusion
+
+This paper investigates a widespread but understudied problem of noisy feedback from LLMs in CoT generation tasks. Motivated by Vapnik's principle, we argue that for LLMs, the simultaneous comparison of two thoughts provides a more robust evaluation compared to individual value evaluations, and thus we propose a pairwise-comparison ToT approach C-ToT, approaching to searching for the most promising intermediate thought. The proposed method directly selects the most promising intermediate thought by pairwise comparison, and incorporates previous thoughts into the comparison to allow for rethinking. To further alleviate the noise in the comparison, we propose two variants of the C-ToT algorithm, and analyze the theoretical properties. Experiments on three real-world mathematical and reasoning tasks show the effectiveness of our proposed algorithm and verify the rationale of the pairwise comparison.
+
+# Acknowledgments
+
+HY was supported by Cisco Faculty Research Award. MS was supported by JST CREST Grant Number JP-MJCR18A2.
+
+# Impact Statement
+
+This research investigates a general problem of CoT generation with any LLM, where we take into account the noise in the feedback of the LLM. Therefore, when using LLMs for complex mathematical or logical reasoning problems, the user could benefit from our study from the aspect of generating a more effective CoT. The consequences of system failure and bias in the data are not applicable.
+
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+
+# A. Implementation Details
+
+In this section, we provide implementation details for all experiments, focusing primarily on the design of the thought generation and comparison prompts for each task.
+
+# A.1. Question Answering
+
+Thought Generator. We use a zero-shot prompt. For each question, we use the same prompt multiple times to generate a specified number of different new thoughts.
+
+```txt
+Prompt = 'Here is a question. You should work on it step by step. Your answer must be only the alphabet of your choice and begin with _____. For example: ____A, which should be at the last line. Q: {question}'
+```
+
+Comparison Prompt. We use multiple different prompts to generate the comparison result at each round. For the QA problem, we use three different prompts. We evaluate the same thought three times by using each prompt once and take the majority as the answer.
+
+```txt
+Prompt 1 = 'You should judge which of the two analysis is better. You must only reply 1 or 2.  
+1: {input_1}  
+2: {input_2}'  
+Prompt 2 = 'Find out which of the two analysis is better. You must only reply 1 or 2.  
+1: {input_1}  
+2: {input_2}'  
+Prompt 3 = 'Compare the two analysis and find which is better. You must only reply 1 or 2.  
+1: {input_1}  
+2: {input_2}'
+```
+
+# A.2. Game of 24
+
+Thought Generator. In this experiment, we use prompts similar to the ToT experiment to generate thoughts. There are two prompts: one is to select two numbers from the remaining list for the next step in the 24-point calculation, and then to add the newly obtained number back into the remaining list of numbers. The other is to generate the total operation formula that results in 24, based on all previous steps, when only one number remains. Both prompts are few-shot.
+
+```txt
+Prompt 1 = 'You should choose two of the input numbers and use basic arithmetic operations (+ - * /) to obtain a new number. The new number should replace those two input numbers. Give me at least 6 possible next steps.  
+Input: 2 8 8 14  
+Possible next steps:  
+2 + 8 = 10 (left: 8 10 14)  
+8 / 2 = 4 (left: 4 8 14)  
+14 + 2 = 16 (left: 8 8 16)  
+2 * 8 = 16 (left: 8 14 16)  
+8 - 2 = 6 (left: 6 8 14)  
+14 - 8 = 6 (left: 2 6 8)  
+14 / 2 = 7 (left: 7 8 8)  
+14 - 2 = 12 (left: 8 8 12)  
+Input: {input}  
+Possible next steps:
+```
+
+```txt
+Prompt 2 = 'Use numbers and basic arithmetic operations (+ - * /) to obtain 24. Each step, you are only allowed to choose two of the remaining numbers to obtain a new number.
+```
+
+```txt
+Input: 4 4 6 8  
+Steps:  
+4 + 8 = 12 (left: 4 6 12)  
+6 - 4 = 2 (left: 2 12)  
+2 * 12 = 24 (left: 24)  
+Answer: (6 - 4) * (4 + 8) = 24  
+Input: 2 9 10 12  
+Steps:  
+12 * 2 = 24 (left: 9 10 24)  
+10 - 9 = 1 (left: 1 24)  
+24 * 1 = 24 (left: 24)  
+Answer: (12 * 2) * (10 - 9) = 24  
+Input: {input}
+```
+
+Comparison Prompt. We use multiple different prompts to generate the comparison result at each round. For the 24 problem, we use three different prompts. All prompts are few-shot. We evaluate the same thought three times by using each prompt once and take the majority as the answer.
+
+```txt
+Prompt 1 = 'I will give you two groups of numbers. The evaluation criteria is if using all of the given numbers with basic arithmetic operations (+ - * /) can reach 24. You should compare the two inputs and decide which input is better. You should only reply 1 or 2.  
+input_1: 2 12  
+2 * 12 = 24  
+input_2: 11 12  
+all arithmetic operations can't get 24  
+Answer: 1  
+input_1: 1 2 4  
+too small  
+input_2: 3 8  
+3 * 8 = 24  
+Answer: 2  
+input_1: 1 12 11  
+1 + 12 + 11 = 24  
+input_2: 12 12  
+12 + 12 = 24  
+Both can reach 24, randomly select one  
+Answer: 1  
+input_1: {input_1}  
+input_2: {input_2}  
+Answer: '
+```
+
+```javascript
+Prompt 2 = 'I will give you two groups of numbers. Tell me which input is better. The better one is more possible to reach 24 by using all of the given numbers with basic arithmetic operations (+ - * /). You should only reply 1 or 2. //same examples input_1: {input_1} input_2: {input_2} Answer: '
+```
+
+```txt
+Prompt  $3 =$  'Here are two groups of numbers. Tell me which input is more possible to use all of the given numbers with basic arithmetic operations (+ - * /) to get 24. You should only reply 1 or 2. Don't add any explanation. //same examples
+```
+
+```txt
+input_1: {input_1}  
+input_2: {input_2}  
+Answer:
+```
+
+# A.3. Sudoku Puzzle
+
+Thought Generator. We use the following prompt to generate thoughts.
+
+```txt
+Prompt = 'This is a {puzzle_size}x{puzzle_size} two-dimensional array represents a matrix, where some numbers are already given, and '\*' represents the numbers that need to be filled in. You should pick 1 or 2 '\*' to fill in a number between 1 to {puzzle_size}. Don't change the given number. Don't complete the whole puzzle immediately until there is only 1 or 2 '\*' left to be filled in. Your answer should just be the same format as the question below. When you answer, begin with \*\*\*. For example: \*\*\*[1, *, *], [*, 1, *], [*, 2, *]] Question: {question}'
+```
+
+Comparison Prompt. We use multiple different prompts to generate the comparison result at each round. For the Sudoku problem, we use three different prompts. All prompts are zero-shot.
+
+```txt
+Prompt 1 = 'You should judge which of the two two-dimensional array better represents a {puzzle_size}x{puzzle_size} Sudooku puzzle. '  $\star$  ' means the value is yet to be decided. You should judge by considering if in each row or column 1 to {puzzle_size} could appear and only appear once. You must only reply 1 or 2.  
+1:{input_1}  
+2:{input_2}'
+```
+
+```txt
+Prompt 2 = 'Find which of the two two-dimensional array better represents a { puzzle_size}x{puzzle_size} Sudoku puzzle. '\* means the value hasn't been decided. The better one should satisfy that in each row or column 1 to { puzzle_size} could appear and only appear once. You must only reply 1 or 2. 1:{input_1} 2:{input_2}'
+```
+
+```txt
+Prompt 3 = 'Which of the two two-dimensional array better represents a { puzzle_size}x{puzzle_size} Sudooku puzzle? ' means the value is yet to be decided. A better one means in each row or column 1 to {puzzle_size} could appear and only appear once. You must only reply 1 or 2. 1:{input_1} 2:{input_2}'
+```
+
+# B. Proofs
+
+We first introduce the following lemma before our main proof.
+
+Lemma 2 (Lemma 2 in (Falahatgar et al., 2017)). Let  $\widetilde{p}(i,j) = p(i,j) - 1/2$  be the additional probability by which  $i$  is preferable to  $j$ . Let  $z^*$  be the maximum in  $Z$  and  $k^*$  be the comparison winner. The comparison algorithm on set  $Z$  uses  $\frac{|Z|}{4\varepsilon^2} \log \frac{2}{\delta}$  comparisons and with probability  $\geq 1 - \delta$ ,
+
+$$
+\widetilde {p} (z ^ {*}, k ^ {*}) \leq \gamma \varepsilon .
+$$
+
+Proof of Lemma 2. To make this paper self-contained, we provide the proofs in (Falahatgar et al., 2017) here. First, we prove that the probability of the direct pairwise comparison process providing a wrong winner is less than  $\delta$ . Let  $\widehat{p}_i^r$  and  $\widehat{c}^r$
+
+denote  $\widehat{p}_i$  and  $\widehat{c}$  respectively after  $r$  number of comparisons. The output of the pairwise comparison will not be  $i$  only if  $\widehat{p}_i^r < \frac{1}{2} + \varepsilon - \widehat{c}^r$  for any  $r < m = \frac{1}{2\varepsilon^2} \log \frac{2}{\delta}$  or if  $\widehat{p}_i < \frac{1}{2}$  for  $r = m$ .
+
+Considering the first case, after  $r$  comparisons, by Chernoff bound,
+
+$$
+\operatorname * {P r} (\widehat {p} _ {i} ^ {r} <   \frac {1}{2} + \varepsilon - \widehat {c} ^ {r}) \leq e ^ {- 2 r (\widehat {c} ^ {r}) ^ {2}} = e ^ {- \log \frac {4 r ^ {2}}{\delta}} = \frac {\delta}{4 r ^ {2}}.
+$$
+
+Using union bound,
+
+$$
+\Pr (\exists r \mathrm {s . t .} \widehat {p} _ {i} ^ {r} \leq \frac {1}{2} + \varepsilon - \widehat {c} ^ {r}) \leq \frac {\delta}{2}.
+$$
+
+Considering the second case, after  $m = \frac{1}{2\varepsilon^2}\log \frac{2}{\delta}$  rounds, by Chernoff bound,
+
+$$
+P r \left(\widehat {p} _ {i} ^ {m} <   \frac {1}{2}\right) \leq e ^ {- 2 m \varepsilon^ {2}} = \frac {\delta}{2}. \tag {1}
+$$
+
+Thus, the probability of each of these events happening is bounded by  $\frac{\delta}{2}$ , and thus the probability of the pairwise comparison process providing a wrong winner is less than  $\delta$ .
+
+As each of the  $|Z|/2$  pairs is compared at most  $\frac{1}{2\varepsilon^2}\log \frac{2}{\delta}$  times, the total comparisons is less than  $\frac{|Z|}{4\varepsilon^2}\log \frac{2}{\delta}$ . Let  $k^*$  be the comparison winner and  $z^*$  be the maximum in  $Z$ . Let  $a$  be the element paired with  $z^*$ . There are two cases:  $\widetilde{p}(z^*, a) \geq \varepsilon$  and  $\widetilde{p}(z^*, a) < \varepsilon$ .
+
+If  $\widetilde{p}(z^*, a) \geq \varepsilon$ , by Eqn. (1) with probability  $\geq 1 - \delta$ ,  $z^*$  will win and hence by definitions of  $z^*$  and  $k^*$ ,  $\widetilde{p}(z^*, k^*) = 0 \leq \gamma \varepsilon$ . Alternatively, if  $\widetilde{p}(z^*, a) < \varepsilon$ , let winner  $(i, j)$  denote the winner between  $i$  and  $j$  when compared for  $\frac{1}{2\varepsilon^2} \log \frac{1}{\delta}$  times. Then,
+
+$$
+r (a) \stackrel {(a)} {\leq} r (\operatorname {w i n n e r} (z ^ {*}, a)) \stackrel {(b)} {\leq} r (k ^ {*}) \stackrel {(c)} {\leq} r (z ^ {*})
+$$
+
+where (a) follows from  $r(a)\leq r(z^{*})$  , (b) and (c) follow from the definitions of  $z^{*}$  and  $k^{*}$  respectively. From strong stochastic transitivity on  $a,k^{*}$  and  $z^{*},\widetilde{p} (z^{*},k^{*})\leq \gamma \widetilde{p} (z^{*},a)\leq \gamma \varepsilon .$
+
+Now we begin to prove Lemma 1.
+
+Proof of Lemma 1. To make this paper self-contained, we also provide the proofs in (Falahatgar et al., 2017) here. We first show that with probability  $\geq 1 - \delta$ , the output of Knockout is an  $\varepsilon$ -maximum. Let  $\varepsilon_{i} = c\varepsilon / 2^{i/3}$  and  $\delta_{i} = \delta / 2^{i}$ . Note that  $\varepsilon_{i}$  and  $\delta_{i}$  are bias and confidence values used in round  $i$ . Let  $b_{i}$  be a maximum element in the set  $Z$  before round  $i$ . Then by Lemma 2, with probability  $\geq 1 - \delta_{i}$ ,
+
+$$
+\widetilde {p} \left(b _ {i}, b _ {i + 1}\right) \leq \frac {c \varepsilon}{2 ^ {i / 3}}. \tag {2}
+$$
+
+By union bound, denote by  $p'$  the probability that Eqn. (2) does not hold for some round  $1 \leq i \leq \log |Z|$ , then we have
+
+$$
+p ^ {\prime} \leq \sum_ {i = 1} ^ {\log | Z |} \delta_ {i} = \sum_ {i = 1} ^ {\log | Z |} \frac {\delta}{2 ^ {i}} \leq \delta .
+$$
+
+With probability  $\geq 1 - \delta$ , Eqn. (2) holds for all  $i$  and by stochastic triangle inequality,
+
+$$
+\widetilde {p} \left(b _ {1}, b _ {\log | Z | + 1}\right) \leq \sum_ {i = 1} ^ {\log | Z |} \widetilde {p} \left(b _ {i}, b _ {i + 1}\right) \leq \sum_ {i = 1} ^ {\infty} \frac {c \varepsilon}{2 ^ {i / 3}} = \varepsilon .
+$$
+
+We now bound the number of comparisons. Let  $n_i = \frac{|Z|}{2^{i-1}}$  be the number of elements in the set at the beginning of round  $i$ . Denote by  $\mathrm{NC}_i$  the number of comparisons at round  $i$ , then we have
+
+$$
+\mathrm {N C} _ {i} \leq \frac {n _ {i}}{2} \cdot \frac {\gamma^ {2} 2 ^ {2 i / 3}}{2 c ^ {2} \varepsilon^ {2}} \cdot \log \frac {2 ^ {i + 1}}{\delta}.
+$$
+
+Hence the number of comparisons in all rounds is
+
+$$
+\begin{array}{l} \sum_ {i = 1} ^ {\log | Z |} \frac {| Z |}{2 ^ {i}} \cdot \frac {\gamma^ {2} 2 ^ {2 i / 3}}{2 c ^ {2} \varepsilon^ {2}} \cdot \log \frac {2 ^ {i + 1}}{\delta} \leq \frac {| Z | \gamma^ {2}}{2 c ^ {2} \varepsilon^ {2}} \sum_ {i = 1} ^ {\infty} \frac {1}{2 ^ {i / 3}} \left(i + \log \frac {2}{\delta}\right) \\ = \frac {| Z | \gamma^ {2}}{2 c ^ {2} \varepsilon^ {2}} \left(\frac {2 ^ {1 / 3}}{c ^ {2}} + \frac {1}{c} \log \frac {2}{\delta}\right) = \mathcal {O} \left(\frac {| Z | \gamma^ {2}}{\varepsilon^ {2}} \log \frac {1}{\delta}\right). \\ \end{array}
+$$
+
+![](images/d54803f95fb1ea2c0f28755c47623e52814f84650300d44c061d2b129ac7c002.jpg)
+
+Now we begin to prove Proposition 1.
+
+Proof of Proposition 1. In each round of comparison, according to Lemma 1, we have the probability of  $1 - \delta$  to output the  $\varepsilon$ -maximum thoughts. Suppose that the thoughts in the shallower layers are more promising than those in the deeper ones, which is often the case in step-by-step reasoning tasks. For example,  $r([{\bf z}_j^1])\geq r([{\bf z}_i^1,{\bf z}^2 ])$  for  $j\in [K],i\notin [K]$ , and  $\mathbf{z}^2\in Z^2$ . When generating the intermediate thoughts in the  $\tau$ -th layer, the probability that the  $\varepsilon$ -maximum thoughts are not selected is  $1 - \delta^{\tau}$ .
+
+Therefore, the probability of missing the  $\varepsilon$ -maximum promising thoughts in the  $\tau$ -th layer is  $1 - \delta^{\tau}$  with at most  $\mathcal{O}\left(\frac{\gamma^2\sum_{i=1}^{T}|Z^i|}{\varepsilon^2}\log \frac{1}{\delta}\right)$  comparisons required for generating the tree.
+
+# C. Additional Experimental Results
+
+# C.1. Experimental Results Summary
+
+First, we present a summary of the experimental results.
+
+We introduce three additional QA datasets—Gsm8k (Cobbe et al., 2021), Coin Flip (OOD) (Wei et al., 2022), and BBH (Srivastava et al., 2023)—along with two more state-of-the-art algorithms for comparison, with implementation details provided below.
+
+Contenders. PoT (Chen et al., 2023): PoT requires in-context samples to guide LLMs generating Python code step-by-step, and the number of samples is a hyperparameter. Since SToT and C-ToT do not always require such samples, we choose appropriate number of samples in PoT to maintain experimental consistency with other methods. We use one in-context sample for all datasets expecting Game of 24, which uses 3 samples.
+
+Self-Refine (Madaan et al., 2023): The parameter of in Self-Refine is the number of iterations of Self-Refine. For each task, we set the number of iterations to 3 so that the number of tokens used is approximately the same. We use the same template as in the original paper and set the number of in-context examples to the same as in the PoT.
+
+We can observe in Table 4 that the proposed C-ToT (Stand.) and C-ToT (Duel.) algorithms achieve the best performance in almost all tasks. The Self-Refine also achieves promising results in QA tasks, while it performs relatively poorly compared to the SToT and C-ToT algorithms in complex tasks that require step-by-step interaction and reasoning, such as Game of 24.
+
+<table><tr><td>Method / Data</td><td>Gsm8k</td><td>Coin flip (OOD)</td><td>BBH</td><td>AQuA</td><td>Game of 24</td><td>Sudoku</td></tr><tr><td>CoT</td><td>68.8 ± 2.5</td><td>56.3 ± 2.1</td><td>67.7 ± 2.6</td><td>42.3 ± 2.5</td><td>4.3 ± 3.2</td><td>44.4 ± 2.3</td></tr><tr><td>SToT</td><td>59.3 ± 2.4</td><td>62.1 ± 2.9</td><td>69.6 ± 2.0</td><td>57.1 ± 2.1</td><td>34.3 ± 3.9</td><td>60.0 ± 3.8</td></tr><tr><td>PoT</td><td>62.2 ± 3.3</td><td>71.1 ± 2.8</td><td>66.7 ± 2.3</td><td>47.5 ± 4.5</td><td>27.2 ± 3.7</td><td>43.1 ± 3.9</td></tr><tr><td>Self-Refine</td><td>67.6 ± 3.2</td><td>64.8 ± 1.6</td><td>69.2 ± 3.7</td><td>56.2 ± 4.1</td><td>16.3 ± 3.9</td><td>52.3 ± 3.0</td></tr><tr><td>C-ToT (Stand.)</td><td>71.3 ± 3.3</td><td>66.2 ± 2.8</td><td>75.7 ± 1.9</td><td>61.4 ± 2.9</td><td>40.0 ± 4.6</td><td>74.4 ± 2.1</td></tr><tr><td>C-ToT (Duel.)</td><td>73.0 ± 2.7</td><td>70.4 ± 2.1</td><td>80.0 ± 2.2</td><td>63.0 ± 2.6</td><td>41.0 ± 3.0</td><td>75.4 ± 3.3</td></tr></table>
+
+Table 4: Performance comparisons on benchmark datasets. On each dataset, 5 test runs were conducted and the average accuracy as well as standard deviation are presented, and the best one is emphasized in bold.
+
+Our method is generally not comparable to GoT (Besta et al., 2023), but the idea of comparison can improve GoT. The GoT divides a complex task into several subtasks, generate thoughts for each subtask and merge them, while the ToT-style
+
+methods aim at step-by-step thinking, where the new generated thought is based on previous ones, so it is hard to split or merge. Thus, they are better suited for different tasks. For example, for sorting, we can divide it into subtasks; while for the Game of 24, the answer should be generated based on previous thoughts, so C-ToT is more proper.
+
+We can use the idea of comparison to improve the solving of subtasks, so we can subsequently improve the GoT. The following is a preliminary study of the sorting task in the GoT paper, remaining all its setups, and comparing the performance with scoring mechanism by LLM and the pairwise comparison mechanism by LLM in subtasks. Accuracy and token usage are reported as in Table 5. 5 test runs were conducted and the average accuracy and the number of tokens (completion tokens / prompt tokens) are reported.
+
+<table><tr><td>Data / Method</td><td>S-GoT</td><td>C-GoT</td></tr><tr><td>32 elements</td><td>90.4%; 1232/12726</td><td>90.5%; 1217/14969</td></tr><tr><td>64 elements</td><td>86.1%; 2592/31118</td><td>86.6%; 2356/35313</td></tr></table>
+
+# C.2. Cost and Efficiency
+
+In Table 6, 7 and 8, we report the token cost and average accuracy comparison of our proposed approaches with other contenders. 5 test runs were conducted and the average accuracy and the number of tokens (completion tokens / prompt tokens) are reported.
+
+We can observe that the token costs of our proposed C-ToT approaches and the contenders are task-specific and generally incomparable. Since we preserve all previous intermediate thoughts in the QA task, the token cost of the C-ToT approaches is higher than that of the S-ToT algorithm, but the performance is simultaneously improved. In the QA and Game of 24, the token costs of the Comp-SToT and S-ToT approaches are comparable, but the Comp-SToT approaches achieve better average accuracy. In the Sudoku puzzle, the token cost of Comp-SToT is lower than that of S-ToT. These results indicate the effectiveness of using the direct pairwise comparison approach to find the most promising intermediate thoughts. In practice, we can further reduce the token cost by using a counter for each intermediate thought to track its comparison frequency. If the comparison count of an intermediate thought exceeds a threshold, we can remove it from the tree.
+
+Table 5: Performance comparisons with GoT.  
+
+<table><tr><td>Method</td><td>Generate/Prompt tokens</td><td>Cost per case</td><td>Accuracy</td></tr><tr><td>CoT</td><td>106/136</td><td>0.0003</td><td>42.3%</td></tr><tr><td>SC-CoT</td><td>1647/2023</td><td>0.0054</td><td>58.4%</td></tr><tr><td>SToS</td><td>1551/5415</td><td>0.0085</td><td>57.1%</td></tr><tr><td>SC-SToS</td><td>2081/13459</td><td>0.018</td><td>57.6%</td></tr><tr><td>Comp-SToS</td><td>1515/6299</td><td>0.0093</td><td>59.0%</td></tr><tr><td>Back-SToS</td><td>1551/8135</td><td>0.011</td><td>58.0%</td></tr><tr><td>C-ToT (Stand.)</td><td>1498/14627</td><td>0.017</td><td>61.4%</td></tr><tr><td>C-ToT (Duel.)</td><td>1649/52044</td><td>0.055</td><td>63.0%</td></tr></table>
+
+Table 6: Average accuracy of different methods with token costs on QA.  
+
+<table><tr><td>Method</td><td>Generate/Prompt tokens</td><td>Cost per case</td><td>Accuracy</td></tr><tr><td>CoT</td><td>99/437</td><td>0.0006</td><td>4.3%</td></tr><tr><td>SC-CoT</td><td>1717/6555</td><td>0.010</td><td>8.0%</td></tr><tr><td>SToS</td><td>1368/12205</td><td>0.015</td><td>34.3%</td></tr><tr><td>SC-SToS</td><td>2284/40825</td><td>0.045</td><td>40.0%</td></tr><tr><td>Comp-SToS</td><td>1309/11963</td><td>0.015</td><td>36.3%</td></tr><tr><td>Back-SToS</td><td>1679/23178</td><td>0.027</td><td>39.0%</td></tr><tr><td>C-ToT (Stand.)</td><td>2452/21003</td><td>0.026</td><td>40.0%</td></tr><tr><td>C-ToT (Duel.)</td><td>2174/60578</td><td>0.065</td><td>41.0%</td></tr></table>
+
+Table 7: Average accuracy of different methods with token costs on Game of 24.
+
+<table><tr><td>Method</td><td>Generate/Prompt tokens</td><td>Cost per case</td><td>Accuracy</td></tr><tr><td>CoT</td><td>431/178</td><td>0.001</td><td>44.4%</td></tr><tr><td>SC-CoT</td><td>6292/2666</td><td>0.015</td><td>47.8%</td></tr><tr><td>SToS</td><td>6309/23933</td><td>0.037</td><td>60.0%</td></tr><tr><td>SC-SToS</td><td>6568/70129</td><td>0.083</td><td>66.7%</td></tr><tr><td>Comp-SToS</td><td>2666/13164</td><td>0.019</td><td>65.6%</td></tr><tr><td>Back-SToS</td><td>4536/21383</td><td>0.030</td><td>72.2%</td></tr><tr><td>C-ToT (Stand.)</td><td>5340/29565</td><td>0.040</td><td>74.4%</td></tr><tr><td>C-ToT (Duel.)</td><td>7148/86425</td><td>0.101</td><td>75.5%</td></tr></table>
+
+# C.3. Ablation Studies
+
+We report the ablation studies on Aqua dataset, while we also observe the same trend in other datasets. 5 test runs were conducted and the average accuracy and the number of tokens (completion tokens / prompt tokens) are reported.
+
+We study the number of intermediate thoughts generated and selected in each round and report the results in Table 9. With a fixed number of thoughts generated each round, selecting more thoughts leads to higher costs, while accuracy may not benefit much. With a fixed number of thoughts selected each round, generating more thoughts leads to a significant increase in accuracy because we can explore more thoughts. These results could benefit the further use of CoT methods.
+
+Table 8: Average accuracy of different methods with token costs on Sodomoku.  
+
+<table><tr><td>Generate m / Select K</td><td>K = 1</td><td>K = 2</td><td>K = 3</td><td>K = 5</td><td>K = 6</td></tr><tr><td>m = 1</td><td>41.6%; 50/515</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>m = 3</td><td>46.7%; 139/1525</td><td>50.2%; 244/2565</td><td>49.8%; 375/3658</td><td>-</td><td>-</td></tr><tr><td>m = 5</td><td>50.1%; 288/2653</td><td>53.5%; 495/4452</td><td>55.0%; 549/5945</td><td>54.8%; 882/9379</td><td>-</td></tr><tr><td>m = 10</td><td>53.0%; 518/5189</td><td>56.6%; 896/8716</td><td>57.3%; 1091/11875</td><td>57.5%; 2120/19473</td><td>57.6%; 2155/20982</td></tr><tr><td>m = 12</td><td>53.8%; 674/6334</td><td>57.4%; 1212/10732</td><td>61.4%; 1498/14627</td><td>61.4%; 1531/16002</td><td>61.5%; 2125/26329</td></tr></table>
+
+We study the threshold for removing intermediate thoughts vs. accuracy and cost and report the results in Table 10. Under the same tree depth, different thresholds show relatively stable performance.
+
+Table 9: Ablation studies on parameters.  
+
+<table><tr><td>Depth d / Threshold Th</td><td>Th= 1</td><td>Th= 2</td><td>Th= 3</td><td>Th= 4</td><td>Th= 5</td></tr><tr><td>d = 3</td><td>58.0%; 1062/11337</td><td>60.7%; 1315/13611</td><td>61.0%; 1478/15111</td><td>61.4%; 1513/15713</td><td>61.4%; 1558/16107</td></tr><tr><td>d = 4</td><td>58.8%; 1426/13792</td><td>60.9%; 1732/18822</td><td>61.4%; 1771/19940</td><td>61.6%; 1804/20003</td><td>61.3%; 2001/20312</td></tr><tr><td>d = 5</td><td>60.0%; 1760/17556</td><td>61.0%; 2143/21006</td><td>61.6%; 2271/23412</td><td>61.7%; 2265/24031</td><td>61.3%; 2425/24755</td></tr></table>
+
+We study the number of comparison and report the results in Table 11. In general, increasing the number of comparison results in better accuracy and higher cost.
