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+# Learning Constraints from Offline Demonstrations via Superior Distribution Correction Estimation
+
+Guorui Quan<sup>1</sup> Zhiqiang Xu<sup>2</sup> Guiliang Liu<sup>1*</sup>
+
+# Abstract
+
+An effective approach for learning both safety constraints and control policies is Inverse Constrained Reinforcement Learning (ICRL). Previous ICRL algorithms commonly employ an online learning framework that permits unlimited sampling from an interactive environment. This setting, however, is infeasible in many realistic applications where data collection is dangerous and expensive. To address this challenge, we propose Inverse Constrained Superior Distribution Correction Estimation (ICSDICE) as an offline ICRL solver. ICSDICE extracts feasible constraints from superior distributions, thereby highlighting policies with expert-exceeding rewards maximization ability. To estimate these distributions, ICSDICE solves a regularized dual optimization problem for safe control by exploiting the observed reward signals and expert preferences. Striving for transferable constraints and unbiased estimations, ICSDICE actively encourages sparsity and incorporates a discounting effect within the learned and observed distributions. Empirical studies show that ICSDICE outperforms other baselines by accurately recovering the constraints and adapting to high-dimensional environments. The code is available at https://github.com/quangr/ICSDICE.
+
+# 1. Introduction
+
+To deploy Reinforcement Learning (RL) algorithms for safety-critical applications, the control policy must adhere to underlying constraints in the environment (Liu et al., 2021). However, in many real-world tasks, due to the inherent
+
+*Corresponding Author  $^{1}$ School of Data Science, The Chinese University of Hong Kong, Shenzhen, Guangdong, 518172, P.R. China  $^{2}$ Machine Learning, Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, UAE. Correspondence to: Guiliang Liu <liuguiliang@cuhk.edu.cn>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+complexity of environmental dynamics, the optimal constraint is often time-varying, context-sensitive, and rooted in human experience. These constraints are challenging to specify manually with prior knowledge and may not be readily available to RL agents in policy learning.
+
+To address these problems, Inverse Constraint Reinforcement Learning (ICRL) considers inferring constraints from expert demonstrations. Existing ICRL algorithms (Scobee & Sastry, 2020; Malik et al., 2021; Liu & Zhu, 2022; Gaurav et al., 2023; Papadimitriou et al., 2023; Qiao et al., 2023; Xu & Liu, 2023) follow an online learning paradigm, where the agent explores and collects experience from an interactive environment. However, boundless exploration often involves unsafe behaviors that potentially violate underlying constraints. Such a shortcoming is fundamental since constraint violation contradicts the primary goal of safe control and may cause significant loss in practical applications, especially for algorithms that require a large number of training samples.
+
+To effectively overcome these limitations, we propose designing an offline ICRL algorithm that relies solely on a fixed dataset for constraint inference. This task is challenging, primarily due to 1) the offline dataset covers only partial knowledge of the training environment, necessitating the algorithm to handle the uncertainty in unvisited states. 2) ICRL algorithms typically depend on executing policy rollout in an online manner to learn the constraint, making it infeasible to transfer to offline settings. 3) Offline ICRL is inherently ill-posed since the ground-truth constraint is not identifiable (Liu et al., 2022) from a limited dataset. How to specify the ICRL objective in a meaningful and practical way remains a critical challenge.
+
+In this work, we address the aforementioned challenges by reformulating offline ICRL into a two-stage optimization problem, integrating policy learning with constraint inference through the estimation of superior distributions. These distributions capture high-risk behaviors that yield greater rewards than those achieved by expert agents. To estimate these distributions from the offline dataset, in the forward stage, we design a dual optimization problem for offline RL under a Constrained Markov Decision Process (CMDP). The problem solver constitutes an effective and meaningful
+
+representation of the superior distributions. Subsequently, in the inverse stage, we update the constraint by assigning costs to the high-risk region sampled from these estimated distributions. We alternatively perform the forward and backward optimizations until superior distributions match expert demonstrations.
+
+In the practical implementation, we propose an Inverse Constrained Superior Distribution Correction Estimation (ICSDICE) algorithm. ICSDICE actively encourages sparsity in the estimation of distributions, thereby capturing the minimum constraint necessary to account for expert preferences. This technique accommodates the ill-posed natures of ICRL problems and substantially improves the generalizability of the learned constraints. To stabilize training, ICSDICE utilizes the discounted probabilities to represent visitation distributions, which provides an unbiased estimation of Lagrangian multiplier.
+
+Through comparative analysis with baseline methods in both discrete and continuous settings, we find ICSDICE can consistently learn a safe policy in conjunction with a sparse cost function. To better illustrate the advantages of ICSDICE, we conduct ablation studies to evaluate its performance under different configurations and evaluate how well the constraints can transfer to new environments.
+
+# 2. Related Work
+
+We review prior works that are most related to our approach.
+
+Inverse Constrained Reinforcement Learning (ICRL). Prior ICRL methods extend the maximum entropy framework (Ziebart et al., 2008) for learned constraints from both the expert demonstrations and the interactive MDP environment. In the discrete state-action space, some recent research (Scobee & Sastry, 2020; McPherson et al., 2021) inferred the constrained sets for recording infeasible state-action pairs in Constrained MDP, but these studies were restricted to the environments with known dynamics. A subsequent work (Malik et al., 2021) extended this approach to continuous state-action spaces with unknown transition models by utilizing neural networks to approximate constraints. To enable constraint inference in stochastic environments, (Papadimitriou et al., 2023) inferred probability distributions over constraints by utilizing the Bayesian framework, and (Baert et al., 2023) incorporate the maximum causal entropy objective (Ziebart et al., 2010) into ICRL. Some recent works explore ICRL under different settings, e.g., (Gaurav et al., 2023) extended ICRL to infer soft constraints. Other works explored ICRL under the multi-agent (Liu & Zhu, 2022; 2023) and the meta learning (Liu & Zhu, 2024) setting. Striving for efficient comparisons, (Liu et al., 2023) established an ICRL benchmark across various RL domains. However, these algorithms primarily target online ICRL that utilizes interactive environments instead of offline datasets.
+
+Offline Reinforcement Learning. Offline RL utilizes a data-driven RL paradigm where the agent learns the control policy exclusively from static datasets of previously collected experiences (Levine et al., 2020). To mitigate the distributional shift between training samples and testing data, previous offline RL solutions commonly involve constraining the learned policy to the data-collecting policy (Fujimoto et al., 2019; Kumar et al., 2019), making conservative estimates of future rewards (Kumar et al., 2020; Yu et al., 2021), and developing uncertainty-aware action selector (Janner et al., 2019; Kidambi et al., 2020). Some recent advancements on Offline RL (Sikchi et al., 2024; Xu et al., 2023) studied a regularized policy optimization problem with convex objective and linear constraints. Some recent DICE-related algorithms (Nachum et al., 2019a;b; Lee et al., 2021) solve offline RL by estimating the visitation distribution corrections term of the optimal policy. A recent IRL work (Yue et al., 2023) considered recovering conservative rewards from offline datasets, but none of these methods has studied the offline constraint inference.
+
+# 3. Problem Formulation
+
+Constrained Reinforcement Learning (CRL). A CRL problem is commonly based on a stationary Constrained Markov Decision Process (CMDP)  $\mathcal{M} \cup \mathrm{c} \coloneqq (\mathcal{S}, \mathcal{A}, p_{\mathcal{T}}, r, \mathrm{c}, \epsilon, \rho_0, \gamma)$  where: 1)  $\mathcal{S}$  and  $\mathcal{A}$  denote the space of states and actions. 2)  $p_{\mathcal{T}} \in \Delta_{\mathcal{S} \times \mathcal{A}}^{\mathcal{S}}$  defines the transition distributions. 3)  $r: \mathcal{S} \times \mathcal{A} \to \mathbb{R}$  and  $\mathrm{c}: \mathcal{S} \times \mathcal{A} \to [0, C_{\max}]$  denotes the reward and cost functions. 4)  $\epsilon \in \mathbb{R}_{+}$  denotes the bound of cumulative costs. 5)  $\rho_0 \in \Delta^{\mathcal{S}}$  denotes the initial states distribution. 6)  $\gamma \in [0, 1)$  is the discount factor. The goal of CRL policy  $\pi \in \Delta_{\mathcal{S}}^{\mathcal{A}}$  is to maximize the expected discounted rewards under known constraints
+
+$$
+\arg \max  _ {\pi} \mathbb {E} _ {p \tau , \pi , \rho_ {0}} \left[ \sum_ {t = 0} ^ {T} \gamma^ {t} r \left(s _ {t}, a _ {t}\right) \right] \tag {1}
+$$
+
+$$
+\text {s . t .} \mathbb {E} _ {p \tau , \pi , \rho_ {0}} \left[ \sum_ {t = 0} ^ {T} \gamma^ {t} c \left(s _ {t}, a _ {t}\right) \right] \leq \epsilon \tag {2}
+$$
+
+Following the setting in (Malik et al., 2021), we are mainly interested in the hard constraints such that  $\epsilon = 0$ . Striving for clarity, we define the CMDP with the known cost as  $\mathcal{M} \cup c$ , and the CMDP without cost (i.e.,  $\mathrm{CMDP}\backslash c$ ) as  $\mathcal{M}$ . Accordingly, the visitation distribution  $d^{\pi} \in \Delta^{S \times \mathcal{A}}$  (i.e., the normalized occupancy measure) produced by policy  $\pi$  can be denoted as:
+
+$$
+d ^ {\pi} (s, a) = (1 - \gamma) \pi (a | s) \sum_ {t = 0} ^ {\infty} \gamma^ {t} p \left(s _ {t} = s \mid \pi\right) \tag {3}
+$$
+
+where  $p(s_{t} = s|\pi)$  defines the probability of arriving state  $s$  at time step  $t$  by performing policy  $\pi$ .
+
+Inverse Constrained Reinforcement Learning. Note that traditional CRL problems often assume the constraint signals  $\mathrm{c}(\cdot)$  are directly observable from the environment, but in real-world problems, instead of observing the constraint signals, we often have access to expert demonstrations  $\mathcal{D}_E$  that adhere to these constraints, and the agent is required to recover the constraint models from the dataset. This task is challenging because various combinations of rewards and constraints can explain the same expert demonstrations. Striving for the identifiability of solutions, ICRL algorithms (Malik et al., 2021; Liu et al., 2023; Papadimitriou et al., 2023) typically assume that reward signals are observable and the goal is to recover only the constraints, in contrast to Inverse Reinforcement Learning (IRL) (Ziebart et al., 2008), which aims to learn rewards from an unconstrained MDP.
+
+From Online to Offline ICRL. Classic ICRL algorithm typically follows an online learning paradigm where the agent iteratively collects experience by interacting with the environment and using that experience for updating constraints and policy. Nevertheless, in many realistic settings, online interaction is impractical, either because data collection is expensive (e.g., in robotics, educational agents, or healthcare) or dangerous (e.g., in autonomous driving, or healthcare). To extend ICRL to the offline setting, we formally define the problem of offline ICRL as follows:
+
+Definition 3.1. (Offline ICRL) In Offline ICRL, we are given an offline dataset  $\mathcal{D}^O = \{s_n^O,a_n^O,r_n^O\}_{n = 1}^{N_O}$ . Within this dataset, expert trajectories are denoted as  $\mathcal{D}^E = \{s_n^E,a_n^E,r_n^E\}_{n = 1}^{N_E}\subset \mathcal{D}^O$ . These expert trajectories are generated by an optimal policy  $\pi^E$  adhering to the unobserved constraints function  $c(s,a)$ . i.e.  $\pi^E = \pi_c^* \coloneqq \arg \max_{\pi}\mathbb{E}_{d^{\pi E}}[r(s,a)]$ , s.t.  $\mathbb{E}_{d^{\pi E}}[c(s,a)] \leq \epsilon$ . An estimated cost function  $\hat{c}$  is a feasible solution to ICRL if and only if the optimal solution  $\pi_c^*$  can reproduce  $d^E$ . It is important to note that we cannot directly use the divergence between the expert distribution and the learned distribution as a feasible constraint for two reasons: 1. The divergence cannot be written as an inner product of a fixed cost function and the occupancy distribution  $d$ . 2. Such a constraint is not incorporated with reward information and thus cannot distinguish between high and low reward states.
+
+In this work, we mainly study hard constraints with  $\epsilon = 0$ . This implies that the feasible set is bounded by the condition  $d(s,a)c(s,a) \leq 0$  for all state-action pairs. Accordingly, the CRL problem (2) can be equivalently formulated as:
+
+$$
+\max  _ {\pi} \mathbb {E} _ {d ^ {\pi}} [ r (s, a) ] \text {s . t .} d ^ {\pi} (s, a) c (s, a) \leq 0 \forall s, a \tag {4}
+$$
+
+# 4. Inverse Constraint Inference via Superior Distribution Correction Estimation
+
+In this section, we introduce our approach to 1) modeling the superior distributions (Section 4.1), 2) solving dual op
+
+timization problem (Section 4.2) to learn the distributions from the offline dataset and 3) inferring constraint (Section 4.3) based on these distributions.
+
+# 4.1. Superior Distributions
+
+To facilitate the cost identification, we formally define the superior distributions.
+
+Definition 4.1. The set of superior distributions, denoted as  $\mathcal{O}$ , is defined as those distributions that generated by certain  $\pi$  and achieve higher cumulative rewards than expert, denoted as  $\mathcal{O} = \{d^{\pi} : \mathbb{E}_{d^{\pi}}[r(s,a)] > \mathbb{E}_{d^{E}}[r(s,a)]\mid d^{\pi}(s,a) = \pi (a|s)\sum_{t = 0}^{\infty}\gamma^{t}P(s_{t} = s|\pi)\}$ .
+
+We now establish the connection between superior distributions  $\mathcal{O}$  and feasible solutions.
+
+Proposition 4.2. The  $c(s, a)$  is a feasible solution for the ICRL problem if and only if 1) For every  $d^* \in \mathcal{O}$ , there exists at least one state-action pair  $(s, a)$  such that  $c(s, a) > 0$  and  $d^*(s, a) > 0$  and 2) if  $d^E(s, a) > 0$ , then  $c(s, a) = 0$ .
+
+Appendix C.3 shows the proof. Intuitively, for any superior distribution  $d^{*} \in \mathcal{O}$ , there's at least one state-action pair  $(s, a)$  with  $d^{*}(s, a) > 0$  that constitutes a violation of the constraints. A direct approach to determine a feasible cost function  $c$  involves initially estimating the superior distributions set  $\mathcal{O}$ , followed by assigning a positive cost to at least one state-action pair in the support of each distribution  $d^{*}$  encompassed by this set.
+
+However, directly constructing the set  $\mathcal{O}$  is impractical, primarily due to 1)  $\mathcal{O}$  may contain an infinite number of distributions, even in discrete state-action space. 2) identifying such sets necessitates solving an offline policy evaluation problem. Learning a low variance estimation of policy values is especially challenging in an offline context.
+
+To circumvent these issues, we explicitly model a cost function  $c(s, a)$  and iteratively update this function by aligning it with the expert demonstration. Specifically, we initialize cost function  $c_0(s, a) = 0$ . At each iteration  $k$ , based on  $c_{k-1}(s, a)$ , we compute a distribution  $d_k^*$  by solving the CRL problem (4). Since it guarantees that  $d^* \in \mathcal{O}$  with the following proposition, we can maintain a subset of superior distributions by collecting the occupancy distribution of solved policies in each round:  $\mathcal{O}_k = \{d_1^*, \ldots, d_k^*\}$ .
+
+Proposition 4.3. If the learned cost function  $c_{k}(s,a)$  satisfies the condition that  $d^{E}(s,a) > 0 \Rightarrow c_{k}(s,a) = 0$ , the distribution generated by the CRL problem (4)  $d^{*}$  is in superior distributions, i.e.,  $d^{*} \in \mathcal{O}$ .
+
+Appendix C.4 shows the proof. In the next step, we update  $c_{k}(s,a)$  to ensure it's a valid cost function based on the distributions from previous  $k$  iterations  $\mathcal{O}_k$ . To achieve this, we increase cost in the support of these sampled distributions so that the next sampled distribution be closer to the expert
+
+distribution. This process terminates when  $d_k^*$  is sufficiently closed to  $d^E$ . Notably, our algorithm requires only solving a forward CRL problem (4) and a cost learning problem. It replaces the challenge of constructing the computationally-intractable set  $\mathcal{O}$  with the task of matching distributions, which can be efficiently solved with a lower variance. In the ideal case, such a process can converge to the expert policy and will converge in a finite of steps in the discrete finite MDPs, as shown in Proposition 4.4.
+
+Proposition 4.4. If the MDP is finite, this process can recover  $d^{E}$  in a finite number of steps when problem 4 can be accurately solved.
+
+It's worth noting that a similar convergence analysis is shown in (Gaurav et al., 2023). However, they don't explicitly introduce the concept of superior distributions and rely on additional assumptions about the MDP.
+
+The concept of superior distributions is key to inverse constraint reinforcement, we introduce the connection of our approach to adversarial imitation learning through the lens of superior distributions in Appendix B.
+
+# 4.2. Offline Identification for Superior Distribution via Dual Optimization
+
+In this section, we introduce the approach to identifying the superior distributions  $d^{*}$  under the offline learning setting. To achieve this goal, we regularize the cumulative rewards in the objective (4) with the offline demonstration. The new objective is given by: (5)
+
+$$
+J (d) = \mathbb {E} _ {d (s, a)} [ r (s, a) ] - \xi_ {r} D _ {f} (d (s, a) | | d ^ {O} (s, a))
+$$
+
+where  $d^{O}$  represents the visitation distribution in the offline dataset  $\mathcal{D}^{\mathrm{O}}$ , and  $D_{f}(\cdot ||\cdot)$  denotes the  $f$ -divergence between two distributions. Intuitively, instead of maximizing only the reward, we augment the cumulative reward with a divergence regularizer to prevent it from deviating beyond the coverage of the offline data. Such an objective guarantees that the agent adheres to a conservative policy.
+
+Since our goal is to identify the occupancy distribution, instead of estimating  $\hat{\pi}$ , we incorporate the Bellman flow constraint to the objective and directly solve the occupancy distribution  $d$  through a convex problem:
+
+$$
+\max  _ {d (s, a) c (s, a) \leq 0, d (s, a) \geq 0} \mathbb {E} _ {d} [ r (s, a) ] - \xi_ {r} D _ {f} (d | | d ^ {O}) \tag {6}
+$$
+
+$$
+\begin{array}{l} \begin{array}{l} \text {s . t .} \sum_ {a \in \mathcal {A}} d (s, a) = (1 - \gamma) \rho_ {0} (s) + \end{array} \\ \gamma \sum_ {s ^ {\prime} \in \mathcal {S}, a ^ {\prime} \in \mathcal {A}} d (s ^ {\prime}, a ^ {\prime}) p (s | s ^ {\prime}, a ^ {\prime}), \forall s \in \mathcal {S} \\ \end{array}
+$$
+
+To derive a solution for the problem (6), we introduce a Lagrangian dual variable  $V$  and construct a dual problem by following (Lee et al., 2021; Sikchi et al., 2024). Appendix C.1 shows the detailed derivation.
+
+$$
+\begin{array}{l} \min  _ {V} \max  _ {d} \mathbb {E} _ {d} [ \delta_ {V} ] - \xi_ {r} D _ {f} (d | | d ^ {O}) + (1 - \gamma) \mathbb {E} _ {\rho_ {0}} [ V ] (7) \\ \text {s . t .} d (s, a) c (s, a) \leq 0 \text {a n d} d (s, a) \geq 0 (8) \\ \end{array}
+$$
+
+where
+
+$$
+\delta_ {V} (s, a) = r (s, a) + \sum_ {s ^ {\prime} \in \mathcal {S}} p _ {\mathcal {T}} \left(s ^ {\prime} \mid s, a\right) \gamma V \left(s ^ {\prime}\right) - V (s)
+$$
+
+We provide the closed-form solution for the inner optimization problem in Equation (7) in the following:
+
+Proposition 4.5. Assume the learned distribution  $d(s, a) > 0$  for all  $(s, a)$  such that  $d^{O}(s, a) > 0$ . The optimal solution for the inner optimization problem in (7), denoted as  $d^{*}$ , is given by:
+
+$$
+d ^ {*} (s, a) = d ^ {O} (s, a) \mathbb {1} _ {c (s, a) = 0} w _ {f} ^ {*} (\delta_ {V} (s, a)) \tag {9}
+$$
+
+Substituting  $d^{*}$  into problem (7), the problem becomes: (10)
+
+$$
+\min  _ {V} \mathbb {E} _ {d ^ {O}} \left[ \xi_ {r} \mathbb {1} _ {c (s, a) = 0} f _ {p} ^ {*} \left(\frac {\delta_ {V} (s , a)}{\xi_ {r}}\right) \right] + (1 - \gamma) \mathbb {E} _ {\rho_ {0}} [ V ]
+$$
+
+where  $w_{f}^{*}$  and  $f_{p}^{*}$  is related to the convex function  $f$  specified in  $f$ -divergence:  $w_{f}^{*}(y) = \max(0, f'^{-1}(\frac{y}{\xi_{r}}))$ ,  $f_{p}^{*}(y) = \mathbb{E}_{w^{*}}[y] - f(w^{*}(y))$ .
+
+The proof is in Appendix C.2. In the representation of superior distributions  $d^{*}$  (Equation 9),  $\omega^{*}$  denotes a correction term in Distribution Correction Estimation (DICE) (Nachum et al., 2019a), and its implementation is vital for controlling the sparsity of the estimated constraint (Section 5.1). At each interaction  $k$ , we estimate a  $d_{k}^{*}$ , and record it to the set  $\mathcal{O}_k = \mathcal{O}_{k - 1}\cup d_k^*$ , with which we learn the cost function as follows.
+
+# 4.3. Constraint Inference from Distribution Set
+
+Given the set of estimated superior distributions  $\mathcal{O}_k$ , we assign high costs to state-action pairs in the support of these distributions (i.e.,  $\{(s,a) \mid d^*(s,a) > 0 \land d^* \in \mathcal{O}_k\}$ ) by following Proposition 4.2. To achieve this goal, we design the following objective to update the cost function.
+
+$$
+c _ {k} = \underset {c} {\arg \min } \mathbb {E} _ {d ^ {*} \in \mathcal {O} _ {k}} \left[ \mathbb {E} _ {d ^ {*} (s, a)} \left[ (c (s, a) - 1) ^ {2}) \right] + \psi (c) \right]
+$$
+
+$$
+\mathrm {s . t .} \mathbb {E} _ {d ^ {E} (s, a)} [ c (s, a) ] = 0 \tag {11}
+$$
+
+where we optimize the cost by adjusting mean cost to 1 for state-action pairs with positive probability in  $d^{*} \in \mathcal{O}$  and  $\psi$  denotes a cost regularizer.
+
+In our experiment, we introduce the L2 norm of the cost function as the cost regularizer, and by utilizing the Lagrange variable  $\lambda_{e}$ , the final objective can be defined as:
+
+$$
+\begin{array}{l} c _ {k} = \underset {c} {\arg \min } \mathbb {E} _ {d ^ {*} \in \mathcal {O} _ {k}} \left[ \mathbb {E} _ {d ^ {*}} \left[ (c (s, a) - 1) ^ {2} + \lambda_ {\mathrm {L} 2} c (s, a) ^ {2} \right] \right] \\ + \lambda_ {e} \mathbb {E} _ {d ^ {E} (s, a)} [ c (s, a) ^ {2} ] \tag {12} \\ \end{array}
+$$
+
+# 5. Practical Implementation
+
+In this section, we present the practical implementation of ICSDICE (see Algorithm 1) by introducing the approach to: 1) Encouraging sparsity in the learned constraints (Section 5.1), 2) Utilizing the discounted visitation distribution to enable the recovery of right sparse superior visitation distributions (Section 5.2), and 3) Extracting policy from the learned distribution (Section 5.3).
+
+# 5.1. Encouraging the Sparsity of Inferred Constraints
+
+The Importance of Sparsity. Similar to IRL, the optimal constraint in ICRL is not uniquely identifiable, indicating that multiple constraints may equivalently explain the expert behaviors. To address this issue, ICRL algorithms aim to learn the most sparse constraint under which the imitation agent can reproduce the behaviors of the expert (Scobee & Sastry, 2020). Additionally, sparsity is critical for guaranteeing the applicability of the learned constraints. This is because the lack of sparsity can lead to learning an excessively restrictive constraint that blocks all the movement not covered by the expert dataset. While such a constraint may accurately reproduce the observed expert behaviors, it lacks practical utility for two reasons: 1) The offline dataset may not cover all potential actions an expert agent could execute (i.e.,  $\{a \mid \pi^E(a|s) > 0\}$ ), thereby resulting in the exclusion of many feasible actions due to overly restrictive constraints. and 2) it cannot be generalized to environments with even minor modifications to rewards or dynamics (see experiments in Section 6.3), which is a critical challenge in bridging the Sim-to-Real gap in practice.
+
+From Cost Sparsity to Distributional Sparsity. In accordance with Proposition 4.2, positive costs are allocated exclusively to the state-action pairs within the set  $\{(s,a) \mid d^{*}(s,a) > 0 \land d^{*} \in \mathcal{O}\}$ . Consequently, when the estimated set superior distributions  $d^{*}$  are sparse, meaning there are relatively few state-action pairs where  $d^{*}(s,a) > 0$ , the positive costs will be assigned to a smaller number of state-action pairs, which efficiently encourages the sparsity of constraint.
+
+To learn a more sparse distribution, we utilize the chi-square distance as the  $f$ -divergence metric  $D_{f}$  in objective (7). Specifically, the corresponding convex function  $f$  and the superior distributions are represented as:
+
+$$
+f (x) = (x - 1) ^ {2} \text {a n d} f ^ {\prime - 1} (x) = 1 + \frac {x}{2} \tag {13}
+$$
+
+$$
+\frac {d ^ {*} (s , a)}{d ^ {O} (s , a)} = w _ {f} ^ {*} (s, a) = \left(1 + \frac {\delta_ {V} (s , a)}{2 \xi_ {r}}\right) _ {+} \tag {14}
+$$
+
+This representation can facilitate sparsity, primarily because in the cost learning step, if the temporal difference  $\delta_V(s,a)\leq -2\xi_r$ , objective (14) ensures  $d^{*}(s,a) = 0$  for this  $(s,a)$ , indicating the following movements will not in
+
+fluence the costs. As a result, positive costs can not be assigned to the low-reward state-action pairs, which encourages the sparsity of  $c(s,a)$ . By reducing  $\xi_r$  (i.e., the tolerant threshold), the impact of this clipping mechanism becomes more influential. In this way, we can control the sparsity of the learned distribution by adjusting  $\xi_r$ . On the other hand, this method makes the cost function only sensitive to regions exhibiting constraint-violating behaviors, particularly those associated with rewards exceeding the expert level.
+
+# 5.2. Utilizing the Discounted Visitation Distribution
+
+While previous DICE-related algorithms (Kostrikov et al., 2020; Lee et al., 2021; Ma et al., 2022a) commonly sample data by following the empirical visitation distribution  $d^{O}$  in the offline dataset  $\mathcal{D}^{O}$ , we propose utilizing a discounted visitation distribution in our training objective (10).
+
+Definition 5.1. Let  $\pi^O$  denote the sampling policy for the offline dataset  $\mathcal{D}^O$ . The discounted visitation distribution is represented as:
+
+$$
+d _ {\gamma} ^ {O} (s, a) = \sum_ {k = 0} ^ {\infty} \gamma^ {k} P (s, a | \rho_ {0}, \pi^ {O}) \tag {15}
+$$
+
+After replacing  $d^{O}(s,a)$  with  $d_{\gamma}^{O}(s,a)$  in our previous objective (10), we obtain a new objective for updating the value functions:
+
+$$
+\begin{array}{l} \mathbb {E} _ {d _ {\gamma} ^ {O}} \left[ \xi_ {r} \mathbb {1} _ {c (s, a) \leq 0} f _ {p} ^ {*} \left(\frac {\delta_ {V} (s , a)}{\xi_ {r}}\right) \right] + (1 - \gamma) \mathbb {E} _ {\rho_ {0} (s)} [ V (s) ] \tag {16} \\ = \mathbb {E} _ {d _ {\gamma} ^ {O}} \left[ \xi_ {r} \mathbb {1} _ {c (s, a) \leq 0} f _ {p} ^ {*} (\frac {\delta_ {V} (s , a)}{\xi_ {r}}) + (1 - \gamma) \frac {\rho_ {0} (s)}{d _ {\gamma} ^ {O} (s)} V (s) \right] \\ \end{array}
+$$
+
+Such a replacement offers several benefits: 1) The discount factor is crucial in defining the occupancy measure; thus, incorporating this factor into the construction of the visitation distribution is essential. 2) It introduces a more principled approach to managing values in absorbing states. To elucidate, we may interpret the left term in the preceding objective as follows:
+
+$$
+\mathbb {E} _ {d _ {\gamma} ^ {O}} \left[ \xi_ {r} \mathbb {1} _ {c (s, a) = 0} f _ {p} ^ {*} \left(\frac {\delta_ {V} (s , a)}{\xi_ {r}}\right) \right] = \tag {17}
+$$
+
+$$
+\begin{array}{l} \sum_ {t = 0} ^ {T - 1} \lambda^ {t} \mathbb {1} _ {c (s _ {t}, a _ {t}) = 0} f _ {p} ^ {*} \left(\frac {\gamma V (s _ {t + 1}) + r (s _ {t} , a _ {t}) - V (s ^ {t})}{\alpha}\right) \\ + \sum_ {t = T} ^ {\infty} \lambda^ {t} f _ {p} ^ {*} \left(\frac {\gamma V (s _ {T}) - V (s _ {T})}{\alpha}\right) \\ \end{array}
+$$
+
+where  $s_T$  denotes an absorbing state occurring at time  $T$ . The value of  $s_T$  is updated to maximize  $\frac{\lambda^T}{1 - \lambda} f_p^* \left[ \frac{(\gamma - 1)}{\alpha} V(s_T) \right]$  as opposite to the previous methods. For example, striving for simplicity, OptiDICE (Lee et al., 2021) enforces  $V(s_T)$  to zero. This conventional approach
+
+is inadequate because  $V$  serves as a Lagrangian multiplier (see objective 6) in the optimization problem.
+
+To demonstrate the importance of our update, we conducted experiments in the Point Maze environment(Fu et al., 2020) to examine whether the algorithm can identify the superior distribution  $d^{*}$  under the offline learning setting. We demonstrated the effects of our modified learning objective against OptiDICE (Lee et al., 2021) and the ablation of our algorithm that utilizes the undiscounted  $d^{O}(s,a)$ . Figure 1 visualize the learned distribution. While our ICS-DICE objective can learn a sparse distribution capturing the shortest path to the destination, both OptiDICE and the ablated ICSDICE objective fail to achieve this goal due to their biased estimate. Besides, we find as  $\xi_{r}$  decreases, the distribution learned by our ICSDICE objective becomes more sparse, which well aligns with our intuition in Section 5.1.
+
+![](images/fa7a81f088468940dc1c4bfac67318b518148d19f30b8a2df25a2157da7e4fe6.jpg)  
+Figure 1: A 2D maze example for comparing different DICE objectives, varying different  $\xi_r$ . We use a green circle to mark the start point, and a red line represents the finish line.
+
+# 5.3. Extracting Policy from the Learned Distribution
+
+Once we learned a matched distribution  $d^{*}$  with  $d^{E}$ , the policy  $\pi$  defined by optimal distribution  $d^{*}$  is given by  $\pi(a|s) = \frac{d(s, a)}{\sum_{a'} d(s, a')}$ . We can extract the policy by maximizing  $\mathbb{E}_{d^{*}(s, a)}[\log \pi(a|s)] = \mathbb{E}_{d^{\mathcal{O}}(s, a)}[\omega^{*}(s, a)\log \pi(a|s)]$ . To stabilize the training, we utilize the advantage-weighted regression (Peters & Schaal, 2007) method as follows:
+
+$$
+\max  \mathbb {E} _ {d ^ {O} (s, a)} \left[ e ^ {\delta_ {V} (s, a)} \log \pi (a | s) \right] \tag {18}
+$$
+
+We also use a max clipping operation on  $\delta_V$  in case its values are too large, which could cause numeric instability during training.
+
+# 6. Empirical Evaluation
+
+# 6.1. Discrete Grid-world Environment
+
+We utilize grid-world environments for evaluating our algorithm. These environments consist of 7x7 discrete maps, and each map has a unique constraint setting (First row in Figure 2). Within these environments, each agent is permitted to perform eight actions: moving up, down, right,
+
+Algorithm 1 Inverse Constrained Superior Distribution Correction Estimation (ICSDICE)
+
+Require: Offline dataset  $\mathcal{D}^O = \{\mathcal{D}^E,\mathcal{D}^{-E}\}$  , Running iterations  $K$  ; Initialize  $c_{0}(\cdot) = 0,\mathcal{O}_{0} = \{\}$
+
+1: for  $k = 1 \dots K$  iterations do  
+2: Learn the value function  $V_{k}$  with the dual optimization objective (16) based on cost function  $c_{k - 1}$ ;  
+3: Solve the distribution  $d_k^*$  with objective (9);  
+4: Update the distribution set  $\mathcal{O}_k = \mathcal{O}_{k - 1}\cup \{d_k^*\}$  
+5: Update the cost function  $c_{k}(s,a)$  with objective (12).  
+6: Extract policy  $\pi$  with objective (18);  
+7: end for
+
+left, or in one of four diagonal directions. The primary objective for every agent is to navigate from the start to the end point, taking the shortest possible route while avoiding specific constrained states. The agent receives a -1 reward for each step taken until it reaches its destination. To enable offline ICRL, we provide an expert dataset  $\mathcal{D}^E$  and a suboptimal dataset  $\mathcal{D}^{\neg E}$  (collected through random walks) for each environment. The size of the sub-optimal dataset is 50 (trajectories) (Check details in Appendix A.1).