+
+Table 10: Ablation studies on the number of thresholds for removing intermediate thoughts.  
+
+<table><tr><td>Dataset / Comparison n</td><td>n = 1</td><td>n = 3</td><td>n = 5</td><td>n = 8</td></tr><tr><td>Aqua</td><td>61.4%; 1498/14627</td><td>63.0%; 1649/52044</td><td>64.3%; 1824/78194</td><td>64.7%; 1997/128802</td></tr></table>
+
+Table 11: Ablation studies on the number of comparisons.

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+# Generating In-Distribution Proxy Graphs for Explaining Graph Neural Networks
+
+Zhuomin Chen<sup>1</sup> Jiaxing Zhang<sup>2</sup> Jingchao Ni<sup>3</sup> Xiaoting Li<sup>4</sup> Yuchen Bian<sup>5</sup> Md Mezbahul Islam<sup>1</sup> Ananda Mohan Mondal<sup>1</sup> Hua Wei<sup>6</sup> Dongsheng Luo<sup>1</sup>
+
+# Abstract
+
+Graph Neural Networks (GNNs) have become a building block in graph data processing, with wide applications in critical domains. The growing needs to deploy GNNs in high-stakes applications necessitate explainability for users in the decision-making processes. A popular paradigm for the explainability of GNNs is to identify explainable subgraphs by comparing their labels with the ones of original graphs. This task is challenging due to the substantial distributional shift from the original graphs in the training set to the set of explainable subgraphs, which prevents accurate prediction of labels with the subgraphs. To address it, in this paper, we propose a novel method that generates proxy graphs for explainable subgraphs that are in the distribution of training data. We introduce a parametric method that employs graph generators to produce proxy graphs. A new training objective based on information theory is designed to ensure that proxy graphs not only adhere to the distribution of training data but also preserve explanatory factors. Such generated proxy graphs can be reliably used to approximate the predictions of the labels of explainable subgraphs. Empirical evaluations across various datasets demonstrate our method achieves more accurate explanations for GNNs.
+
+![](images/6a40a0d06ee56ad842055a3afd0a568e6b47952aaaeb56d6c4b0b255e5fc6eef.jpg)  
+(a) Explanation process
+
+![](images/96f242702dbafcf6c606056c5b47bf3dffbed85cc9db43c38b3734ac5920aa6c.jpg)  
+(b) OOD problem  
+Figure 1. Examples of the explanation process and out-of-distribution problem. (a) is the explanation process for a graph learning model. The original graph  $G$  undergoes an explanation process, resulting in an explanation graph  $G'$  that highlights the most significant features and relationships; (b) shows the explanation graph  $G'$  is out of distribution where the GNN is trained.
+
+# 1. Introduction
+
+Graph Neural Networks (GNNs) have emerged as a pivotal technology for handling graph-structured data, demonstrating remarkable performance in various applications including node classification and link prediction (Kipf & Welling, 2017; Hamilton et al., 2017; Velicković et al., 2018; Scarselli et al., 2008). Their growing use in critical sectors such as healthcare and fraud detection has escalated the need for explainability in their decision-making processes (Wu et al., 2022; Li et al., 2022; Zhang et al., 2024). To meet this demand, a variety of explanation methods have been recently developed to interpret the behavior of GNN models. These methods primarily concentrate on identifying a subgraph that significantly impacts the model's prediction for a particular instance (Ying et al., 2019; Luo et al., 2020).
+
+A prominent approach to explain GNNs involves the Graph Information Bottleneck (GIB) principle (Wu et al., 2020). This principle focuses on extracting a compact yet informative subgraph from the input graph, ensuring that this subgraph retains sufficient information for the model to maintain its original prediction. A key aspect of the GIB approach is evaluating the predictive capability of such a subgraph. Typically, this is accomplished by feeding the subgraph into the GNN model and comparing its prediction against that of the complete input graph.
+
+Although it is intuitively correct, the underlying assumption of the aforementioned approach - GNN model can
+
+make accurate predictions on explanation subgraphs – may not always hold. As shown in Figure 1, explanation subgraphs can significantly deviate from the distribution of original graphs, leading to an Out-Of-Distribution (OOD) issue (Zhang et al., 2023b; Fang et al., 2023c;a; Zheng et al., 2024). For instance, in the MUTAG dataset (Debnath et al., 1991), each graph represents a molecule, with nodes symbolizing atoms and edges indicating chemical bonds. The molecular graphs in this dataset usually contain hundreds of edges. In contrast, the  $NO_2$  functional group, identified as a key subgraph influencing positive mutagenicity in a molecule, comprises merely 2 edges. This stark contrast in structural properties leads to a significant difference in the distributions of explanation subgraphs and original graphs. Since the model was trained with original graphs, the reliability of predictions on subgraphs is undermined due to the distribution shifting problem.
+
+Several pioneering studies have attempted to address this distributional challenge (Fang et al., 2023c; Zhang et al., 2023b;a). For example, CGE regards the GNN model as a transparent, fully accessible system. It considers the GNN model as a teacher network and employs an additional "student" network to predict the labels of explanation subgraphs (Fang et al., 2023c). As another example, MixupExplainer (Zhang et al., 2023b) generates a mixed graph for the explanation by blending it with a non-explanatory subgraph from a different input graph. This method posits that the mixup graph aligns with the distribution of the original input graphs. However, this claim is predicated on a rather simplistic assumption that the explanation and non-explanatory subgraphs are independently drawn. However, in real-world applications, these methods often face practical limitations. The dependence on a "white box" model in CGE and the oversimplified assumptions in MixupExplainer are not universally applicable. Instead, GNN models are usually given as "black boxes", and the graphs in real-life applications do not conform to strict independence constraints, highlighting the need for more versatile and realistic approaches to the OOD problem.
+
+In response to these challenges, in this work, we introduce an innovative concept of proxy graphs to the realm of explainable GNNs. These proxy graphs are designed to be both explanation-preserving and distributionally aligned with the training data. By this means, the predictive capabilities of explanation subgraphs can be reliably inferred from the corresponding proxy graphs. We begin with a thorough investigation into the feasible conditions necessary for generating such in-distributed proxy graphs. Leveraging our findings, we further propose a novel architecture that incorporates variational graph auto-encoders to produce proxy graphs. Specifically, we utilize a graph auto-encoder to reconstruct the explainable subgraph and another variational auto-encoder to generate a non-explanatory subgraph.
+
+A proxy graph is then obtained by combining two output subgraphs of auto-encoders. We delineate our main contributions as follows:
+
+- We systematically analyze and address the challenge of the OOD issue in explainable GNNs, which is pivotal for enhancing the reliability and interpretability of GNNs in real-world applications.  
+- We introduce an innovative parametric method that incorporates graph auto-encoders to produce in-distributed proxy graphs that are both situated in the original data distribution and preserve essential explanation information. This facilitates more precise and interpretable explanations in GNN applications.  
+- Through comprehensive experiments on various real-world datasets, we substantiate the effectiveness of our proposed approach, showcasing its practical utility and superiority in producing explanations.
+
+# 2. Notations and Preliminary
+
+# 2.1. Notations and Problem Formulation
+
+We denote a graph  $G$  from an graph set  $\mathcal{G}$  by a triplet  $(\mathcal{V},\mathcal{E};\boldsymbol {X})$ , where  $\mathcal{V} = \{v_{1},v_{2},\dots,v_{n}\}$  is the node set and  $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$  is the edge set.  $\boldsymbol {X}\in \mathbb{R}^{n\times d}$  is the node feature matrix, where  $d$  is the feature dimension and the  $i$ -th row is the feature vector associated with node  $v_{i}$ . The adjacency matrix of  $G$  is denoted by  $\pmb {A}\in \{0,1\}^{n\times n}$ , which is determined by the edge set  $\mathcal{E}$  such that  $A_{ij} = 1$  if  $(v_{i},v_{j})\in \mathcal{E}$ ,  $A_{ij} = 0$ , otherwise. In this paper, we focus on the graph classification task, and node classification can be converted to computation graph classification problem (Ying et al., 2019; Luo et al., 2020). Specifically, for graph classification task, each graph  $G$  is associated with a label  $Y\in \mathcal{V}$ . The to-be-explained GNN model  $f(\cdot)$  has been well-trained to classify  $G$  into its class, i.e.,  $f:\mathcal{G}\mapsto \{1,2,\dots,|\mathcal{V}|\}$ .
+
+Following the existing works (Ying et al., 2019; Luo et al., 2020; Yuan et al., 2022; Huang et al., 2024), the explanation methods under consideration in this paper are model/task agnostic and treat GNN models as black boxes — i.e., the so-called post-hoc, instance-level explanation methods. Formally, our research problem is described as follows:
+
+Problem 1 (Post-hoc Instance-level GNN Explanation). Given a to-be-explained GNN model  $f(\cdot)$  and a set of graphs  $\mathcal{G}$ , the goal of post-hoc instance-level explanation is to learn a parametric function so that for an arbitrary graph  $G \in \mathcal{G}$ , it finds a compact subgraph  $G^{*} \subseteq G$  that can "explain" the prediction  $f(G)$ . The parametric mapping  $\Psi_{\psi}: \mathcal{G} \mapsto \mathcal{G}^{*}$  is called an explanation function, where  $\mathcal{G}^{*}$  is the alphabet of  $G^{*}$ , and  $\psi$  is the parameter of the explanation function.
+
+# 2.2. Graph Information Bottleneck as the Objective Function
+
+The Information Bottleneck (IB) principle, foundational in learning dense representations, suggests that optimal representations should balance minimal and sufficient information for predictions (Tishby et al., 2000). This concept has been adapted for GNNs through the Graph Information Bottleneck (GIB) approach, consolidating various post-hoc GNN explanation methods like GNNExplainer (Ying et al., 2019) and PGExplainer (Luo et al., 2020). The GIB framework aims to find a subgraph  $G^{*}$  to explain a GNN model's prediction on a graph  $G$  as (Zhang et al., 2023b):
+
+$$
+G ^ {*} = \underset {G ^ {\prime}} {\arg \min } I (G, G ^ {\prime}) - \alpha I (Y, G ^ {\prime}), \tag {1}
+$$
+
+where  $G'$  is a candidate explanatory subgraph,  $Y$  is the label, and  $\alpha$  is a balance parameter. The first term is the mutual information between the original graph  $G$  and the explanatory subgraph  $G'$  and the second term is negative mutual information of  $Y$  and the explanatory subgraph  $G'$ . At a high level, the first term encourages detecting a small and dense subgraph for explanation, and the second term requires that the explanatory subgraph is label preserving.
+
+Due to the intractability of mutual information between  $Y$  and  $G'$  in equation 1, some existing works (Wu et al., 2020; Miao et al., 2022) derived a parameterized variational lower bound of  $I(Y, G')$ :
+
+$$
+I \left(Y, G ^ {\prime}\right) \geq \mathbb {E} _ {G ^ {\prime}, Y} \left[ \log P \left(Y \mid G ^ {\prime}\right) \right] + H (Y), \tag {2}
+$$
+
+where the first term measures how well the subgraphs predict the labels. A higher value indicates that the subgraphs are, on average, more predictive of the correct labels. The second term  $H(Y)$  quantifies the amount of inherent unpredictability or variability in the labels. By introducing equation 2 to equation 1, a tractable upper bound is used as the objective:
+
+$$
+G ^ {*} = \underset {G ^ {\prime}} {\arg \min } I (G, G ^ {\prime}) - \alpha \mathbb {E} _ {G ^ {\prime}, Y} [ \log P (Y | G ^ {\prime}) ], \tag {3}
+$$
+
+where  $H(Y)$  is omitted due to its independence to  $G'$ .
+
+The OOD Problem in GIB. In the existing research, the estimation of  $P(Y|G')$  is typically achieved by applying the to-be-explained model  $f(\cdot)$  to the input graph  $G'$  (Ying et al., 2019; Miao et al., 2022). A critical assumption in these approaches involves the GNN model's ability to accurately predict candidate explanation subgraphs  $G'$ . This assumption, however, overlooks the OOD problem, where the distribution of explanation subgraphs significantly deviates from that of the original training graphs (Zhang et al., 2023b; Fang et al., 2023a;c;b; Zheng et al., 2024; Amara
+
+et al., 2023). Formally, let  $P_{\mathcal{G}}$  be the distribution of the training graphs, and  $P_{\mathcal{G}'}$  be the distribution of the explanation subgraphs. The core issue in the GIB objective function is caused by  $P_{\mathcal{G}} \neq P_{\mathcal{G}'}$ . This distributional disparity undermines the predictive reliability of the model  $f(\cdot)$ , trained on  $P_{\mathcal{G}}$  when applied to subgraphs from  $P_{\mathcal{G}'}$ . As a result, the predictive power of explanations provided by  $f(G')$  is an unreliable approximation of  $P(Y|G')$  in equation 3.
+
+# 3. Graph Information Bottleneck with Proxy Graphs
+
+In this section, we propose the concept of proxy graphs to mitigate the aforementioned OOD issue. A proxy graph not only retains the label information present in  $G'$  but also conforms to the distribution of the original graph dataset. Specifically, we assume that proxy graphs  $\tilde{G}$  are drawn from the distribution  $P_{\mathcal{G}}$  and reformulate the estimation of  $P(Y|G')$  by marginalizing over the distribution of proxy graphs as follows:
+
+$$
+P (Y | G ^ {\prime}) = \mathbb {E} _ {\tilde {G} \sim P _ {\mathcal {G}}} [ P (Y | \tilde {G}) \cdot P (\tilde {G} | G ^ {\prime}) ]. \tag {4}
+$$
+
+In equation 4, we address the OOD challenge by predicting  $Y$  using a proxy graph  $\tilde{G}$  instead of directly using  $G'$ . This approach is particularly effective when the conditional probability  $P(Y|\cdot)$  is approximated by the model  $f(\cdot)$ . To facilitate the maximization of the likelihood as outlined in equation 4, we further approximate  $P(\tilde{G}|G')$  with a parameterized function, denoted as  $Q_{\phi}(\tilde{G}|G')$ . The formal representation is thus given by
+
+$$
+P \left(Y \mid G ^ {\prime}\right) = \mathbb {E} _ {\tilde {G} \sim P _ {\mathcal {G}}} \left[ P \left(Y \mid \tilde {G}\right) \cdot Q _ {\phi} \left(\tilde {G} \mid G ^ {\prime}\right) \right], \tag {5}
+$$
+
+where  $\phi$  denotes model parameters.
+
+Given the combinatorial complexity inherent in graph structures, it is computationally infeasible to enumerate all potential proxy graphs  $\tilde{G}$  from the unknown distribution  $P_{\mathcal{G}}$  which is necessary for calculating equation 5. To overcome this challenge, we propose to approximate  $P_{\mathcal{G}}$  with the parameterized function  $Q_{\phi}(\tilde{G}|G')$ , and estimate equation 5 with a Monte Carlo estimator that samples proxy graphs from  $Q_{\phi}(\tilde{G}|G')$ . That is
+
+$$
+P (Y | G ^ {\prime}) = \mathbb {E} _ {\tilde {G} \sim Q _ {\phi} (\tilde {G} | G ^ {\prime})} [ P (Y | \tilde {G}) ], \tag {6}
+$$
+
+with constraints
+
+$$
+Q _ {\phi} (\tilde {G} | G ^ {\prime}) \approx P _ {\mathcal {G}}, \quad H (Y | \tilde {G}) \approx H (Y | G ^ {\prime}), \tag {7}
+$$
+
+where the first constraint ensures that  $\tilde{G}$  is sampled from a distribution that approximates  $P_{\mathcal{G}}$ , effectively addressing the OOD challenge. The second constraint guarantees that the label information preserved in  $\tilde{G}$  is similar to that in  $G^{\prime}$ .
+
+Therefore, combining equation 3, equation 6 and equation 7, our proxy graph-induced objective function becomes:
+
+$$
+\begin{array}{l} \underset {G ^ {\prime}} {\arg \min } I (G, G ^ {\prime}) - \alpha \mathbb {E} _ {G ^ {\prime}, Y} [ \log \mathbb {E} _ {\tilde {G} \sim Q _ {\phi} (\tilde {G} | G ^ {\prime})} [ P (Y | \tilde {G}) ] ] \\ \text {s . t .} Q _ {\phi} (\tilde {G} \mid G ^ {\prime}) \approx P _ {\mathcal {G}}, H (Y | \tilde {G}) \approx H (Y \mid G ^ {\prime}). \tag {8} \\ \end{array}
+$$
+
+Bi-level optimization. In equation 8, we formulate a joint optimization loss function that aims to identify the optimal explanation alongside its corresponding proxy graphs. Building upon this, we refine the framework by replacing the first constraint with a distributional distance measure, specifically the Kullback-Leibler (KL) divergence, between  $Q_{\phi}(\tilde{G}|G^{\prime})$  and  $P_{\mathcal{G}}$ . This leads to the development of a bi-level optimization model. Formally, the model is expressed as follows:
+
+$$
+\begin{array}{l} \underset {G ^ {\prime}} {\arg \min } I (G, G ^ {\prime}) - \alpha \mathbb {E} _ {G ^ {\prime}, Y} [ \log \mathbb {E} _ {\tilde {G} \sim Q _ {\phi^ {*} (\tilde {G} | G ^ {\prime})}} [ P (Y | \tilde {G}) ] ] \\ \text{where}\phi^{*} = \operatorname *{arg  min}_{\phi}\mathrm{KL}(Q_{\phi}(\tilde{G} |G^{\prime}),P_{\mathcal{G}}), \\ s. t. H (Y | \tilde {G}) \approx H (Y | G ^ {\prime}). \tag {9} \\ \end{array}
+$$
+
+# 3.1. Derivation of Outer Optimization
+
+For the outer optimization objective, we elaborate on instantiating the first term  $I(G, G')$ . Akin to the original graph, we denote the explanation subgraph  $G'$  by  $(\mathcal{V}, \mathcal{E}', X)$ , whose adjacency matrix is  $A'$ . We follow (Miao et al., 2022) to include a variational approximation distribution  $R(G')$  for the distribution  $P(G')$ . Then, we obtain its upper bound as follows:
+
+$$
+I (G, G ^ {\prime}) \leq \mathbb {E} _ {G} [ \mathrm {K L} (P (G ^ {\prime} | G) | | R (G ^ {\prime})) ]. \tag {10}
+$$
+
+We follow the Erdős–Rényi model (Erdős et al., 1960) and assume that each edge in  $G'$  has a probability of being present or absent, independently of the other edges. Specifically, we assume that the existence of an edge  $(u, v)$  in  $G'$  is determined by a Bernoulli variable  $A_{u, v}' \sim \mathrm{Bern}(\pi_{uv}')$ . Thus,  $P(G'|G)$  can be decomposed as  $P(G'|G) = \prod_{(u, v) \in \mathcal{E}} P(A_{uv}'|\pi_{uv}')$ . The bound is always true for any prior distribution  $R(G')$ . We follow an existing work and assume that in the prior distribution, the existence of an edge  $(u, v)$  in  $G'$  is determined by another Bernoulli variable  $A_{u, v}'' \sim \mathrm{Bern}(r)$ , where  $r \in [0, 1]$  is a hyper-parameter (Miao et al., 2022), independently to the graph  $G$ . We have  $R(G') = P(|\mathcal{E}|) \prod_{(u, v) \in \mathcal{E}} P(A_{uv}'')$ . Thus, the KL divergence between  $P(G'|G)$  and the above
+
+marginal distribution  $R(G^{\prime})$  becomes:
+
+$$
+\begin{array}{l} \mathrm {K L} (P (G ^ {\prime} | G) | | R (G ^ {\prime})) \\ = \sum_ {(u, v) \in \mathcal {E}} \pi_ {u v} ^ {\prime} \log \frac {\pi_ {u v} ^ {\prime}}{r} + \left(1 - \pi_ {u v} ^ {\prime}\right) \log \frac {1 - \pi_ {u v} ^ {\prime}}{1 - r} + \text {C o n s t .} \tag {11} \\ \end{array}
+$$
+
+By replacing  $I(G, G')$  with the above tractable upper bound, we obtain the loss function, denoted by  $\mathcal{L}_{\mathrm{exp}}$ , for training the explainer  $\Psi(\cdot)$ .
+
+# 3.2. Derivation of Inner Optimization
+
+For the inner optimization objective, we implement the first constraint by minimizing the distribution distance between  $Q_{\phi}(\hat{G}|G')$  and  $P_{\mathcal{G}}$ . Under the Erdős–Rényi assumption, the distribution loss is equivalent to the cross-entropy loss between  $\tilde{G}$  given  $G'$  and  $G$  over the full adjacency matrix (Chen et al., 2023b). Considering that  $G$  is usually sparse rather than a fully connected graph, in practice, we adopt a weighted version to emphasize more connected node pairs (Wang et al., 2016). Formally, we have the following distribution loss.
+
+$$
+\mathcal {L} _ {\text {d i s t}} = \frac {\beta}{| \mathcal {E} |} \sum_ {(u, v) \in \mathcal {E}} \log \left(\tilde {p} _ {u v}\right) + \frac {1}{| \bar {\mathcal {E}} |} \sum_ {(u, v) \in \bar {\mathcal {E}}} \log \left(1 - \tilde {p} _ {u v}\right), \tag {12}
+$$
+
+where  $\bar{\mathcal{E}}$  is the set of node pairs that are unconnected in  $G$ ,  $\tilde{p}_{uv}$  is the probability of node pair  $(u,v)$  in  $\tilde{G}$ , and  $\beta$  is a hyper-parameter to for the trade-off between connected and unconnected node pairs.
+
+The second constraint requires the mutual information of  $Y$  and  $\tilde{G}$  is the same as that of  $Y$  and  $G'$ . Due to the OOD problem, it is non-trivial to directly compute  $P(Y|G')$  or  $H(Y|G')$ . Instead, we implement this constraint with a novel graph generator that  $\tilde{G}$  is obtained by combining  $G'$  and a non-explanatory subgraph (Zhang et al., 2023b). In practice, we implement the non-explanatory part by perturbing the residual subgraph  $G - G'$ . The intuition is that if an explanation comprises label information, it is unlikely to change the prediction by manipulating the remaining non-explanatory part, which is widely adopted in the literature (Fang et al., 2023a; Zheng et al., 2024).
+
+# 4. The Proxy Explainer
+
+Based on our novel GIB with proxy graphs, in this section, we introduce a straightforward yet theoretically robust instantiation, named ProxyExplainer. As shown in Figure 2, the architecture of our model comprises two key modules: the explainer and the proxy graph generator. The explainer takes the original graph  $G$  as its input and outputs a subgraph  $G'$  as an explanation, which is optimized through the outer objective in equation 9. The proxy graph generator
+
+![](images/e79c0274d96fa8168ad7cd556a156bdb1d70b87fabb8c9fa09fa3adf11fa0467.jpg)  
+Figure 2. ProxyExplainer consists of two components: an Explainer and a Proxy Graph Generator. The Explainer takes a graph  $G$  as input and produces an explainable subgraph  $G'$ , The proxy generator creates in-distribution proxy graphs that preserve the label information in  $G'$ . The proxy generator consists of a graph auto-encoder (GAE) and a variational graph auto-encoder (VGAE).
+
+generates an in-distributed proxy graph, which is optimized with the inner objective.
+
+# 4.1. The Explainer
+
+For the sake of optimization, we follow the existing works to relax the element in  $\mathbf{A}$  from binary values to continuous values in the range [0, 1] (Luo et al., 2020; 2024). We adopt a generative explainer due to its effectiveness and efficiency (Chen et al., 2023a). Specifically, we decompose the to-be-explained model  $f(\cdot)$  into two functions that  $f_{\mathrm{enc}}(\cdot)$  learns node representations and  $f_{\mathrm{cls}}(\cdot)$  predicts graph labels based on node embeddings. Formally, we have  $\mathbf{Z} = f_{\mathrm{enc}}(\mathbf{X},\mathbf{A})$  and  $\hat{Y} = f_{\mathrm{cls}}(\mathbf{Z})$ , where  $\mathbf{Z}$  is the node representation matrix and  $\hat{Y}$  is prediction. Routinely,  $f_{\mathrm{cls}}(\cdot)$  consists of a pooling layer followed by a classification layer and  $f_{\mathrm{enc}}(\cdot)$  consists of the other layers. Following the independence assumption in Section 3.1, we approximate Bernoulli distributions with binary concrete distributions (Maddison et al., 2017; Jang et al., 2017). Specifically, the probability of sampling an edge  $(u,v)$  is computed by an MLP parameterized by  $\psi$ , denoted by  $g_{\psi}(\cdot)$ . Formally, we have
+
+$$
+\omega_ {u v} = g _ {\psi} ([ \boldsymbol {z} _ {u}; \boldsymbol {z} _ {v} ]),
+$$
+
+$$
+\epsilon \sim \operatorname {U n i f o r m} (0, 1), \tag {13}
+$$
+
+$$
+A _ {u v} ^ {\prime} = \sigma \left(\left(\log \epsilon - \log (1 - \epsilon) + \omega_ {u v}\right) / \tau\right),
+$$
+
+where  $[z_u;z_v]$  is the concatenation of node representations  $z_{u}$  and  $z_{v},\epsilon$  is a independent variable,  $\sigma (\cdot)$  is the Sigmoid function, and  $\tau$  is a temperature hyper-parameter for approximation. According to (Luo et al., 2020), the parameter  $\pi_{uv}^{\prime}$  of Bernoulli distribution in equation 11 can be obtained by  $\pi_{uv}^{\prime} = \frac{\exp(\omega_{uv})}{1 + \exp(\omega_{uv})}$ .
+
+# 4.2. The Proxy Graph Generator
+
+As shown in our analysis in Section 3.2, we demonstrate the synthesis of a proxy graph through the amalgamation of the explanation subgraph  $G'$  and the perturbation of its non-explanatory subgraph, represented as  $G^{\Delta} = (\mathcal{V}, \mathcal{E}^{\Delta}, X)$ . We define the edge set of  $G^{\Delta}, \mathcal{E}^{\Delta}$ , as the differential set  $\mathcal{E} - \mathcal{E}'$ . Correspondingly, the adjacency matrix  $A^{\Delta}$  is derived through  $A - A'$ . Building upon this foundation, and as illustrated in Figure 2, our proposed framework introduces a dual-structured mechanism comprising two distinct graph auto-encoders(GAE) (Bank et al., 2023). To be more specific, we first include an encoder to learn a latent matrix  $Z'$  according to  $A'$  and  $X'$ , and a decoder network recovers  $A'$  based on  $Z'$ . Formally, we have:
+
+$$
+\boldsymbol {Z} ^ {\prime} = \operatorname {E N C} _ {1} \left(\boldsymbol {A} ^ {\prime}, \boldsymbol {X}\right), \quad \tilde {\boldsymbol {A}} ^ {\prime} = \operatorname {D E C} \left(\boldsymbol {Z} ^ {\prime}\right), \tag {14}
+$$
+
+where  $\mathrm{ENC}_1$  and DEC are the encoder network and decoder network. Our framework is flexible to the choices of these two networks.  $\tilde{\boldsymbol{A}}^{\prime}\in \mathbb{R}^{n\times n}$  is the reconstructed adjacency matrix.