+
+Constraint Visualization. Figure 2 illustrates the information captured by different methods from an offline dataset. We find ICSDICE learns sparse constraints that are most similar to the ground-truth ones. When it comes to comparison methods, Offline IL (i.e., Inverse Q-learning (AlHafez et al., 2023)) learns a Q function representing reward. We observe that the value function in each state learned by the imitation model fails to accurately capture the constraint, especially in settings 1 and 4. This highlights that the constraint cannot be captured by traditional IRL algorithms. The method, Offline IL + rewards, is an extension of Inverse Q learning, where we incorporate rewards into the Q-learning process by modeling the learned rewards as  $\bar{r}(s,a) = r(s,a) + c(s,a)$ , where  $r(s,a)$  represents the known rewards. This enables the cost signals to be extracted from the learned Q function. For additional details, please refer to Appendix A.4. We observe that after incorporating rewards, the learned value function becomes overly conservative and thus can only represent part of cost information. This indicates that incorporating reward signals into IRL is insufficient to learn a well-defined cost function.
+
+Additionally, Figure 3 visualizes the learned superior distribution  $d_k^*$  and the learned cost function  $c_k$  from the learning process. Our method successfully samples sparse  $d^*$  from  $\mathcal{O}$  and effectively learns a sparse cost function by capturing sparse superior distributions.
+
+# 6.2. High-Dimensional Continuous Environment
+
+We extend MuJoCo environments (Todorov et al., 2012) for evaluating ICRL algorithms by incorporating predefined
+
+![](images/e928a96813b2a81f7a8e0d6e2cfc62a78925f19f1dc3606ff83f3aad0a7b29e1.jpg)  
+Figure 2: Visualizing the constraints captured by different methods based on an offline dataset. Blue, red mark the starting, target states.
+
+![](images/d4da1264ed8bf96733660493d8b0a5ef8e24f9ce528e04eef84a9abd0f123582.jpg)  
+Figure 3: Visualizing the support of superior distributions (top) and constraints (bottom) in the setting 1 from the beginning (left) until the end (right) of training.
+
+constraints into each environment. Inspired by (Liu et al., 2023), the first type of constraint, marked by HalfCheetah(Obstacle), prohibits the robot from moving backward, although it's relatively easier for the robot to move backward than forward. The second type of constraint, denoted as Ant(LS) and Walker(LS), limits the robot's maximum speed of moving forward. The third type of constraint, denoted as Ant(Blocked) and Hopper(Blocked), restricts the robot leg's angular velocity and thus constricts the size of each move. For further details, please refer to Appendix A.2.
+
+Experiment Setting. By following (Malik et al., 2021), we adopt the following evaluation metrics: 1) Constraint Violation Rate, which assesses the likelihood for a policy to violate a constraint within a given trajectory, and 2) Feasible Cumulative Rewards, which calculates the total rewards accumulated by the agent before violating any constraints. We run experiments with 5 different seeds and present the mean  $\pm$  std results for each algorithm. Appendix A.5 reports the detailed settings and random seeds.
+
+Comparison Methods. Due to the lack of offline ICRL baselines, we compare ICSDICE with the following baselines: 1) LS-IQ follows the implementation of Least Square Inverse Q-Learning (Al-Hafez et al., 2023) method that infers reward value functions from offline data to imitate expert policy. 2) SMODICE refers to the recently proposed
+
+SMODICE (Ma et al., 2022b) algorithm that leverages the dual and offline reward function to control. 3) Behavior Cloning. As these methods do not utilize reward information, we introduce two additional baselines: 1) OptiDICE-Constraint replaces our DICE objective in Section 5.2 with OptiDICE. 2) SMODICE-Constraint Based on SMODICE, we incorporate rewards information by adding environment rewards to the SMODICE's learned discriminator reward. For additional details, please refer to Appendix A.4.
+
+Table 1 shows the evaluation results, and Figure 7 illustrates the corresponding learning curve is shown in. Across all environments, ICSDICE achieves leading performance in imitating high-performing policies offline while maintaining policy safety. The offline IL methods cannot ensure lower cost even when incorporated with rewards information. This limitation is understandable since these methods are not specifically designed for safety concerns.
+
+One intriguing observation is that OptiDICE-Constraint performs comparably to our method. Given the tricks applied by OptiDICE-c (normalization and soft-chi divergence), it's not surprising that OptiDICE-c may perform slightly better than our algorithm in the task of recovering policy. However, it's important to note that this improvement comes at the cost of learning a dense superior distribution and the inaccurate constraint. To understand the difference to our ICSDICE, we investigate whether these ICRL methods have learned the true cost by utilizing the ROC (Receiver Operating Characteristic) curve to illustrate the relationship between the learned cost and the ground truth cost. As shown in Figure 5, although OptiDICE-Constraint exhibits similar performance in policy learning, our ICSDICE achieves notably higher performance in cost identification. This is because OptiDICE-Constraint tends to have a high false positive rate due to its biased estimation of the distribution (See Section 5.2). Additionally, as  $\xi_r$  decreases, the curve shifts to the left, indicating the learned constraint becomes more sparse and thus the false positive errors become smaller. For more evidence, Figure 8 visualizes the learned constraints.
+
+# 6.3. Generalizability of Constraints
+
+To examine whether the learned constraint can be transferred to new configurations, we apply our methods and OptiDICE-Constraint to learn cost functions in the limited-speed Ant environment. Then, we modify the environment setting by rewarding the velocity in a distinct direction. We utilize the forward objective (10) to learn policy based on these costs under new rewards. Ideally, the transfer behaviors should move in the rewarded direction while adhering to the same constraint on the x-axis velocity. For more details, please refer to Appendix A.3. Additionally, we used a zero-cost function (denoted as w/o cost) for comparison.
+
+Figure 4 shows the learning curve of cumulative rewards
+
+Table 1: MuJoCo testing performance. We report average cumulative rewards and costs over 10 runs. Bold denotes safe methods with a violation rate below  $5\%$ , gray denotes unsafe methods with a violation rate above  $5\%$ , and blue highlights methods whose standard deviation range includes the average performance of the top-performing method. The percentage in the dataset column indicates how many percent of expert transitions are known as expert demonstrations.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Task</td><td colspan="2">BC</td><td colspan="2">LS-IQ</td><td colspan="2">SMODICE</td><td colspan="2">SMODICE-c</td><td colspan="2">OptiDICE-c</td><td colspan="2">Our Method</td></tr><tr><td>cost ↓</td><td>reward ↑</td><td>cost ↓</td><td>reward ↑</td><td>cost ↓</td><td>reward ↑</td><td>cost ↓</td><td>reward ↑</td><td>cost ↓</td><td>reward ↑</td><td>cost ↓</td><td>reward ↑</td></tr><tr><td rowspan="5">100% expert</td><td>Ant(Blocked)</td><td>0.04±0.02</td><td>876±138</td><td>0.40±0.12</td><td>-63±208</td><td>0.33±0.07</td><td>1,410±153</td><td>0.53±0.03</td><td>1,763±180</td><td>0.01±0.00</td><td>3,070±91</td><td>0.01±0.00</td><td>3,073±103</td></tr><tr><td>Ant(LS)</td><td>0.00±0.00</td><td>1,106±93</td><td>0.07±0.06</td><td>-1,232±1,616</td><td>0.06±0.05</td><td>1,341±50</td><td>0.03±0.04</td><td>1,361±67</td><td>0.00±0.00</td><td>1,368±3</td><td>0.00±0.00</td><td>1,340±44</td></tr><tr><td>HalfCheetah(Obstacle)</td><td>0.30±0.40</td><td>731±693</td><td>0.04±0.05</td><td>2,175±775</td><td>0.13±0.11</td><td>3,565±345</td><td>0.30±0.24</td><td>3,829±661</td><td>0.03±0.06</td><td>2,749±597</td><td>0.04±0.04</td><td>2,315±740</td></tr><tr><td>Hopper(Blocked)</td><td>0.40±0.05</td><td>158±78</td><td>0.07±0.02</td><td>350±242</td><td>0.23±0.12</td><td>2,024±657</td><td>0.19±0.21</td><td>1,609±907</td><td>0.00±0.00</td><td>1,255±576</td><td>0.02±0.02</td><td>1,017±341</td></tr><tr><td>Walker(LS)</td><td>0.01±0.02</td><td>-7±5</td><td>0.17±0.09</td><td>603±203</td><td>0.52±0.19</td><td>2,334±238</td><td>0.18±0.15</td><td>1,871±155</td><td>0.01±0.01</td><td>1,538±283</td><td>0.01±0.02</td><td>1,587±308</td></tr><tr><td rowspan="5">50% expert</td><td>Ant(Blocked)</td><td>0.03±0.01</td><td>1,012±258</td><td>0.35±0.16</td><td>-14±70</td><td>0.04±0.03</td><td>2,856±266</td><td>0.18±0.07</td><td>2,574±284</td><td>0.01±0.00</td><td>3,069±141</td><td>0.01±0.00</td><td>3,025±101</td></tr><tr><td>Ant(LS)</td><td>0.00±0.00</td><td>1,047±88</td><td>0.10±0.00</td><td>-125±187</td><td>0.25±0.24</td><td>1,516±234</td><td>0.22±0.13</td><td>1,422±106</td><td>0.00±0.00</td><td>1,361±9</td><td>0.00±0.00</td><td>1,364±5</td></tr><tr><td>HalfCheetah(Obstacle)</td><td>0.08±0.11</td><td>938±807</td><td>0.03±0.04</td><td>2,097±365</td><td>0.22±0.13</td><td>3,796±170</td><td>0.21±0.13</td><td>3,693±434</td><td>0.01±0.02</td><td>2,086±1,153</td><td>0.00±0.01</td><td>1,657±723</td></tr><tr><td>Hopper(Blocked)</td><td>0.46±0.20</td><td>209±77</td><td>0.07±0.11</td><td>245±146</td><td>0.26±0.27</td><td>1,589±819</td><td>0.33±0.12</td><td>1,166±692</td><td>0.04±0.03</td><td>1,327±885</td><td>0.02±0.01</td><td>1,101±714</td></tr><tr><td>Walker(LS)</td><td>0.00±0.00</td><td>-9±3</td><td>0.02±0.03</td><td>123±146</td><td>0.23±0.27</td><td>1,922±209</td><td>0.49±0.26</td><td>2,191±408</td><td>0.01±0.03</td><td>1,645±73</td><td>0.05±0.07</td><td>1,168±516</td></tr></table>
+
+![](images/cf4e703c61b1627f60dc75a5d05c6fcf005f63ae7ad6a38e75278d98e4f9ab9d.jpg)  
+Figure 4: Generalization performance of learned costs. The black dashed line marks the velocity constraint.
+
+and costs in the first two plots, and the velocity distribution sampled from the different transfer policies in the third plot. We observe that, with a zero-cost function, the policy can achieve high rewards but also has a high violation rate. The OptiDICE-Constraint method's learned cost resulted in the agent being constrained to only move forward and achieve the lowest return. This occurred because it masked unnecessary states, leading to an overly conservative cost and ultimately failing during transformation. In contrast, our methods successfully enabled the agent to follow the constraint while moving in the direction aligning with the rewards. These results demonstrate that our method can effectively learn a transferable cost function.
+
+# 6.4. Realistic Environments
+
+To evaluate the practical applicability of our algorithm, we examined its performance in various traffic scenarios represented by highway driving datasets HighD (Krajewski et al., 2018). Additionally, we utilized the commonroad-RL (Wang et al., 2021) environments to extract relevant features from driving scenarios.
+
+We first conducted an experiment to determine if our algorithm could effectively identify and adhere to constraints related to vehicle velocity. We set the constraint as Car Velocity  $\leq 40\mathrm{m / s}$ . Table 2 presents the performance of different algorithms. Our methods outperformed the baseline BC and OptiDICE-c in terms of both cumulative returns and Area Under the Curve (AUC).
+
+To evaluate the performance on more challenging tasks in
+
+volving high-dimensional inputs like images, we replaced the input to the constraint network with raw images and modified the network structure to use a CNN instead of an MLP. We designed a constraint related to car distance (Car distance  $\geq 20\mathrm{m}$ ) and expected our method to learn this distance information from the pixels. We tested the learned cost function on a split test dataset, which included scenarios not seen during training. Our algorithm learned costs that could transfer to these settings, achieving an AUC of 0.75, while the OptiDICE-c baseline only achieved 0.69. Additionally, we visualized the images assigned with the highest cost in 5 scenarios in Figure 6. It can be clearly observed that our method assigns high costs to close car distances, while the OptiDICE-c baseline fails to do so.
+
+Table 2: Performance under HighD speed constraint setting.  
+
+<table><tr><td>Metric</td><td>BC</td><td>OptiDice-c</td><td>Our Method</td><td>Expert</td></tr><tr><td>Return</td><td>-2.3 ± 2.3</td><td>-1.6 ± 3.7</td><td>10.7 ± 1.5</td><td>14.0</td></tr><tr><td>Cost</td><td>1.4%</td><td>12%</td><td>6.5%</td><td>0.8%</td></tr><tr><td>AUC</td><td>NA</td><td>0.76 ± 0.02</td><td>0.81 ± 0.01</td><td>NA</td></tr></table>
+
+# 6.5. Ablation Study
+
+In this experiment, we conduct ablations on our algorithm's design choice. We evaluate the performance of the learned policy by cumulative return and violation rate, and assess the learned cost discriminator by comparing the Area under the ROC curve. We compare the following variants:  $w/o$  discount: Replace the discounted distribution with the empirical distribution.  $w/o$  constraint: Remove the constraint learning step.
+
+Table 3: Performance of different variants of ICSDICE.  
+
+<table><tr><td rowspan="2">Task</td><td colspan="2">Our Method</td><td colspan="2">w/o discount</td><td colspan="2">w/o constraint</td></tr><tr><td>cost ↓</td><td>reward ↑</td><td>cost ↓</td><td>reward ↑</td><td>cost ↓</td><td>reward ↑</td></tr><tr><td>Ant(Blocked)</td><td>0.01</td><td>3,073</td><td>0.05</td><td>2,894</td><td>0.21</td><td>1,692</td></tr><tr><td>Ant(LS)</td><td>0.00</td><td>1,340</td><td>0.00</td><td>1,369</td><td>0.18</td><td>1,381</td></tr><tr><td>HalfCheetah(Obstacle)</td><td>0.04</td><td>2,315</td><td>0.07</td><td>3,332</td><td>0.30</td><td>4,343</td></tr><tr><td>Hopper(Blocked)</td><td>0.02</td><td>1,017</td><td>0.02</td><td>1,009</td><td>0.32</td><td>2,091</td></tr><tr><td>Walker(LS)</td><td>0.01</td><td>1,587</td><td>0.13</td><td>1,256</td><td>0.29</td><td>1,942</td></tr></table>
+
+The results in Table 3 and Table 4 indicate that, by using the
+
+![](images/edf65d87acc166306bc05bbf8e4ba918eb596b20cb6bacb645595dce4e0e041a.jpg)  
+Figure 5: Receiver Operating Characteristic (ROC) Curve illustrating the accuracy of the learned cost. A curve closer to the top-left corner indicates higher accuracy.
+
+Table 4: Area under the ROC Curve (AUC) of different variants of ICSDICE.  
+
+<table><tr><td>Task</td><td>Our Method</td><td>w/o discount</td><td>OptiDice-c</td></tr><tr><td>Ant(LS)</td><td>0.94±0.01</td><td>0.98±0.01</td><td>0.60±0.04</td></tr><tr><td>Ant(Blocked)</td><td>0.91±0.01</td><td>0.88±0.01</td><td>0.54±0.08</td></tr><tr><td>Hopper(Blocked)</td><td>0.63±0.04</td><td>0.63±0.03</td><td>0.59±0.01</td></tr><tr><td>HalfCheetah(Obstacle)</td><td>0.98±0.02</td><td>0.77±0.05</td><td>0.97±0.02</td></tr><tr><td>Walker(LS)</td><td>0.93±0.02</td><td>0.86±0.00</td><td>0.80±0.03</td></tr></table>
+
+discounted distribution, our objectives can strike the right trade-off between the performance of the learned policy and the learned cost discriminator. Our approach performs very well on both the recovered policy and the discriminator performance compared to other ablation variants.
+
+# 7. Conclusion
+
+In this paper, we solve an offline ICRL by proposing a ICS-DICE algorithm. As the first attempt, ICSDICE derives superior distributions from the dual objective of regularized policy learning under a CMDP and proposes the two-stage optimization method for updating both policies and constraints. To enhance practical applicability, ICSDICE effectively addresses insufficient sparsity and biased estimation. Empirical results demonstrate the performance of ICSDICE in various settings. A promising avenue for future research involves extending our method to accommodate uncertainty raised in the offline dataset and conducting a rigorous theoretical study for analyzing the convergence rate and sample efficiency of ICSDICE.
+
+# Acknowledgements
+
+This work is supported in part by the National Key R&D Program of China under grant No.2022ZD0116004, Guangdong-Shenzhen Joint Research Fund under grant 2023A1515110617, GuangDong Basic and Applied Basic Research Foundation under grant 2024A1515012103, Shenzhen Fundamental Research Program (General Program) under grant JCYJ20230807114202005, Shenzhen Science
+
+and Technology Program ZDSYS20211021111415025, and Guangdong Provincial Key Laboratory of Mathematical Foundations for Artificial Intelligence (2023B1212010001).
+
+# Impact Statement
+
+Our approach significantly advances the ability to acquire behavioral insights from datasets, furthering the progress of user-centric artificial intelligence systems. While there are numerous potential societal implications stemming from our research, we believe that none require specific emphasis in this context.
+
+# References
+
+Al-Hafez, F., Tateo, D., Arenz, O., Zhao, G., and Peters, J. Ls-iq: Implicit reward regularization for inverse reinforcement learning. arXiv preprint arXiv:2303.00599, 2023.  
+Baert, M., Mazzaglia, P., Leroux, S., and Simoens, P. Maximum causal entropy inverse constrained reinforcement learning. arXiv preprint arXiv:2305.02857, 2023.  
+Fu, J., Kumar, A., Nachum, O., Tucker, G., and Levine, S. D4rl: Datasets for deep data-driven reinforcement learning. arXiv preprint arXiv:2004.07219, 2020.  
+Fujimoto, S., Meger, D., and Precup, D. Off-policy deep reinforcement learning without exploration. In International Conference on Machine Learning (ICML), volume 97, pp. 2052-2062, 2019.  
+Garg, D., Chakraborty, S., Cundy, C., Song, J., and Ermon, S. Iq-learn: Inverse soft-q learning for imitation. In Neural Information Processing Systems (NeurIPS), pp. 4028-4039, 2021.  
+Gaurav, A., Rezaee, K., Liu, G., and Poupart, P. Learning soft constraints from constrained expert demonstrations. In International Conference on Learning Representations (ICLR), 2023.
+
+Ho, J. and Ermon, S. Generative adversarial imitation learning. In Neural Information Processing Systems (NeurIPS), pp. 4565-4573, 2016.  
+Janner, M., Fu, J., Zhang, M., and Levine, S. When to trust your model: Model-based policy optimization. In Neural Information Processing Systems (NeurIPS), pp. 12498-12509, 2019.  
+Kidambi, R., Rajeswaran, A., Netrapalli, P., and Joachims, T. Morel: Model-based offline reinforcement learning. In Neural Information Processing Systems (NeurIPS), 2020.  
+Kostrikov, I., Nachum, O., and Tompson, J. Imitation learning via off-policy distribution matching. In International Conference on Learning Representations (ICLR), 2020.  
+Krajewski, R., Bock, J., Kloeker, L., and Eckstein, L. The highd dataset: A drone dataset of naturalistic vehicle trajectories on german highways for validation of highly automated driving systems. In 2018 21st International Conference on Intelligent Transportation Systems (ITSC), pp. 2118-2125, 2018. doi: 10.1109/ITSC.2018.8569552.  
+Kumar, A., Fu, J., Soh, M., Tucker, G., and Levine, S. Stabilizing off-policy q-learning via bootstrapping error reduction. In Neural Information Processing Systems (NeurIPS), pp. 11761-11771, 2019.  
+Kumar, A., Zhou, A., Tucker, G., and Levine, S. Conservative q-learning for offline reinforcement learning. In Neural Information Processing Systems (NeurIPS), 2020.  
+Lee, J., Jeon, W., Lee, B., Pineau, J., and Kim, K.-E. Optidice: Offline policy optimization via stationary distribution correction estimation. In International Conference on Machine Learning (ICML), pp. 6120-6130, 2021.  
+Levine, S., Kumar, A., Tucker, G., and Fu, J. Offline reinforcement learning: Tutorial, review, and perspectives on open problems. CoRR, abs/2005.01643, 2020.  
+Liu, G., Luo, Y., Schulte, O., and Poupart, P. Uncertainty-aware reinforcement learning for risk-sensitive player evaluation in sports game. In Neural Information Processing Systems (NeurIPS), 2022.  
+Liu, G., Luo, Y., Gaurav, A., Rezaee, K., and Poupart, P. Benchmarking constraint inference in inverse reinforcement learning. In International Conference on Learning Representations (ICLR), 2023.  
+Liu, S. and Zhu, M. Distributed inverse constrained reinforcement learning for multi-agent systems. In Neural Information Processing Systems (NeurIPS), 2022.  
+Liu, S. and Zhu, M. Learning multi-agent behaviors from distributed and streaming demonstrations. In Advances in Neural Information Processing Systems (NeurIPS), 2023.
+
+Liu, S. and Zhu, M. Meta inverse constrained reinforcement learning: Convergence guarantee and generalization analysis. In International Conference on Learning Representations (ICRL), 2024.  
+Liu, Y., Halev, A., and Liu, X. Policy learning with constraints in model-free reinforcement learning: A survey. In International Joint Conference on Artificial Intelligence (IJCAI), pp. 4508-4515, 2021.  
+Ma, J. Y., Yan, J., Jayaraman, D., and Bastani, O. Offline goal-conditioned reinforcement learning via  $f$ -advantage regression. Neural Information Processing Systems (NeurIPS), 35:310-323, 2022a.  
+Ma, Y., Shen, A., Jayaraman, D., and Bastani, O. Versatile offline imitation from observations and examples via regularized state-occupancy matching. In International Conference on Machine Learning (ICML), pp. 14639–14663, 2022b.  
+Malik, S., Anwar, U., Aghasi, A., and Ahmed, A. Inverse constrained reinforcement learning. In International Conference on Machine Learning (ICML), pp. 7390-7399, 2021.  
+McPherson, D. L., Stocking, K. C., and Sastry, S. S. Maximum likelihood constraint inference from stochastic demonstrations. In IEEE Conference on Control Technology and Applications, (CCTA), pp. 1208-1213, 2021.  
+Nachum, O., Chow, Y., Dai, B., and Li, L. Dualdice: Behavior-agnostic estimation of discounted stationary distribution corrections. In Neural Information Processing Systems (NeurIPS), pp. 2315–2325, 2019a.  
+Nachum, O., Dai, B., Kostrikov, I., Chow, Y., Li, L., and Schuurmans, D. Algaedice: Policy gradient from arbitrary experience. arXiv preprint arXiv:1912.02074, 2019b.  
+Papadimitriou, D., Anwar, U., and Brown, D. S. Bayesian methods for constraint inference in reinforcement learning. Transactions on Machine Learning Research, 2023.  
+Peters, J. and Schaal, S. Reinforcement learning by reward-weighted regression for operational space control. In International Conference on Machine Learning (ICML), pp. 745-750, 2007.  
+Qiao, G., Liu, G., Poupart, P., and Xu, Z. Multi-modal inverse constrained reinforcement learning from a mixture of demonstrations. In Neural Information Processing Systems (NeurIPS), 2023.  
+Scobee, D. R. R. and Sastry, S. S. Maximum likelihood constraint inference for inverse reinforcement learning. In International Conference on Learning Representations (ICLR), 2020.
+
+Sikchi, H., Zheng, Q., Zhang, A., and Niekum, S. Dual RL: Unification and new methods for reinforcement and imitation learning. In International Conference on Learning Representations, ICLR, 2024.  
+Todorov, E., Erez, T., and Tassa, Y. Mujoco: A physics engine for model-based control. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 5026-5033, 2012.  
+Wang, X., Krasowski, H., and Althoff, M. Commonroad-rl: A configurable reinforcement learning environment for motion planning of autonomous vehicles. In 2021 IEEE International Intelligent Transportation Systems Conference (ITSC), pp. 466-472. IEEE, 2021.  
+Xu, H., Jiang, L., Li, J., Yang, Z., Wang, Z., Chan, W. K. V., and Zhan, X. Offline RL with no OOD actions: In-sample learning via implicit value regularization. In International Conference on Learning Representations, ICLR, 2023.  
+Xu, S. and Liu, G. Uncertainty-aware constraint inference in inverse constrained reinforcement learning. In The Twelfth International Conference on Learning Representations, 2023.  
+Yu, T., Kumar, A., Rafailov, R., Rajeswaran, A., Levine, S., and Finn, C. COMBO: conservative offline model-based policy optimization. In Neural Information Processing Systems (NeurIPS), pp. 28954-28967, 2021.  
+Yue, S., Wang, G., Shao, W., Zhang, Z., Lin, S., Ren, J., and Zhang, J. CLARE: conservative model-based reward learning for offline inverse reinforcement learning. In International Conference on Learning Representations, ICLR, 2023.  
+Ziebart, B. D., Maas, A. L., Bagnell, J. A., and Dey, A. K. Maximum entropy inverse reinforcement learning. In AAAI Conference on Artificial Intelligence, pp. 1433-1438, 2008.  
+Ziebart, B. D., Bagnell, J. A., and Dey, A. K. Modeling interaction via the principle of maximum causal entropy. In International Conference on Machine Learning (ICML), pp. 1255-1262, 2010.
+
+# A. Implementation and Environment Details
+
+# A.1. Discrete Environments
+
+Our discrete environments use a  $7 \times 7$  gridworld map, with four different settings for constraint learning. In each setting, we assign a starting point where the agent is initially located and an absorbing point, which also functions as a terminate state. We employ value iteration to obtain the expert demonstration, and we shape the reward as  $\hat{r} = r - 100c$  by subtracting a cost term to ensure that the demonstration is safe. For offline data collection, we sampled from random walks starting at grids [1,1], [1,5], [5,1], and [5,5]. To enrich the dataset with additional information, the data collect agent was instructed to choose a state it had not visited previously whenever making a move. To obtain the value function recovered by inverse imitation learning, we employed inverse q learning(Garg et al., 2021).
+
+# A.2. Continuous Environments
+
+Our virtual environments are based on Mujoco. We utilize  $c(s, a)$ , derived from a state function  $\mathbf{c}(s')$  representing state safety, with  $c(s, a) = \mathbb{E}_{s' \sim P(s'|s, a)}[\mathbf{c}(s')]$ . We provide more details about the virtual environments as follows:
+
+1. HalfCheetah(Obstacle) This environments are taken from (Liu et al., 2023). In this environment, The agent controls a robot that can move faster backward than forward. The reward is based on the distance it moves between the current and previous time steps, along with a penalty related to the magnitude of the input action. We have defined a constraint that restricts movement in the region where the X-coordinate is less than or equal to -3, allowing the robot to move forward only.
+
+# 2. Ant (Blocked) and Hopper (Blocked)
+
+In this environment, the agent controls a robot to move forward and obtain rewards related to moving distance. However, we enforce a constraint on the agent's leg angular velocity to prevent a large force on the ground. The limits are 1 for Ant and 0.3 for Hopper.
+
+# 3. Ant (LS) and Hopper (LS)
+
+In this environment, the agent controls a robot to move forward and obtain rewards related to moving distance. However, it must not exceed a speed limit (0.5 for Ant and 1 for Hopper), resulting in obtaining fewer rewards compared to no constraint.
+
+To obtain offline data, we first build a suboptimal dataset using two 200,000-step early stop SAC agents. We use the original reward and reward-shaping (modifying the reward by subtracting a cost) as reward signals and collect the replay buffers between 100,000 and 200,000 steps. Together, we obtain a suboptimal dataset with 200,000 steps.
+
+For the expert dataset, we sample 50 trajectories from a PPO-lag algorithm. When sampling from the offline dataset, we maintain a balanced approach by selecting half of the samples from the suboptimal dataset and half from the expert trajectories.
+
+# A.3. Transfer Environments
+
+In the Ant environment, we train an agent with the constraint  $v_{x} < 1.0$  as the expert policy. For the transfer rewards, we modify the direction of the velocity in the x-axis in the original rewards  $v_{x}$  to be the velocity along a -45 degree angle, denoted as  $v_{-45^{\circ}}$ . For offline data, we employ PPO to train a policy that maximizes the transfer rewards, and PPO-lag to train a policy that maximizes rewards while adhering to the specified constraint. We sample 50 trajectories from each trained policy and merge together the data in  $D^{O}$  as offline data.
+
+# A.4. Incorporating Rewards in Inverse Reinforcement Learning
+
+Since offline inverse reinforcement learning does not assume the existence of reward information, we construct a variant for inverse RL by modeling the reward model  $\bar{r}$  in inverse RL with  $\bar{r} = r + c$ , where  $r$  is the ground truth reward function, and  $c$  is the cost function to be estimated. To be more specific, we incorporate the reward information in the following formula:
+
+LS-IQ: We replace the original Q-learning objective (objective (20) in (Al-Hafez et al., 2023)) with
+
+$$
+\mathcal {L} _ {\mathrm {l s i q - o}} (Q) = \alpha \mathbb {E} _ {d _ {\pi_ {E}}} \left[ \left(Q (s, \Gamma_ {\omega} (s, s ^ {\prime})) - Q _ {\max } + r (s, a)\right) ^ {2} \right] + \bar {\alpha} \mathbb {E} _ {d _ {\pi}} \left[ \left(Q (s, a) - \left(r _ {\min } + r (s, a) + \gamma \mathbb {E} _ {s ^ {\prime} \sim P (\cdot | s, a)} [ V ^ {*} (s ^ {\prime}) ]\right)\right) ^ {2} \right], \tag {19}
+$$
+
+SMODICE: We add the ground truth reward function SMODICE's learned discriminator reward  $R(s, a)$  (objective (14) in (Ma et al., 2022b)) with
+
+$$
+R _ {\text {n e w}} (s, a) = R (s, a) + r (s, a) \tag {20}
+$$
+
+# A.5. Implementation Details
+
+Across our set of environments, we maintain the same hyperparameters, except for the temperature parameter  $\xi_r$ . We select the parameter from the set [0.01, 0.001, 0.0001, 0.00001], choosing the one with the highest mean return from safe results (violation rate  $< 0.05$ ) and the lowest violation rate when all parameters are unsafe. We utilize a two-layer MLP with 256 hidden units and a ReLU activation function. The complete set of hyperparameters used in our experiments is presented in Table 5.
+
+<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Batch Size</td><td>256</td></tr><tr><td>Policy Learning Rate</td><td>3e-4</td></tr><tr><td>Value Learning Rate</td><td>3e-4</td></tr><tr><td>L2 Regularization Coefficient</td><td>0.0005</td></tr><tr><td>Random Seed α</td><td>1 2 3 4 5</td></tr></table>
+
+Table 5: Hyperparameters for our experiments.
+
+# A.6. Experimental Equipment and Infrastructures
+
+We ran the experiment on a cluster that has multiple RTX 3090 GPUs, each with 24 GB of memory. There is only one running node. With the aforementioned resources, running one seed in the virtual environment takes about 40 minutes.
+
+# B. Compared to Adversarial Imitation Learning
+
+As proposed by (Ho & Ermon, 2016), the idea behind adversarial imitation learning is to learn a cost(reward) function to discriminate expert behavior from others:
+
+$$
+\max _ {r} \min _ {\pi \in \Pi} L (\pi , r) = \mathbb {E} _ {d ^ {E}} [ c (s, a) ] - \mathbb {E} _ {d ^ {\pi}} [ c (s, a) ] - H (\pi)
+$$
+
+In our ICRL setting, we can view our tasks as discriminating expert behavior from behavior within a smaller set, i.e. superior distributions. Because of information provided by predefined rewards, the feasible region of such max-min question is smaller and thus makes it become an easier problem to solve. This implies the potential for better policy recovery and a more transferable cost function.
+
+# C. Proof and Derivation
+
+# C.1. Deriving Lagrangian of Problem (6)
+
+By introducing the Lagrange multiplier  $V$  and  $\lambda_{d}$ . The Lagrangian of problem (6) can be:
+
+$$
+\begin{array}{l} \mathbb {E} _ {d} [ r (s, a) ] - \xi_ {r} D _ {f} (d | | d ^ {O}) - V (s) (\sum_ {a \in \mathcal {A}} d (s, a) - (1 - \gamma) \rho_ {0} (s) - \gamma \sum_ {(s ^ {\prime}, a ^ {\prime})} d (s ^ {\prime}, a ^ {\prime}) p (s | s ^ {\prime}, a ^ {\prime})) \\ = \mathbb {E} _ {d} \left[ r (s, a) + \gamma \sum_ {\left(s ^ {\prime}, a ^ {\prime}\right)} d \left(s ^ {\prime}, a ^ {\prime}\right) p \left(s \mid s ^ {\prime}, a ^ {\prime}\right)) V \left(s ^ {\prime}\right) - V (s) \right] + (1 - \gamma) \mathbb {E} _ {\rho_ {0}} [ V ] - \xi_ {r} D _ {f} (d | | d ^ {O}) \\ = \mathbb {E} _ {d} [ \delta_ {V} ] - \xi_ {r} D _ {f} (d | | d ^ {O}) + (1 - \gamma) \mathbb {E} _ {\rho_ {0}} [ V ] \\ \end{array}
+$$
+
+# C.2. Proof of Proposition 4.5
+
+Proof. In order to solve:
+
+$$
+\max  _ {d} \mathbb {E} _ {d} [ \delta_ {V} - c ] - \xi_ {r} D _ {f} (d | | d ^ {O}) + (1 - \gamma) \mathbb {E} _ {\rho_ {0}} [ V ] \tag {21}
+$$
+
+$$
+\text {s . t .} d (s, a) c (s, a) \leq 0 \text {a n d} d (s, a) \geq 0 \forall s, a \tag {22}
+$$
+
+Assume  $d(s, a) > 0$  implies  $d^{O}(s, a) > 0$ . We introduce a new variable  $w(s, a) = \frac{d(s, a)}{d^{O}(s, a)}$  and redefine our optimization objective as follows:
+
+$$
+\max  _ {w} \mathbb {E} _ {d ^ {O}} [ w (s, a) \delta_ {V} - \xi_ {r} f (w (s, a)) ] + (1 - \gamma) \mathbb {E} _ {\rho_ {0}} [ V (s) ] \tag {23}
+$$
+
+$$
+s. t. \omega (s, a) c (s, a) \leq 0 \text {a n d} \omega (s, a) \geq 0 \tag {24}
+$$
+
+Without the constraint on  $d(s, a)$ , the maximization of the first part is equivalent to finding the convex conjugate of the function  $f$ , denoted as  $f^{*}(y) = \max_{x} (xy - f(y))$ . However, we must consider the non-negativity constraint  $d(s, a) \geq 0$  and feasible constraint  $\omega(s, a)c(s, a) \leq 0$ .