+
+To introduce perturbations into the non-explanatory segment, our approach employs a Variational Graph AutoEncoder (VGAE), a generative model adept at creating varied yet structurally coherent graph data. This capability is pivotal in generating nuanced variations of the non-explanatory subgraph. The VGAE operates by first encoding the non-explanatory subgraph  $G^{\Delta}$  into a latent probabilistic space, characterized by Gaussian distributions. This process is articulated as:
+
+$$
+\boldsymbol {\mu} ^ {\Delta} = \operatorname {E N C} _ {1} (\boldsymbol {A} ^ {\Delta}, \boldsymbol {X}), \quad \boldsymbol {\sigma} ^ {\Delta} = \operatorname {E N C} _ {2} (\boldsymbol {A} ^ {\Delta}, \boldsymbol {X}), \tag {15}
+$$
+
+where  $\mathrm{ENC}_1$  and  $\mathrm{ENC}_2$  are encoder networks that learn the mean  $\mu^{\Delta}$  and variance  $\sigma^{\Delta}$  of the Gaussian distributions, respectively. Following this, the latent representations  $Z^{\Delta}$  are sampled from these distributions, ensuring that each generated instance is a unique variation of the original. The decoder network then reconstructs the perturbed non-explanatory subgraph from these sampled latent representations:
+
+$$
+\boldsymbol {Z} ^ {\Delta} \sim \mathcal {N} (\boldsymbol {\mu} ^ {\Delta}, \operatorname {d i a g} (\boldsymbol {\sigma} ^ {\Delta}) ^ {2}), \quad \tilde {\boldsymbol {A}} ^ {\Delta} = \mathrm {D E C} (\boldsymbol {Z} ^ {\Delta}), \tag {16}
+$$
+
+where  $\tilde{A}^{\Delta} \in \mathbb{R}^{n \times n}$  represents the adjacency matrix of the perturbed non-explanatory subgraph. This novel use of VGAE facilitates the generation of diverse yet representative perturbations, crucial for enhancing the interpretability of explainers in our proxy graph framework. The adjacency matrix of a proxy graph is then obtained by
+
+$$
+\tilde {\boldsymbol {A}} = \tilde {\boldsymbol {A}} ^ {\prime} + \tilde {\boldsymbol {A}} ^ {\Delta}. \tag {17}
+$$
+
+Loss function. To train the proxy graph generator, we introduce a standard Gaussian distribution as the prior for
+
+the latent space representations in the VGAE, specifically  $\mathbf{Z} \sim \mathcal{N}(0,\mathcal{I})$ , where  $\mathcal{I}$  represents the identity matrix. Then, the loss function is as follows.
+
+$$
+\mathcal {L} _ {\text {p r o x y}} = \mathcal {L} _ {\text {d i s t}} + \lambda \mathcal {L} _ {\mathrm {K L}}, \tag {18}
+$$
+
+where  $\mathcal{L}_{\mathrm{dist}}$  is equivalent to cross-entropy between  $\tilde{G}$  and  $G$  (Chen et al., 2023b).  $\mathcal{L}_{\mathrm{KL}}$  represents the KL divergence between the distribution of the latent representations  $Z^{\Delta}$  and the assumed Gaussian prior. This term is crucial for regulating the variational aspect of the VGAE, ensuring that the generated perturbations are meaningful and controlled.  $\lambda$  is a hyper-parameter.
+
+Alternate Training. To train the explainer and the proxy graph generator networks, we follow existing works (Zheng et al., 2024) to use an alternate training schedule that trains the proxy graph generator network  $M$  times and then trains the explainer network once.  $M$  is a hyper-parameter determined by grid search. The detailed algorithm description of our model is shown in Appendix B.
+
+# 5. Related Work
+
+GNN Explanation. The goal of explainability in GNNs is to ensure transparency in graph-based tasks. Recent works have been directed towards elucidating the rationale behind GNN predictions. These explanation methods can be broadly classified into two categories: instance-level and model-level approaches (Yuan et al., 2022). In this study, we focus on instance-level explanations, which aim to clarify the specific reasoning behind individual predictions made by GNNs. These methods are critical for understanding the decision-making process on a case-by-case basis to enhance the explainability of GNNs. For example, GNNExplainer (Ying et al., 2019) excludes certain edges and node features to observe the changes in prediction. However, its single-instance focus limits its applicability to provide a global understanding of the to-be-explained model (Chen et al., 2023a). PGExplainer (Luo et al., 2020; 2024) introduces a parametric neural network to learn edge weights. Thus, once training is complete, it can explain new graphs without retraining. ReFine (Wang et al., 2021) integrates a pre-training phase that focuses on class comparisons and is fine-tuned to refine context-specific explanations. GStarX (Zhang et al., 2022) assigns an importance score to each node by calculating the Hamiache and Navarro values of the structure to obtain explanatory subgraphs. GFlowExplainer (Li et al., 2023) uses a generator to construct a TD-like flow matching condition to learn a policy for generating explanations by adding nodes sequentially.
+
+Distribution Shifting in Explanations. The distribution shifting problem in post-hoc explanations has been increasingly recognized in explainable AI fields (Chang et al., 2019;
+
+Qiu et al., 2022). For example, FIDO (Chang et al., 2019) works on enhancing image classifier explanations, focusing on relevant contextual details that agree with the training data's distribution. A recent study tackles the distribution shifting problem in image explanations by introducing a module that assesses the similarity between altered data and the original dataset distribution (Qiu et al., 2022). In the graph domain, an ad-hoc strategy to mitigate distribution shifting is to initially reduce the size constraint coefficient during the explanation process (Fang et al., 2023b). Mixup-Explainer (Zhang et al., 2023b) and RegExplainer (Zhang et al., 2023a) propose non-parametric solutions by mixing up the explanation subgraph with a non-explainable part from another graph. However, these methods operates under the assumption that the explanatory and non-explanatory subgraphs in mixed graphs are independent, which may not hold in many real-life graphs.
+
+# 6. Experiments
+
+We present empirical results that illustrate the effectiveness of our proposed method. These experiments are mainly designed to explore the following research questions:
+
+- RQ1: Can the proposed framework outperform other baselines in identifying explanations for GNNs?  
+- RQ2: Is the distribution shifting severe in explanation subgraphs? Can the proposed approach alleviate that?  
+- RQ3: How does each component of ProxyExplainer impact the overall performance in generating explanations?
+
+# 6.1. Experimental Settings
+
+To evaluate the performance of ProxyExplainer, we use six benchmark datasets with ground-truth explanations. These include four real-world datasets: MUTAG (Kazius et al., 2005), Benzene (Sanchez-Lengeling et al., 2020), Alkane-Carbonyl (Sanchez-Lengeling et al., 2020), and Fluoride-Carbonyl (Sanchez-Lengeling et al., 2020), along with two synthetic datasets: BA-2motifs (Luo et al., 2020) and BA-3motifs (Chen et al., 2023b). We take GradCAM (Pope et al., 2019), GNNExplainer(Ying et al., 2019), PGExplainer(Luo et al., 2020), ReFine (Wang et al., 2021), and MixupExplainer (Zhang et al., 2023b) for comparison. We follow the experimental setting in previous works (Ying et al., 2019; Luo et al., 2020; Sanchez-Lengeling et al., 2020) to train a Graph Convolutional Network (GCN) model (Kipf & Welling, 2017) with three layers. Experiments on another representative GNN, Graph Isomorphism Network (GIN) (Xu et al., 2019), can be found in Appendix D.3. We use the Adam optimizer (Kingma & Ba, 2014) with the inclusion of a weight decay  $5e - 4$ . Detailed information
+
+Table 1. Explanation accuracy in terms of AUC-ROC on edges.  
+
+<table><tr><td></td><td>MUTAG</td><td>Benzene</td><td>Alkane-Carbonyl</td><td>Fluoride-Carbonyl</td><td>BA-2motifs</td><td>BA-3motifs</td></tr><tr><td>GradCAM</td><td>0.727±0.000</td><td>0.740±0.000</td><td>0.448±0.000</td><td>0.694±0.000</td><td>0.714±0.000</td><td>0.709±0.000</td></tr><tr><td>GNNExplainer</td><td>0.682±0.009</td><td>0.485±0.001</td><td>0.551±0.003</td><td>0.574±0.002</td><td>0.644±0.007</td><td>0.511±0.002</td></tr><tr><td>PGExplainer</td><td>0.832±0.032</td><td>0.793±0.054</td><td>0.660±0.036</td><td>0.702±0.018</td><td>0.734±0.117</td><td>0.796±0.010</td></tr><tr><td>ReFine</td><td>0.612±0.004</td><td>0.606±0.002</td><td>0.768±0.001</td><td>0.571±0.000</td><td>0.698±0.001</td><td>0.629±0.005</td></tr><tr><td>MixupExplainer</td><td>0.863±0.103</td><td>0.611±0.032</td><td>0.811±0.006</td><td>0.706±0.013</td><td>0.906±0.059</td><td>0.859±0.019</td></tr><tr><td>ProxyExplainer</td><td>0.977±0.009</td><td>0.845±0.036</td><td>0.934±0.005</td><td>0.758±0.068</td><td>0.935±0.008</td><td>0.960±0.008</td></tr></table>
+
+Table 2. Explanation accuracy in terms of AP on MUTAG and BA-2motifs.  
+
+<table><tr><td></td><td>MUTAG</td><td>BA-2motifs</td></tr><tr><td>GradCAM</td><td>0.247±0.000</td><td>0.664±0.000</td></tr><tr><td>GNNExplainer</td><td>0.232±0.001</td><td>0.608±0.004</td></tr><tr><td>PGExplainer</td><td>0.611±0.024</td><td>0.682±0.117</td></tr><tr><td>ReFine</td><td>0.227±0.001</td><td>0.619±0.002</td></tr><tr><td>MixupExplainer</td><td>0.647±0.083</td><td>0.787±0.073</td></tr><tr><td>ProxyExplainer</td><td>0.756±0.107</td><td>0.839±0.036</td></tr></table>
+
+Table 3. Fidelity evaluation on MUTAG and BA-2motifs.  
+
+<table><tr><td></td><td colspan="2">MUTAG</td><td colspan="2">BA-2motifs</td></tr><tr><td></td><td>\(Fid_{\alpha_1,+}\)↑</td><td>\(Fid_{\alpha_2,-}\)↓</td><td>\(Fid_{\alpha_1,+}\)↑</td><td>\(Fid_{\alpha_2,-}\)↓</td></tr><tr><td>GradCAM</td><td>0.004±0.000</td><td>0.162±0.000</td><td>0.072±0.000</td><td>0.107±0.000</td></tr><tr><td>GNNExp.</td><td>0.031±0.001</td><td>0.148±0.001</td><td>0.057±0.002</td><td>0.132±0.001</td></tr><tr><td>PGExp.</td><td>0.034±0.011</td><td>0.148±0.005</td><td>0.065±0.017</td><td>0.126±0.009</td></tr><tr><td>ReFine</td><td>0.003±0.000</td><td>0.160±0.001</td><td>0.060±0.005</td><td>0.125±0.001</td></tr><tr><td>MixupExp.</td><td>0.037±0.006</td><td>0.146±0.003</td><td>0.074±0.005</td><td>0.112±0.003</td></tr><tr><td>ProxyExp.</td><td>0.040±0.002</td><td>0.145±0.001</td><td>0.086±0.003</td><td>0.106±0.002</td></tr></table>
+
+regarding datasets and baselines is delineated in Appendix C.
+
+To evaluate the quality of explanations, we approach the explanation task as a binary classification of edges. Edges that are part of ground truth subgraphs are labeled as positive, while all others are deemed negative. The importance weights given by the explanation methods are interpreted as prediction scores. An effective explanation technique is one that assigns higher weights to edges within the ground truth subgraphs compared to those outside of them. We utilize the AUC-ROC for quantitative assessment (Ying et al., 2019; Luo et al., 2020).
+
+# 6.2. Quantitative Evaluation (RQ1)
+
+To answer RQ1, we compare the proposed method, ProxyExplainer, to other baselines. Each experiment was conducted 10 times using random seeds, and the average AUC scores as well as standard deviations are presented in Table 1.
+
+The results demonstrate that ProxyExplainer provides the
+
+most accurate explanations across all datasets. Specifically, it improves the AUC scores by an average of  $10.6\%$  on real-world datasets and  $7.5\%$  on synthetic datasets over the leading baselines. Comparisons with baseline methods highlight the advantages of our proposed explanation framework. Besides, ProxyExplainer captures underlying explanatory factors consistently across diverse datasets. For instance, MixupExplainer exhibits proficiency on the synthetic BA-2motifs dataset but performs poorly on the real-world Benzene dataset. The reason is that MixupExplainer relies on the independence assumption of explanation and non-explanation subgraphs, which may not hold in real-world datasets. In contrast, ProxyExplainer consistently demonstrates high performance across different datasets, showcasing its robustness and adaptability.
+
+Considering the importance of precision for the positive class in our context, we further adopt AP to evaluate the performance. As shown in Table 2, with AP scores, ProxyExplainer consistently outperforms the other baselines in two benchmark datasets MUTAG and BA-2motifs.
+
+Additionally, some existing works, such as SubgraphX (Yuan et al., 2021), adopt faithfulness-based metrics for evaluation. However, these metrics are problematic due to the OOD problem (Zheng et al., 2024; Amara et al., 2023). So we use the robust fidelity metrics  $(\text{Fid}_{\alpha_1, + }, \text{Fid}_{\alpha_2, - })$  as described in (Zheng et al., 2024) with default settings  $(\alpha_1 = 0.1, \alpha_2 = 0.9)$  to evaluate model faithfulness. Table 3 demonstrates that ProxyExplainer is consistently outperforms all baselines in both  $\text{Fid}_{\alpha_1, + }$  and  $\text{Fid}_{\alpha_2, - }$ .
+
+# 6.3. Alleviating Distribution Shifts (RQ2)
+
+In this section, we assess ProxyExplainer's ability to generate in-distribution proxy graphs. Due to the intractable of direct computation, we follow the previous work (Chen et al., 2023b) to utilize Maximum Mean Discrepancy (MMD) between distributions of multiple graph statistics, including degree distributions, clustering coefficients, and spectrum distributions, between the generated proxy graphs and original graphs. Specifically, we utilize Gaussian Earth Mover's Distance kernel when computing MMDs. Smaller MMD
+
+Table 4. MMD results between the ground truth explanations and original graphs (GT); PGExplainer explanations and original graphs (PGE); proxy graphs in our methods and original graphs (Proxy).  
+
+<table><tr><td></td><td colspan="3">MUTAG</td><td colspan="3">Benzene</td><td colspan="3">Alkane-Carbonyl</td></tr><tr><td>Metric</td><td>GT</td><td>PGE</td><td>Proxy</td><td>GT</td><td>PGE</td><td>Proxy</td><td>GT</td><td>PGE</td><td>Proxy</td></tr><tr><td>Deg.</td><td>0.614</td><td>0.468</td><td>0.123</td><td>0.843</td><td>0.393</td><td>0.236</td><td>0.872</td><td>0.665</td><td>0.177</td></tr><tr><td>Clus.</td><td>0.003</td><td>0.003</td><td>0.009</td><td>0.009</td><td>0.002</td><td>0.004</td><td>0.011</td><td>0.011</td><td>0.011</td></tr><tr><td>Spec.</td><td>0.414</td><td>0.341</td><td>0.186</td><td>0.295</td><td>0.163</td><td>0.101</td><td>0.596</td><td>0.447</td><td>0.049</td></tr><tr><td>Sum.</td><td>1.032</td><td>0.813</td><td>0.317</td><td>1.147</td><td>0.558</td><td>0.341</td><td>1.479</td><td>1.123</td><td>0.237</td></tr><tr><td></td><td colspan="3">Fluoride-Carbonyl</td><td colspan="3">BA-2motifs</td><td colspan="3">BA-3motifs</td></tr><tr><td>Metric</td><td>GT</td><td>PGE</td><td>Proxy</td><td>GT</td><td>PGE</td><td>Proxy</td><td>GT</td><td>PGE</td><td>Proxy</td></tr><tr><td>Deg.</td><td>0.638</td><td>0.488</td><td>0.196</td><td>0.759</td><td>0.496</td><td>0.060</td><td>0.541</td><td>0.149</td><td>0.092</td></tr><tr><td>Clus.</td><td>0.012</td><td>0.012</td><td>0.012</td><td>0.447</td><td>0.463</td><td>0.584</td><td>0.262</td><td>0.382</td><td>0.245</td></tr><tr><td>Spec.</td><td>0.351</td><td>0.315</td><td>0.100</td><td>0.245</td><td>0.256</td><td>0.091</td><td>0.217</td><td>0.063</td><td>0.062</td></tr><tr><td>Sum.</td><td>1.000</td><td>0.815</td><td>0.308</td><td>1.451</td><td>1.215</td><td>0.735</td><td>1.020</td><td>0.594</td><td>0.399</td></tr></table>
+
+values indicate similar graph distributions. For comparison, we also include the Ground truth explanations and the ones generated by PGExplainer.
+
+The results are shown in Table 4. "GT" denotes the MMDs between the ground truth explanations and original graphs. "PGE" represents the MMDs between explanations generated by PGExplainer and original graphs. "Proxy" denotes the MMDs between proxy graphs in our method and original graphs. We have the following observations. First, the MMDs between ground truth and original graphs are usually large, verifying our motivation that a model trained on original graphs may not have correct predictions on the OOD explanation subgraphs. Second, the explanations generated by a representative work, PGExplainer, are often OOD from original graphs, indicating that the original GIB-based objective function may be sub-optimal. Third, in most cases, proxy graphs generated by our method are with smaller MMDs, demonstrating their in-distribution property.
+
+# 6.4. Ablation Studies (RQ3)
+
+In this section, we conduct ablation studies to investigate the roles of different components. Specifically, we consider the following variants of ProxyExplainer: (1) w/o  $G^{\Delta}$ : in this variant, we remove the non-explanatory subgraph generator (VGAE), which is the bottom half as shown in Figure 2; (2) w/o  $\mathcal{L}_{\mathrm{KL}}$ : in this variant, we remove the KL divergence from the training loss in ProxyExplainer; (3) w/o  $\mathcal{L}_{\mathrm{dist}}$ : in this variant, we remove the distribution loss from ProxyExplainer. The results of the ablation study on MUTAG and BA-2motifs are reported in Figure 3.
+
+Figure 3 illustrates a notable performance drop for all variants, indicating that each component contributes positively to the effectiveness of ProxyExplainer. Especially, in the real-life dataset MUTAG, without the in-distribution constraint, w/o  $\mathcal{L}_{\mathrm{dist}}$  is much worse than ProxyExplainer, indicating the vital role of in-distributed proxy graphs in our
+
+![](images/0b1d66053df82b784db082b3b23d7d28a8282664251f4a0df6cfab5d512be87d.jpg)  
+Figure 3. Ablation Studies on MUTAG and BA-2motifs.
+
+framework. Extensive ablation studies on other datasets can be found in the Appendix D.2.
+
+# 6.5. Case Studies
+
+In this part, we conduct case studies to qualitatively compare the performances of ProxyExplainer against others. We adopt examples from Benzene and BA-2motifs in this part. We show visualization results in Figure 4 and Figure 5. Explanations are highlighted with bold black and bold orange edges, respectively. From the results, our ProxyExplainer stands out by generating more compelling explanations compared to baselines. Specifically, ProxyExplainer maintains clarity without introducing irrelevant edges and exhibits more concise results compared to alternative methods. The visualized performance underscores ProxyExplainer's ability to provide meaningful and focused subgraph explanations. More visualizations on these two datasets can be found in the Appendix D.5.
+
+# 7. Conclusion
+
+In this paper, we systematically investigate the OOD problem in the de facto framework, GIB, for learning explanations in GNNs, which is highly overlooked in the literature. To address this issue, we extend the GIB by innovatively including in-distributed proxy graphs. On top of that, we derive a tractable objective function for practical implementations. We further present a new explanation framework that utilizes two graph auto-encoders to generate proxy graphs. We conduct comprehensive empirical studies on both benchmark synthetic datasets and real-life datasets to demonstrate the effectiveness of our method in alleviating the OOD problem and achieving high-quality explanations. There are several topics we will investigate in the future. First, the OOD problem also exists in obtaining model-level explanations and counterfactual explanations. We will extend our
+
+![](images/af0c2330ce3fe8e5ad873dfa6ee7597990addc6cdc744289c30b6de18d1916e6.jpg)  
+(a) Ground Truth
+
+![](images/ea1f94a18f534031990089870794d21b0a876f21876e1f442d7209194e0cada3.jpg)  
+(b) ProxyExplainer
+
+![](images/6523c7584423569f8f9437aefa874ce22d79804f3bd9273f086548fe37849dc7.jpg)  
+(c) MixupExplainer
+
+![](images/ea40d356680fbb1232f7039bf37adb054a34dfbc191a2b076008a1b1baa85244.jpg)  
+(d) PGExplainer
+
+![](images/4e48565005a65000729635f9c984a347eef174884b5b2b4db0afd082dc919a21.jpg)  
+(e) GNNExplainer
+
+![](images/cb26f73851cb00c141cefc4185dab949599b592a3f4b4e7449c87657712f512a.jpg)  
+Figure 4. Visualization of explanation results from different explanation models on Benzene. The generated explanations are highlighted with bold orange edges.  
+(a) Ground Truth  
+Figure 5. Visualization of explanation results from different explanation models on BA-2motifs. The generated explanations are highlighted with bold black edges.
+
+![](images/c23463453ac4ec5868780f2da5a876ae112efb348052a075831793da48fe07e9.jpg)  
+(b) ProxyExplainer
+
+![](images/4ebd0d6c88a420cf54318084076512566efc2e1cc79520d8eb7d2b2e32366018.jpg)  
+(c) MixupExplainer
+
+![](images/39999e69bda92848e744903c4185c51c0db798692df36598e409491beb84bee8.jpg)  
+(d) PGExplainer
+
+![](images/3721ce9b63ca7878da1988fe12a65896b5fd1a949b97725bf689a22f48754d00.jpg)  
+(e) GNNExplainer
+
+method to these research problems. Second, we will also analyze explainable learning methods with proxies in other data structures, such as image, language, and time series.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
+
+# Acknowledgments
+
+This project was partially supported by NSF grant IIS-2331908. The views and conclusions contained in this paper are those of the authors and should not be interpreted as representing any funding agencies.
+
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+
+# A. Notations
+
+In Table 5, we summarized the main notations we used and their descriptions in this paper.
+
+Table 5. Symbols and their descriptions.  
+
+<table><tr><td>Symbols</td><td>Descriptions</td></tr><tr><td>G</td><td>A set of graphs</td></tr><tr><td>G, V, E</td><td>Graph instance, node set, edge set</td></tr><tr><td>vi</td><td>The i-th node</td></tr><tr><td>X</td><td>Node feature matrix</td></tr><tr><td>A</td><td>Adjacency matrix</td></tr><tr><td>Z</td><td>Node representation matrix</td></tr><tr><td>Y</td><td>Label of graph G</td></tr><tr><td>y</td><td>A set of labels</td></tr><tr><td>G*</td><td>Optimal explanatory subgraph</td></tr><tr><td>G*</td><td>A set of G*</td></tr><tr><td>G&#x27;</td><td>Candidate explanatory subgraph</td></tr><tr><td>GΔ</td><td>Non-explanatory graph</td></tr><tr><td>G</td><td>Proxy graph of G&#x27; with a fixed distribution</td></tr><tr><td>h, h&#x27;, h</td><td>Graph embeddings</td></tr><tr><td>d</td><td>Dimension of node feature</td></tr><tr><td>f(·)</td><td>To-be-explained GNN model</td></tr><tr><td>Ψψ(·)</td><td>Explanation function</td></tr><tr><td>ψ</td><td>Parameter of the explanation function</td></tr><tr><td>PG</td><td>Distribution of original training graphs</td></tr><tr><td>PG&#x27;</td><td>Distribution of explanation subgraphs</td></tr><tr><td>Qφ</td><td>Parameterized function of P(G|G&#x27;)</td></tr><tr><td>φ</td><td>Model parameters of Qφ</td></tr><tr><td>φ*</td><td>Optimal φ</td></tr><tr><td>α</td><td>Balance parameter between I(G, G&#x27;) and I(Y, G&#x27;)</td></tr><tr><td>E</td><td>The set of node pairs that are unconnected in G</td></tr><tr><td>puv</td><td>Probability of node pair (u, v) in G</td></tr><tr><td>β</td><td>A hyper-parameter to get a trade-off between connected and unconnected node pairs</td></tr><tr><td>fenc(·)</td><td>The front part of GNNs that learns node representations</td></tr><tr><td>fcls(·)</td><td>The back part of GNNs that predicts graph labels based on node embeddings</td></tr><tr><td>σ(·)</td><td>Sigmoid function</td></tr><tr><td>τ</td><td>Temperature hyper-parameter for approximation</td></tr><tr><td>λ</td><td>A hyper-parameter in Proxy loss function</td></tr><tr><td>Ldist</td><td>Distribution loss between G and G</td></tr><tr><td>KL</td><td>KL divergence between distribution of ZΔ and its prior</td></tr><tr><td>Lproxy</td><td>Proxy loss</td></tr><tr><td>Lexp</td><td>Explainer loss</td></tr></table>
+
+# B. Algorithm
+
+We take as input a set of graphs  $\mathcal{G} = \{G_i\}_{i=0}^N$ . For each graph  $G_i$ , we use an explainer model to identify a subgraph  $G_i'$  as the explanation and then compute the non-explanatory graph  $G_i^\Delta$ , which is obtained by removing edges in  $G_i$  that exist in  $G_i'$ . We use GAE to reconstruct a subgraph, denoted by  $\tilde{G}_i'$  from the subgraph  $G_i'$  and another VGAE to generate a new subgraph, denoted by  $\tilde{G}_i^\Delta$ , from the non-explainable subgraph  $G_i^\Delta$ . The proxy graph,  $\tilde{G}_i$ , is obtained by combining them, whose adjacency matrix can be denoted by  $\tilde{A}_i = \tilde{A}_i' + \tilde{A}_i^\Delta$ . Here  $\tilde{A}_i'$  and  $\tilde{A}_i^\Delta$  are adjacency matrices of  $\tilde{G}_i'$  and  $\tilde{G}_i^\Delta$ , respectively. We alternatively train the explainer model and proxy graph generator as shown in Algorithm 1.
+
+Algorithm 1 Algorithm of ProxyExplainer  
+Input: A set of graphs  $\mathcal{G} = \{G_i\}_{i=0}^N$ , with each  $G_i = (\mathcal{V}_i, \mathcal{E}_i, X_i)$ , a pretrain to-be-explained model  $f(\cdot)$ , hyper parameters  $\alpha, \lambda, M$ , epochs  $E$ . Initialize an explainer function  $\Psi_{\psi}(\cdot)$  epoch  $\leftarrow 0$  while epoch  $< E$  do for  $G_i \in \mathcal{G}$  do  $G_i' \gets \Psi_{\psi}(G_i)$ $G_i' \gets G_i - G_i'$  Compute  $\tilde{A}_i'$  with equation 14 Compute  $Z^{\Delta}$  and  $\tilde{A}_i^{\Delta}$  with equation 16 Compute proxy loss  $\mathcal{L}_{\mathrm{proxy}}$  with equation 18 Update parameters in proxy graph generator with backpropagation if epoch%M == 0 then Compute explainer loss  $\mathcal{L}_{\mathrm{exp}}$  Update parameters in the explainer with backpropagation end if end for epoch  $\leftarrow$  epoch + 1 end while
+
+# C. Full Experimental Setups
+
+We conduct all experiments on a Linux machine with 8 Nvidia A100-PCIE GPUs, each with 40GB of memory. The CUDA version is 12.4 and the driver version is 550.54.15. We use Python 3.9 and Torch 2.0.1 in our project. The code is available at https://github.com/realMoana/ProxyExplainer.
+
+# C.1. Datasets
+
+MUTAG (Kazius et al., 2005). MUTAG dataset includes 4,337 molecular graphs, each of them is classified into two groups based on its mutagenic effect on the Gram-negative bacterium S. Typhimurium. This classification comes from several specific virulence gland groups with mutagenicity in molecular mapping by Kazius et al.(Kazius et al., 2005).
+
+Benzene (Sanchez-Lengeling et al., 2020). Benzene consists 12,000 molecular graphs from the ZINC15(Sterling & Irwin, 2015) database, which can be classified into two classes. The main goal is to determine if a Benzene ring is exited in each molecule. In cases where multiple Benzene rings are present, each Benzene ring is treated as a distinct explanation.