+
+We can form the Lagrangian function:
+
+$$
+\mathcal {L} (V, \lambda^ {c}, \lambda) = \mathbb {E} _ {d ^ {o}} [ w (s, a) \delta_ {V} - \xi_ {r} f (w (s, a)) ] + (1 - \gamma) \mathbb {E} _ {\rho_ {0}} [ V (s) ] + \sum_ {s, a} (\lambda (s, a) \omega (s, a) - \lambda_ {c} (s, a) c (s, a) \omega (s, a))
+$$
+
+The KKT condition of this problem is:
+
+1. Primal feasibility  $\omega(s, a)c(s, a) \leq 0$ ,  $\omega^{*}(s, a) \geq 0$ $\forall s, a$  
+2.Dual feasibility  $\lambda_c^* (s,a)\geq 0,\lambda^* (s,a)\geq 0\forall s,a$  
+3. Stationarity  $\frac{\partial\mathcal{L}}{\partial V} = d^{O}(s,a)(-f^{\prime}(\omega^{*}(s,a)) + \delta_{V}(s,a) + \lambda^{*}(s,a) - \lambda_{c}^{*}(s,a)c(s,a)) = 0\forall s,a$  
+4.Complementary Slackness  $\omega^{*}(s,a)\lambda^{*}(s,a) = 0,\lambda_{c}^{*}(s,a)\omega^{*}(s,a)c(s,a) = 0$
+
+Using Complementary Slackness, we have two cases:
+
+Case  $1:\lambda_c^* (s,a) = 0$ , which means the solution without introducing constraint is feasible. Following a similar proof given in (Sikchi et al., 2024)(Refer to equation 43 in the appendix of the cited paper for more details), we solve the optimal solution with the stationary equation and first complementary slackness equation:
+
+$$
+\omega^ {*} (s, a) = \omega^ {r} (s, a) = \max (0, f ^ {\prime - 1} (\frac {\delta_ {V} (s , a)}{\xi_ {r}}))
+$$
+
+and  $\omega^r (s,a)$  satisfies primal feasibility  $\omega^{*}(s,a)c(s,a) = 0$
+
+Case2:  $\lambda_c^*(s, a) \neq 0$ , which means  $\omega^*(s, a)c(s, a) = 0$ . Which means we need to solve the following question. Note that such a solution is possible if only there exists some x s.t.  $f'(x) = 0$ .
+
+$$
+\omega^ {*} (s, a) = \max  (0, f ^ {\prime - 1} \left(\frac {\delta_ {V} (s , a) - \lambda_ {c} ^ {*} (s , a) c (s , a)}{\xi_ {r}}\right)) \tag {25}
+$$
+
+$$
+\omega^ {*} (s, a) c (s, a) = 0 \tag {26}
+$$
+
+Summarizing these two cases and replace  $\omega^{*}$  with  $d^{*}$ , we have
+
+$$
+d ^ {*} (s, a) = d ^ {O} (s, a) \mathbb {1} _ {c (s, a) = 0} \max  \left(0, f ^ {\prime - 1} \left(\frac {\delta_ {V} (s , a)}{\xi_ {r}}\right)\right) \tag {27}
+$$
+
+# C.3. Proof of Proposition 4.2
+
+Proof.  $\Rightarrow$ : If  $d = d^{E}$  solve the problem. Then the  $c(s, a) = 0$  for all  $d^{E}(s, a) > 0$ , otherwise the  $d^{E}$  is not feasible. And if there exists a  $d \in O$  s.t.  $d(s, a)c(s, a) = 0 \forall s, a$ , then  $d$  is feasible and  $d$  is optimal than  $d^{r}$  in primal problem, which leads to a contradiction.
+
+"←": If  $\forall d \in O$  there is at least one  $(s, a)$ , s.t.  $dc \neq 0$ , then all  $d \in O$  are all not feasible. Since  $c(s, a) = 0 \forall (s, a)s.t.d^{E}(s, a) > 0$ $d^{E}$  is feasible, so  $d^{E}$  is optimal.
+
+# C.4. Proposition of Maintaining Optimality
+
+Proposition C.1. If the learned cost function  $c_{k}(s,a)$  satisfies the condition that  $d^{E}(s,a) > 0 \Rightarrow c_{k}(s,a) = 0$ , the distribution generated by the CRL problem (4)  $d^{*}$  is in superior distributions, i.e.,  $d^{*} \in \mathcal{O}$ .
+
+Proof. If  $d^{*} \notin \mathcal{O}$ , then  $\mathbb{E}_{d^{*}(s,a)}[r(s,a)] < \mathbb{E}_{d^{E}(s,a)}[r(s,a)]$ . But since  $d^{E}(s,a) > 0 \Rightarrow c_{k}(s,a) = 0$ ,  $d^{E}$  is a feasible solution for CRL problem (4), which contradicts to the optimality of  $d^{*}$ .
+
+Proposition C.2. For a finite state space  $\mathcal{S}$  and a finite action space  $\mathcal{A}$ , if  $d^{*}$  is the only optimal solution under the constraint, we initialize  $c_{0}(s,a) = 0$ . By alternately solving for  $d_{i}^{*}$  and updating  $c_{i}$  to project at any  $d_{i}^{*}(s,a)$  such that at least  $c^{*}(s,a)$  is set to 0 when  $d_{i}^{*}(s,a) > 0$ , while keeping  $c_{i}(s,a) = 0$  for all  $d^{E}(s,a) > 0$ .
+
+Then, the algorithm will converge to  $d^{E}$ . During the learning process, the visiting distribution  $d_{k}^{*}$  maintains  $J(d_{k}^{*}) \geq J(d^{E})$ .
+
+Proof. We know  $J(d) \geq J(d^{E})$  since  $d^{E}$  is always a candidate for a solution in each round of the update. We now prove that if  $d_{(k)}^* \neq d^{E}$ , we can always find some  $s, a$  such that  $d_{(k)}^*(s, a) > 0$  and  $d^{E}(s, a) > 0$ . First, we observe that for any policy  $d$  that visits fewer state-action pairs than  $d^{E}$ , i.e., if  $\{(s, a) : d(s, a) > 0\} \subset \{(s, a) : d^{E}(s, a) > 0\}$ , we have  $J(d) < J(d^{E})$ . Otherwise,  $d$  is feasible, which would contradict the uniqueness of optimality of  $d^{E}$ . Thus, we can always find some  $(s, a)$  in  $\{(s, a) : d(s, a) > 0\}$  but not in  $\{(s, a) : d^{E}(s, a) > 0\}$ .
+
+From this, we also know that if the algorithm converges, it will always converge to  $d^{E}$ . We notice that for every round of updating, the number of zeros in  $c_{(k)}(s,a)$  is increasing. Since in every round  $k$ , the optimal visitation distribution  $d_{(k)}^* (s,a)$  with respect to  $V_{(k)}^*$  satisfies  $d_{(k)}^* (s,a)c_{(k - 1)}(s,a) = 0$ , and in round  $k$ , we must set some  $c_{(k)}(s,a)$  to be above 0 for  $d_{(k)}^* (s,a) > 0$ .
+
+Now, the spaces  $\mathcal{S}$  and  $\mathcal{A}$  are finite. We know the algorithm will converge to  $d^{E}$  since there are only a finite number of combinations of  $(s, a)$ .
+
+# D. Experiments
+
+This section contains additional results.
+
+# D.1. Cost Visualization in HighD Dataset
+
+We visualize the images with the highest assigned cost in each scenario using different methods. Notably, our method assigns higher costs to cars that are much closer in the test dataset, while OptiDICE-c failed to do so.
+
+![](images/5df204295137381ec5a19d3963ea05895f3835885961b3e1d9496152671e8968.jpg)
+
+![](images/42b3117263e2db911a8988889de119be5f0ab770260eb60223886b19d018b930.jpg)
+
+![](images/2ebf801e264c2b8f73b7bd2a0b221192d4f2de2cc1ec2f92e00b58b0d9e0743b.jpg)
+
+![](images/ce528346f83f1c6e2aad2fe5fe826c2fe439a7bd39565b2cb90dd54dcb43e32e.jpg)
+
+![](images/4a75ef837afc6dc3430deb9500bbf8fa41cd40ff8e993ad283932a75c1381e27.jpg)
+
+![](images/d7e043394468c88a9058dd94cf0ef81b704c7c3808cec90a73a78d3765b45a0d.jpg)
+
+![](images/3636d4ee54b9ad52a0a4418c981da39e436bb59f1b601fb0cf16e99276b44010.jpg)
+
+![](images/550f46e6333336108fc1778e00f43c4601edc6318fd97e821ec3ffc47d79b855.jpg)
+
+![](images/deb2a9990ba4789036639a3963ca51a6af47a0bb5df96ce1acf4d411c49d8017.jpg)  
+Figure 6: The red car represents the ego car, the blue cars are the other vehicles the agent is trying to maintain distance from, and the yellow block is the destination.
+
+![](images/18c1d79c316792073575f820c5ba09785c12c818b330e2257707360578216384.jpg)
+
+# D.2. Training Curves for Baseline
+
+# D.3. Results in MuJoCo
+
+![](images/99ff45da907b095ddee9050bdfe76ae3853d575f4f2119b1a7ffadb55a27a8d3.jpg)
+
+![](images/6ab23874d5d37115ef31b8a44efb681648a3f7ebd6c16fc0b69b43da7af7147c.jpg)
+
+![](images/11bcb4d40df45ac156581df840d4b3a57bc4bc9103af3250b69c94613302a553.jpg)
+
+![](images/4a124a3bb651f4e63a22f4cb85160179ff62b8df82bc0162bf9aaac94fb8b2b7.jpg)
+
+![](images/726ac4cfd8448188ba6f59b0cdd9cd4f89d55341f272873409f3276b46ed2690.jpg)
+
+Figure 7: The cumulative rewards and the costs from the evaluation during training.  
+![](images/4f4aff062460904fbc3f8979aa9ecc1a17390832bea1dc52def81ae2eacd20fc.jpg)  
+IQ-LS SMODice-c SMODice OptiDice-c Our Method Expert Offline
+
+![](images/f0780cd9c4b614b53f0cc993bc5ae21962f604c8e545bca13c0a1fea627ee6ce.jpg)
+
+![](images/76cfcab0601fc5b1d89f264d8ab983240ac2f3d46738e5b1f76bfa7ffab732c7.jpg)
+
+![](images/2a3a681164b3fbfcf7aef85d5386427e5db65a85512d34970fb3e60b7d41b3b9.jpg)
+
+![](images/2764cfaeb01cd7cd5fb708accf93de2f943f291c50c14ba3c529f0541f1bb9bb.jpg)
+
+![](images/403d53508a5789581a9993726aabd859c443ae78dce23b4cfdb92287ef6efa73.jpg)
+
+![](images/a4bf0b5fb7110be489c3a0f5ce07a1cd0035965a79d13875693fa54e7ba7e6d8.jpg)
+
+![](images/cabed9c013e8c6b630cc607a0fdd7b2cd054dbb14e53f03af5283a336ad69642.jpg)  
+Figure 8: Recovered cost results: The x-axis represents the cost dimension of the state, and the y-axis represents the learned normalized cost. The plot is jittered along the y-axis for visualization.
+
+![](images/4319a623d86bd3fc4866cb464fa0fece96d341d10ab8880a8c979822335d03b5.jpg)

2024/Learning Constraints from Offline Demonstrations via Superior Distribution Correction Estimation/images.zip

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2024/Learning Coverage Paths in Unknown Environments with Deep Reinforcement Learning/5fdda607-092a-4005-b8f9-ceaa7f621422_origin.pdf

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2024/Learning Coverage Paths in Unknown Environments with Deep Reinforcement Learning/full.md

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+# Learning Coverage Paths in Unknown Environments with Deep Reinforcement Learning
+
+Arvi Jonnarth  $^{12}$  Jie Zhao  $^{3}$  Michael Felsberg  $^{14}$
+
+# Abstract
+
+Coverage path planning (CPP) is the problem of finding a path that covers the entire free space of a confined area, with applications ranging from robotic lawn mowing to search-and-rescue. When the environment is unknown, the path needs to be planned online while mapping the environment, which cannot be addressed by offline planning methods that do not allow for a flexible path space. We investigate how suitable reinforcement learning is for this challenging problem, and analyze the involved components required to efficiently learn coverage paths, such as action space, input feature representation, neural network architecture, and reward function. We propose a computationally feasible egocentric map representation based on frontiers, and a novel reward term based on total variation to promote complete coverage. Through extensive experiments, we show that our approach surpasses the performance of both previous RL-based approaches and highly specialized methods across multiple CPP variations.
+
+# 1. Introduction
+
+Task automation is an ever growing field, and as automated systems become more intelligent, they are expected to solve increasingly challenging problems. One such problem is coverage path planning (CPP), where a path is sought that covers the entire free space of a confined area. It is related to the traveling salesman and covering salesman problems (Galceran & Carreras, 2013), and known to be NP-hard (Arkin et al., 2000). If the environment is known a priori, including the extent of the area and the location and geometry of all obstacles, an optimal path can be planned offline
+
+$^{1}$ Linköping University.  $^{2}$ Husqvarna Group.  $^{3}$ Dalian University of Technology.  $^{4}$ Co-affiliation: University of KwaZulu-Natal. Correspondence to: Arvi Jonnarth <arvi.jonnarth@liu.se>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+(Huang, 2001). However, if the environment is initially unknown, or at least partially unknown, the path has to be planned online, e.g. by a robot, using sensors to simultaneously map the environment. In this case, an optimal path cannot generally be found (Galceran & Carreras, 2013).
+
+Coverage path planning is an essential part in a wide variety of robotic applications, where the range of settings define different variants of the CPP problem. In applications such as robotic lawn mowing (Cao et al., 1988), snow removal (Choset, 2001), and vacuum cleaning (Yasutomi et al., 1988), the region that is covered is defined by an attached work tool. In contrast, in applications such as search-and-rescue (Jia et al., 2016), autonomous underwater exploration (Hert et al., 1996), and aerial terrain coverage (Xu et al., 2011), the covered region is defined by the range of a sensor. Additionally, the space of valid paths is limited, induced by the motion constraints of the robot.
+
+We aim to develop a task- and robot-agnostic method for online CPP in unknown environments. This is a challenging task, as intricacies in certain applications may be difficult to model, or even impossible. In the real world, unforeseen discrepancies between the modeled and true environment may occur, e.g. due to wheel slip. A further aspect that needs to be considered is the sensing capability of the agent, e.g. if it is not omnidirectional, the agent needs to rotate itself to explore a different view. Incompleteness of the present model must be assumed to account for open world scenarios. Thus, we approach the problem from the perspective of learning through embodiment. Reinforcement learning (RL) lends itself as a natural choice, where an agent can interact with the world and adapt to specific environment conditions, without the need for an explicit model. Concretely, we consider the case where an RL model directly predicts the control signals for an agent from sensor data. Besides the benefit of adapting to specific environment characteristics, this approach does not constrain the path space in terms of pre-defined patterns in cellular decomposition methods (Choset & Pignon, 1998), or primitive motions in grid-based methods (Gabriely & Rimon, 2002). In other words, it allows for flexibility in the path space, as shown in Figure 1.
+
+We analyze the different components in RL to efficiently learn coverage paths. To enable learning every valid path, a
+
+![](images/8df1bd6463666c38df3d9621a1d9302fb3896d5946c60fade7b827e42ea7aca8.jpg)  
+Figure 1. Learned paths for exploration (left) and lawn mowing (right), including the start (red triangle) and end position (green square).
+
+![](images/307c765b9c38e3c813846926af001c896f29017505485d6c9cbbd0349471f41c.jpg)
+
+discrete action space based on a grid world is insufficient, as it excludes a large set of paths. A continuous action space is necessary, where the model predicts control signals for an agent. To decide which action to perform, the agent needs information about its pose and the currently known state of the environment. Following Chen et al. (2019a), we use egocentric maps. This provides a suitable observation space, as the egocentric maps efficiently encode the pose in an easily digestible manner. For a scalable input representation for the RL model, we propose to use multiple maps in different scales, with lower resolutions for the larger scales. With this approach, a square of size  $n \times n$  can be represented in  $\mathcal{O}(\log n)$  memory complexity. We make this multi-scale approach viable through frontier maps that preserve information about the existence of non-covered space at any scale. A convolutional network architecture is a natural choice as it efficiently exploits the spatial correlation in the maps. We propose to group the maps by scale and convolve them separately, as their spatial positions do not correspond to the same location. Besides intuitive positive reward terms based on covering new ground (Chen et al., 2019a; Chaplot et al., 2020), we propose a novel reward term based on total variation, which guides the agent not to leave small holes of non-covered free space. Our code implementation can be found online<sup>1</sup>, and our contributions are summarized as follows:
+
+- We propose an end-to-end deep reinforcement learning approach for the continuous online CPP problem.  
+- We make multi-scale maps viable through frontier maps, which can represent non-visited space in any scale.  
+- We propose a network architecture that exploits the structure of the multi-scale maps.  
+- We introduce a novel total variation reward term for reducing holes of non-covered space, improving convergence.  
+- Finally, we conduct extensive experiments and ablations to validate the effectiveness of our approach.
+
+# 2. Related Work
+
+Coverage path planning has previously been approached from the perspective of classical robotics, as well as reinforcement learning. Both approaches require a suitable map representation of the environment. Below, we summarize the related works within these topics.
+
+Planning-based methods. Cellular decomposition methods, such as boustrophedon cellular decomposition (BCD) (Choset & Pignon, 1998) and Morse decomposition (Acar et al., 2002), subdivide the area into non-overlapping cells free of obstacles. Each cell is covered with a pre-defined pattern, e.g. a back-and-forth motion. The cells are then connected using transport paths, which inevitably lead to overlap with the covered cells. Grid-based methods, such as Spiral-STC (Gabriely & Rimon, 2002) and the backtracking spiral algorithm (BSA) (Gonzalez et al., 2005), decompose the environment into uniform grid cells, such as rectangles (Zelinsky et al., 1993) or triangles (Oh et al., 2004), with a comparable size to the agent. The coverage path is defined by primitive motions between adjacent cells. The simplicity of these methods is attractive, but they heavily constrain the space of available paths. Frontier-based methods keep track of the boundary between covered and non-covered free space, i.e. the frontier. Segments of the frontier are clustered into nodes, where one node is chosen as the next short-term goal. Different criteria for choosing which node to visit next have been explored, such as the distance to the agent (Yamauchi, 1997), the path in a rapidly exploring random tree (RRT) (Umari & Mukhopadhyay, 2017), or the gradient in a potential field (Yu et al., 2021). These methods are mainly tailored for robot exploration, where the notion of a frontier is suitable. Compared to planning-based methods, RL can adapt to specific environment characteristics.
+
+RL-based methods. Many learning-based methods combine classical components with reinforcement learning. For known environments, Chen et al. (2019b) present Adaptive Deep Path, where they apply RL to find the cell order in
+
+BCD. Discrete methods (Piardi et al., 2019; Kyaw et al., 2020) utilize the simplicity of grid-based approaches, where an RL model predicts movement primitives, e.g. to move up, down, left, or right, based on a discrete action space. However, this constrains the space of possible paths, and may result in paths that are not possible to navigate by a robot in the real world, which is continuous by nature. Thus, we consider a continuous action space, which has been used successfully for the related navigation problem (Tai et al., 2017). We utilize soft actor-critic (SAC) RL (Haarnoja et al., 2018a) due to its sample efficiency for continuous spaces. Other works combine RL with a frontier-based approach, either by using an RL model to predict the cost of each frontier node (Niroui et al., 2019), or to predict the control signals for navigating to a chosen node (Hu et al., 2020). Different from previous approaches, which use RL within a multi-stage framework, we predict continuous control signals end-to-end to fully utilize the flexibility of RL. To the best of our knowledge, we are the first to do so for CPP.
+
+Map representation. To perform coverage path planning, the environment needs to be represented in a suitable manner. Similar to previous work, we discretize the map into a 2D grid with sufficiently high resolution to accurately represent the environment. This map-based approach presents different choices for the input feature representation. Saha et al. (2021a) observe such maps in full resolution, where the effort for a  $d \times d$  grid is  $\mathcal{O}(d^2)$ , which is infeasible for large environments. Niroui et al. (2019) resize the maps to keep the input size fixed. However, this means that the information in each grid cell varies between environments, and hinders learning and generalization for differently sized environments. Meanwhile, Shrestha et al. (2019) propose to learn the map in unexplored regions. Instead of considering the whole environment at once, other works (Heydari et al., 2021; Saha et al., 2021b) observe a local neighborhood around the agent. While the computational cost is manageable, the long-term planning potential is limited as the space beyond the local neighborhood cannot be observed. For example, if the local neighborhood is fully covered, the agent must pick a direction at random to explore further. To avoid the aforementioned limitations, we use multiple maps in different scales, similar to Klamt & Behnke (2018).
+
+# 3. Learning Coverage Paths
+
+Before presenting our approach, we first define the online CPP problem in Section 3.1, after which we formulate it as a Markov decision process in Section 3.2. Then, we present our RL-based approach in terms of observation space in Section 3.3, action space in Section 3.4, reward function in Section 3.5, and agent architecture in Section 3.6.
+
+# 3.1. Problem Definition and Delineations
+
+Without prior knowledge about the geometry of a confined area, the goal is to navigate in it with an agent with radius  $r$  to visit every point of the free space. The free space includes all points inside the area that are not occupied by obstacles. A point is considered visited when the distance between the point and the agent is less than the coverage radius  $d$ , and the point is within the field-of-view of the agent. This definition unifies variations where  $d \leq r$ , which we define as the lawn mowing problem, and where  $d > r$ , which we refer to as exploration. To interactively gain knowledge about the environment, and for mapping its geometry, the agent needs some form of sensor. Both ranging sensors (Hu et al., 2020) and depth cameras (Chen et al., 2019a) have been utilized for this purpose. Following Xu et al. (2022), we choose a simulated 2D light detection and ranging (lidar) sensor that can detect obstacles in fixed angles relative to the agent, although our proposed framework is not limited to this choice. Based on the pose of the agent, the detections are transformed to global coordinates, and a map of the environment is continuously formed. As the focus of this paper is to learn coverage paths, and not to solve the localization problem, we assume known pose up to a certain noise level. However, our method may be extended to the case with unknown pose with the use of an off-the-shelf SLAM method. Finally, while our method can be further extended to account for moving obstacles and multi-agent coverage, they are beyond the scope of this paper.
+
+# 3.2. Coverage Path Planning as a Markov Decision Process
+
+We formulate the CPP problem as a partially observable Markov decision process (POMDP) with discrete time steps  $t \in [1..T]$ , continuous actions  $a_{t} \sim \pi(a_{t}|o_{t})$  made by the agent according to policy  $\pi$ , observations and rewards  $o_{t}, r_{t} \sim p(o_{t}, r_{t}|s_{t}, a_{t-1})$ , and states  $s_{t} \sim p(s_{t}|s_{t-1}, a_{t-1})$  provided by the environment. We use a neural network for the policy, and the goal for the agent is to maximize the expected discounted reward  $\mathbb{E}\left(\sum_{t=1}^{T} \gamma^{t}r_{t}\right)$  with discount factor  $\gamma$ . In our case, the state  $s_{t}$  consists of the geometry of the area, location and shape of all obstacles, visited points, and the pose of the agent. The observation  $o_{t}$  contains the area and obstacle geometries mapped until time step  $t$ , as well as visited points, agent pose, and sensor data. Based on the observation, the model predicts control signals for the agent. The reinforcement learning loop is depicted in Figure 2(a), where observations, actions, and rewards are collected into a replay buffer, from which training batches are drawn for gradient backpropagation.
+
+![](images/c7591263d50af528f9e436f8d38b1f3611c5ba0c40d03e5c645a3fc3993b9c39.jpg)  
+Figure 2. (a) Agent-environment interaction: The observation consists of multi-scale maps from (b) and lidar detections, based on which the model predicts continuous control signals for an agent. (b) Illustration of coverage, obstacle, and frontier maps in multiple scales: This example shows  $m = 4$  scales with a scale factor of  $s = 2$ . All scales are centered at the agent, and discretized into the same pixel resolution, resulting in the multi-scale maps  $M_{c}$ ,  $M_{o}$ , and  $M_{f}$ , of size  $8 \times 8 \times 4$  here.
+
+![](images/4265e7cb96a5d364dbbd599eb7628d7d66df6d93c719e3f58c97c82235d9fc8d.jpg)
+
+# 3.3. Observation Space
+
+To efficiently learn coverage paths, the observation space needs to be mapped into a suitable input feature representation for the neural network policy. To this end, we represent the visited points as a coverage map, and the mapped obstacles and boundary of the area as an obstacle map. The maps are discretized into a 2D grid with sufficiently high resolution to accurately represent the environment. To represent large regions in a scalable manner, we propose to use multi-scale maps, which was necessary for large environments, see Appendix A.1. We make this viable through frontier maps that preserve information about the existence of non-covered space, even in coarse scales.
+
+Multi-scale maps. Inspired by multi-layered maps with varying scale and coarseness levels for search-and-rescue (Klamt & Behnke, 2018), we propose to use multi-scale maps for the coverage and obstacles to solve the scalability issue. We extract multiple local neighborhoods with increasing size and decreasing resolution, keeping the grid size fixed. We start with a local square crop  $M^{(1)}$  with side length  $d_{1}$  for the smallest and finest scale. The multiscale map representation  $M = \{M^{(i)}\}_{i=1}^{m}$  with  $m$  scales is constructed by cropping increasingly larger areas based on a fixed scale factor  $s$ . Concretely, the side length of map  $M^{(i)}$  is  $d_{i} = sd_{i-1}$ . The resolution for the finest scale is chosen sufficiently high such that the desired level of detail is attained in the nearest vicinity of the agent, allowing it to perform precise local navigation. At the same time, the large-scale maps allow the agent to perform long-term planning, where a high resolution is less important. This multi-scale map representation can completely contain an area of size  $d \times d$  in  $\mathcal{O}(\log d)$  number of scales. The total number of grid cells is  $\mathcal{O}(wh\log d)$ , where  $w$  and  $h$  are the fixed width and height of the grids, and do not depend on
+
+$d$ . This is a significant improvement over a single fixed-resolution map with  $\mathcal{O}(d^2)$  grid cells. For the observation space, we use a multi-scale map  $M_{c}$  for the coverage and  $M_{o}$  for the obstacles. These are illustrated in Figure 2(b).
+
+Frontier maps. When the closest vicinity is covered, the agent needs to make a decision where to explore next. However, the information in the low-resolution large-scale maps may be insufficient. For example, consider an obstacle-cluttered region where the obstacle boundaries have been mapped. A low coverage could either mean that some parts are non-covered free space, or that they are occupied by obstacles. These cases cannot be distinguished if the resolution is too low. As a solution to this problem, we propose to encode a multi-scale frontier map  $M_{f}$ , which we define in the following way. In the finest scale, a grid cell that has not been visited is a frontier point if any of its neighbours have been visited. Thus, a frontier point is non-visited free space that is reachable from the covered area. A region where the entire free space has been visited does not induce any frontier points. In the coarser scales, a grid cell is a frontier cell if and only if it contains a frontier point. In this way, the existence of frontier points persists through scales. Thus, regions with non-covered free space can be deduced in any scale, based on this multi-scale frontier map representation.
+
+Egocentric maps. As the movement is made relative to the agent, its pose needs to be related to the map of the environment. Following Chen et al. (2019a), we use egocentric maps which encode the pose by representing the maps in the coordinate frame of the agent. Each multi-scale map is aligned such that the agent is in the center of the map, facing upwards. This allows the agent to easily map observations to movement actions, instead of having to learn an additional mapping from a separate feature representation of its position, such as a 2D one-hot map (Theile et al., 2020).
+
+Sensor observations. To react on short-term obstacle detections, we include the sensor data in the input feature representation. The depth measurements from the lidar sensor are normalized to  $[0,1]$  based on its maximum range, and concatenated into a vector  $S$  (see Appendix A.6 for a discussion on implications of this choice).
+
+# 3.4. Action Space
+
+We let the model directly predict the control signals for the agent. This allows it to adapt to specific environment characteristics, while avoiding a constrained path space, different from a discrete action space. We consider an agent that independently controls the linear velocity  $v$  and the angular velocity  $\omega$ , although the action space may seamlessly be changed to specific vehicle models. To keep a fixed output range, the actions are normalized to  $[-1, 1]$  based on the maximum linear and angular velocities, where the sign of the velocities controls the direction of travel and rotation.
+
+A continuous action space is of course not the only choice, but it is more realistic than a discrete one. There are many reasons why we chose continuous actions. (1) This allows us to model a continuous pose that is not constrained by the grid discretization, and thus not constraining the path space. (2) The agent can adapt and optimize its path for a specific kinematic model. In Section 4 we find that our approach works well in a continuous setting, which hints that it may also work in other continuous control spaces for specific kinematic models. The same conclusion could not be drawn if a discrete action space was used. (3) Our specific choice of linear and angular velocities applies directly to a wide variety of robots, e.g. differential drive systems and the Ackermann kinematic model (by using a constraint on the angular velocity). Thus, sim-to-real transfer is more likely compared to a discrete action space. (4) Compared to grid-based discrete actions, a continuous action space introduces additional sources for error, both regarding dynamics and localization, which are more realistic.
+
+# 3.5. Reward Function
+
+As the goal is to cover the free space, a reward based on covering new ground is required. Similar to Chaplot et al. (2020) and Chen et al. (2019a), we define a reward term  $R_{\mathrm{area}}$  based on the newly covered area  $A_{\mathrm{new}}$  in the current time step that was not covered previously. To control the scale of this reward signal, we first normalize it to [0, 1] based on the maximum area that can be covered in each time step, which is related to the maximum speed  $v_{\mathrm{max}}$ . We subsequently multiply it with the scale factor  $\lambda_{\mathrm{area}}$ , which is the maximum possible reward in each step, resulting in the reward
+
+$$
+R _ {\mathrm {a r e a}} = \lambda_ {\mathrm {a r e a}} \frac {A _ {\mathrm {n e w}}}{2 r v _ {\mathrm {m a x}} \Delta t}, \tag {1}
+$$
+
+![](images/4dd170c3201d1792ab2522578f7b55bc77ac5aeebcb7f0452c358c825bfb6aab.jpg)  
+Figure 3. The area reward  $R_{\mathrm{area}}$  is based on the maximum possible area that can be covered in each time step.
+
+where  $r$  is the agent radius and  $\Delta t$  is the time step size. See Figure 3 for an illustration.
+
+By only maximizing the coverage reward in (1), the agent is discouraged from overlapping its previous path, as this reduces the reward in the short term. This might lead to holes or stripes in the coverage, which we observed in our experiments (see Appendix A.4), where the agent would leave a small gap when driving adjacent to a previously covered part. These leftover parts can be costly to cover later on for reaching complete coverage. Covering the thin stripes afterward only yields a minor reward, resulting in slow convergence towards an optimal path. To reduce the leftover parts, we propose a reward term based on minimizing the total variation (TV) of the coverage map. Minimizing the TV corresponds to reducing the boundary of the coverage map, and thus leads to fewer holes. Given a 2D signal  $x$ , the discrete isotropic total variation, which has been used for image denoising (Rudin et al., 1992), is expressed as
+
+$$
+V (x) = \sum_ {i, j} \sqrt {\left| x _ {i + 1 , j} - x _ {i , j} \right| ^ {2} + \left| x _ {i , j + 1} - x _ {i , j} \right| ^ {2}}. \tag {2}
+$$
+
+We consider two variants of the TV reward term, a global and an incremental. For the global TV reward  $R_{\mathrm{TV}}^{\mathrm{G}}$ , the agent is given a reward based on the global coverage map  $C_t$  at time step  $t$ . To avoid an unbounded TV for large environments, it is scaled by the square root of the covered area  $A_{\mathrm{covered}}$ , as this results in a constant TV for a given shape of the coverage map, independent of scale. The incremental TV reward  $R_{\mathrm{TV}}^{\mathrm{I}}$  is based on the difference in TV between the current and the previous time step. A positive reward is given if the TV is decreased, and vice versa. The incremental reward is scaled by the maximum possible increase in TV in a time step, which is twice the traveled distance. The global and incremental rewards are respectively given by
+
+$$
+R _ {\mathrm {T V}} ^ {\mathrm {G}} (t) = - \lambda_ {\mathrm {T V}} ^ {\mathrm {G}} \frac {V \left(C _ {t}\right)}{\sqrt {A _ {\text {c o v e r e d}}}}, \tag {3}
+$$
+
+$$
+R _ {\mathrm {T V}} ^ {\mathrm {I}} (t) = - \lambda_ {\mathrm {T V}} ^ {\mathrm {I}} \frac {V (C _ {t}) - V (C _ {t - 1})}{2 v _ {\mathrm {m a x}} \Delta t}, \qquad (4)
+$$
+
+![](images/c18402d3dee50cfcefb0a54f4575e638df56746ac41b771e89aff681f9127f94.jpg)  
+Figure 4. Our proposed SGCNN architecture consists of convolution (CONV) and fully connected (FC) layers. The scales of the multi-scale maps are convolved separately as their spatial positions are not aligned in the grid. x3/x4 refer to the number of layers.