+
+Alkane-Carbonyl (Sanchez-Lengeling et al., 2020). The Alkane-Carbonyl dataset consists of a total of 4,326 molecule graphs that are categorized into two different classes. Positive samples refer to molecules that have both alkane and carbonyl functional groups. The ground-truth explanation includes alkane and carbonyl functional groups in a given molecule.
+
+Fluoride-Carbonyl (Sanchez-Lengeling et al., 2020). The Fluoride-Carbonyl dataset has 8,671 molecular graphs. The ground-truth explanation is based on the particular combination of fluoride atoms and carbonyl functional groups present in each molecule.
+
+BA-2motifs (Luo et al., 2020). The BA-2motifs dataset consists 1,000 synthetic graphs, each derived from a basic Barabasi-Albert (BA) model. The dataset is divided into two categories: one part of the graphs add patterns that mimic the structure of a house, and the remaining integrate five-node cyclic patterns. The classification of these graphs depends on the specific patterns.
+
+BA-3motifs (Chen et al., 2023b). BA-3motifs is an extended dataset inspired by the BA-2motifs and contains 3,000 synthetic graphs. Each base graph is accompanied by one of three different patterns: house, cycle, or grid.
+
+# C.2. Baselines
+
+To evaluate our model, we well-train the GCN model in to ensure that it takes good performance in graph classification tasks. The results are displayed in Table 7. For a comprehensive comparison, we incorporate various post-hoc explanation methods, including GradCAM, GNNExplainer, PGExplainer, ReFine, and MixupExplainer.
+
+Table 6. Statistics of molecule datasets for graph classification with ground-truth explanations.  
+
+<table><tr><td></td><td>MUTAG</td><td>Benzene</td><td>Alkane-Carbonyl</td><td>Fluoride-Carbonyl</td><td>BA-2motifs</td><td>BA-3motifs</td></tr><tr><td>Graphs</td><td>4,337</td><td>12,000</td><td>4,326</td><td>8,671</td><td>1,000</td><td>3,000</td></tr><tr><td>Average nodes</td><td>29.15</td><td>20.58</td><td>21.13</td><td>21.36</td><td>25.00</td><td>21.92</td></tr><tr><td>Average edges</td><td>60.83</td><td>43.65</td><td>44.95</td><td>45.37</td><td>25.48</td><td>29.51</td></tr><tr><td>Node features</td><td>14</td><td>14</td><td>14</td><td>14</td><td>10</td><td>4</td></tr><tr><td>Original graph</td><td>C O O H N</td><td>C O</td><td>C O</td><td>C O F</td><td></td><td></td></tr><tr><td>Ground truth explanation</td><td>NH2, NO2</td><td>Benzene Ring</td><td>Alkane,C=O</td><td>F-,C=O</td><td>House, cycle</td><td>House,cycle,grid</td></tr></table>
+
+Table 7. The graph-classification task performances of the GCN model  
+
+<table><tr><td></td><td>MUTAG</td><td>Benzene</td><td>Alkane-Carbonyl</td><td>Fluoride-Carbonyl</td><td>BA-2motifs</td><td>BA-3motifs</td></tr><tr><td>Train Acc</td><td>0.850</td><td>0.930</td><td>0.979</td><td>0.951</td><td>0.999</td><td>0.997</td></tr><tr><td>Val Acc</td><td>0.834</td><td>0.927</td><td>0.986</td><td>0.956</td><td>1.0</td><td>0.997</td></tr><tr><td>Test Acc</td><td>0.804</td><td>0.915</td><td>0.975</td><td>0.951</td><td>1.0</td><td>0.977</td></tr></table>
+
+- GradCAM (Pope et al., 2019). This method utilizes gradients as a weighting mechanism to merge various feature maps. It operates on heuristic assumptions and cannot elucidate node classification models.  
+- GNNExplainer (Ying et al., 2019). It learns soft masks for edges and node features, and aims to elucidate predictions through mask optimization. GNNExplainer integrates these masks with the original graph through element-wise multiplifications. The masks are optimized by enhancing the mutual information between the original graph and the modified graph prediction results.  
+- PGExplainer (Luo et al., 2020). This method extends the idea of GNNExplainer by assuming that the graph is a random Gilbert graph. PGExplainer generates each edge embedding by combining the embeddings of its constituent nodes, then uses these edge embeddings to determine a Bernoulli distribution to indicate whether to mask an edge or not, and utilizes a Gumbel-Softmax approach to model the Bernoulli distribution for end-to-end training.  
+- ReFine (Wang et al., 2021). ReFine identifies the edge probabilities for the entire category by maximizing the mutual information and contrastive loss between categories. In fine-tuning, it uses the edge probabilities from the previous stage to sample edges, and find explanations that maximize mutual information for specific instances.  
+- MixupExplainer (Zhang et al., 2023b). This method combines original explanatory subgraphs with randomly sampled, label-independent base graphs in a non-parametric way to mitigate the common OOD issue which found in previous methods.
+
+# D. Extensive Experiments
+
+# D.1. Extensive Distribution Analysis
+
+As shown in previous work (Zhang et al., 2023b), another way to approximate the distribution differences of graphs is to compare their vector embeddings. Here, we adopt the same setting and use Cosine similarity and Euclidean distance to intuitively approximate the similarities between graph distributions. Table 8 presents the computed Cosine similarity and Euclidean distance between the distribution embeddings of the original graph  $\pmb{h}$ , the ground truth explanation subgraph  $\pmb{h}'$  and the generated proxy graph  $\tilde{\pmb{h}}$ . Notably, our proxy graph embedding  $\tilde{\pmb{h}}$  exhibits higher Cosine similarity scores and lower Euclidean distance with the original graph embedding  $\pmb{h}$  compared to the ground truth explanation embedding  $\pmb{h}'$ . We observe an average improvement of  $19.5\%$  in Cosine similarity and  $35.6\%$  in Euclidean distance. Particularly in the BA-2motifs dataset, there is a significant improvement of  $60.4\%$  in Cosine similarity and  $51.8\%$  in Euclidean distance. These findings underscore the effectiveness of our ProxyExplainer method in mitigating distribution shifts caused by induction bias in the to-be-explained GNN model  $f(\cdot)$ , thereby enhancing explanation performance.
+
+Table 8. The Cosine similarity score and Euclidean distance between the distribution embeddings of the original graph  $\mathbf{h}$ , explanation subgraph  $\mathbf{h}'$ , and our proxy graph  $\tilde{\mathbf{h}}$  on different datasets.  
+
+<table><tr><td></td><td>MUTAG</td><td>Benzene</td><td>Alkane-Carbonyl</td><td>Fluoride-Carbonyl</td><td>BA-2motifs</td><td>BA-3motifs</td></tr><tr><td>Avg. Cosine(h,h&#x27;)↑</td><td>0.883</td><td>0.835</td><td>0.889</td><td>0.904</td><td>0.571</td><td>0.686</td></tr><tr><td>Avg. Cosine(h,tilde h)↑</td><td>0.985</td><td>0.905</td><td>0.938</td><td>0.908</td><td>0.916</td><td>0.918</td></tr><tr><td>Avg. Euclidean(h,h&#x27;)↓</td><td>0.975</td><td>1.010</td><td>0.940</td><td>0.806</td><td>1.210</td><td>1.199</td></tr><tr><td>Avg. Euclidean(h,tilde h)↓</td><td>0.368</td><td>0.767</td><td>0.719</td><td>0.779</td><td>0.583</td><td>0.613</td></tr></table>
+
+# D.2. Extensive Ablation Study
+
+In Figure 6, we conduct a comprehensive ablation study across all datasets to examine the impact of different components. The findings illustrate the effectiveness of these components and their positive contribution to our ProxyExplainer.
+
+![](images/d296e8ac7cbaf013bbd9d3c28d92cc7abbddee2b47e31bbfd5cdc5810c48ac0b.jpg)
+
+![](images/f74915da5e3ed27e218a8f6d0ea09d8342a40adfc7cb67186dae3eb7794ceb49.jpg)
+
+![](images/bc74991c50eb927e72620125ae82ac5959acbf594cc454fcf055afc6d7808c9f.jpg)
+
+![](images/88defe994c9f5f2bea7ae8e7156e724d3c0aff775711bea4c02706437a434c33.jpg)  
+(a) MUTAG  
+(d) Fluoride-Carbonyl
+
+![](images/7979db29f0d8a5f05065ec14c9e1ec06ce6ec476f12a165906b19a1bf9c68640.jpg)  
+(b) Benzene  
+(e) BA-2motifs  
+Figure 6. Ablation study on all datasets.
+
+![](images/51415cd046c47166aa8e4f163207f0106eaab4e1e1a6ce690311d42bcdb6b13f.jpg)  
+(c) Alkane-Carbonyl  
+(f) BA-3motifs
+
+# D.3. Extensive Experiment with GIN
+
+In order to show the robustness of our ProxyExplainer in explaining different GNN models, we first pre-train GIN on two real-world datasets and two synthetic datasets to ensure that GIN has the ability to classify graphs accurately. The results are displayed in Figure 7. Then we use both baseline methods in the previous experiment and our method ProxyExplainer to provide explanations for the pre-trained GIN model. As seen in Figure 8, it is noticeable that ProxyExplainer achieves the best performance on these datasets. Specifically, it improves the AUC scores of  $2.9\%$  on Alkane-Carbonyl,  $7.9\%$  on Fluoride-Carbonyl,  $45.9\%$  on BA-2motifs, and  $6.1\%$  on BA-3motifs. From the analysis, we can see that our ProxyExplainer can identify accurate explanations across different datasets.
+
+![](images/05d4165b1af38fd0ff698f6f23d9c12d66f6f645dc2f590ee92aa8300c86fca5.jpg)  
+Figure 7. The graph-classification task performances of the GIN model.
+
+![](images/edebe441ea337fee7126ffe4a0cd9ea9e2597e7203d6ddfc050132b41b2436d3.jpg)  
+(a) Alkane-Carbonyl
+
+![](images/dd127c34385ff0c2136b284fefd33b63b6858a11e667932aa08bc8d72306ccec.jpg)  
+(b) Fluoride-Carbonyl
+
+![](images/971a3f8d4817b89146aac046d6d14ec6b045d8e79c67ffc04e838f3fd23aed06.jpg)  
+(c) BA-2motifs
+
+![](images/ab20908370061e9cfa594e4a78a72a4f6fd7a1e4bdc01496677b59f961a6a4b9.jpg)  
+(d) BA-3motifs  
+Figure 8. Explanation accuracy in terms of AUC-ROC on edges based on GIN.
+
+# D.4. Parameter Sensitive Analysis
+
+In this section, we analyze the influence of parameters including  $\lambda$ , which controls the KL divergence during the proxy graph generation process, and the dimension  $D$  of node latent embedding. We vary  $\lambda$  from 0.01 to 1.0. For  $D$ , we vary it among
+
+{32, 64, 128, 256, 512, 1024}. We only change the value of a specific parameter while fixing the remaining parameters to their respective optimal values.
+
+Figure 9 shows the performance of our ProxyExplainer with respect to  $\lambda$  on two real-world datasets (MUTAG and Alkanecarbonyl) and two synthetic datasets (BA-2motifs and BA-3motifs). From Figure 9, we can see that ProxyExplainer consistently outperforms the best baseline, MixupExplainer, for  $\lambda \in [0.25,1.0]$ . This indicates that ProxyExplainer is stable. Figure 10 presents how the dimension of node latent embedding affects performance in ProxyExplainer, the results show that ProxyExplainer can reach the best performance at  $D = 512$ .
+
+![](images/1d08640ca82fbf2f869bf042477453416d62241c19f209a3dc9c06cf2c25fa0e.jpg)  
+(a) MUTAG
+
+![](images/173c7b79861ae98533e038bb05561a9bf51bc2e25e0edb5b214fd4c2384893bf.jpg)
+
+![](images/a316f80d68d8ef1fe75430d32b71cd50c308704508b7c35b3412f601dbae3e44.jpg)  
+(b) Alkane-Carbonyl
+
+![](images/53412378ea01bf267dc4d569a8e1bc8025892ac71080bce26a8c76d684ef5356.jpg)
+
+![](images/768382e95df7cca7c1af2e2c312d6259febd1c96f3c7b1e793f779a6fb275dc5.jpg)  
+(c) BA-2motifs
+
+![](images/7142efd12e83bffc6366353427fcbd293d397e336ecdd81d127cf04f555f05ae.jpg)  
+Figure 9. Parameter analysis of  $\lambda$  on four datasets. The left side of each graph shows the explanation performance. The right side shows the Distance Analysis between  $h$  and  $\tilde{h}$ .
+
+![](images/6d0b0859330346b05d13c3c8f78609542593a6400cc5a38873ae4592ed94fb64.jpg)  
+(d) BA-3motifs
+
+![](images/52d4fe6008d0adbb742fa6c5f18dabcc8098f9400b9d9597021177854b88ed76.jpg)
+
+![](images/55d7d4be62ce637b8fd40d3b78a124aa79df75056a8d1e26868fb4a8ff648246.jpg)  
+(a) MUTAG
+
+![](images/a1d751284676f5c5ca64b3e5833e60f4610e954857880df4f29180714ea2a256.jpg)
+
+![](images/126fb1db5ea856b13ba385329ae60fb9247c0449669bfbd21329ffdedf931c1a.jpg)  
+(b) Alkane-Carbonyl
+
+![](images/013a70a9fb83184959a92304ad70b214b16fdde1db233850f32a430d95f08932.jpg)
+
+![](images/8049246a7e30d40d042dd25d4bc8b24b586438b74aaf0af1e2b7f5744604efe0.jpg)  
+(c) BA-2motifs
+
+![](images/b1c4765a911b6b0c401ef9916de6a88548678c94dd4f3265c9817a5357f64dbb.jpg)  
+Figure 10. Parameter analysis of the dimension of node latent embedding. The left side of each graph shows the explanation performance and the right side displays the Cosine score and Euclidean distance between  $\pmb{h}$  and  $\bar{\pmb{h}}$ .
+
+![](images/30b35465dd5b66802fe915f0e70cba10b6d3adb2166b327f851ee6c85c354573.jpg)  
+(d) BA-3motifs
+
+![](images/afe1891e67f803aad46b0d2708f8947133d1c5ad62049206bf6597ad611d6383.jpg)
+
+# D.5. Extensive Case Study
+
+We show more visualization examples of explanation graphs on BA-2motifs, BA-3motifs, and Benzene in Figure 11, Figure 12, and Figure 13, respectively.
+
+![](images/ee0b7fd616cc8396306cdf9fe242cc37ab2d9adcf4ed65d69602cc127485212d.jpg)
+
+![](images/8f7258060ed44690611ff8d58233091ea96665e2105367b3328843f1ba4ff7d2.jpg)
+
+![](images/3c3aedd28bb6876aed363aa65b0c69013ebc27beff9074039bad0caeecc4731a.jpg)
+
+![](images/938139a2f8673fb3aee8e8b9744648c847b05a3fdce266f3cdf582565e518f18.jpg)
+
+![](images/1245e3400db483bd8b1e716997a074fd6ad22a178cde253c8a0e79a17f6577c6.jpg)
+
+![](images/46d07eab3975bc14368faba08a6dd2a24e1b531c0c843c11e1783a7bb5004106.jpg)
+
+![](images/2b11229bbd71f827faace022eb2293c1bbac3a7a82a197195be6c5f5635358c1.jpg)
+
+![](images/0ea4fb3dff531d6b4569c3dfcf0cdfcbc6a058a34461b80b0171d0d8148d340f.jpg)
+
+![](images/f6f23a0c783a7d4fdd79ed3f097a57a61f3f07cbb2bade544a8d76aee01cb47b.jpg)
+
+![](images/96a4a1b0b243bfcdecf95e1e8510877564948cba88b9a27711c7d08405554fa5.jpg)
+
+![](images/59688ddba7ea01136fbe88a9acf8c54132de049ba43d3ba8612b3f80bd0d07b0.jpg)  
+(a) Ground Truth
+
+![](images/37976f3142874ca81f12461ba9ee39d3cfbe11ad8af704685d49b7827a166813.jpg)  
+(b) ProxyExplainer
+
+![](images/c5ca61459c22763a9aaab17b92b3b0ef4064af1633bdddddc4cf983ce20eba2e.jpg)  
+(c) MixupExplainer
+
+![](images/458592fb93c3fd9198df05b71e75e66dd45de599ae7760a03d346a585e66b54e.jpg)  
+(d) PGExplainer
+
+![](images/4db0d5ad507971124b3810aa4a4e568108f281dd1a7c851896d8d7a37e109030.jpg)  
+(e) GNNExplainer
+
+![](images/065f3ef3ea696b839494cacdd0cef6e3d2006386967cd32dc904ac469298e523.jpg)
+
+![](images/38e273e45a27ccc9cefaa4087080c7257e3c9d15ca2cdcd499a1d16b1531cbfe.jpg)  
+Figure 11. Visualization of explanation on BA-2motifs.
+
+![](images/c20d8701a83ca9757d1c6ab150e7f2e684db9c1a9964196b71ac1a38b3a8a671.jpg)
+
+![](images/a5a60394b2f1b82948000640bef2792a5dd7a2aef970e9e71f3136ad024d720a.jpg)
+
+![](images/5461218781a6900e90b8e5cef32e24282d7ba665be30084f720d3c8d13661239.jpg)
+
+![](images/f3846a31a9c9534df2da3c26d8d13c18d617ba51726cb0a6a0dd81fa74d3fb69.jpg)
+
+![](images/4e47897c59c398de2f0c2bf53daf18b7862d7246d060ef351b72feb846374637.jpg)
+
+![](images/bb47a52ca586f14676cd4fecbf806ca082baed6a571802d07c203a95d141820c.jpg)
+
+![](images/6580f1f40b90246ab7373e45faeb82302d8e75fc5baeedb437db6146988fc0c6.jpg)
+
+![](images/4099aaef25bc3194215062fddf4a834284579e9befc03846f62f9d46abbdf201.jpg)
+
+![](images/06770bee3e222b0c900a95a3178ab784c1e6e65af807adc02b4a10f1c2f31f60.jpg)  
+(a) Ground Truth
+
+![](images/cf7e4f0e7e03786d5912eafddf37982976aee984f6f222b6df03b7ffe38d8cc2.jpg)  
+(b) ProxyExplainer
+
+![](images/9f9d28d019a78a79c3e468565fd6dc2199c753c8c283bb60075761d3a5632034.jpg)  
+(c) MixupExplainer
+
+![](images/e3663b940a7b2f3e344555edb0ab7d50ce6d7da688f60d8682c78abb58b64c76.jpg)  
+(d) PGExplainer
+
+![](images/be319b74c3894a45c4ccc47a6217717113dbe854a0fe7e1124ebf054ce5c1e85.jpg)  
+(e) GNNExplainer  
+Figure 12. Visualization of explanation on BA-3motifs.
+
+![](images/395d3a2bb16faabd75abb0f844a5f33ed3d72ddd039c27f5a83c083edc70d0f7.jpg)  
+Figure 13. Visualization of explanation on Benzene.

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+# Generative Active Learning for Long-tailed Instance Segmentation
+
+Muzhi Zhu  $^{1*}$  Chengxiang Fan  $^{1*}$  Hao Chen  $^{1}$  Yang Liu  $^{1}$  Weian Mao  $^{2,1}$  Xiaogang Xu  $^{3,1}$  Chunhua Shen  $^{1,4}$
+
+# Abstract
+
+Recently, large-scale language-image generative models have gained widespread attention and many works have utilized generated data from these models to further enhance the performance of perception tasks. However, not all generated data can positively impact downstream models, and these methods do not thoroughly explore how to better select and utilize generated data. On the other hand, there is still a lack of research oriented towards active learning on generated data. In this paper, we explore how to perform active learning specifically for generated data in the long-tailed instance segmentation task. Subsequently, we propose BSGAL, a new algorithm that online estimates the contribution of the generated data based on gradient cache. BSGAL can handle unlimited generated data and complex downstream segmentation tasks effectively. Experiments show that BSGAL outperforms the baseline approach and effectually improves the performance of long-tailed segmentation.
+
+# 1. Introduction
+
+Data is one of the driving forces behind the development of artificial intelligence. In the past, securing high-quality data was a time-consuming and laborious task. Yet, a large amount of high-quality data is crucial for a model to achieve breakthrough performance. Therefore, many active learning methods have emerged to explore the most informative samples from massive unlabeled data to achieve better model performance with minimal annotation costs. Currently, the rapid development of generative models has made it possible to obtain massive amounts of high-quality data, including long-tailed data, at a relatively low cost. In the field of visual perception, there have been many works utilizing
+
+*Equal contribution 1Zhejiang University, China 2The University of Adelaide, Australia 3The Chinese University of Hong Kong, China 4Ant Group. Correspondence to: Hao Chen <haochen CAD@zju.edu.cn>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+generated data to improve perception tasks, including classification (Azizi et al., 2023), detection (Chen et al., 2023), and segmentation (Wu et al., 2023b). However, they often directly use generated samples as mixed training data (Yang et al., 2023) or as data augmentation (Zhao et al., 2023) without exploring how to better filter and utilize the data for downstream models.
+
+On the other hand, existing data mining and filtering methods, such as active learning, have only been validated on real data and are not suitable for generated data, as there are differences between generated data and real data regarding characteristics and usage scenarios. The differences are mainly as follows: 1) Existing data analysis methods are aimed at a limited data pool, while the scale of generated data is almost infinite. 2) Active learning is often carried out under a specified annotation budget. Thanks to the development of conditional generative models, the annotation cost of generated data can be almost negligible. However, this results in an unclear and uncertain quality of annotation in generated data compared to expert annotations. 3) There are differences in the distribution between real data and generated data, while the target data of previous methods do not have obvious distribution differences.
+
+In response to the aforementioned challenges, we propose a novel problem called "Generative Active Learning for Long-tailed Instance Segmentation" (see Figure 1), which investigates how to utilize generated data effectively for downstream tasks. We focus on long-tailed instance segmentation for three main reasons. First, data collection for long-tailed categories is exceedingly arduous, and how to do classification well is currently a focus in the segmentation field (Kirillov et al., 2023; Li et al., 2023a; Yuan et al., 2024; ?: Zou et al., 2023; Liu et al., 2023). Given that generated data has demonstrated the potential to alleviate this difficulty (Zhao et al., 2023; Xie et al., 2023; Fan et al., 2024), it is necessary to introduce generated data for long-tail segmentation tasks. Second, the quality requirements for generated data in long-tailed instance segmentation tasks are very high, and not all generated data can have a promoting effect. Therefore, it is necessary to further explore how to screen generated data. Third, this task itself is very comprehensive and challenging, which has more guiding significance for migration to real-world scenarios.
+
+![](images/72aa04e41c9d7bba0e4ca658eacd046fe19e3de2f3dfa18988c37e3b787faba6.jpg)  
+(a) Traditional Active Learning
+
+![](images/1b389ade06293479cdbb7963a5166c713f89f99f033e41e0d5f749b52939bb46.jpg)  
+(b) Generative Active Learning  
+Figure 1. Comparison between Traditional Active Learning and Generative Active Learning frameworks. (a) Traditional Active Learning relies on a human oracle, therefore the annotation is accurate but with a limited budget, so the model is required to select the most informative unlabeled data. (b) Generative Active Learning, which relies on a generative oracle, has an unlimited labeled pool. However, the quality of annotation varies greatly, so the model must judiciously accept data.
+
+Here, our objective is to design a generative active learning pipeline targeted at long-tailed image segmentation tasks. Inspired by data-influence analysis methods (Ling, 1984; Koh and Liang, 2017), we first use the change of the loss value to provide an estimation of the contribution of a single generated instance in the ideal case, as discussed in Section 4.1. Employing the first-order Taylor expansion, we introduce an approximate contribution estimation function based on the gradient dot product, which avoids repeated calculations on the test set in the offline setting. Based on this technique, we conduct a toy experiment on CIFAR-10 (Krizhevsky et al., 2009) in Section 5.1.1 along with a qualitative analysis of each sample on the LVIS dataset in Section 5.2.2, which preliminarily verify the feasibility of our approach. Subsequently, in Section 4.2 we explore how to apply this evaluation function to the actual segmentation training process. We propose the Batched Streaming Generative Active Learning algorithm (BSGAL), which allows for online acceptance or rejection of each batch of generated data. Additionally, based on the first-order gradient approximation, we maintain a gradient cache based on momentum updates to enable a more stable contribution estimation. Finally, experiments are carried out on the LVIS dataset, establishing that our method outperforms both unfiltered or CLIP-filtered counterparts under various backbones. Notably, in the long-tailed category, there is an over  $10\%$  improvement in  $AP_r$  (Gupta et al., 2019). In addition, We conduct a series of ablation experiments to delve into the particulars of our algorithmic design, including the choice of loss, the way of contribution estimation, and the sampling strategy of the test set. To summarize, our primary contributions are detailed below:
+
+- We introduce a novel problem called "Generative Ac
+
+tive Learning for Long-tailed Instance Segmentation": how to design an effective method focused on the successful using generated data, aimed at enhancing the performance of downstream segmentation tasks. Existing data analysis methods are neither directly applicable to generated data nor have they been affirmed as efficient for such data.
+
+- We propose a batched streaming generative active learning method (BSGAL) based on gradient cache to estimate generated data contribution. This pipeline can adapt to the actual batched segmentation training process, handle unlimited generative data online, and effectively enhance the performance of the model.  
+- We carry out experiments on the LVIS dataset and demonstrate that our method outperforms both unfiltered and CLIP-filtered methods. Our model surpasses the baseline by  $+1.2$  on  $AP_{box}$  and  $+3.62$  on  $AP_{r}$ . We also conduct more comprehensive analysis experiments on the design or hyperparameters of our method.
+
+# 2. Related work
+
+# 2.1. Generative Data Augmentation
+
+Generative data augmentation (GDA) refers to using generative models to synthesize additional data for augmentation. With the continuous improvement in the capabilities of generative models (Goodfellow et al., 2020; Sahara et al., 2022; Rombach et al., 2022b), GDA has become a popular technique for improving model performance. Several works use GDA in perceptual tasks such as classification (Feng et al., 2023; Zhang et al., 2023; Azizi et al., 2023), detection (Zhao et al., 2023; Chen et al., 2023), and segmentation (Li et al., 2023b; Wu et al., 2023b;a; Xie et al., 2023). Early works (Zhang et al., 2021; Li et al., 2022) involve the use of generative models like Generative Adversarial Networks (GANs) (Goodfellow et al., 2020) to generate additional training data. With the evolution of diffusion models, recent works (Azizi et al., 2023; Li et al., 2023b; Wu et al., 2023b; Yang et al., 2023; Zhao et al., 2023) favor using high-quality diffusion models such as Imagen (Saharia et al., 2022) and Stable Diffusion (Rombach et al., 2022b) for data generation. X-Paste (Zhao et al., 2023) has proven the strategy of using copy-paste to be more effective than directly using generated data for mixed training, and for the first time demonstrated that using generated data can enhance the performance of segmentation models on the long-tailed segmentation dataset LVIS (Gupta et al., 2019). Therefore, we consider it as the baseline for our work. However, while previous works have investigated the effects of GDA on different tasks, there has been limited exploration on how to better filter and utilize generative data for downstream perception models.
+
+# 2.2. Active Learning and Data Analysis
+
+Analysis of the information or contribution of data samples to a model has been extensively studied long before the advent of deep learning. Among them, two fields are most relevant to our work, one is active learning, and the other is training data influence analysis.
+
+Active learning (Ren et al., 2021) mainly focuses on how to explore the most informative samples from massive unlabeled data to achieve better model performance with minimal annotation costs. Generally speaking, active learning can be divided into two categories. One is uncertainty-based active learning, which measures the uncertainty of samples by the posterior probability of the predicted category (Lewis and Catlett, 1994; Lewis, 1995; Goudjil et al., 2018) or the entropy of the predicted distribution (Joshi et al., 2009; Luo et al., 2013), and then selects the most uncertain samples for annotation. The other is diversity-based active learning, which is based on clustering (Nguyen and Smeulders, 2004) or core-set (Sener and Savarese, 2018) methods. They attempt to mine the most representative samples from the data to achieve minimal annotation costs. Recently, active learning in deep learning also tends to adopt a batch-based sample querying method (Ash et al., 2020), which is consistent with our work. The most relevant work to our work is VeSSAL (Saran et al., 2023), which does batched active learning in a streaming setting and samples in a gradient space. Another relatively related work (Mahapatra et al., 2018) trains a GAN on medical images, using the GAN to generate more data for active learning.