+
+where  $\lambda_{\mathrm{TV}}^{\mathrm{G}}$  and  $\lambda_{\mathrm{TV}}^{\mathrm{I}}$  are reward scaling parameters to make sure that  $|R_{\mathrm{TV}}| < |R_{\mathrm{area}}|$  on average. Otherwise, the optimal behaviour is simply to stand still.
+
+To avoid obstacle collisions, a negative reward  $R_{\mathrm{coll}}$  is given each time the agent collides with an obstacle. Finally, a small constant negative reward  $R_{\mathrm{const}}$  is given in each time step to encourage fast execution. Thus, our final reward function reads
+
+$$
+R = R _ {\mathrm {a r e a}} + R _ {\mathrm {T V}} ^ {\mathrm {G}} + R _ {\mathrm {T V}} ^ {\mathrm {I}} + R _ {\mathrm {c o l l}} + R _ {\mathrm {c o n s t}}. \quad (5)
+$$
+
+During training, each episode is terminated when the agent reaches a pre-defined goal coverage, or when it has not covered any new space in  $\tau$  consecutive time steps.
+
+# 3.6. Agent Architecture
+
+Due to the multi-modal nature of the observation space, we use a map feature extractor  $g_{m}$ , a sensor feature extractor  $g_{s}$ , and a fusing module  $g_{f}$ . The map and sensor features are fused, resulting in the control signal prediction
+
+$$
+(v, \omega) = g _ {f} \left(g _ {m} \left(M _ {c}, M _ {o}, M _ {f}\right), g _ {s} (S)\right). \tag {6}
+$$
+
+We consider three network architectures, a simple multilayer perceptron (MLP), a standard convolutional neural network (CNN), and our proposed scale-grouped convolutional neural network (SGCNN) that independently processes the different scales in the maps, see Figure 4. The MLP is mainly used as a benchmark to evaluate the inductive priors in the CNN-based architectures. For the MLP, the feature extractors,  $g_{m}$  and  $g_{s}$ , are identity functions that simply flatten the inputs, and the fusing module consists of three fully connected (FC) layers. The CNN-based architectures use convolutional layers in the map feature extractor followed by a single FC layer. In SGCNN, we group the maps in  $M_{c}$ ,  $M_{o}$  and  $M_{f}$  by scale, as the pixel positions between different scales do not correspond to the same world coordinate. Each scale is convolved separately using grouped
+
+convolutions. This ensures that each convolution kernel is applied to grids where the spatial context is consistent across channels. The sensor feature extractor is a single FC layer, and the fusing module consists of three FC layers. Full details can be found in Appendix B.2.
+
+# 4. Experiments
+
+In this section, we first describe the implementation details regarding training, environment setup, and evaluation in Section 4.1. Subsequently, we evaluate our method in three settings; both omnidirectional and non-omnidirectional exploration in Section 4.2, and on the lawn mowing task in Section 4.3. Finally, we evaluate the robustness to noise in Section 4.4 and provide ablation studies in Section 4.5.
+
+# 4.1. Implementation Details
+
+Agent and training details. We use soft actor-critic (SAC) RL (Haarnoja et al., 2018a) with automatic temperature tuning (Haarnoja et al., 2018b). The actor and critic networks share the network architecture, but not any weights. We train for 8M iterations with learning rate  $10^{-5}$ , batch size 256, replay buffer size  $5 \cdot 10^{5}$ , discount factor  $\gamma = 0.99$ , and a minimal noise level. The simulated lidar field-of-view was  $360^{\circ}$  in omnidirectional exploration, and  $180^{\circ}$  in the other two settings. The coverage radius was set to the lidar range in both exploration tasks, which was  $7\mathrm{m}$  and  $3.5\mathrm{m}$  in omnidirectional and non-omnidirectional exploration respectively. In the lawn mowing case, the coverage radius was  $0.15\mathrm{m}$ , same as the agent radius. For full details on the physical dimensions, see Appendix B.1. The training time for one agent varied between 100 to 150 hours on a T4 GPU and a 6226R CPU.
+
+Environment details. For the multi-scale maps, we use  $m = 4$  scales with  $32 \times 32$  pixel resolution, a scale factor of  $s = 4$ , and 0.0375 meters per pixel for the finest scale, which corresponds to 4 pixels per agent radius. Thus, the maps span a square with side length  $76.8\mathrm{m}$ . We progressively increase the difficulty of the environment during training, starting from simple maps and increase their complexity over time. The episodes are terminated when a goal coverage rate between  $90 - 99\%$  is reached, depending on the training progression. For full details on the progressive training, see Appendix B.5. Based on initial experiments, we find suitable reward parameters. The episodes are prematurely truncated if  $\tau = 1000$  consecutive steps have passed without the agent covering any new space. We set the maximum coverage reward  $\lambda_{\mathrm{area}} = 1$ , the incremental TV reward scale  $\lambda_{\mathrm{TV}}^{\mathrm{I}} = 0.2$  for exploration and  $\lambda_{\mathrm{TV}}^{\mathrm{I}} = 1$  for lawn mowing, the collision reward  $R_{\mathrm{coll}} = -10$ , and the constant reward  $R_{\mathrm{const}} = -0.1$ . The global TV reward scale was set to  $\lambda_{\mathrm{TV}}^{\mathrm{G}} = 0$  as it did not contribute to a performance gain in our ablations in Section 4.5.
+
+![](images/013de6ae066d2ce225e6163ffbb9af1b884dafc941a106f232045c1f72b7890c.jpg)  
+Figure 5. Examples of exploration maps (a-c), lawn mowing maps (d-f), and randomly generated maps (g-h).
+
+Table 1. Time in seconds for reaching  ${90}\%$  and  ${99}\%$  coverage in Explore-Bench.  
+
+<table><tr><td rowspan="2">Map → Method ↓</td><td colspan="2">Loop</td><td colspan="2">Corridor</td><td colspan="2">Corner</td><td colspan="2">Rooms</td><td colspan="2">Comb. 1</td><td colspan="2">Comb. 2</td><td colspan="2">Total</td></tr><tr><td>T90</td><td>T99</td><td>T90</td><td>T99</td><td>T90</td><td>T99</td><td>T90</td><td>T99</td><td>T90</td><td>T99</td><td>T90</td><td>T99</td><td>T90</td><td>T99</td></tr><tr><td>Distance frontier</td><td>124</td><td>145</td><td>162</td><td>169</td><td>210</td><td>426</td><td>159</td><td>210</td><td>169</td><td>175</td><td>230</td><td>537</td><td>1054</td><td>1662</td></tr><tr><td>RRT frontier</td><td>145</td><td>180</td><td>166</td><td>170</td><td>171</td><td>331</td><td>176</td><td>211</td><td>141</td><td>192</td><td>249</td><td>439</td><td>1048</td><td>1523</td></tr><tr><td>Potential field frontier</td><td>131</td><td>152</td><td>158</td><td>162</td><td>133</td><td>324</td><td>156</td><td>191</td><td>165</td><td>183</td><td>224</td><td>547</td><td>967</td><td>1559</td></tr><tr><td>Active Neural SLAM</td><td>190</td><td>214</td><td>160</td><td>266</td><td>324</td><td>381</td><td>270</td><td>315</td><td>249</td><td>297</td><td>588</td><td>755</td><td>1781</td><td>2228</td></tr><tr><td>Ours</td><td>89</td><td>101</td><td>98</td><td>128</td><td>80</td><td>291</td><td>82</td><td>99</td><td>87</td><td>93</td><td>120</td><td>349</td><td>556</td><td>1061</td></tr></table>
+
+Map geometry. To increase the variation of encountered scenarios, we use both a fixed set of maps and randomly generated maps, see Figure 5 for examples. The fixed maps provide both simple, and more challenging scenarios that a CPP method is expected to solve. The randomly generated maps provide orders of magnitude more variation during training, and avoid the problem of overfitting to the fixed maps. They are created by randomizing grid-like floor plans with obstacles. For more details, see Appendix B.4.
+
+Evaluation and comparison. We measure the times  $T_{90}$  and  $T_{99}$  to reach 90% and 99% coverage, respectively. For inference times and collision statistics, see Appendices A.2 and A.3. We use separate maps for evaluation that are not seen during training, see Appendix B.7 for a full list. We compare our approach across three CPP variations, to a total of seven previous methods that are tailored for either exploration or the lawn mowing task. For exploration, we evaluate our method on Explore-Bench (Xu et al., 2022), which is a recent benchmark that implements challenging environments, where four methods have been evaluated by the authors. These include three frontier-based methods, namely a distance-based frontier method (Yamauchi, 1997), an RRT-based frontier method (Umari & Mukhopadhyay, 2017), and a potential field-based frontier method (Yu et al., 2021). The fourth method is an RL-based approach, where Xu et al. (2022) train an RL model to determine a global goal based on active neural SLAM (Chaplot et al., 2020). Finally, we reimplement a fifth method by Hu et al. (2020) that uses RL to navigate to a chosen frontier node. For the lawn mowing task, we compare with the backtracking spiral algorithm (BSA) (Gonzalez et al., 2005), which is a common benchmark for this CPP variation. Additionally, we implement a baseline that combines A* (Hart et al., 1968) with a travelling salesman problem (TSP) solver on nodes in a grid-like configuration. This has previously been
+
+proposed for the lawn mowing task in the offline setting for small environments (Bormann et al., 2018). To make it feasible for our larger environments, a heuristic was required for distant nodes, and periodic replanning was used for the online case. For full details, see Appendix B.6. We provide videos showcasing the learned paths of our approach. $^{2}$
+
+# 4.2. Exploration
+
+Omnidirectional exploration. In omnidirectional exploration, the agent observes its surroundings in all directions through a  $360^{\circ}$  field-of-view lidar sensor. To evaluate our method on this CPP variation, we use Explore-Bench (Xu et al., 2022), which includes six environments; loop, corridor, corner, rooms, combination 1 (rooms with corridors), and combination 2 (complex rooms with tight spaces), which can be found in Appendix B.7. The results are presented in Table 1. Our approach surpasses the performance of both the frontier-based methods and Active Neural SLAM. This shows that learning control signals end-to-end with RL is in fact a suitable approach to CPP. Furthermore, Figure 1 shows that the agent has learned an efficient exploration path in a complex and obstacle-cluttered environment.
+
+Non-omnidirectional exploration. We further evaluate our method on non-omnidirectional exploration, with a  $180^{\circ}$  lidar field-of-view. We compare with the recent frontier-based RL method of Hu et al. (2020), which was trained specifically for this setting. The coverage over time is presented in Figure 6. Our method outperforms the frontier-based RL approach, demonstrating that end-to-end learning of control signals is superior to a multi-stage approach for adapting to a specific sensor setup.
+
+Table 2. Time in minutes for reaching  ${90}\%$  and  ${99}\%$  coverage on the lawn mowing task.  
+
+<table><tr><td rowspan="2">Setting</td><td rowspan="2">Map → Method ↓</td><td colspan="2">Map 1</td><td colspan="2">Map 2</td><td colspan="2">Map 3</td><td colspan="2">Map 4</td><td colspan="2">Map 5</td><td colspan="2">Map 6</td><td colspan="2">Total</td></tr><tr><td>T90</td><td>T99</td><td>T90</td><td>T99</td><td>T90</td><td>T99</td><td>T90</td><td>T99</td><td>T90</td><td>T99</td><td>T90</td><td>T99</td><td>T90</td><td>T99</td></tr><tr><td rowspan="2">Offline</td><td>TSP</td><td>45</td><td>49</td><td>45</td><td>51</td><td>43</td><td>48</td><td>55</td><td>61</td><td>27</td><td>30</td><td>110</td><td>122</td><td>325</td><td>361</td></tr><tr><td>BSA</td><td>30</td><td>35</td><td>29</td><td>35</td><td>31</td><td>36</td><td>34</td><td>41</td><td>17</td><td>23</td><td>88</td><td>100</td><td>229</td><td>270</td></tr><tr><td rowspan="2">Online</td><td>TSP</td><td>62</td><td>69</td><td>70</td><td>77</td><td>67</td><td>75</td><td>70</td><td>78</td><td>37</td><td>41</td><td>142</td><td>158</td><td>448</td><td>498</td></tr><tr><td>Ours</td><td>44</td><td>60</td><td>40</td><td>50</td><td>43</td><td>49</td><td>40</td><td>69</td><td>25</td><td>32</td><td>118</td><td>149</td><td>310</td><td>409</td></tr></table>
+
+Table 3. Coverage time in minutes for different levels of noise in the lawn mowing task.  
+
+<table><tr><td colspan="3">Noise standard deviation</td><td colspan="2">Time</td></tr><tr><td>Position</td><td>Heading</td><td>Lidar</td><td>T90</td><td>T99</td></tr><tr><td>0.01 m</td><td>0.05 rad</td><td>0.05 m</td><td>310</td><td>409</td></tr><tr><td>0.02 m</td><td>0.1 rad</td><td>0.1 m</td><td>338</td><td>486</td></tr><tr><td>0.05 m</td><td>0.2 rad</td><td>0.2 m</td><td>317</td><td>434</td></tr></table>
+
+![](images/0c478e572ba0c5c782d67fdbaa392b4b61cf07fd51fc0bc73cf828a59696828e.jpg)  
+Figure 6. Coverage over time in nonomnidirectional exploration.
+
+![](images/c4286a6e76b103269e33fff18eabc904fda5df9134393e296a08a849edb1e606.jpg)  
+Figure 7. Exploration coverage for different number of scales.
+
+# 4.3. Lawn Mowing
+
+For the lawn mowing problem, we compare with the backtracking spiral algorithm (BSA) (Gonzalez et al., 2005). Note however that BSA is an offline method and does not solve the mapping problem. As such, we do not expect our approach to outperform it in this comparison. Instead, we use it to see how close we are to a good solution. We further compare with an offline and online version of the TSP-based solution with  $\mathbf{A}^*$  (Bormann et al., 2018). Table 2 shows  $T_{90}$  and  $T_{99}$  for six maps numbered 1-6, which can be found in Appendix B.7. Compared to offline BSA, our method takes  $35\%$  and  $51\%$  more time to reach  $90\%$  and  $99\%$  coverage respectively. This is an impressive result considering the challenge of simultaneously mapping the environment. Moreover, our approach outperforms the online version of the TSP-based solution, and even surpasses the offline version for  $90\%$  coverage. The limiting factors for TSP are likely the grid discretization and suboptimal replanning, which lead to overlap, see Appendix B.6.
+
+# 4.4. Robustness to Noise
+
+As the real world is noisy, we evaluate how robust our method is to noise. We apply Gaussian noise to the position, heading, and lidar measurements during both training and evaluation. Thus, both the maps and the sensor data perceived by the agent contain noise. We consider three levels of noise and present  $T_{90}$  and  $T_{99}$  for the lawn mowing task in Table 3. The result shows that, even under high levels of noise, our method still functions well. Our approach surpasses the TSP solution under all three noise levels.
+
+Table 4. Coverage (\%) at 1500 and 1000 seconds for mowing (Mow) and exploration (Exp) respectively, comparing agent architecture (NN), TV rewards, and frontier map observation  $(M_f)$ .  
+
+<table><tr><td></td><td colspan="4">Settings</td><td colspan="2">Coverage</td></tr><tr><td></td><td>\( R_{\text{TV}}^I \)</td><td>\( R_{\text{TV}}^G \)</td><td>\( M_f \)</td><td>NN</td><td>Mow</td><td>Exp</td></tr><tr><td>①</td><td>✓</td><td></td><td>✓</td><td>MLP</td><td>81.4</td><td>27.3</td></tr><tr><td>②</td><td>✓</td><td></td><td>✓</td><td>CNN</td><td>93.2</td><td>88.4</td></tr><tr><td>③</td><td></td><td></td><td>✓</td><td>SGCNN</td><td>85.1</td><td>91.5</td></tr><tr><td>④</td><td>✓</td><td></td><td></td><td>SGCNN</td><td>72.6</td><td>80.6</td></tr><tr><td>⑤</td><td>✓</td><td>✓</td><td>✓</td><td>SGCNN</td><td>96.6</td><td>89.0</td></tr><tr><td>⑥</td><td>✓</td><td></td><td>✓</td><td>SGCNN</td><td>97.8</td><td>93.0</td></tr></table>
+
+# 4.5. Ablation Study
+
+In Table 4, we explore the impact of different components of our approach via a series of ablations. Since some baselines struggled to reach  $90\%$  or  $99\%$  coverage, we provide the coverage at fixed times, instead of  $T_{90}$  and  $T_{99}$ .
+
+TV rewards. We find that the incremental TV reward has a major impact in the lawn mowing problem, increasing the coverage from  $85.1\%$  to  $97.8\%$  (③ vs. ⑥). It also affects exploration, but not to the same extent. This is reasonable, as the total variation in the coverage map is lower in this case. Moreover, as global TV has less temporal variation, and behaves more like a constant reward, it turns out not to be beneficial for CPP (⑤ vs. ⑥).