+
+Training data influence analysis (Hammoudeh and Lowd, 2022) explores the relationship between training data samples and model performance, which can be divided into retraining-based (Ling, 1984; Roth, 1988; Feldman and Zhang, 2020) and gradient-based (Koh and Liang, 2017; Yeh et al., 2018). The most typical retraining-based method is Leave-One-Out (Ling, 1984; Jia et al., 2021), which measures the contribution of a sample to the model by removing a sample from the training set and then retraining the model. However, this method is obviously impractical for modern large-scale datasets. Therefore, many gradient-based methods have emerged recently, which use gradients to approximate the change of loss, such as using first-order Taylor expansion or Hessian matrix, to estimate the influence of samples. The most relevant work to ours is TracIn (Pruthi et al., 2020), which implements heuristic dynamic estimation through first-order gradient approximation and stored checkpoints. Unlike our work, the ultimate goal of TracIn is to estimate and filter out unlabeled samples in the training set through self-influence. Moreover, TracIn is only applicable to small-scale classification datasets, it is difficult to migrate to larger and complex tasks like segmentation, let alone handle nearly infinite generated data. Our work
+
+# Algorithm 1 Pipeline for copy-paste baseline
+
+Require: labeled real data  $\mathcal{R}$ , generated data  $\mathcal{G}$ , batch size  $B$ , number of iterations  $T$ , pretrained segmentation network  $f$  with parameters  $\theta$ , maximum number of paste instances  $K$  for each image
+
+1: for  $t = 1$  to  $T$  do
+
+2: Sample a batch of real data  $\mathcal{R}_b\in \mathcal{R}$
+
+3: for  $\mathbf{I}_r\in \mathcal{R}_b$  do
+
+4: Get a random number  $k$  from  $[0, K]$
+
+5: Sample  $k$  instances from  $\mathcal{G}$  in a class-balanced way to get  $\widehat{\mathbf{I}}_r$ ,  $\widehat{\mathbf{Y}}_r = \text{Copypaste}(\mathcal{G}_b, \mathbf{I}_r)$
+
+6: end for
+
+7: Train  $f$  on this augmented data  $\widehat{\mathcal{R}}_b$  and update  $\theta$
+
+8: end for
+
+9: return: Final segmentation network  $f$  with parameters  $\theta$
+
+succeeds in designing an automated pipeline for utilizing generated data to enhance downstream perception tasks.
+
+Most importantly, the above work is all done on relatively simple classification tasks, and only a few works have explored more complex perception tasks such as detection (Shrivastava et al., 2016; Liu et al., 2021) and segmentation (Jain and Grauman, 2016; Vezhnevets et al., 2012; Casanova et al., 2020), but they are all aimed at real data. Our work is the first to explore the generated data on the complex perception task of long-tail instance segmentation.
+
+# 3. Preliminary
+
+Formally, in generated data augmented instance segmentation tasks, we have a set of labeled real data  $\mathcal{R} = \{(\mathbf{I}_r,\mathbf{Y}_r)\}$  and a set of generated data (with noisy label)  $\mathcal{G} = \{(\mathbf{I}_g,\mathbf{Y}_g)\}$  where  $\mathbf{I}$  is the image and  $\mathbf{Y}$  is the label for instance segmentation. Our goal is to effectively utilize the existing data  $\mathcal{R}$  and  $\mathcal{G}$  to train a segmentation network  $f$  parameterized by  $\theta$  to achieve optimal performance on unseen test data  $\mathcal{U} = \{(\mathbf{I}_u,\mathbf{Y}_u)\}$ , that is, minimize the loss on the test set  $L_{\mathcal{U}}(\theta) = \sum_{(\mathbf{I}_u,\mathbf{Y}_u)\in \mathcal{U}}\ell (\mathbf{I}_u,\mathbf{Y}_u;\theta)$ , where  $\ell$  represents the loss of the segmentation network. The most direct way to utilize the two types of data is to perform joint training, and X-Paste proves that using the copy-paste (Ghiasi et al., 2021) method to add instances from  $\mathbf{I}_g$  to real images  $\mathbf{I}_r$  can achieve better results. Although our method is universal, we build it upon a copy-paste baseline. For the convenience of subsequent description, we record this operation as Copypaste such that  $\widehat{\mathbf{I}}_r$ ,  $\widehat{\mathbf{Y}}_r = \text{Copypaste}(\mathcal{G}_b,\mathbf{I}_r)$ , where  $\mathcal{G}_b\in \mathcal{G}$  is a subset sampled from  $\mathcal{G}$ ,  $\widehat{\mathbf{I}}_r$  and  $\widehat{\mathbf{Y}}_r$  represent the image and the label obtained after pasting the instance in  $\mathcal{G}_b$  to  $\mathbf{I}_r$ .
+
+As shown in Algorithm 1, it displays the overall process of our baseline, which does not consider the different impacts each sample could impose on the model. In other words,
+
+our aim is to identify a function,  $\phi(g, \theta)$ , capable of gauging the contribution of any given generated sample  $g \in \mathcal{G}$  to the current model  $f$ . Then, via this scoring mechanism, we can filter and retain the most helpful samples for the model and simultaneously discard those that are useless or even harmful to the model.
+
+# 4. Our method
+
+# 4.1. Estimation of Contribution in the Ideal Scenario
+
+Here we first provide the ideal estimation of the contribution of an independent sample. Assuming that a target test set  $\mathcal{U}$  is given, the contribution of  $g\in \mathcal{G}$  to  $f$  can be measured by calculating the change in the loss of  $f$  on  $\mathcal{U}$ , that is,
+
+$$
+\phi (g, \theta) = L _ {\mathcal {U}} (\theta) - L _ {\mathcal {U}} (\theta + \Delta \theta_ {g}) \tag {1}
+$$
+
+where  $\Delta \theta_{g}$  represents the one-step gradient update on  $g$ , that is,
+
+$$
+\Delta \theta_ {g} = - \alpha \nabla_ {\theta} \ell (\mathbf {I} _ {g}, \mathbf {Y} _ {g}; \theta) \tag {2}
+$$
+
+where  $\alpha$  represents the learning rate.
+
+Moreover, we can employ the classic first-order Taylor expansion to approximate  $\phi(g,\theta)$ , which has also been widely used in previous work (Pruthi et al., 2020; He et al., 2023). Note that, it is possible to use more sophisticated methods here, e.g., Koh and Liang (2017).
+
+Lemma 4.1. The loss of a network  $f$  on a dataset  $\mathcal{U}$  can be approximated by a first-order approximation:
+
+$$
+L _ {\mathcal {U}} (\theta + \Delta \theta) = L _ {\mathcal {U}} (\theta) + \Delta \theta^ {T} \nabla_ {\theta} L _ {\mathcal {U}} (\theta) + O (\Delta \theta^ {2}) (3)
+$$
+
+So the contribution of  $g$  to  $f$  can be approximated by:
+
+$$
+\phi (g, \theta) \approx \phi_ {a} (g, \theta) = \alpha \nabla_ {\theta} \ell (\mathbf {I} _ {g}, \mathbf {Y} _ {g}; \theta) ^ {T} \nabla_ {\theta} L _ {\mathcal {U}} (\theta) \tag {4}
+$$
+
+Remark 4.2. Advantages of first-order approximation: In the case of a fixed test set  $\mathcal{U}$ , it eliminates the need for multiple forward computations on  $\mathcal{U}$ , requiring only one forward computation and one backward propagation. so it brings the possibility expand the scale of  $\mathcal{U}$ . Besides, the fact that there is no need to update the weights makes our subsequent designs more flexible.
+
+Remark 4.3. (Test set) In real-world scenarios, the test set  $\mathcal{U}$  is typically unknown. A straightforward solution might be to reserve a part of the training set  $\mathcal{R}$  as an unseen test set. However, this approach poses issues for datasets with significant category imbalance, such as LVIS, where certain long-tailed categories only appear once in the training set. In light of this, we resort to a strategy that involves sampling a subset of  $\mathcal{R}$  to serve as the test set  $\mathcal{U}$  for each iteration. See Section 5.2.2 for more details.
+
+# 4.2. Batched Streaming Generative Active Learning
+
+While we have proposed an ideal estimation of contribution in the preceding section, it is not directly transferable to a real segmentation training process. The specific reasons are as follows:
+
+1. In the previous discussion, we posit that  $g \in \mathcal{G}$  is an independent sample, yet we actually paste multiple instances from  $\mathcal{G}$  to  $I_r$ . This results in reciprocal influences among these instances, as well as interactions with  $I_r$ .  
+2. We conduct batch data training, eliminating the possibility of estimating each instance individually, as this would provoke excessive computation.  
+3. Our model undergoes constant updates. Therefore, even the same sample's contribution to the model varies under different training stages. Furthermore, given the near-infinite data pool, the entire training process closely resembles a streaming process(Saran et al., 2023). After each data entry, we must decide whether to include this data in the present update.
+
+For the first and second points, we redefine the contribution function  $\phi(g,\theta)$  so that it can simultaneously consider the mutual influence between multiple instances and the real image  $I_r$ , and can also adapt to batch data training.
+
+Definition 4.4. Specifically, for a certain iteration  $t$ , we introduce a batch of data  $\mathcal{R}_b$  and the corresponding instances  $\mathcal{G}_b$  to be pasted. We will paste  $\mathcal{G}_b$  to each image in  $\mathcal{R}_b$  in a predefined random way to get  $\widehat{\mathcal{R}}_b = \mathcal{R}_b \oplus \mathcal{G}_b$ . Then our contribution function  $\phi(\mathcal{G}_b, \mathcal{R}_b, \theta)$  can be defined as
+
+$$
+\phi \left(\mathcal {G} _ {b}, \mathcal {R} _ {b}, \theta\right) = L _ {U} \left(\theta + \Delta \theta_ {\mathcal {R} _ {b}}\right) - L _ {U} \left(\theta + \Delta \theta_ {\widehat {\mathcal {R}} _ {b}}\right) \tag {5}
+$$
+
+where  $\Delta \theta_{\widehat{\mathcal{R}}_b}$  represents a one-step gradient update on  $\widehat{\mathcal{R}}_b$ :
+
+$$
+\Delta \theta_ {\widehat {\mathcal {R}} _ {b}} = - \alpha \nabla_ {\theta} L _ {\widehat {\mathcal {R}} _ {b}} (\theta) \tag {6}
+$$
+
+Likewise,  $\Delta \theta_{\mathcal{R}_b}$  represents a one-step gradient update on  $\mathcal{R}_b$
+
+In response to the third point, based on Definition 4.4, we can propose an algorithm called Batched Streaming Generative Active Learning (BSGAL), as shown in Algorithm 2.
+
+The basic idea is to calculate the loss and gradient of the model on  $\widehat{\mathcal{R}}_b$  and  $\mathcal{R}_b$  respectively and use the two updated models to calculate the loss on  $\mathcal{U}_b$ . Then measure the contribution of  $\mathcal{G}_b$  with the difference of two losses. Ultimately, we decide whether to accept this batch of generated data.
+
+As pointed out in Remark 4.2, there are some advantages of using one-order approximation. Building upon Lemma 4.1,
+
+Algorithm 2 Batched streaming generative active learning (BSGAL)  
+```txt
+Require: labeled real data  $\mathcal{R}$ , generated data  $\mathcal{G}$ , batch size  $B$ , number of iterations  $T$ , pretrained segmentation network  $f$  with parameters  $\theta$ , maximum number of paste instances  $K$  for each image, one step learning rate  $\alpha$ , contribution threshold  $\tau$
+```
+
+1: for  $t = 1$  to  $T$  do  
+2: Sample a batch of real data  $\mathcal{R}_b\in \mathcal{R}$  with batch size  $B_{accept}$ .  
+3: for  $(\mathbf{I}_r,\mathbf{Y}_r)\in \mathcal{R}_b$  do  
+4: Get a random number  $k$  from  $[0, K]$  
+5: Sample  $k$  instances in a class-balanced way from  $\mathcal{G}$  to get  $\hat{\mathbf{I}}_r$ ,  $\hat{\mathbf{Y}}_r = \text{Copypaste}(\mathcal{G}_b, \mathbf{I}_r)$  
+6: end for  
+7: Merge all the augmented image and label pairs to get  $\widehat{\mathcal{R}}_b = \mathcal{R}_b\oplus \mathcal{G}_b$  
+8: Calculate loss on  $\mathcal{R}_b$  and  $\widehat{\mathcal{R}}_b$  and get the gradient  $\nabla_{\theta}L_{\mathcal{R}_b}(\theta)$  and  $\nabla_{\theta}L_{\widehat{\mathcal{R}}_b}(\theta)$  
+9: Sample a batch of data  $\mathcal{U}_b\in \mathcal{R}$  as test set with batch size  $B_{test}$  
+10: Calculate the contribution  $\phi (\mathcal{G}_b,\mathcal{R}_b,\theta)$  using Equation (5)  
+11: if  $\phi (\mathcal{G}_b,\mathcal{R}_b,\theta) > \tau$  then  
+12: Train  $f$  on this augmented data  $\widehat{\mathcal{R}}_b$  and update  $\theta$  
+13: else  
+14: Train  $f$  on this real data  $\mathcal{R}_b$  and update  $\theta$  
+15: end if  
+16: end for  
+17: return: Final segmentation network  $f$  with parameters  $\theta$
+
+we can further approximate  $\phi (\mathcal{G}_b,\mathcal{R}_b,\theta)$  as
+
+$$
+\phi \left(\mathcal {G} _ {b}, \mathcal {R} _ {b}, \theta\right) \approx \alpha \nabla_ {\theta} \left(L _ {\hat {\mathcal {R}} _ {b}} (\theta) - L _ {\mathcal {R} _ {b}} (\theta)\right) ^ {T} \nabla_ {\theta} L _ {U} (\theta) \tag {7}
+$$
+
+So the Algorithm 2 can be further simplified by using Equation (7) in Line 10.
+
+Remark 4.5. (Batch size) It is crucial to note that three distinct batch sizes here, one is the batch size  $B_{train}$  for model update, one is the batch size  $B_{accept}$  for calculating the data contribution to determine whether to accept, and the other is the batch size  $B_{test}$  of the test set formed by sampling. These three are not necessarily identical. Ideally, a smaller  $B_{accept}$  is preferable – when the batch size is reduced to 1, it allows for a more accurate per-image estimation. In the actual implementation process, we execute this algorithm independently on each GPU, except for summing up all losses when updating the model. Consequently, the batch size  $B_{train} = B_{accept} \times \#GPU$ .
+
+Usually, due to the limitation of GPU memory,  $\mathcal{U}_b$  can only take a very small batch size  $B_{test}$ , which will lead to the instability and inaccuracy of the estimation of  $\phi (\mathcal{G}_b,\mathcal{R}_b,\theta)$ . Thanks to the one-order approximation, we can consider
+
+Algorithm 3 Contribution estimation based on gradient cache  
+```latex
+Require: gradient  $\nabla_{\theta}L_{\mathcal{R}_b}(\theta)$ , gradient  $\nabla_{\theta}L_{\widehat{\mathcal{R}}_b}(\theta)$ , momentum coefficient  $\beta$ , current iteration  $t$ , model parameters  $\theta$ , sampled test set  $\mathcal{U}_b$
+```
+
+1: Calculate loss on  $\mathcal{U}_b$  to get  $L_{\mathcal{U}_b}$  and get the grad  $\nabla_{\theta}L_{\mathcal{U}_b}(\theta)$  
+2: if  $t == 1$  then  
+3: Initialize grad cache  $\mathbf{C} = \nabla_{\theta}L_{\mathcal{U}_b}(\theta)$  
+4: else  
+5: Update the grad cache with the grad of current batch  $\mathbf{C} = \beta \mathbf{C} + (1 - \beta)\nabla_{\theta}L_{\mathcal{U}_b}(\theta)$  
+6: end if  
+7: Calculate the contribution of  $\mathcal{G}_b$  as  $\phi (\mathcal{G}_b,\mathcal{R}_b,\theta) = \alpha \nabla_{\theta}(L_{\widehat{\mathcal{R}}_h}(\theta) - L_{\mathcal{R}_b}(\theta))^T\mathbf{C}$  
+8: return:  $\phi(\mathcal{G}_b, \mathcal{R}_b, \theta)$
+
+more test data when calculating  $\phi (\mathcal{G}_b,\mathcal{R}_b,\theta)$ . Specifically, we can keep a grad cache to record the grad obtained on other batches in the previous iterations. Every time we get the current batch  $\nabla_{\theta}L_{\mathcal{U}_b}(\theta)$ , we will update the grad cache in a momentum way. Then when calculating  $\phi (\mathcal{G}_b,\mathcal{R}_b,\theta)$  we use the grad in the grad cache to replace the grad of the current batch, which is equivalent to expanding the scale of  $\mathcal{U}_b$ , thereby achieving a more stable estimation.
+
+However, this batch size  $B_{test}$  is not the bigger the better. Our subsequent experiments prove that if we use the grad cache to approximate the entire training set  $\mathcal{R}$  as  $\mathcal{U}_b$ , it will lead to a decrease in the performance of the model. We believe that this is caused by overfitting. When we approximate the entire training set  $\mathcal{R}$  as  $\mathcal{U}_b$ , we can only screen out samples similar to the training set  $\mathcal{R}$ , thereby inhibiting the diversity of the data. Diversity is also an important factor considered in many active learning works. We believe that the design here parallels the idea of diversity-based active learning (Sener and Savarese, 2018; Geifman and El-Yaniv, 2017) using a core-set, with the only difference being that core-set methods mine the most representative samples from unlabeled data, while we randomly sample some from the labeled real data to act as a kind of "core-set". In our task, the diversity of generated data is particularly crucial for long-tailed categories, which can effectively bridge the gap between scarce training data and real-world distribution. Therefore, we need to balance the stability of the estimation and the diversity of the data. Thus, we use the momentum method to update the grad cache, which can ensure the stability of the estimation to a certain extent, and at the same time will not inhibit the diversity of the data.
+
+The modified contribution estimation algorithm for the final BSGAL is shown in Algorithm 3.
+
+Remark 4.6. (Extension to offline learning) Offline learning needs to satisfy the following two assumptions: 1. The generated data is limited or small in scale. 2. The model pa
+
+rameters will not change significantly during the fine-tuning process. Our method can also be easily extended to the case of offline learning. Specifically, we only need to use a fixed model, and then use the entire dataset  $\mathcal{R}$  as the test set  $\mathcal{U}$ . Forward and backward once on the entire dataset, we can get the gradient  $\nabla_{\theta}L_{\mathcal{U}}(\theta)$  for the entire dataset. Then we can use this gradient to calculate the contribution of each generated sample. Using Equation (4), we can estimate the contribution of each generated sample.
+
+# 5. Experiments
+
+First, we perform some analytical experiments in an offline setting (as discussed in Remark 4.6) to verify the feasibility of our method and also to facilitate a better understanding of our method for readers. Then, we conduct the main experiments under the online setting, compared with our baseline. Key ablation studies are also conducted to substantiate the efficiency of our method. Detailed information about the implementation can be found in Appendix A.
+
+# 5.1. Offline Setting
+
+# 5.1.1. CIFAR-10
+
+In this section, we conduct a toy experiment on CIFAR-10 (Krizhevsky et al., 2009) to verify our method. The original CIFAR-10 dataset comprises five splits within the training set, each containing 10,000 images, and a test set equally housing 10,000 images. We use the first split in the training set as our training set  $\mathcal{R}$ , and the remaining 4 splits are added with noise of different scales (40,100,200,400) to simulate the generated data  $\mathcal{G}$ . We use the model trained only on the first split to perform offline mining and then use 1000 images in the first split as test set  $\mathcal{U}$ . By estimating the contribution of each sample, we can draw the distribution of the contribution of samples on different splits.
+
+As shown in Figure 2, it is observable that with the escalating scale of noise, the distribution of contributions progressively shifts to the left. This indicates that excessive noise tends to negatively impact the model. Note that the split with a noise of 0 is our training set, so we can see that the contribution values of these samples are concentrated around zero. In other words, these samples can no longer bring positive effects to the model because they have been fully utilized in previous training. This observation is consistent with some previous active learning work (Cai et al., 2013; Ash et al., 2021; Saran et al., 2023), where they also estimate the amount of information or the difficulty level of samples through gradients. However, they do not consider the positive or negative contributions but only select samples with larger absolute values. We further conduct quantitative experiments, as shown in Table 1, to prove that using our method to select data can effectively improve the
+
+![](images/a7a5ba63e582fd94a75d9e415978694a77be93fa3ab8c90534e4726bb4e583da.jpg)  
+Figure 2. The distribution of contributions under different noise scales.
+
+Table 1. Using our method to select samples brings improvement to the model.  
+
+<table><tr><td>Training set</td><td>Accuracy (avg ± std)</td></tr><tr><td>only R</td><td>86.28 ± 0.55</td></tr><tr><td>R + all G</td><td>86.71 ± 0.21</td></tr><tr><td>R + selected G</td><td>87.61 ± 0.20</td></tr></table>
+
+performance of the model.
+
+# 5.1.2. LVIS
+
+We further carry out offline experiments on the generated data of LVIS categories, qualitatively examining the efficacy of our method. We calculate the gradient  $\nabla_{\theta}L_{\mathcal{U}}(\theta)$  preemptively on the LVIS training set with a trained model.
+
+This gradient then serves to estimate each instance's contribution. Subsequently, we rank these instances in decreasing order of their contribution, facilitating per-image analysis. As an illustrative example, we use a 'bun' category from the LVIS, because we discover that Stable Diffusion does not perform optimally within this category, often leading to confusion between 'bun' and 'bunny', thereby resulting in the generation of ambiguous data. As depicted in Figure 3, it can be observed that the instances having the most significant contributions are nearly unambiguous, whereas the instances with minimal contributions are mostly incorrect, resulting in rabbit images being generated. Therefore, through our method, we can effectively filter out the generated data with ambiguity.
+
+To verify the indispensability of online learning, we first use the offline method to filter the generated data for training and compare it with our baseline. As shown in Figure 4, the offline method can only bring a slight improvement to the
+
+Table 2. Main results on LVIS. "  $^+$  CLIP" means using CLIP to filter the generated data.  
+
+<table><tr><td>Method</td><td>Backbone</td><td>APbox</td><td>APmask</td><td>APrbox</td><td>APrmask</td><td>APcbox</td><td>APcmask</td><td>APfbox</td><td>APfmask</td></tr><tr><td>Baseline</td><td>R50</td><td>34.20</td><td>30.39</td><td>24.33</td><td>22.21</td><td>33.23</td><td>29.57</td><td>39.63</td><td>34.89</td></tr><tr><td>Baseline + CLIP</td><td>R50</td><td>34.35</td><td>30.70</td><td>25.99</td><td>24.38</td><td>32.83</td><td>29.41</td><td>39.71</td><td>34.92</td></tr><tr><td>Ours</td><td>R50</td><td>35.40</td><td>31.56</td><td>27.95</td><td>25.43</td><td>34.14</td><td>30.56</td><td>40.07</td><td>35.37</td></tr><tr><td>Baseline</td><td>Swin-L</td><td>49.57</td><td>43.85</td><td>44.87</td><td>39.66</td><td>49.74</td><td>44.64</td><td>51.46</td><td>44.82</td></tr><tr><td>Baseline + CLIP</td><td>Swin-L</td><td>49.80</td><td>44.51</td><td>45.28</td><td>40.62</td><td>49.33</td><td>44.96</td><td>52.30</td><td>45.72</td></tr><tr><td>Ours</td><td>Swin-L</td><td>50.47</td><td>44.85</td><td>47.55</td><td>42.37</td><td>50.43</td><td>45.47</td><td>51.79</td><td>45.26</td></tr></table>
+
+![](images/2cf773abf945918bf7f58b0effa176b9869a638d772d0f4090837b6e48aa3cb5.jpg)  
+Figure 3. The best and worst samples found using our contribution estimation function for a LVIS class 'bun'.  
+Figure 4. Performance of the model under different iterations.
+
+final model performance. In addition, in the early stage of model training, this performance improvement is still quite obvious, but with the training process, this performance improvement gradually diminishes. We conjecture that this trend is likely due to the offline contribution estimation's reliance on the initial model, and as the model undergoes training, the parameters change significantly, which leads to the inaccuracy of the offline contribution estimation. Therefore, the necessity arises for online contribution estimation.
+
+# 5.2. Online Setting
+
+# 5.2.1. MAIN RESULTS
+
+To validate the effectiveness of our method in handling long-tailed segmentation tasks, we perform experiments on the LVIS (Gupta et al., 2019) dataset. A strong baseline — X-Paste (Zhao et al., 2023), is compared with our method. We further examine the impact of the usage (or non-usage) of CLIP (Radford et al., 2021), as mentioned in their paper, for filtering generated data. Given that X-Paste does not open source the generated data used, we have re-implemented the data generation pipeline and generated thousands of images
+
+![](images/cde8cec0fdb697ed84bea93a5259227a2501fd7784ec680cdc614553e3ad1412.jpg)
+
+for each category of LVIS. The specific generation details are shown in the appendix. In addition, original X-Paste also uses retrieval to obtain extra real data and uses Copy-Paste to augment the real training data. Since the primary focus of our methodology is on the selection of generated data, to mitigate the influences from those additional factors, we choose to refrain from using these tricks.
+
+As shown in Table 2, following X-Paste, the segmentation architecture we used is CenterNet2 (Zhou et al., 2021), and we test two different backbones, ResNet50 (He et al., 2016) and Swin-L (Liu et al., 2022). It can be observed that the impact of filtration via CLIP is rather subtle, with the  $AP^{box}$  witnessing an increment of merely  $0.1 \sim 0.2$  points. Furthermore, we also find that CLIP filtering shows a more significant improvement on  $AP^{mask}$ . Conversely, our method continues to deliver substantial improvements across all categories when contrasted with the use of CLIP filtration. This improvement is especially notable for long-tailed categories,  $AP_{r}^{box}$  increases by  $2 \sim 2.3$ , and  $AP_{r}^{mask}$  increases by  $1 \sim 1.7$ .
+
+# 5.2.2. ABLATION
+
+Table 3. Comparison of different  ${L}_{\text{test }}$  on contribution estimation.  
+
+<table><tr><td>Loss</td><td>APbox</td><td>APmask</td><td>APrbox</td><td>APrmask</td></tr><tr><td>cls</td><td>35.24</td><td>31.49</td><td>28.14</td><td>25.74</td></tr><tr><td>cls stage0</td><td>34.94</td><td>31.23</td><td>26.34</td><td>24.27</td></tr><tr><td>all</td><td>34.98</td><td>30.94</td><td>26.91</td><td>23.87</td></tr></table>
+
+Table 4. Comparison of different algorithms we designed.  
+
+<table><tr><td>Method</td><td>APbox</td><td>APmask</td><td>APrbox</td><td>APrmask</td><td>APcbox</td><td>APcmask</td><td>APfbox</td><td>APfmask</td></tr><tr><td>Loss estimate (Algorithm 2)</td><td>35.11</td><td>31.29</td><td>27.47</td><td>25.10</td><td>33.55</td><td>30.14</td><td>40.20</td><td>35.30</td></tr><tr><td>Grad estimate (Equation (7))</td><td>35.24</td><td>31.49</td><td>28.14</td><td>25.74</td><td>33.67</td><td>30.31</td><td>40.12</td><td>35.34</td></tr><tr><td>Grad cache (Algorithm 3)</td><td>35.40</td><td>31.56</td><td>27.95</td><td>25.43</td><td>34.14</td><td>30.56</td><td>40.07</td><td>35.37</td></tr><tr><td>Grad cache (global)</td><td>35.07</td><td>31.38</td><td>26.84</td><td>24.82</td><td>33.74</td><td>30.36</td><td>40.18</td><td>35.40</td></tr></table>
+
+Loss. The algorithm in Section 4.2 deploys two distinct types of loss. First is the loss used to train and finally update the model,  $L_{train}$ , which is subject to the original network design. In our case, We utilize the loss in CenterNet2 (Zhou et al., 2021). For the instance segmentation task, the actual  $L_{train}$  is composed of multiple tasks and multiple stages of Loss. Specifically, it can be written in the following form<sup>1</sup>:
+
+$$
+L _ {\text {t r a i n}} = \sum_ {i = 1} ^ {S} \left(L _ {\text {c l s}} ^ {i} + L _ {\text {b o x}} ^ {i} + L _ {\text {m a s k}} ^ {i}\right) \tag {8}
+$$
+
+where  $S$  represents the number of stages,  $L_{cls}^{i}$ ,  $L_{box}^{i}$ ,  $L_{mask}^{i}$  represent the classification loss, regression loss, and segmentation loss of the  $i$ -th stage, respectively.
+
+Second is the test loss  $L_{test}$  used to estimate the contribution, which does not need to be consistent with  $L_{train}$ . Considering that the main purpose of introducing generated data is to improve classification on long-tailed categories,  $L_{cls}$  may play a dominant role. Thus, we compare the effects of different  $L_{test}$  on contribution estimation in Table 3.
+
+In this experiment, we are based on Equation (7), where "cls" denotes using all stages of  $L_{cls}$  "cls stage0" denotes only using the first stage of  $L_{cls}$ , and "all" denotes using all Loss in  $L_{train}$ . The findings indicate that if only the  $L_{cls}$  of first stage is used, the efficiency of contribution estimation diminishes. Additionally, utilizing all losses within  $L_{train}$  does not surpass the performance of solely exploiting  $L_{cls}$ , which also verifies our assumption that  $L_{cls}$  dominates in contribution estimation, and other losses may cause interference. Therefore, our final decision is to rely singularly on  $L_{cls}$  for estimating contribution.
+
+Contribution estimation. We are interested in whether the three algorithms proposed in Section 4.2 are effective. Therefore, we conduct comparative experiments here, and the specific results are presented in Table 4.
+
+We observe that there's not a significant difference in performance among the three algorithms, all of which demonstrated efficacy when compared to the baseline. Overall, Algorithm 3 has the best effect, which is also the algorithm we finally adopted. Hence, our proposed method of using a larger test set  $\mathcal{U}_b$ , ensuring a smoother and more stable contribution estimation by updating the grad cache with
+
+momentum, is proven effective. Compared with Grad estimate and Loss estimate, it can be proved that the first-order approximation will not bring significant performance loss.
+
+As discussed in Section 4.2, there exists a trade-off between the stability of the estimation and the diversity of the filtered data. To verify this, we additionally added the fourth experiment, where we use global average pooling to estimate the contribution. Specifically, We modify the original update method  $\mathbf{C} = \beta \mathbf{C} + (1 - \beta)\nabla_{\theta}L_{\mathcal{U}_b}(\theta)$  to be related to the current iteration  $t$ ,  $\mathbf{C} = \frac{t - 1}{t}\mathbf{C} + \frac{1}{t}\nabla_{\theta}L_{\mathcal{U}_b}(\theta)$ . That is,  $\mathbf{C} = \frac{1}{t}\sum_{i = 1}^{t}\nabla_{\theta_i}L_{\mathcal{U}_b^i}(\theta_i)$ .
+
+However, this modification incurs a slight performance drop. We believe that this is due to enforcing the generated data to align with the distribution of the entire training set and suppressing the diversity of the data. For long-tailed categories with relatively few real data, this diversity is quite significant. That's why using global average pooling,  $AP_{r}^{box}$  and  $AP_{r}^{mask}$  have a significant drop, while  $AP_{f}^{mask}$  exhibits an improvement. Correspondingly, Grad estimate, which solely uses the current batch to estimate the contribution of the test set, is most conducive to ensuring data diversity, so the performance of  $AP_{r}^{box}$  and  $AP_{r}^{mask}$  is the best.
+
+Table 5. Comparison of different sampling strategies for  ${\mathcal{U}}_{b}$  .  
+
+<table><tr><td>Strategy</td><td>APbox</td><td>APmask</td><td>APrbox</td><td>APrmask</td></tr><tr><td>all classes</td><td>35.21</td><td>31.38</td><td>26.75</td><td>24.21</td></tr><tr><td>pasted classes</td><td>35.40</td><td>31.56</td><td>27.95</td><td>25.43</td></tr><tr><td>all images</td><td>35.15</td><td>31.26</td><td>26.59</td><td>23.75</td></tr></table>
+
+Sampling strategy. We compare three different sampling test set strategies: 1. Sample from all classes: Sample uniformly from all categories, and then sample uniformly from the image pool corresponding to the sampled category. 2. Sample from pasted classes: Sample uniformly from the categories in the generated data  $\mathcal{G}_b$  used in this batch, and then sample uniformly from the image pool corresponding to the sampled category. 3. Sample from all images: Sample uniformly from all image pools.
+
+As indicated in Table 5, it is evident that the approach of uniformly sampling from pasted categories delivers the most effective performance, thus we finalize on this sampling strategy. Especially in  $AP_{r}^{box}$  and  $AP_{r}^{mask}$ , the improvement of this sampling strategy is the most obvious. Compared with sampling from all images, sampling from class
+
+Table 6. Comparison of random batch-level dropout and our method.  
+
+<table><tr><td>Method</td><td>APbox</td><td>APmask</td><td>APrbox</td><td>APrmask</td></tr><tr><td>Random Dropout</td><td>34.73</td><td>30.96</td><td>25.05</td><td>22.69</td></tr><tr><td>Our method</td><td>35.40</td><td>31.56</td><td>27.95</td><td>25.43</td></tr></table>
+
+can better ensure the balance of categories, thereby enhancing the impact on rare categories. As for the comparison with sampling from all classes, sampling from pasted classes is more directional, so the estimation of the contribution is more accurate, leading to a boost in overall performance.
+
+Random batch-level dropout. Our algorithm is essentially to accept or reject the generated data on a batch-by-batch basis. Therefore, when the contribution evaluation of the data is completely invalid, our algorithm degenerates into a random batch-level dropout. To verify that it is not this random dropout that brings performance improvement, we conduct a random batch-level Dropout experiment with the same acceptance rate. Table 6 shows that although random dropout will also bring a slight improvement, compared with our method, there is still a very obvious gap, which shows that the improvement brought by our method is not entirely due to random dropout.
+
+# 6. Conclusion
+
+In this paper, we propose a new problem, how to design an effective method to realize the effective screening and utilization of generated data, to further improve the performance of downstream perception tasks. To address this problem, we propose a gradient-based generated data contribution estimation method and embed it into the actual training process. We design a complete pipeline that can automatically generate data to improve the performance of downstream perception tasks. Experiments prove that our method can achieve better performance than unfiltered or CLIP-filtered methods on long-tailed segmentation tasks.
+
+# Impact Statement
+
+Our goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
+# Acknowledgements
+
+This work was supported by the National Key R&D Program of China (No. 2022ZD0118700).
+
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+
+# A. Implementation Details
+
+# A.1. Dataset
+
+We choose LVIS (Gupta et al., 2019) as the dataset for our experiments. LVIS is a large-scale instance segmentation dataset, comprising approximately 160,000 images with over 2 million high-quality instance segmentation annotations across 1203 real-world categories. The dataset is further divided into three categories: rare, common, and frequent, based on their occurrence across images. Instances marked as 'rare' appear in 1-10 images, 'common' instances appear in 11-100 images, whereas 'frequent' instances appear in more than 100 images. The overall dataset exhibits a long-tail distribution, closely resembling the data distribution in the real world, and is widely applied under multiple settings, including few-shot segmentation (Liu et al., 2023) and open-world segmentation (Zhu et al., 2023; Wang et al., 2022). Therefore, we believe that selecting LVIS allows for a better reflection of the model's performance in real-world scenarios. We use the official LVIS dataset splits, with about 100,000 images in the training set and 20,000 images in the validation set.
+
+# A.2. Data Generation
+
+Our data generation and annotation process is consistent with Zhao et al. (2023), and we briefly introduce it here. We first use StableDiffusion V1.5 (Rombach et al., 2022a) (SD) as the generative model. For the 1203 categories in LVIS (Gupta et al., 2019), we generate 1000 images per category, with image resolution  $512 \times 512$ . The prompt template for generation is "a photo of a single {CATEGORY_NAME}" We use U2Net (Qin et al., 2020), SelfReformer (Yun and Lin, 2022), UFO (Su et al., 2023), and CLIPseg (Lüddecke and Ecker, 2022) respectively to annotate the raw generative images, and select the mask with the highest CLIP score as the final annotation. To ensure data quality, images with CLIP scores below 0.21 are filtered out as low-quality images. During training, we also employ the instance paste strategy provided by Zhao et al. (2023) for data augmentation. For each instance, we randomly resize it to match the distribution of its category in the training set. The maximum number of pasted instances per image is set to 20.
+
+In addition, to further expand the diversity of generated data and make our research more universal, we also use other generative models, including DeepFloyd-IF (Shonenkov et al., 2023) (IF) and Perfusion (Tewel et al., 2023) (PER), with 500 images per category per model. For IF, we use the pre-trained model provided by the author, and the generated images are the output of Stage II, with a resolution of  $256 \times 256$ . For PER, the base model we use is StableDiffusion V1.5. For each category, we fine-tune the model using the images cropped from the training set, with 400 fine-tuning steps. We use the fine-tuned model to generate images.
+
+Table 7. Comparison of different generated data.  
+
+<table><tr><td>Method</td><td>Generated Data</td><td>APbox</td><td>APmask</td><td>APrbox</td><td>APrmask</td><td>APcbox</td><td>APcmask</td><td>APfbox</td><td>APfmask</td></tr><tr><td>Baseline</td><td>SD</td><td>34.00</td><td>30.33</td><td>24.48</td><td>22.65</td><td>32.71</td><td>29.27</td><td>39.62</td><td>34.89</td></tr><tr><td>Baseline</td><td>SD + IF</td><td>34.15</td><td>30.39</td><td>26.12</td><td>23.76</td><td>32.40</td><td>28.97</td><td>39.62</td><td>34.89</td></tr><tr><td>Baseline</td><td>SD + IF + Per</td><td>34.20</td><td>30.39</td><td>24.33</td><td>22.21</td><td>33.23</td><td>29.57</td><td>39.63</td><td>34.89</td></tr><tr><td>BSGAL</td><td>SD</td><td>34.82</td><td>31.21</td><td>26.76</td><td>24.84</td><td>33.28</td><td>30.01</td><td>40.08</td><td>35.34</td></tr><tr><td>BSGAL</td><td>SD + IF</td><td>35.13</td><td>31.34</td><td>26.83</td><td>24.32</td><td>33.92</td><td>30.57</td><td>40.13</td><td>35.29</td></tr><tr><td>BSGAL</td><td>SD + IF + Per</td><td>35.40</td><td>31.56</td><td>27.95</td><td>25.43</td><td>34.14</td><td>30.56</td><td>40.07</td><td>35.37</td></tr></table>
+
+We also explore the effect of using different generated data on the model performance (see Table 7). We can see that based on the original StableDiffusion V1.5, using other generative models can bring some performance improvement, but this improvement is not obvious. Specifically, for specific frequency categories, we found that IF has a more significant improvement for rare categories, while PER has a more significant improvement for common categories. This is likely because IF data is more diverse, while PER data is more consistent with the distribution of the training set. Considering that the overall performance has been improved to a certain extent, we finally adopt the generated data of  $\mathrm{SD} + \mathrm{IF} + \mathrm{PER}$  for subsequent experiments.
+
+# A.3. Model Training
+
+Follow Zhao et al. (2023), We use CenterNet2 (Zhou et al., 2021) as our segmentation model, with ResNet-50 (He et al., 2016) or Swin-L (Liu et al., 2022) as the backbone. For ResNet-50, the maximum training iteration is set to 90,000 and the model is initialized with weights first pretrained on ImageNet-22k then finetuned on LVIS (Gupta et al., 2019), as Zhao
+
+![](images/38edc23e4e6fcb14eed55529878b63b47ab55f31586f3256eca02de2959ef887.jpg)  
+Figure 5. Model performances when using different amount of generated data.
+
+et al. (2023) did. And we use 4 Nvidia 4090 GPUs with a batch size of 16 during training. As for Swin-L, the maximum training iteration is set to 180,000 and the model is initialized with weights pretrained on ImageNet-22k, since our early experiments show that this initialization can bring a slight improvement compared to the weights trained with LVIS. And we use 4 Nvidia A100 GPUs with a batch size of 16 for training. Besides, due to the large number of parameters of Swin-L, the additional memory occupied by saving the gradient is large, so we actually use the algorithm in Algorithm 2.
+
+The other unspecified parameters also follow the same settings as X-Paste (Zhao et al., 2023), such as the AdamW (Loshchilov and Hutter, 2017) optimizer with an initial learning rate of  $1\mathrm{e} - 4$
+
+# A.4. Data Amount
+
+In this work, we have generated over 2 million images. Figure 5 shows the model performances when using different amount of generated data(1%,10%,40%,70%,100%). Overall, as the amount of generated data increases, the performance of the model also improves, but there is also some fluctuation. Our method is always better than the baseline, which proves the effectiveness and robustness of our method.
+
+# A.5. Contribution Estimation
+
+As mentioned in Section 4.2, we use  $\nabla_{\theta}(L_{\widehat{\mathcal{R}}_b}(\theta) - L_{\mathcal{R}_b}(\theta))^T\nabla_{\theta}L_{\mathcal{U}_b}(\theta)$  to estimate the contribution of  $\mathcal{G}_b$
+
+Here we actually consider both the direction and the magnitude of the gradient. It is worth mentioning that many previous works actually mainly consider the magnitude of the gradient, for example, the data with large gradient magnitude has more information and should be annotated (Cai et al., 2013) or the wrong outlier data should be filtered (Pruthi et al., 2020).
+
+If we only consider the direction, it is equivalent to normalizing each gradient first, and then calculating, then our calculation formula becomes  $\mathcal{G}_b = \frac{\nabla_\theta(L_{\widehat{\mathcal{R}}_b}(\theta) - L_{\mathcal{R}_b}(\theta))^T\nabla_\theta L_{\mathcal{U}_b}(\theta)}{\|\nabla_\theta(L_{\widehat{\mathcal{R}}_b}(\theta) - L_{\mathcal{R}_b}(\theta))\|_2\|\nabla_\theta L_{\mathcal{U}_b}(\theta)\|_2}$ .
+
+Thus, we essentially calculate the cosine similarity. Then we conducted an experimental comparison, as shown in Table 8,
+
+Table 8. Comparison of using grad normalization or not.  
+
+<table><tr><td>Normalize</td><td>APbox</td><td>APmask</td><td>APrbox</td><td>APrmask</td><td>APcbox</td><td>APcmask</td><td>APfbox</td><td>APfmask</td></tr><tr><td>×</td><td>35.05</td><td>31.27</td><td>26.78</td><td>24.34</td><td>33.79</td><td>30.32</td><td>40.10</td><td>35.39</td></tr><tr><td>√</td><td>35.40</td><td>31.56</td><td>27.95</td><td>25.43</td><td>34.14</td><td>30.56</td><td>40.07</td><td>35.37</td></tr></table>
+
+![](images/7c51735fc369a366bae554993625a8764257f1db74523e522288c0840a6d0c93.jpg)  
+Figure 6. Illustration of noisy images exhibiting various noise scales and categories. Each row, from top to bottom, signifies different noise levels, specifically 0, 40, 100, 200, and 400, respectively. All images are sourced from the CIFAR-10 dataset.
+
+we can see that if we normalize the gradient, our method will have a certain improvement. In addition, since we need to keep two different thresholds, it is difficult to ensure the consistency of the acceptance rate. So we adopt a dynamic threshold strategy, pre-set an acceptance rate, maintain a queue to save the contribution of the previous iter, and then dynamically adjust the threshold according to the queue, so that the acceptance rate stays at the pre-set acceptance rate.
+
+# A.6. Toy Experiment
+
+The following are the specific experimental settings implemented on CIFAR-10: We employed a simple ResNet18 as the baseline model and conducted training over 200 epochs, and the accuracy after training on the original training set is  $93.02\%$ . The learning rate is set at 0.1, utilizing the SGD optimizer. A momentum of 0.9 is in effect, with a weight decay of 5e-4. We use a cosine annealing learning rate scheduler. The constructed noisy images are depicted in Figure 6. A decline in image quality is observed as the noise level escalates. Notably, when the noise level reaches 200, the images become significantly challenging to identify. For Table 1, we use Split1 as  $R$ , while  $G$  consists of 'Split2 + Noise40', 'Split3 + Noise100', 'Split4 + Noise200',
+
+# A.7. A Simplification Only Forward Once
+
+In Section 4.2, we actually need one more forward on  $\mathcal{R}_b$  compared to our baseline. However, we can simplify it to only one forward. The specific reason is that as mentioned in Table 3, we only use the classification loss,  $L_{cls}$  this loss is actually the sum of the cross entropy loss of each instance, and whether this instance is generated or real is known during the training process. so the loss can be further distangled as  $L_{cls} = L_{real} + L_{gen} + L_{neg}$ , where  $L_{real}$  is the loss of real instances,  $L_{gen}$  is the loss of generated instances, and  $L_{neg}$  is the loss of negative instances. Consequently, we can use  $\nabla_{\theta}L_{gen}$  to replace  $\nabla_{\theta}(L_{\widehat{\mathcal{R}}_b}(\theta) - L_{\mathcal{R}_b}(\theta))$
+
+# B. More ablations
+
+Momentum Coefficient  $\beta$ . In Algorithm 3, we introduce a momentum coefficient  $\beta$  to update the grad cache. Here we explore the effect of different  $\beta$  on the model performance. A larger beta signifies a greater focus on global information, while a smaller beta indicates a higher attention to the current test batch  $\mathcal{U}_b$ . Detailed results are presented in Table 9. Observations suggest that when  $\beta$  is 0.1, the performance is the best, which is also the  $\beta$  we finally adopted.
+
+Table 9. Comparison of different  $\beta$  for updating grad cache.  
+
+<table><tr><td>β</td><td>APbox</td><td>APmask</td><td>APrbox</td><td>APrmask</td></tr><tr><td>0.05</td><td>35.14</td><td>31.24</td><td>27.24</td><td>24.50</td></tr><tr><td>0.10</td><td>35.40</td><td>31.56</td><td>27.95</td><td>25.43</td></tr><tr><td>0.30</td><td>34.84</td><td>31.18</td><td>26.12</td><td>24.32</td></tr><tr><td>0.50</td><td>34.87</td><td>31.03</td><td>25.80</td><td>22.88</td></tr><tr><td>0.80</td><td>34.50</td><td>30.72</td><td>24.37</td><td>21.89</td></tr></table>
+
+Contribution threshold  $\tau$ . In Algorithm 3, we incorporate a contribution threshold  $\tau$ , intended for filtering the produced data. Here we investigate the impact of varying values of  $\tau$  on the model's performance. The larger  $\tau$  implies a stricter filtration of the generated data, while the smaller  $\tau$  signifies a looser filtering of the generated data. The specific results are shown in Table 10. We can see the performance is optimal when  $\tau$  equals -0.05, which is also the  $\tau$  we eventually settle on for our final model.
+
+Table 10. Comparison of different  $\tau$  for filtering generated data.  
+
+<table><tr><td>τ</td><td>APbox</td><td>APmask</td><td>APBoxr</td><td>APrmask</td></tr><tr><td>-0.10</td><td>34.68</td><td>30.80</td><td>26.60</td><td>24.40</td></tr><tr><td>-0.05</td><td>35.40</td><td>31.56</td><td>27.95</td><td>25.43</td></tr><tr><td>0.00</td><td>34.72</td><td>30.98</td><td>24.96</td><td>22.79</td></tr><tr><td>0.05</td><td>34.29</td><td>30.55</td><td>23.75</td><td>21.71</td></tr></table>
+
+Online learning vs. Offline learning We compare online learning and offline learning under different iterations. The result is shown in Figure 9.
+
+# C. Discussion
+
+# C.1. Comparing with existing methods
+
+Table 11. Comparing with existing methods  
+
+<table><tr><td>Method</td><td>Data Scale</td><td>Downstream</td><td>Cost</td><td>Quality</td><td>Domain Diff</td></tr><tr><td>Traditional Active Learning</td><td>Limited</td><td>✓</td><td>High</td><td>✓</td><td>✓</td></tr><tr><td>Generated Data Filtering Methods</td><td>Unlimited</td><td>×</td><td>Low</td><td>×</td><td>×</td></tr><tr><td>Generative Active Learning</td><td>Unlimited</td><td>✓</td><td>Low</td><td>×</td><td>✓</td></tr></table>
+
+We've drawn the Table 11, analyzing our setting compared to previous active learning or generative data filtration methods. We've conducted analysis from aspects of data scale, whether it's oriented towards downstream tasks, label quality, labeling costs, and whether there exists domain difference (between generated and real data).
+
+# C.2. Analysis of the computational cost
+
+We recorded the training duration and GPU memory usage for training 90,000 iterations with 4 Nvidia 4090 GPUs. It can be observed that our method based on Grad cache increases the GPU memory usage compared to Loss estimate, but it significantly reduces the training time. Compared with our Baseline, the additional time and memory overheads are within an acceptable range.
+
+Table 12. Analysis of the computational cost of different algorithms  
+
+<table><tr><td>Methods</td><td>Total Time</td><td>Max Memory per GPU</td></tr><tr><td>Baseline</td><td>17h</td><td>6534M</td></tr><tr><td>Loss estimate (Algorithm 2)</td><td>31h</td><td>7114M</td></tr><tr><td>Grad cache (Algorithm 3)</td><td>21h</td><td>9836M</td></tr></table>
+
+# C.3. Future work
+
+We hope that this paper can provide more inspiration to the academic community on how to utilize generated data and how to design better data analysis methods. It should be pointed out that our method is not limited to specific tasks or specific model architectures. In this work, for the convenience of comparison with the baseline, we use the same dataset and model architecture as the baseline. We hope that in future work, we can further verify it on more tasks and model architectures. At the same time, we can also design more flexible and controllable evaluation functions to better utilize generated data. For example, in this paper, when filtering the data with a gradient, there is a trade-off between diversity and consistency. For rare categories in the data, due to the small number of real data itself, diversity should be considered more, while for common categories, due to the large number of real data itself, consistency should be considered more. Therefore, in the future, we can consider adopting a dynamic strategy for different categories. In the long run, our current research is done under the premise of a fixed generative model. A more ideal situation is to involve the generative model in this loop, further optimizing the generative model based on the downstream model's feedback, to achieve a true "generative model in the loop".
+
+# D. Visualization
+
+# D.1. Selected and Discarded Samples
+
+We show some samples selected and discarded by our method in Figure 7. Our proposed method is able to select high-quality samples (best sample) while filtering out low-quality samples (worst sample), which can effectively improve the data learning efficiency of the model. For example, our method is capable of identifying accurately segmented data for applesauce. In cases where applesauce is not present in the generated raw image or is not encompassed within the segmentation mask, our method can discard such samples. For alarm clocks, our method tends to choose images with more complex appearances.
+
+# D.2. Instance Augmentation
+
+We present some augmented data in Figure 8. By randomly pasting generated samples onto the LVIS training set, we effectively enrich the complexity of the scenes and thus increase the model's learning efficiency on the generated data.
+
+![](images/31f03ba62728abcb4689ff172f8c4b2d06d4e9d7657021205fcdd7c5d53fb12e.jpg)  
+Figure 7. Examples of selected and discarded samples.
+
+![](images/62671480a0cce2ba1fa8b609b56e92ae43ce218caa245ab1825373c43941746e.jpg)
+
+![](images/27c6c01516bcb1330a869b937c14b12c33eaa1b4407be8641ac6b12c596c305c.jpg)
+
+![](images/1e9d154c900e07d2ba6b29ba578d385519769bfad7cea6b5fa973ec27407dc9c.jpg)
+
+![](images/d4d2cc3188caaac70887b4b4cc3448f122f5512502dac07a8eba274d4f95b931.jpg)
+
+![](images/6a9f4f65b4d8702794644395bcff742418dac0b1ffa06745de704c55ba5ffdbd.jpg)
+
+![](images/a2b1a6b9b7361e3812bbbe66a5e2866ead1ac0e2c0ab5f4bc8f9cd73bff65ba4.jpg)
+
+![](images/8ec8bc2f5d556973aebcb3598e36a6990b0ec74987a16e52e7e637c7c92b7408.jpg)
+
+![](images/202893fdc6b533624292839232624e7a10da1beca735597c414dcf0c0ebef543.jpg)
+
+![](images/cf6e73a38b2f4ced76631eae1ba06c4783b9677eb43daaa09c64fb69a011db7e.jpg)
+
+![](images/604d3873beb61c931d36f3ba9a150b40ef6584c83c32832067f7664c2c1a9490.jpg)
+
+![](images/c32294b4e7b4f772b0a69e21451a06f262f3b3754cac798e0c2c67f2f9e76997.jpg)
+
+![](images/e9ed2cd3aef5be9e9b866a6c64cac8b0eafafacde034c71641b70fb6102fa86f.jpg)
+
+![](images/7e2b97617894640ec13aaabef43a14bcbebc08415a79003eac5480f70ddcaf64.jpg)
+
+![](images/be4b38fcc2925087ae37ba14b10f4c27df52328ecf8104e56d80b0865b70178e.jpg)
+
+![](images/94dd024797eaeb3f948a2c7f7958bd7bacf6cb04728f1168aad69a29f24a932f.jpg)
+
+![](images/7861649470afc961144a4eb16d4aeb924012b06ef4a77055ab3935d771c972d5.jpg)
+
+![](images/f1b71b319970c13fb7bffa4a00d5f834af2069dff5d2fed5ee0e9a59169e8749.jpg)
+
+![](images/95296124182a604134fcdb07fe772cb160fbeb884b5c3f4a02aef43ac5c6dce8.jpg)
+
+![](images/43aae3b0b874838c28211c5bbdad7bec90eae8fb693f34d1285e74141c335c7a.jpg)
+
+![](images/c5ea8c3598663f4d912acbbc53a5bf46b5678fac602ae6a009faa49dd6d339aa.jpg)
+
+![](images/0f93a20f20a48679652f437c346ba64e3eca2667a3f8130a5cde54fae0ceace5.jpg)
+
+![](images/76235af62127c6a81343b630842f54eae9f68a75b1962e5a2cb36a3601becbf5.jpg)
+
+![](images/924ceeda8cac471beded76ed00fa486170a7fbac86c3d99a023741e373093a6f.jpg)
+
+![](images/9833766d1b2ebdc09bd10be212b91f52efecf74d5c7ac1b89c2889a6b4ce8773.jpg)
+
+![](images/b0f16dc369be51a8f00b296f60388d2657d9711c733aa5c1b8ab6f0de0467995.jpg)
+
+![](images/3c87f7fbe4792bae0027a1e7ed6909d0f6855f0e56d6cc0fe8ef79229af74881.jpg)  
+Figure 8. Examples of augmented data.
+
+![](images/958d4d7761f28fdcb8480ae7f042295f829a81fc4c9542a2bf368813efee4fc7.jpg)
+
+![](images/54d1efe4ffdec196ac445bfecac9505a7fe9a67f055fcb72f66028c32f20f95f.jpg)
+
+![](images/10b7bf7c87bc51fab486e21081a076489908e821c8caaa0ecc04dfe8d610a092.jpg)
+
+![](images/50efdea4d52e96d677a5c16f215a8982983c6ff1481a60d58141318ecec5c82a.jpg)  
+Figure 9. Performance of the model under different iterations.

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+# Generative Conditional Distributions by Neural (Entropic) Optimal Transport
+
+Bao Nguyen<sup>12</sup> Binh Nguyen<sup>3</sup> Hieu Trung Nguyen<sup>14</sup> Viet Anh Nguyen<sup>1</sup>
+
+# Abstract
+
+Learning conditional distributions is challenging because the desired outcome is not a single distribution but multiple distributions that correspond to multiple instances of the covariates. We introduce a novel neural entropic optimal transport method designed to effectively learn generative models of conditional distributions, particularly in scenarios characterized by limited sample sizes. Our method relies on the minimax training of two neural networks: a generative network parametrizing the inverse cumulative distribution functions of the conditional distributions and another network parametrizing the conditional Kantorovich potential. To prevent overfitting, we regularize the objective function by penalizing the Lipschitz constant of the network output. Our experiments on real-world datasets show the effectiveness of our algorithm compared to state-of-the-art conditional distribution learning techniques. Our implementation can be found at https://github.com/nguyennocbaocmt02/GENTLE.
+
+# 1. Introduction
+
+The conditional distribution of a response variable given the covariate information is a key quantity for numerous tasks in machine learning and data science. For example, knowing the conditional distribution of a patient's health outcome (the response) given the patient's characteristics (the covariate) improves risk assessment and prediction accuracy and allows personalized interventions wherein healthcare providers can tailor treatments based on individual patient profiles. It is also a useful tool to evaluate the effects of a policy in economics (Díaz, 2020; Athey & Imbens, 2015). Access to a response's conditional probability
+
+<sup>1</sup>The Chinese University of Hong Kong <sup>2</sup>VinUniversity Department of Mathematics, National University of Singapore <sup>4</sup>VinAI Research. Correspondence to: Bao Nguyen <bao.nn2@vinuni.edu.vn>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+distribution also leads to an estimate of the treatment effect without explicit, additional experiments.
+
+Unfortunately, learning the relationship between the response and the covariate is challenging, especially in high-dimensional covariate space or complex dependency settings. Many simple and interpretable methods, such as kernel density estimation, may struggle to capture the complexity of the data-generating conditional distribution accurately (Athey et al., 2021). On the other hand, there are overly complex (deep neural network) models that perform well on the training data but generalize poorly to new data; this overfitting behavior triggers unreliable predictions and interpretations. Additionally, performing inferences on these overly complex models can be time-consuming and environmentally unfriendly. Alternatively, using additional expert inputs, one can build physically informed simulation models to represent the real physical world. However, running simulations can be time-consuming, making them ill-suited for real-time decision-making.
+
+Another critical obstacle underlying conditional distribution learning is that we are not simply learning one distribution; instead, we need to learn a collection of distributions, one conditional distribution for each possible value of the covariate information. This challenge intensifies because, in reality, the data is scarce. This scarcity is particularly prevalent in healthcare, where privacy concerns or budget constraints prohibit the collection and dissemination of samples. As an illustrative example, we will consider the Lalonde-Dehjia-Wahba (LDW) (LaLonde, 1986; Dehejia & Wahba, 1999) dataset, which is a widely used dataset for analyzing the average treatment effects (Athey et al., 2021). The LDW dataset comprises 16,177 samples aggregated from actual experiments and the Current Population Survey. Each observed sample includes nine covariate attributes of a person and a scalar response reporting the individual's earnings in 1978. Since two persons may have the exact attributes, it happens that for a specific covariate value, there are multiple observations of the response. The histogram in Figure 1 shows the count numbers per distinct covariates for the LDW dataset. We observe that more than 13,000 covariates have only one response. This is an extreme case where we must learn the conditional distribution with a single data point observed at that covariate. On the other hand, we also observe several popular covari
+
+ates with many responses: the most popular covariate has 47 responses. This again induces unbalanced sampling in which the number of responses is not balanced across different covariate values. A naive solution to fix this problem could be under-sampling (remove data) the more frequent covariate and over-sampling (create synthetic data) for the rare covariate to balance the dataset using popular methods such as SMOTE (Chawla et al., 2002), SMOTER (Torgo et al., 2013) and SMOGN (Branco et al., 2017). Unfortunately, these methods do not scale well for large datasets with more than thousands of covariates.
+
+![](images/7b18db15641ac49977fac6aecffdbe5c7668c882b39a29f4c820d1fb43083407.jpg)  
+Figure 1. Histogram count of Number of observed responses for distinct covariate values in the LDW dataset (LaLonde, 1986). This dataset has over 13,000 covariates with only one response.
+
+The current literature on conditional distribution learning reveals a notable gap in methodologies that can effectively address the aforementioned difficulties. Specifically, our experimental section shows that CWGAN (Athey et al., 2021) suffers from mode collapse and fails to recover the data-generating conditional distributions.
+
+# 1.1. Problem Statement and Contributions
+
+We consider  $X$  a multi-dimensional covariate and  $Y$  a one-dimensional response. For any specific value  $x$ , the data-generating conditional distribution of  $Y|X = x$  is denoted by  $\mu_{x}$ . Our goal is to learn these conditional distributions  $\mu_{x}$  uniformly over all  $x$  from a finite dataset  $\mathcal{D} = \{(x_i, \{y_{i,k}\}_{k=1,\dots,N_i}) \in \mathbb{R}^d \times \mathbb{R}\}$ . Without any loss of generality, we suppose that the covariates  $x_{i}$  are distinct, and for each  $x_{i}$ ,  $\{y_{ik}\}_{k=1,\dots,N_i}$  are  $N_{i}$  independent samples drawn from  $\mu_{x_{i}}$ .
+
+More importantly, we aim to learn a generative model of the conditional distributions, which can allow us to draw new, unseen samples. This task is more challenging than learning specific finite-dimensional statistics of the conditional distributions, such as learning conditional expecta
+
+tions or quantiles, especially under a small training sample size. We reemphasize that conditional distribution learning is challenging: our goal is not simply learning one distribution, but instead, a collection of distributions – one conditional distribution for each possible value of the covariate information. This challenge intensifies because of data scarcity. This difficulty is often encountered in real-world deployment, for instance, in health care and finance.
+
+Contributions. We introduce GENTLE, a generative neural transport learning model for conditional distributions. GENTLE is represented by a neural network  $T_{\theta}$  that takes the covariate  $x$  and a uniform  $(0,1)$  random variable  $U$  as input, and it outputs  $T_{\theta}(x,U)$  with a distribution that is close to  $\mu_x$ , the data-generating conditional distribution of  $Y|X = x$ . Due to the low sample size,  $T_{\theta}$  can overfit, and we propose learning  $\theta$  by minimizing the sum of the fitness and the overfitting regularization trade-off. The overfitting regularizer ensures that  $T_{\theta}$  is sufficiently smooth in the covariate  $x$ , and we use an entropic optimal transport distance to measure the discrepancy of the generative distribution outputs. To this end, we create a second network  $v_{\phi}$  to approximate the conditional potential associated with the semi-dual formulation of the entropic optimal transport distance. This leads to a min-max optimization problem over the pair of generative-conditional potential network parameters  $(\theta, \phi)$ . This min-max problem is then solved using a state-of-the-art gradient descent-ascent algorithm.
+
+Using real-life datasets, we empirically validate that GENTLE performs better than state-of-the-art methods such as conditional Wasserstein GANs across multiple performance metrics (Wasserstein metric and Kolmogorov-Smirnov metric). In particular, even for the case of LDWCPS data that exhibits severe imbalances in observed responses, GENTLE still demonstrates a remarkable ability to generate observations that follow ground truth distributions closely, quantitatively, and qualitatively.
+
+This paper unfolds as follows: Section 2 provides an overview of the existing literature. Section 3 reviews the technical background on optimal transports and the tools we will utilize. Section 4 presents our approach to generative conditional distributions. The numerical results are provided in Section 5.
+
+Notations. We write the capital letter  $X$  for the random variable and the case letter  $x$  for a specific realization of  $X$ . For simplicity, we use  $Y(x)$  to denote the conditional random variable  $Y|X = x$ . For a measurable map  $f: \mathbb{R} \to \mathbb{R}$  and any probability measure  $\mathbb{P}$  on  $\mathbb{R}$ , we use  $f_{\#} \mathbb{P}$  to denote the pushforward measure of  $\mathbb{P}$  under  $f$ . The (standard) uniform distribution on  $(0,1)$  is denoted  $\mathcal{U}(0,1)$ , and we use  $\mathbb{U}$  to represent the probability measure associated with this uniform distribution. We denote  $N = \sum_{i} N_{i}$  as the total number of observations in the training dataset.
+
+# 2. Related Works
+
+Metamodeling is a computational approach that efficiently captures the relationship between input variables and key output statistics like means or quantiles. This is crucial for real-time simulation applications such as personalized decision making (Shen et al., 2021) and online risk monitoring (Jiang et al., 2020). Traditional metamodeling techniques, while effective, often require pre-specification of relevant statistics, limiting their flexibility. To address this, generative metamodeling has emerged as a promising technique that estimates conditional distributions quickly and dynamically. Generative metamodeling differs from traditional approaches by simulating a wider range of possible outcomes, providing a more comprehensive analysis of potential scenarios. Athey et al. (2021) propose using a Wasserstein Generative Adversarial Model (WGAN, Arjovsky et al. 2017) to simulate the distribution of outcomes conditioned on covariates, offering a novel approach to outcome simulation. Similarly, Hong et al. (2023) propose quantile-regression-based generative metamodeling (QRGMM), another method of conditional distribution learning. QRGMM leverages the conditional quantile regression technique to provide a more detailed understanding of the distribution tails, which is crucial in risk assessment and decision-making processes. These advancements in generative metamodeling can significantly enhance the accuracy and applicability of real-time simulations, paving the way for more personalized and precise decision-making tools in various fields.
+
+Synthetic evaluation data. Previous works (van Breugel et al., 2024; Parikh et al., 2022) leverage the deep generative model to approximate conditional distribution for synthetic data generation. (Shrivastava & Tuzel, 2019) parameterize a conditionally Gaussian model with a Deep Neural Network to model error distribution for each state of the object.
+
+Neural Optimal Transport is a recent emergent framework that proposes to learn the optimal transport plan by deep neural networks. Various existing works in this direction include parameterization of the OT map as the gradient of an input convex neural network (Jacob et al., 2019; Makkuva et al., 2020; Bunne et al., 2022; 2023); or solutions of convex-concave problems (Korotin et al., 2022; Gushchin et al., 2022; Choi et al., 2023). However, these formulations are primarily suited to computer vision or computational biology applications. To the best of our knowledge, our paper is the first work attempting to formulate the neural OT framework into a conditional distribution learning task.
+
+Learning and decision-making with conditional distributions also attracted tremendous attention in artificial intelligence and operations research communities.
+
+Nguyen et al. (2020b) studied the robust parameter estimation of conditional distributions under the parametric exponential family of distributions, while Nguyen et al. (2020a) studied the non-parametric Wasserstein robust estimation of conditional statistics. Making conditional decisions based on observed covariates is a well-known problem in finance (Nguyen et al., 2024) and operations research (Esteban-Pérez & Morales, 2022); interested readers are invited to the survey by Sadana et al. (2024) for recent developments.
+
+# 3. Background
+
+We focus on a one-dimensional response space for  $Y$ . The optimal transport distance between two distributions supported on  $\mathbb{R}$  is formally defined as follows.
+
+Definition 3.1 (Optimal Transport Distance, Villani 2009). Given two probability measures  $\mu$  and  $\nu$  supported on  $\mathbb{R}$  with finite second moments, the optimal transport (2-Wasserstein) distance between them is defined as
+
+$$
+\mathrm {W} _ {2} ^ {2} (\mu , \nu) \stackrel {\text {d e f .}} {=} \inf  _ {\pi \in \Pi (\mu , \nu)} \int_ {\mathbb {R} \times \mathbb {R}} | y - y ^ {\prime} | ^ {2} \mathrm {d} \pi (y, y ^ {\prime}), \tag {1}
+$$
+
+where  $\Pi (\mu ,\nu)$  denotes the set of probability distributions on  $\mathbb{R}^2$  with marginals  $\mu$  and  $\nu$ , respectively.
+
+Optimal transport in 1d. In one dimension, the 2-Wasserstein distance admits a closed form expression (Bobkov & Ledoux, 2019, Theorem 2.10): for two measures  $\mu$  and  $\nu$  with cumulative density functions (CDF)  $F_{\mu}$  and  $F_{\nu}$ , respectively, we have
+
+$$
+\begin{array}{l} \mathrm {W} _ {2} ^ {2} (\mu , \nu) = \int_ {0} ^ {1} | F _ {\mu} ^ {- 1} (t) - F _ {\nu} ^ {- 1} (t) | ^ {2} \mathrm {d} t \tag {2} \\ = \mathbb {E} _ {U \sim \mathcal {U} (0, 1)} \left[ | F _ {\mu} ^ {- 1} (U) - F _ {\nu} ^ {- 1} (U) | ^ {2} \right]. \\ \end{array}
+$$
+
+Entropic-regularized optimal transport. The recent popularity in applications of OT to machine learning is largely due to its efficient computation by adding negative entropy of the transport plan as a form of regularization to the linear OT program (1). This problem is called entropic optimal transport (EOT), which is smooth and strongly convex, therefore has a unique minimizer:
+
+$$
+\mathrm {W} _ {2, \varepsilon} ^ {2} (\mu , \nu) \stackrel {\text {d e f .}} {=} \min  _ {\pi \in \Pi (\mu , \nu)} \int_ {\mathbb {R} ^ {2}} | y - y ^ {\prime} | ^ {2} \mathrm {d} \pi - \varepsilon H (\pi). \tag {3}
+$$
+
+Cuturi (2013) showed that the minimization problem (3) can be solved by Sinkhorn's algorithm (a type of matrix scaling algorithm, also called iterative proportional fitting procedure, Kullback 1968) that can be implemented at large scale and is analytically tractable. In addition to its computational advantages, problem (3) can also be interpreted as a specific instance of the Schrödinger bridge
+
+problem, which has a rich history in physics. For an in-depth theoretical introduction to EOT, readers can refer to Nutz (2021) or Léonard (2014).
+
+A nice property of the EOT problem (3) is that it allows an equivalent smooth unbounded semi-dual formulation (Genevay et al., 2016, Proposition 2.1)
+
+$$
+\begin{array}{l} \min _ {\pi \in \Pi (\mu , \nu)} \int_ {\mathbb {R} ^ {2}} | y - y ^ {\prime} | ^ {2} \mathrm {d} \pi - \varepsilon H (\pi) \\ = \sup  _ {v (\cdot) \in \mathcal {C}} \int_ {\mathbb {R}} v ^ {\varepsilon} (y) \mathrm {d} \mu + \int_ {\mathbb {R}} v \left(y ^ {\prime}\right) \mathrm {d} \nu - \varepsilon , \tag {4} \\ \end{array}
+$$
+
+where  $\mathcal{C}$  is the space of all continuous functions from  $\mathbb{R}$  to  $\mathbb{R}$ , and  $v^{\varepsilon}$  is defined as
+
+$$
+v ^ {\varepsilon} (y) = - \varepsilon \log \left(\int_ {\mathbb {R}} \exp \left(\frac {v \left(y ^ {\prime}\right) - \left| y - y ^ {\prime} \right| ^ {2}}{\varepsilon}\right) \mathrm {d} \nu\right). \tag {5}
+$$
+
+The function  $v$  is called Kantorovich potential, along with its smoothed  $c$ -transform  $v^{\varepsilon}$ , with  $c$  being the absolute difference bi-function. Using the semi-dual form is that we need to optimize over only one function  $v$  associating with  $\nu$ , and we can obtain  $v^{\varepsilon}$  as the potential associating with  $\mu$ .
+
+# 4. Methodologies
+
+We propose to learn a mapping  $T: \mathbb{R}^d \times (0,1) \to \mathbb{R}$  such that uniformly over all  $x$ ,  $T(x,U) = Y(x)$  in distribution when  $U \sim \mathcal{U}(0,1)$ . The existence of this mapping  $T(x,U)$  is guaranteed by the following celebrated result.
+
+Lemma 4.1 (Noise Outsourcing, Austin 2015). Suppose that  $\mathcal{X}$  and  $\mathcal{Y}$  are standard Borel spaces and that  $(X,Y)$  is an  $(\mathcal{X} \times \mathcal{Y})$ -valued random variable. Then, there are random variables  $U \sim \mathcal{U}(0,1)$  coupled with  $\mathcal{X}$  and  $\mathcal{Y}$  and a Borel function  $T: \mathcal{X} \times (0,1) \to \mathcal{Y}$  such that  $U$  is independent from  $X$  and
+
+$$
+(X, Y) = (X, T (X, U)) \quad a l m o s t s u r e l y.
+$$
+
+In general, the mapping  $T$  is not unique. Following the neural OT framework (Korotin et al., 2022; Makkuva et al., 2020; Bunne et al., 2022), we use a neural network to parametrize a possible instance of  $T(x,U)$ , and we denote this network as  $T_{\theta}(x,U)$ . For any specific value of the parameter  $\theta$ , the network outputs a probability measure  $T_{\theta}(x,\cdot)_{\#}\mathbb{U}$ , which represents the distribution of  $T_{\theta}(x,U)$  with  $U\sim \mathcal{U}(0,1)$ . We propose the following regularized loss to find the optimal map  $T_{\theta}$ :
+
+$$
+\min  _ {\theta} \operatorname {F i t} (\theta) + \lambda \operatorname {R e g} (\theta), \tag {6}
+$$
+
+where the first term is a measure of the fitness of the map, the second is a regularization to prevent overfitting, and  $\lambda \geq 0$  is the regularization weight. We explain the detailed formulation of each term and its rationale in the next sections.
+
+# 4.1. Fitness Measure
+
+Motivated by the OT distance in (2), a natural choice to measure the discrepancy between the network outputs and the conditional data-generating distributions is by integrating the discrepancy of the respective inverse CDFs over all possible values  $x$ , i.e., by evaluating
+
+$$
+\begin{array}{l} \mathbb {E} _ {X \sim \mathcal {D}, U \sim \mathcal {U} (0, 1)} [ | F _ {\mu_ {X}} ^ {- 1} (U) - T _ {\theta} (X, U) | ^ {2} ] \\ = \mathbb {E} _ {X \sim \mathcal {D}} \left[ \mathbb {E} _ {U \sim \mathcal {U} (0, 1)} \left[ \left| F _ {\mu_ {X}} ^ {- 1} (U) - T _ {\theta} (X, U) \right| ^ {2} \right] \right]. \\ \end{array}
+$$
+
+Above,  $F_{\mu_x}$  is the cumulative distribution function of  $Y(x)$ , and the outer expectation is taken over the (uniform) sampling of all distinct values of  $X$  in the training dataset  $\mathcal{D}$ . Unfortunately, the inverse CDF  $F_{\mu_X}^{-1}$  can be unbounded, and the magnitude of  $F_{\mu_X}^{-1}$  explodes to infinity for tail events whenever  $U$  is to close to the boundary 0 or 1. Hence, learning  $T_{\theta}$  with the above metric is not stable. To improve stability, we use the following fitness function
+
+$$
+\operatorname {F i t} (\theta) \stackrel {\text {d e f .}} {=} \mathbb {E} _ {X \sim \mathcal {D}} \left[ \mathbb {E} _ {U \sim \mathcal {U} (0, 1)} \left[ | U - F _ {\mu_ {X}} \left(T _ {\theta} (X, U)\right) | ^ {2} \right] \right]. \tag {8}
+$$
+
+In practice, we do not have access to the real cumulative distribution of  $Y(x)$ . However, we have a set of responses sampled from  $Y(x)$ , so we propose to use a simple method such as Kernel Density Estimator (KDE) with the kernel bandwidth of  $h$  to have a smooth conditional density from responses of the covariate  $x$  in the training dataset. The kernel bandwidth plays a vital role in our objective function, and tuning this parameter is important. In theory, our KDE smoothness using the Gaussian kernel is strongly related to the smoothed Wasserstein distance. From Nietert et al. (2021, Lemma 1 and Corollary 1), the smooth-2-Wasserstein distance is continuous and monotonically non-increasing in the Gaussian kernel bandwidth. This means that the larger the bandwidth (the more smoothing applied), the smaller the fitness term will become. In practice, we could implement an automatic kernel bandwidth selection such as the Scott (Scott, 1979) or the Silverman (Silverman, 2018) methods. In our implementation, the value of the bandwidth is selected by a grid search to minimize the Wasserstein distance on the validation set.
+
+# 4.2. Regularization Measure
+
+Minimizing the  $\operatorname{Fit}(\theta)$  term can lead to overfitting due to the low sample sizes for many values of  $x_{i}$ . To combat this problem, we add a regularization term to promote the transfer learning across different values of  $x_{i}$ . Consider the CPS dataset, we examine the relationship between the covariate distance  $\| x_{i} - x_{j}\|$  versus the empirical Wasserstein distance between the conditional random variables  $Y(x_{i})$  and  $Y(x_{j})$ . For better accuracy, we select only pairs  $(x_{i},x_{j})$  such that both  $N_{i}$  and  $N_{j}$  are greater than 18. The scatter plot in Figure 2 indicates a positive correlation between
+
+![](images/32ed63d700a911bb31400d3cc80e35428b435ca79c15e0834c5a3c467c91cb71.jpg)  
+Figure 2. Positive correlation between the Wasserstein distance between  $Y(x_{i})$  and  $Y(x_{j})$  and the covariate distance  $\| x_{i} - x_{j}\|$ . Only covariates  $x$  with more than 18 observations are selected.
+
+the covariate distance and the Wasserstein distance of the conditional distributions. We remark that we observed the same trend in other datasets.
+
+From this empirical evidence, we can assume that if two covariates  $x_{i}$  and  $x_{j}$  are similar, their respective conditional distributions of  $Y(x_{i})$  and  $Y(x_{j})$  should also be similar. We emphasize that relying on this assumption to design the regularization term does not restrict the generalization of our framework. This assumption, in general, holds for the case where the data-generating process follows the form  $Y = f(X) + \mathrm{noise}$ , where  $f$  is a continuous function. This is a classic setting in distribution learning, a more general form of linear assumption. Moreover, classic algorithms for supervised and unsupervised learning, such as kNN or k-clustering, also require this assumption to work. We can, therefore, assume a Lipschitz condition that there exists a finite positive number  $L$  such that
+
+$$
+W _ {2} ^ {2} (\mu_ {x _ {i}}, \mu_ {x _ {j}}) \leq L \| x _ {i} - x _ {j} \| \qquad \forall i, j,
+$$
+
+where  $\mu_{x}$  is the data-generating conditional probability measure for  $Y(x)$ . If the generative network  $T_{\theta}$  matches sufficiently well the conditional distributions, then it is reasonable to translate the above Lipschitz assumption into a regularization term: let  $\mathcal{E}$  be a set of pairs  $(x_{i}, x_{j})$  for which we would like to impose Lipschitz conditions, we can regularize using the sum of the pairwise entropic OT distance:
+
+$$
+\sum_ {\left(x _ {i}, x _ {j}\right) \in \mathcal {E}} \mathrm {W} _ {2, \varepsilon} ^ {2} \left(T _ {\theta} \left(x _ {i}, \cdot\right) _ {\#} \mathbb {U}, T _ {\theta} \left(x _ {j}, \cdot\right) _ {\#} \mathbb {U}\right). \tag {9}
+$$
+
+The semi-dual form of the EOT (4) leads to
+
+$$
+\sum_ {\left(x _ {i}, x _ {j}\right) \in \mathcal {E}} \Big (\max  _ {v _ {i j} (\cdot) \in \mathcal {C}} \int_ {\mathbb {R}} v _ {i j} ^ {\varepsilon} (y) \mathrm {d} \mathbb {P} _ {\theta , j} + \int_ {\mathbb {R}} v _ {i j} (y) \mathrm {d} \mathbb {P} _ {\theta , i} - \varepsilon \Big).
+$$
+
+Above, we have used the shorthand  $\mathbb{P}_{\theta ,i} = T_{\theta}(x_i,\cdot)_{\#}\mathbb{U},$  the pushforward measure of the uniform measure under the
+
+map  $T_{\theta}(x_i,\cdot)$ . Each maximization problem inside the sum is an infinite-dimensional optimization problem, in which we need to search over all potentials  $v_{ij}$  continuous. To alleviate this computational burden, a naive approach is to use a neural network  $v_{\phi}(x_i,x_j,\cdot)\approx v_{ij}(\cdot)$ , where  $\phi$  is a learnable parameter. By omitting the constant  $\varepsilon$  and switching the order of the summation and the maximization operator, we can regularize using
+
+max
+
+$$
+\sum_ {(x _ {i}, x _ {j}) \in \mathcal {E}} \left(\int_ {\mathbb {R}} v _ {\phi} ^ {\varepsilon} (x _ {i}, x _ {j}, y) \mathrm {d} \mathbb {P} _ {\theta , j} + \int_ {\mathbb {R}} v _ {\phi} (x _ {i}, x _ {j}, y) \mathrm {d} \mathbb {P} _ {\theta , i}\right).
+$$
+
+A downside of this parametrization is that it requires both  $x_{i}$  and  $x_{j}$  as input to  $v_{\phi}$  to output the potential. This leads to redundancy in the parametrization. To combat this problem, we propose a single input network  $v_{\phi}(x_i,y)$  to approximate  $v_{ij}(y)$ , the potential associating with  $\mathbb{P}_{\theta ,i}$ . Further, we compute the smoothed transform  $v_{\phi}^{\varepsilon}(x_i,y)$
+
+$$
+\begin{array}{l} v _ {\phi} ^ {\varepsilon} (x _ {i}, y) \stackrel {{\text {d e f .}}} {{=}} \\ - \varepsilon \log \left(\int_ {\mathbb {R}} \exp \left(\frac {v _ {\phi} \left(x _ {i} , y ^ {\prime}\right) - \left| y - y ^ {\prime} \right| ^ {2})}{\varepsilon}\right) \mathrm {d} \mathbb {P} _ {\theta , i}\right), \\ \end{array}
+$$
+
+and use  $v_{\phi}^{\varepsilon}(x_i, y)$  to approximate  $v_{ij}^{\varepsilon}(y)$ , the potential associating with  $\mathbb{P}_{\theta,j}$ . The above single input parametrization no longer takes  $x_j$  as input; thus, to make the network well-defined, we need disambiguation by imposing the following condition: for each distinct value  $x_i$ , there is at most one pair  $(x_i, x_j)$  that belongs to  $\mathcal{E}$ . If this condition on  $\mathcal{E}$  holds, then we can uniquely identify the  $x_j$  such that  $v_{\phi}^{\varepsilon}(x_i, y)$  is serving as the potential.
+
+Thus, for a valid set  $\mathcal{E}$ , we propose the following regularization term:
+
+$$
+\operatorname {R e g} (\theta) \stackrel {\text {d e f .}} {=} \max  _ {\phi} \mathcal {R} (\theta , \phi), \tag {10a}
+$$
+
+where  $\mathcal{R}(\theta ,\phi)$  is
+
+$$
+\mathcal {R} (\theta , \phi) \stackrel {\text {d e f .}} {=} \tag {10b}
+$$
+
+$$
+\sum_ {(x _ {i}, x _ {j}) \in \mathcal {E}} \left(\int_ {\mathbb {R}} v _ {\phi} ^ {\varepsilon} (x _ {i}, y) \mathrm {d} \mathbb {P} _ {\theta , j} + \int_ {\mathbb {R}} v _ {\phi} (x _ {i}, y) \mathrm {d} \mathbb {P} _ {\theta , i}\right).
+$$
+
+In this way,  $\operatorname{Reg}(\theta)$  is the sum of the semi-dual neural entropic OT distances of all pairwise distributions specified by  $\mathcal{E}$ .
+
+To conclude this section, we elaborate on a reasonable approach to building  $\mathcal{E}$ . Ideally,  $\mathcal{E}$  should capture neighborhood information about which pair  $(x_i, x_j)$  is close to each other; further, it should satisfy our disambiguation criterion that for each distinct value  $x_i$ , there is at most one pair  $(x_i, x_j)$  that belongs to  $\mathcal{E}$ . A reasonable choice of  $\mathcal{E}$  that is
+
+also computationally efficient is to build  $\mathcal{E}$  as the directed edge set of a minimum spanning tree of all  $x$ . Here, the spanning tree minimizes the sum of distance  $\| x_i - x_j\|$  between two connected nodes, and we can employ Kruskal's algorithm (Kruskal, 1956). For the direction of the edge: at each iteration, when we add  $x_i$  to the incumbent minimum spanning tree, then  $x_i$  is chosen as the tail of the edge, and the other endpoint of the adding edge (which already belongs to the incumbent tree) is chosen as the head of the edge. This construction will satisfy the disambiguation criterion for the set  $\mathcal{E}$ . We emphasize that the graph-based approach to identify which  $x_j$  is associated with  $x_i$  in the regularization term is not restricted only to a discrete set of covariates. This is because the graph-based approach is only employed during the training when the number of covariates is finite. In inference phases, the model can give the distributional prediction to any covariates.
+
+Remark 4.2. In (8), we use the unregularized OT distance as a discrepancy function for the fit term  $\mathrm{Fit}(\theta)$  because therein, we are measuring the discrepancy between a fixed target (a uniform distribution  $U$ ) and a quantity that depends on the generative network output  $T_{\theta}(X,U)$  and the data-generating CDFs  $F_{\mu_X}$ . This contrasts with the regularization term  $\mathrm{Reg}(\theta)$ , where we need to compare two distributions that are both outputs of the generative network  $T_{\theta}$ . Using the unregularized OT distance for the  $\mathrm{Reg}(\theta)$  will require evaluating the CDFs of  $T_{\theta}$ . To alleviate this difficulty, we use the semi-dual form of EOT, where we can parametrize a single conditional potential network to evaluate all the pairwise EOT distances between covariate pairs in  $\mathcal{E}$ .
+
+# 4.3. Min-Max Optimization with Smooth Gradient Descent-Ascent
+
+By substituting the form of the regularization term (10) into (6), we obtain the min-max optimization problem:
+
+$$
+\min  _ {\theta} \max  _ {\phi} \operatorname {F i t} (\theta) + \lambda \mathcal {R} (\theta , \phi). \tag {11}
+$$
+
+Naively optimizing problem (11) will result in a common pitfall in optimization of training GANs: gradient descent-ascent (GDA) on nonconvex-nonconcave objective is, in general, hard to converge (Heusel et al., 2017; Lin et al., 2020; Zheng et al., 2023). To alleviate this problem, we employ the state-of-the-art first-order algorithm for GDA (Zheng et al., 2023) with a convergence guarantee. First, for parameters  $r_1 \neq r_2$ ,  $r_1, r_2 > 0$ , we add two extra regularization (smoothing) terms with auxiliary variables  $p$  and  $q$  into the loss function. This leads to the augmented objective
+
+$$
+\begin{array}{l} \mathcal {L} (\theta , \phi , p, q) \\ \stackrel {\text {d e f .}} {=} \operatorname {F i t} (\theta) + \lambda \mathcal {R} (\theta , \phi) + \frac {r _ {1}}{2} \| \theta - p \| _ {2} ^ {2} - \frac {r _ {2}}{2} \| \phi - q \| _ {2} ^ {2}. \tag {12} \\ \end{array}
+$$
+
+Including the quadratic terms enhances the smoothness of the primal and dual update, facilitating a more balanced trade-off between the gradient updates in our nonconvex-nonconcave setting. In practice, we must evaluate this loss with training samples, detailed in Algorithm 1. Our training procedure is summarized in Algorithm 2. This doubly-smoothed GDA update is guaranteed to converge to a stationary point (Zheng et al., 2023, Theorem 2) as long as the max term  $\mathcal{R}(\theta ,\phi)$  satisfies Kurdyka-Lojasiewicz condition (a nonsmooth extension of the more popular Polyak-Lojasiewicz condition).
+
+In general, we resort to a validation procedure to choose the appropriate parameters  $h, \lambda, \varepsilon, r_1, r_2$ , and the optimization algorithm's learning rates  $(\alpha, \beta, \gamma, \delta)$ . Details on the parameters grid are given in the Appendix A.
+
+# Algorithm 1 Empirical loss evaluation
+
+1: Input: data  $(x_{i},\hat{F}_{x_{i}})$ , network  $T_{\theta}$  and  $v_{\phi}$ , auxiliary variables  $p,q$ , batch size  $B$ , set of pair covariates  $\mathcal{E}$ , regularization parameter  $\varepsilon, \lambda, r_1, r_2$  
+2: Output: loss estimate  $\widehat{\mathcal{L}} (\theta ,\phi ,p,q)$  
+3: Initialize: take a batch of  $B$  training nodes  $\{x_{i}^{(b)}\}_{b\in [B]}$  
+4: for each  $(x_{i}^{(b)}, x_{j}^{(b)}) \in \mathcal{E}$  do  
+5: Sample  $U^{(b)}\sim \mathcal{U}[0,1]$  
+6: for  $m = 1,\dots ,M$  do  
+7: Sample  $U^{(b,m)}\sim \mathcal{U}[0,1]$  
+8: end for  
+9: end for  
+10: Compute following (8):
+
+$$
+\widehat {\mathrm {F i t}} (\theta) \gets \frac {1}{B} \sum_ {b} \left[ U ^ {(b)} - \hat {F} _ {x _ {i} ^ {(b)}} (T _ {\theta} (x _ {i} ^ {(b)}, U ^ {(b)})) \right] _ {2} ^ {2}
+$$
+
+11: Compute following (10):
+
+$$
+\begin{array}{l} \sum_ {\left(x _ {i} ^ {(b)}, x _ {j} ^ {(b)}, U ^ {(b, m)}\right)} \left[ v _ {\phi} ^ {\varepsilon} \left(x _ {i} ^ {(b)}, T _ {\theta} \left(x _ {j} ^ {(b)}, U ^ {(b, m)}\right)\right) \right. \\ \left. \left. + v _ {\phi} \left(x _ {i} ^ {(b)}, T _ {\theta} \left(x _ {i} ^ {(b)}, U ^ {(b, m)}\right)\right) \right] / (M \times B) \right. \\ \end{array}
+$$
+
+12: Compute following (12):
+
+$$
+\begin{array}{l} \widehat {\mathcal {L}} (\theta , \phi , p, q) \\ \leftarrow \widehat {\operatorname {F i t}} (\theta) + \lambda \hat {\mathcal {R}} (\theta , \phi) + \frac {r _ {1}}{2} \| \theta - p \| _ {2} ^ {2} - \frac {r _ {2}}{2} \| \phi - q \| _ {2} ^ {2} \\ \end{array}
+$$
+
+# 5. Experiments
+
+Datasets. To benchmark our proposed method GENTLE, we use the following two datasets and the same preprocessing steps as in Athey et al. (2021).
+
+# Algorithm 2 Training of the Minimax loss (12)
+
+1: Input: data  $\mathcal{D} = (x_{i},\{y_{ik}\}_{k = 1,\dots ,N_{i}})$ , network  $T_{\theta}$  and  $v_{\phi}$ , learning rate  $\alpha, \beta$ , extrapolation parameters  $\gamma, \delta$  
+2: Initialize:  $\theta^0, \phi^0, p^0, q^0$  
+3: for each number of training iteration  $t$  do  
+4:  $\theta^{t + 1}\gets \theta^t -\alpha \nabla_\theta \widehat{\mathcal{L}} (\theta^t,\phi^t,p^t,q^t)$  
+5:  $\phi^{t + 1}\gets \phi^t +\beta \nabla_\phi \widehat{\mathcal{L}} (\theta^{t + 1},\phi^t,p^t,q^t)$  
+6:  $p^{t + 1} \gets p^t + \gamma (\theta^{t + 1} - p^t)$  
+7:  $q^{t + 1}\gets q^t +\delta (\phi^{t + 1} - q^t)$  
+8: end for
+
+The LDW-CPS dataset, constructed by LaLonde (1986); Dehejia & Wahba (1999), is widely used in studies of Average Treatment Effects. This dataset contains 16177 samples from the real-world experiment and the Current Population Survey. Each covariate  $x_{i}$  includes the individual's eight attributes: two earnings measure (earning 74, earning 75), two indicators for ethnicity (black, hispanic), marital status (married), age (age), two education measures (education, nodegree), and a binary variable for whether this individual receives treatment (is_treated). The response  $y_{i}$  is earning 78, this individual's yearly income in 1978. This dataset has limited samples for each covariate, shown in Figure 1. To achieve better testing evaluation and model selection, we choose covariates based on their frequency for the test and validation sets. Specifically, covariates with a frequency higher than 30 are included in the test set, while those with a frequency higher than 20 but less than or equal to 30 are included in the validation set. According to this split strategy, the training set contains 15,255 samples corresponding to 13,745 covariates, the validation set contains 488 samples corresponding to 19 covariates, and the test set contains 434 samples corresponding to 20 covariates.
+
+We also apply frameworks to a practical simulator called the Esophageal Cancer Markov Chain Simulation Model (ECM). This simulator enables us to simulate patients' quality-adjusted life years (QALYs) from their current treatment until death. The approach to generate data for training and testing follows the authors in Hong et al. (2023). Each covariate  $x_{i}$  consists of five attributes which are Barrett, aspirinEffect, statinEffect, drugIndex, and initialAge. The response feature for this dataset is QALY (quality-adjusted life years). Because we have access to the simulator, we can sample infinite responses for each covariate. We follow the dataset generation in Hong et al. (2023) in which they use the simulator to uniformly generate  $2 \times 10^{4}$  different covariates for the train set, and each covariate has one response. The validation set has 200 covariates that are different from those in
+
+the train set, while the test set also has 200 covariates that are different from both the train and validation sets. Each covariate has  $10^{4}$  responses.
+
+Baselines. We compare our method against state-of-the-art baselines, including CWGAN (Athey et al., 2021), WGAN-GP (Gulrajani et al., 2017), MGAN (Baptista et al., 2020), and CDSB (Shi et al., 2022). We keep the architecture of  $T_{\theta}(x,U)$  and  $v_{\phi}$  simple with seven fully connected layers with ReLU activation.
+
+We denote  $D_{\mathrm{train}}$ ,  $D_{\mathrm{val}}$ , and  $D_{\mathrm{test}}$  as the set of covariates in the training, validation, and test sets, respectively. While the covariates in the training set are used for training the model parameters, the ones in the validation test are utilized to select hyperparameters. For each distinct  $x \in D_{\mathrm{test}}$ , the set of actual respective responses is denoted as  $Y_{GT}(x)$ , while we use GENTLE, CWGAN, WGAN-GP, MGAN, and CDSB to generate  $K = 10^4$  approximating points for  $Y_{\mathrm{GT}}(x)$ , denoted as  $Y_{\mathrm{GENTLE}}(x)$ ,  $Y_{\mathrm{CWGAN}}(x)$ ,  $Y_{\mathrm{WGAN-GP}}(x)$ ,  $Y_{\mathrm{MGAN}}(x)$ ,  $Y_{\mathrm{CDSB}}(x)$  respectively. We then compute the average Wasserstein distance and Kolmogorov-Smirnov statistic (Kolmogorov, 1933) over  $D_{\mathrm{test}}$  between  $Y_{\mathrm{GT}}(x)$  and the outputs  $Y_{\mathrm{GENTLE}}(x)$ ,  $Y_{\mathrm{CWGAN}}(x)$ ,  $Y_{\mathrm{WGAN-GP}}(x)$ ,  $Y_{\mathrm{MGAN}}(x)$ , and  $Y_{\mathrm{CDSB}}(x)$ , as shown in Table 1.
+
+Table 1. Wasserstein distance and K-S statistic on generated  $\hat{Y}(x)$  using different methods vs. corresponding ground truth. The symbol  $\pm$  represents the standard deviation.  
+
+<table><tr><td>DATASET</td><td>METHOD</td><td>WD(↓)</td><td>KS(↓)</td></tr><tr><td rowspan="5">LDW-CPS</td><td>GENTLE</td><td>3767.62 (±1337.56)</td><td>0.48 ± 0.05</td></tr><tr><td>CWGAN</td><td>13371.99 (±3637.66)</td><td>0.88 ± 0.09</td></tr><tr><td>MGAN</td><td>4096.75 (±1004.5)</td><td>0.91 ± 0.19</td></tr><tr><td>WGAN-GP</td><td>7828.66 (±2306.00)</td><td>0.99 ± 0.02</td></tr><tr><td>CDSB</td><td>4970.66 (±1094.74)</td><td>0.56 ± 0.14</td></tr><tr><td rowspan="5">ECM</td><td>GENTLE</td><td>0.97 ± 0.14</td><td>0.21 ± 0.02</td></tr><tr><td>CWGAN</td><td>4.42 ± 2.01</td><td>0.94 ± 0.05</td></tr><tr><td>MGAN</td><td>1.09 ± 0.59</td><td>0.55 ± 0.13</td></tr><tr><td>WGAN-GP</td><td>1.59 ± 0.77</td><td>0.66 ± 0.27</td></tr><tr><td>CDSB</td><td>2.04 ± 0.43</td><td>0.47 ± 0.12</td></tr></table>
+
+Results. We plot the qualitative results of estimated densities for different covariates  $X$  in Figure 3 and 4. Overall, synthetic observations generated by GENTLE exhibit a strong resemblance to the ground truth distributions in both datasets, while the ones generated by baselines deviate significantly from the data-generating distributions. For each covariate, the distributions generated by CWGAN seem to collapse into the mean of the training data distribution, while those of MGAN and WGAN-GP tend to shrink
+
+towards the mean of the conditional data distribution of the covariate. Although CDSB does not suffer from this collapse tendency, it still fails to match the data-generating distributions accurately. This is consistent with the quantitative results in Table 1, which show that the data generated by GENTLE have not only smaller Wasserstein distance (WD) and Kolmogorov-Smirnov (KS) values but also smaller standard deviations. This suggests GENTLE's superior performance compared to the other baselines.
+
+# 5.1. Ablation Study on Training Loss Terms
+
+We explore the impact of removing specific components from the loss function (12) by considering two ablation versions. The first version omits the smoothing term in the loss function  $\mathcal{L}$  by setting  $r_1 = r_2 = 0$ , while the second version omits the regularization term by setting  $\lambda = 0$ . When we exclude the regularization term, the model generates uniformly distributed responses in the LDW-CPS dataset, as illustrated in Figure 6. We hypothesize that the absence of  $\mathrm{Reg}(\theta)$  compromises the model's ability to enforce the Lipschitz condition between the proximity in the covariates and responses space. This absence significantly hampers the method's performance in the dataset with an unbalanced observation of covariates like LDW-CPS. This problem does not exist in the ECM dataset which covariates are uniformly drawn from the simulation, so the performance of this version in ECM just slightly decreases. On the other hand, removing the smoothing term leads the model to get trapped in bad local optima, learning only simple two-peak patterns in both two datasets, as depicted in Figure 7 and Figure 8. This becomes more apparent when examining Table 2, where the KS statistic of this version surpasses that of the main method. This improvement may be attributed to the model's inclination to match high-density regions exclusively, lacking overall generalization. In summary, the results in Table 2 highlight that both ablation versions exhibit significantly worse performance than the official method, particularly regarding Wasserstein distance. This underscores the importance of the regularization and smoothing terms in maintaining the model's robustness and enhancing its overall performance.
+
+# 6. Conclusions
+
+This paper introduces a neural generative model for conditional distributions, whose parameters can be learned from limited data. Our method demonstrates remarkable efficacy in generating distributions over response spaces from given inputs. This is exemplified by successfully applying conditional dataset synthesis tasks on LDW-CPS and ECM datasets, where our framework exhibits superior performances.
+
+An exciting extension of the current work could involve
+
+Table 2. Wasserstein distance and K-S statistic on generated  $\hat{Y}(x)$  using different ablation versions of GENTLE. The version GENTLE w/o SMO does not use the smoothing term while GENTLE w/o REG does not use the regularization term. The symbol  $\pm$  represents the standard deviation.  
+
+<table><tr><td>DATASET</td><td>METHOD</td><td>WD(↓)</td><td>KS(↓)</td></tr><tr><td rowspan="3">LDW-CPS</td><td>GENTLE</td><td>3767.62 (±1337.56)</td><td>0.48 ± 0.05</td></tr><tr><td>GENTLE w/o REG</td><td>13625.83 (±562.10)</td><td>0.57 ± 0.04</td></tr><tr><td>GENTLE w/o SMO</td><td>6075.02 (±1340.75)</td><td>0.41 ± 0.04</td></tr><tr><td rowspan="3">ECM</td><td>GENTLE</td><td>0.97 ± 0.14</td><td>0.21 ± 0.02</td></tr><tr><td>GENTLE w/o REG</td><td>1.13 ± 0.47</td><td>0.29 ± 0.04</td></tr><tr><td>GENTLE w/o SMO</td><td>1.80 ± 1.28</td><td>0.30 ± 0.05</td></tr></table>
+
+exploring the application of our proposed method to dynamic or time-series data. Incorporating temporal dependencies could enhance the model's ability to capture evolving conditional distributions over time, enabling more accurate predictions in dynamic environments. Additionally, extending the setting of our approach beyond one-dimensional responses could further broaden its scope and impact. Finally, exploring interpretability techniques to elucidate the underlying mechanisms driving the learned conditional distributions could enhance understanding of the model's predictions in real-world decision-making scenarios.
+
+Acknowledgments. The work of Binh Nguyen is supported by Singapore's Ministry of Education grant A0004595-00-00. Viet Anh Nguyen gratefully acknowledges the generous support from the CUHK's Improvement on Competitiveness in Hiring New Faculties Funding Scheme and the CUHK's Direct Grant Project Number 4055191.
+
+# Impact Statement
+
+Our paper presents an advancement in the field of machine learning and also has potential societal implications. By improving the understanding and capabilities of conditional distribution modeling, our framework can contribute to various real-world applications. For instance, it can aid in personalized treatment planning in healthcare by modeling conditional distributions of patient responses to different therapies. In finance, it can enhance risk assessment by predicting conditional distributions of asset returns under varying market conditions. By enabling more accurate and nuanced predictions in diverse domains, our research can positively impact decision-making processes, resource allocation, and, ultimately, the well-being of individuals and society.
+
+![](images/2ef7822bd95191795abaaddc0c805c9df4f5d6d7f80e0be00282d28113227f18.jpg)
+
+![](images/c5ef323c31f6cd75bc6c875ee1e4c8ed033a79ca18c235b15e3498156fae7c85.jpg)
+
+![](images/53bbad6bc1506b477024f16cd27ab8d73ebbcd4b8b29c88b9da1b1a7fb1ff2e6.jpg)
+
+![](images/342a5fb4651403b349e9423b642348b8a7094150d9607fa24274c9bd80d97712.jpg)  
+Figure 3. The qualitative results of methods on the LDW-CPS dataset. The density graph 'GT', 'GENTLE' (ours), 'CWGAN', 'WGAN-GP', 'MGAN', 'CDSB' is constructed by applying kernel density estimate (KDE) on  $Y_{\mathrm{GT}}(x)$ ,  $Y_{\mathrm{GENTLE}}(x)$ ,  $Y_{\mathrm{CWGAN}}(x)$ ,  $Y_{\mathrm{WGAN-GP}}(x)$ ,  $Y_{\mathrm{MGAN}}(x)$ ,  $Y_{\mathrm{CDSB}}(x)$ , respectively. Each subfigure is for a different value  $x$  on the test set. We observe that GEN-TLE can produce the true distribution much more effectively than baselines.
+
+![](images/d2664c5cb82a043d3ff41aac8973435112bd350bd7f0546ae56ed4a0aa0b2fbe.jpg)
+
+![](images/caabb5cc1963b45e83528f96a7c35ee749ad3aac1b998f3373561e34336ffd75.jpg)
+
+![](images/24a8b1f67960ccc68b441e118dc9e3fa46afc63c3deabcd574a0dae3f5133704.jpg)
+
+![](images/67ca8de6f1a8d7c1d473640344a065a6de1843ae789aba6c329cb46ffd88dc65.jpg)
+
+![](images/e2c0e6cce9053b66860b337ceabc9ad451b2d1b3591849930b7b5ec2ab93ca2a.jpg)
+
+![](images/4a43aef02931b9bd96efd5058b601a67af58f0e767711055d62c729cbbc9acf1.jpg)  
+Figure 4. The qualitative results of methods on the ECM dataset. The density graph 'GT', 'GENTLE' (ours), 'CWGAN', 'WGAN-GP', 'MGAN', 'CDSB' is constructed by applying kernel density estimate (KDE) on  $Y_{\mathrm{GT}}(x)$ ,  $Y_{\mathrm{GENTLE}}(x)$ ,  $Y_{\mathrm{CWGAN}}(x)$ ,  $Y_{\mathrm{WGAN-GP}}(x)$ ,  $Y_{\mathrm{MGAN}}(x)$ ,  $Y_{\mathrm{CDSB}}(x)$ , respectively. Each subfigure is for a different value  $x$  on the test set. We observe that GENTLE can produce the true distribution much more effectively than baselines.
+
+![](images/7e2a3ec168b03536ad205906a494eb77e79c9435c3babec1e36860f2e5d0a0e9.jpg)
+
+![](images/639f5a1158feeb6e32b16691b86b238ef048a54ff62259b65694c39d173fd06c.jpg)
+
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+
+# A. Additional Experimental Details
+
+# A.1. Parameter Tuning
+
+We heuristically fix the learning rates  $\alpha = \beta = 0.001$  as a reasonable value for almost every simple multilayer perception model. For other parameters, we conducted a grid search on minimizing the WD metric of the validation set. We found that the best combination for our parameters in the LDW-CPS dataset is KDE bandwidth of  $h = 0.3$ ,  $\varepsilon = 1.0$ ,  $\lambda = 0.4$ ,  $r_1 = 3.0$ ,  $r_2 = 2.0$ ,  $\gamma = 0.5$ ,  $\delta = 0.7$ . Regarding the ECM dataset, the best combination is  $h = 0.2$ ,  $\varepsilon = 1.0$ ,  $\lambda = 0.4$ ,  $r_1 = 3.0$ ,  $r_2 = 2.0$ ,  $\gamma = 0.5$ ,  $\delta = 0.7$ . We fix these parameter values for all our experiments in the main paper.
+
+# A.2. Comparison between GENTLE and Baselines on Other Metrics
+
+In this section, we report two additional metrics, including Mean Squared Error (MSE) and  $R^2$  score, to evaluate the performance of GENTLE and baselines. Tables 3 and 4 represent the average results of algorithms across ten different seed numbers in terms of these two metrics. We observe that GENTLE outperforms baselines on both MSE and  $R^2$  criteria. Although MGAN has comparable results to GENTLE in terms of MSE and  $R^2$ , the qualitative results shown in Figures 3 and 4 and the statistical metrics of WD and KS in Tables 1 reveals its trend of collapsing to the mean of the ground truth distribution.
+
+Table 3. MSE and  $R^2$  on generated  $\hat{Y}(x)$  using different methods vs. corresponding ground truth on the LDW-CPS dataset. The symbol  $\pm$  represents the standard deviation.  
+
+<table><tr><td>Method</td><td>MSE (×106)(↓)</td><td>R2(↑)</td></tr><tr><td>GENTLE</td><td>3.70 ± 0.73</td><td>0.92 ± 0.05</td></tr><tr><td>CWGAN</td><td>108.43 ± 42.93</td><td>-1.83 ± 0.13</td></tr><tr><td>MGAN</td><td>3.82 ± 0.71</td><td>0.92 ± 0.09</td></tr><tr><td>WGAN-GP</td><td>59.64 ± 9.65</td><td>-0.59 ± 0.07</td></tr><tr><td>CDSB</td><td>8.54 ± 1.21</td><td>0.77 ± 0.08</td></tr></table>
+
+Table 4. MSE and  $R^2$  on generated  $\hat{Y}(x)$  using different methods vs. corresponding ground truth on the ECM dataset. The symbol  $\pm$  represents the standard deviation.  
+
+<table><tr><td>Method</td><td>MSE (↓)</td><td>R2(↑)</td></tr><tr><td>GENTLE</td><td>0.62 ± 0.32</td><td>0.96 ± 0.05</td></tr><tr><td>CWGAN</td><td>24.30 ± 4.36</td><td>0.00 ± 0.13</td></tr><tr><td>MGAN</td><td>1.24 ± 0.22</td><td>0.91 ± 0.10</td></tr><tr><td>WGAN-GP</td><td>2.11 ± 0.33</td><td>0.86 ± 0.08</td></tr><tr><td>CDSB</td><td>2.01 ± 0.39</td><td>0.89 ± 0.04</td></tr></table>
+
+# A.3. Additional Experimental Results
+
+In Figure 5, we plot the value of  $T_{\theta}(x,U)$  for an increasing grid of  $U \sim \mathcal{U}(0,1)$ . We observe that the output of  $T_{\theta}$  also increased with  $U$ , empirically confirming that  $T_{\theta}$  is an increasing function, which conforms with the fact that  $T_{\theta}(x,U)$  is an approximation of an increasing function of  $U$ .
+
+![](images/fde31d5e475f170dbba5b6effe9d488fe6f3cc35fab0f274d7fcee44dd944454.jpg)  
+(a) GENTLE trained on LDW-CPS
+
+![](images/3a89c47cbf2e9677496f78b395da42ab901d9a8a7cf56e78a80e3022c5e3f18b.jpg)  
+(b) GENTLE trained on ECM  
+Figure 5. Empirical evidence for the monotonically increasing property of the learned network  $T_{\theta}(x,U)$  in the variable  $U$ .
+
+![](images/4f93c096811597c7cf44ea7a4d955498e3c40cf68bb5a581da8dd42520fcd0d7.jpg)
+
+![](images/d0aa08d4e18ff470203b23c6374c66e8f8e08d44a9abf0e9ab7c92d361aa3d4b.jpg)
+
+![](images/ad2b2e3bd93dc0a9eb822b83d1bccd632b4fe891d75a2252a22729b997c8c140.jpg)
+
+![](images/52a3c682ab1c6a0c34649c09b519f7bd9b91016940c8cd0fd831c4ac8d6a76bb.jpg)  
+Figure 6. Qualitative result of the GENTLE version without regularization term on the LDW-CPS dataset. Each subfigure represents the results of a different covariate value in the test set. The learned distribution of this version tends to span over the range of the response term uniformly.
+
+![](images/03541ef79fe9c5583b585ba68236bc4d925f38b8986a6ad628dc5c49cfa1bcfa.jpg)
+
+![](images/cf1bf7a58a7dd5f88e023af0a2f243d28006dad93769f91647e7988a131c8dbd.jpg)
+
+![](images/cdcfd43efa1e839f9f995e6ae12a34158157e1f5420c789582e008179c4d327d.jpg)
+
+![](images/49ebb09b0c8beccb020e0f8b834acfabf9571130e589fcb02a262bf104d353ae.jpg)
+
+![](images/64ed8785e4a2ea946ed1d84f3b1fa2f87099ba3447be1d017d62b8b9803266a7.jpg)
+
+![](images/d762c98fa849a72c5ff20070967100a414d362e01284295eb723a9c6674c1032.jpg)  
+Figure 7. Qualitative result of the GENTLE version without smoothing term on the LDW-CPS dataset. Each subfigure represents the results of a different covariate value in the test set.
+
+![](images/2979fb86dfd4c18a6b01c9c1b17ed0627cb4be2f25b885d548e2daa39901f7f7.jpg)
+
+![](images/b6468a43ebb2b8c80e62049e460b772e08c230df8b0b48c206a0a7fa3f9b8fcf.jpg)
+
+![](images/10ba92ddc6d302f0f5d101b055f01b7d527583a102f6c959da5d79dcbe904264.jpg)
+
+![](images/98a8fb444421967ec607122bea13ef302b105de582a7ff28e0e04c8da60f20f8.jpg)
+
+![](images/38061b5862c257f001956ef967e305aad18c1552dec1e44dcde6ea0b914fd5ad.jpg)
+
+![](images/ba263272fe13bd089b31f0d29c9067d8ddbc4cef53849479fcc0ba9b8a244153.jpg)  
+Figure 8. Qualitative result of the GENTLE version without smoothing term on the ECM dataset. Each subfigure represents the results of a different covariate value in the test set.
+
+![](images/928348236fd46f79e60de5b36b5b98bda9c9ef05147ccae5e35a77449041c8cd.jpg)
+
+![](images/5c199af4a1d1875f860dbc3ce6b540c1cabecba57973e260382a92771ad63e01.jpg)

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