+
+Agent architectures. The higher model capacity of the CNN architecture enables a better understanding of the environment, and outperforms the MLP baseline (2 vs. 1). Our proposed architecture (SGCNN, 6), which groups the
+
+![](images/f12ecb06ae226f68d2a4173b48eb973544a94d6e07bcfc6c3b7c9b7346607685.jpg)  
+Figure 8. Learned paths on the lawn mowing task after different number of training iterations.
+
+![](images/e3c3b4ddaefa0613c87e7add77c12a317704030ce3567bf5a1403d40d7626a89.jpg)
+
+![](images/9ceb15955483a60f2702b10c8225f72c15759ff025575cb33da0e5cacfdee249.jpg)
+
+![](images/c3d984c60fc4b7f49bbef9f9a69f11710da0909711ff4932d5955b04236003ed.jpg)
+
+![](images/6a67033b7d1aadb63aba7ace4c9c20c1e4f5c1ef4418cb76bc71413ed8c94dd5.jpg)
+
+![](images/12e0a78dee40199b6cf8eeac21baa9a4507188475bbc997f4d7c39df80308d8d.jpg)  
+Figure 9. Learned paths for a fully trained agent on the lawn mowing task (a-c) and exploration (d-e).
+
+![](images/668b22e5c4165216e7edd0eed5dbd7d6a44236065e4765b7fa47240a8855c317.jpg)
+
+![](images/27b52057bec468fbf73515ca179676db655fefb6e57a49791036201da4734cc6.jpg)
+
+![](images/0cc33d202aec48c3148a94a7c644777d87a52820c7eccf0cd7c8ec7245ce2928.jpg)
+
+![](images/368a2d60022e3dd3a87def66d2b885c270dcd37ced8fa0706042c10fda9b1af9.jpg)
+
+different scales and convolves them separately, further improves the performance compared to a naive CNN, which treats all scales as the same spatial structure.
+
+Multi-scale frontier maps. We find that our proposed frontier map representation is crucial for achieving a high performance with the multi-scale map approach (④ vs. ⑥). Without this input feature, too much information is lost in the coarser scales, thus hindering long-term planning.
+
+Number of scales. In Figure 7, we compare the coverage for different number of scales on the exploration task. It shows that only using one or two scales is not sufficient, where at least three scales are required to represent a large enough environment to enable long-term planning. The discrepancy was not as large for the lawn mowing problem, as it is more local in nature with a lower coverage radius.
+
+# 4.6. Qualitative Results
+
+In Figure 8, we show learned paths during different stages of the training process. Initially, the agent performs constant actions, first with a low steering input, and later with a high steering input, presumably to avoid collisions. After 100k iterations, the agent can already cover an empty environment, but it leaves small patches that it struggles to cover efficiently. After 1M iterations, it leaves fewer small patches, but still struggles to efficiently cover the final few percentages of the area. In the later stages, the agent perfects its abilities of not leaving any small patches uncovered, to quickly cover the final remaining regions, and to reduce overlap. Figure 9 shows learned paths for a fully trained agent, both on the lawn mowing task and on exploration.
+
+# 5. Conclusions
+
+We present a method for online coverage path planning in unknown environments, based on a continuous end-to-end deep reinforcement learning approach. The agent implicitly solves three tasks simultaneously, namely mapping of the environment, planning a coverage path, and navigating the planned path while avoiding obstacles. For a scalable solution, we propose to use multi-scale maps, which is made viable through frontier maps that preserve the existence of non-covered space in coarse scales. We propose a novel reward term based on total variation, which significantly improves coverage time when applied locally, but not globally. The result is a task- and robot-agnostic CPP method that can be applied to different applications, which has been demonstrated on the exploration and lawn mowing tasks.
+
+# Acknowledgements
+
+This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP), funded by the Knut and Alice Wallenberg (KAW) Foundation. The work was funded in part by the Vinnova project, human-centered autonomous regional airport, Dnr 2022-02678. The computational resources were provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS), partially funded by the Swedish Research Council through grant agreement no. 2022-06725, and by the Berzelius resource, provided by the KAW Foundation at the National Supercomputer Centre (NSC).
+
+# 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 which we feel must be specifically highlighted here.
+
+# References
+
+Acar, E. U., Choset, H., Rizzi, A. A., Atkar, P. N., and Hull, D. Morse decompositions for coverage tasks. The international journal of robotics research, 21(4):331-344, 2002.  
+Arkin, E. M., Fekete, S. P., and Mitchell, J. S. Approximation algorithms for lawn mowing and milling. Computational Geometry, 17(1-2):25-50, 2000.  
+Bengio, Y., Louradour, J., Collobert, R., and Weston, J. Curriculum learning. In Proceedings of the 26th annual international conference on machine learning, pp. 41-48, 2009.  
+Bormann, R., Jordan, F., Hampp, J., and Hägele, M. Indoor coverage path planning: Survey, implementation, analysis. In 2018 IEEE International Conference on Robotics and Automation (ICRA), pp. 1718-1725. IEEE, 2018.  
+Cao, Z. L., Huang, Y., and Hall, E. L. Region filling operations with random obstacle avoidance for mobile robots. Journal of Robotic Systems, 5(2):87-102, 1988.  
+Chaplot, D. S., Gandhi, D., Gupta, S., Gupta, A., and Salakhutdinov, R. Learning to explore using active neural slam. In International Conference on Learning Representations, 2020.  
+Chen, T., Gupta, S., and Gupta, A. Learning exploration policies for navigation. In International Conference on Learning Representations, 2019a.  
+Chen, X., Tucker, T. M., Kurfess, T. R., and Vuduc, R. Adaptive deep path: Efficient coverage of a known environment under various configurations. In 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3549-3556, 2019b. doi: 10.1109/IROS40897.2019.8967793.  
+Choset, H. Coverage for robotics-a survey of recent results. Annals of mathematics and artificial intelligence, 31:113-126, 2001.  
+Choset, H. and Pignon, P. Coverage path planning: The boustrophedon cellular decomposition. In Field and service robotics, pp. 203-209. Springer, 1998.  
+Gabriely, Y. and Rimon, E. Spiral-stc: An on-line coverage algorithm of grid environments by a mobile robot.
+
+In Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No. 02CH37292), volume 1, pp. 954-960. IEEE, 2002.  
+Galeran, E. and Carreras, M. A survey on coverage path planning for robotics. Robotics and Autonomous Systems, 61(12):1258-1276, 2013.  
+Gonzalez, E., Alvarez, O., Diaz, Y., Parra, C., and Bustacara, C. Bsa: A complete coverage algorithm. In proceedings of the 2005 IEEE international conference on robotics and automation, pp. 2040-2044. IEEE, 2005.  
+Haarnoja, T., Zhou, A., Abbeel, P., and Levine, S. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. In International conference on machine learning, pp. 1861-1870. PMLR, 2018a.  
+Haarnoja, T., Zhou, A., Hartikainen, K., Tucker, G., Ha, S., Tan, J., Kumar, V., Zhu, H., Gupta, A., Abbeel, P., et al. Soft actor-critic algorithms and applications. arXiv preprint arXiv:1812.05905, 2018b.  
+Hart, P. E., Nilsson, N. J., and Raphael, B. A formal basis for the heuristic determination of minimum cost paths. IEEE transactions on Systems Science and Cybernetics, 4(2):100-107, 1968.  
+Hert, S., Tiwari, S., and Lumelsky, V. A terrain-covering algorithm for an auv. Autonomous Robots, 3:91-119, 1996.  
+Heydari, J., Saha, O., and Ganapathy, V. Reinforcement learning-based coverage path planning with implicit cellular decomposition. arXiv preprint arXiv:2110.09018, 2021.  
+Hu, J., Niu, H., Carrasco, J., Lennox, B., and Arvin, F. Voronoi-based multi-robot autonomous exploration in unknown environments via deep reinforcement learning. IEEE Transactions on Vehicular Technology, 69 (12):14413-14423, 2020.  
+Huang, W. H. Optimal line-sweep-based decompositions for coverage algorithms. In Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No. 01CH37164), volume 1, pp. 27-32. IEEE, 2001.  
+Jia, D., Wermelinger, M., Diethelm, R., Krusi, P., and Hutter, M. Coverage path planning for legged robots in unknown environments. In 2016 IEEE international symposium on safety, security, and rescue robotics (SSRR), pp. 68-73. IEEE, 2016.  
+Justesen, N., Torrado, R. R., Bontrager, P., Khalifa, A., Togelius, J., and Risi, S. Illuminating generalization in deep reinforcement learning through procedural level
+
+generation. In NeurIPS Workshop on Deep Reinforcement Learning, 2018.  
+Klamt, T. and Behnke, S. Planning hybrid driving-stepping locomotion on multiple levels of abstraction. In 2018 IEEE International Conference on Robotics and Automation (ICRA), pp. 1695-1702. IEEE, 2018.  
+Kyaw, P. T., Paing, A., Thu, T. T., Mohan, R. E., Le, A. V., and Veerajagadheswar, P. Coverage path planning for decomposition reconfigurable grid-maps using deep reinforcement learning based travelling salesman problem. IEEE Access, 8:225945-225956, 2020.  
+Narvekar, S., Peng, B., Leonetti, M., Sinapov, J., Taylor, M. E., and Stone, P. Curriculum learning for reinforcement learning domains: A framework and survey. The Journal of Machine Learning Research, 21(1):7382-7431, 2020.  
+Niroui, F., Zhang, K., Kashino, Z., and Nejat, G. Deep reinforcement learning robot for search and rescue applications: Exploration in unknown cluttered environments. IEEE Robotics and Automation Letters, 4(2):610-617, 2019.  
+Oh, J. S., Choi, Y. H., Park, J. B., and Zheng, Y. F. Complete coverage navigation of cleaning robots using triangular-cell-based map. IEEE Transactions on Industrial Electronics, 51(3):718-726, 2004.  
+Piardi, L., Lima, J., Pereira, A. I., and Costa, P. Coverage path planning optimization based on q-learning algorithm. In AIP Conference Proceedings, volume 2116, pp. 220002. AIP Publishing LLC, 2019.  
+Rudin, L. I., Osher, S., and Fatemi, E. Nonlinear total variation based noise removal algorithms. Physica D: nonlinear phenomena, 60(1-4):259-268, 1992.  
+Saha, O., Ganapathy, V., Heydari, J., Ren, G., and Shah, M. Efficient coverage path planning in initially unknown environments using graph representation. In 2021 20th International Conference on Advanced Robotics (ICAR), pp. 1003-1010. IEEE, 2021a.  
+Saha, O., Ren, G., Heydari, J., Ganapathy, V., and Shah, M. Deep reinforcement learning based online area covering autonomous robot. In 2021 7th International Conference on Automation, Robotics and Applications (ICARA), pp. 21-25. IEEE, 2021b.  
+Shrestha, R., Tian, F.-P., Feng, W., Tan, P., and Vaughan, R. Learned map prediction for enhanced mobile robot exploration. In 2019 International Conference on Robotics and Automation (ICRA), pp. 1197-1204. IEEE, 2019.
+
+Tai, L., Paolo, G., and Liu, M. Virtual-to-real deep reinforcement learning: Continuous control of mobile robots for mapless navigation. In 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 31-36. IEEE, 2017.  
+Theile, M., Bayerlein, H., Nai, R., Gesbert, D., and Caccamo, M. Uav coverage path planning under varying power constraints using deep reinforcement learning. In 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 1444-1449. IEEE, 2020.  
+Umari, H. and Mukhopadhyay, S. Autonomous robotic exploration based on multiple rapidly-exploring randomized trees. In 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 1396-1402. IEEE, 2017.  
+Xu, A., Viriyasuthee, C., and Rekleitis, I. Optimal complete terrain coverage using an unmanned aerial vehicle. In 2011 IEEE International conference on robotics and automation, pp. 2513-2519. IEEE, 2011.  
+Xu, Y., Yu, J., Tang, J., Qiu, J., Wang, J., Shen, Y., Wang, Y., and Yang, H. Explore-bench: Data sets, metrics and evaluations for frontier-based and deep-reinforcement-learning-based autonomous exploration. In 2022 International Conference on Robotics and Automation (ICRA), pp. 6225-6231. IEEE, 2022.  
+Yamauchi, B. A frontier-based approach for autonomous exploration. In Proceedings 1997 IEEE International Symposium on Computational Intelligence in Robotics and Automation CIRA'97. Towards New Computational Principles for Robotics and Automation', pp. 146-151. IEEE, 1997.  
+Yasutomi, F., Yamada, M., and Tsukamoto, K. Cleaning robot control. In Proceedings. 1988 IEEE International Conference on Robotics and Automation, pp. 1839-1841. IEEE, 1988.  
+Yu, J., Tong, J., Xu, Y., Xu, Z., Dong, H., Yang, T., and Wang, Y. Smmr-exlore: Submap-based multi-robot exploration system with multi-robot multi-target potential field exploration method. In 2021 IEEE International Conference on Robotics and Automation (ICRA), pp. 8779-8785. IEEE, 2021.  
+Zelinsky, A., Jarvis, R. A., Byrne, J., Yuta, S., et al. Planning paths of complete coverage of an unstructured environment by a mobile robot. In Proceedings of international conference on advanced robotics, volume 13, pp. 533-538, 1993.
+
+# Appendix
+
+This appendix contains an extended analysis in Section A and additional implementation details in Section B.
+
+# A. Extended Analysis
+
+# A.1. The Necessity of Multi-scale Maps
+
+While the multi-scale maps can represent large regions efficiently, they are even necessary for large-scale environments. In large environments, using a single map is not feasible as the computational cost is  $\mathcal{O}(n^2)$ . For example, with our current setup with 4 maps that span an area of size  $76.8 \times 76.8 \, \mathrm{m}^2$ , the training step time is  $40 \, \mathrm{ms}$ . With a single map spanning only  $19.2 \times 19.2 \, \mathrm{m}^2$  using the same pixel resolution as the finest scale, the training step time is  $2500 \, \mathrm{ms}$ . This would increase the total training time by roughly two orders of magnitude. Going beyond this size resulted in memory problems on a T4 16GB GPU. Thus, a multi-scale map is necessary for large-scale environments.
+
+# A.2. Inference Time Analysis
+
+We evaluate the inference time of our method on two different systems in Table 5. System 1 is a high-performance computing cluster, with 16-core 6226R CPUs, and T4 GPUs. However, only 4 CPU cores and one GPU were allocated for the analysis. System 2 is a laptop computer, with a 2-core i5-520M CPU, without a GPU. The times are compared between the mowing and exploration tasks, where the inference time is lower for the mowing task since a smaller local neighborhood is used for updating the global coverage map. As our network is fairly lightweight it can run in real time, i.e.  $\geq 20$  frames per second, even on a low-performance laptop without a GPU. All map updates are performed locally, resulting in high scalability, reaching real-time performance on the cluster node, and close to real-time performance on the laptop.
+
+Table 5. Inference time in milliseconds, on a high-performance computing cluster and a laptop, for the model forward pass, other components (map updates, observation creation etc.), and in total.  
+
+<table><tr><td rowspan="2">Task</td><td colspan="3">Cluster node 6226R CPU, T4 GPU</td><td colspan="3">Laptop i5-520M CPU, No GPU</td></tr><tr><td>model</td><td>other</td><td>total</td><td>model</td><td>other</td><td>total</td></tr><tr><td>Exploration</td><td>2</td><td>19</td><td>21</td><td>5</td><td>57</td><td>62</td></tr><tr><td>Mowing</td><td>2</td><td>3</td><td>5</td><td>5</td><td>9</td><td>14</td></tr></table>
+
+# A.3. Collision Frequency
+
+In most real-world applications, it is vital to minimize the collision frequency, as collisions can inflict harm on humans, animals, the environment, and the robot. To gain insights into the collision characteristics of our trained agent, we tracked the collision frequency during evaluation. For exploration, it varied between once every 100-1000 seconds, which is roughly once every 50-500 meters. For lawn mowing, it varied between once every 150-250 seconds, which is roughly once every 30-50 meters. These include all forms of collisions, even low-speed side collisions. The vast majority of collisions were near-parallel, and we did not observe any head on collisions. The practical implications are very different between these cases.
+
+# A.4. Impact of the TV Reward on Learned Paths
+
+Figure 10 shows learned paths both with and without the incremental TV reward. The TV reward reduces small leftover regions which are costly to cover later on.
+
+# A.5. Additional Rewards that Did Not Work
+
+In addition to the reward terms described in Section 3.5, we experimented with a goal coverage reward, and an episode truncation reward. Both were added at the end of every episode. The goal coverage reward was a positive reward that was given once the agent reached the predefined goal coverage. The truncation reward was a negative reward that was given in
+
+![](images/00050a1a43c037796d9d9b575d29f9aa7358761dc7e936998f54cf81390d342a.jpg)  
+Figure 10. The total variation reward term improves path quality. The figure shows the path (yellow) without TV reward (left), and with TV reward (right), including the covered region (green), lidar rays (blue lines) and frontier points (magenta).
+
+![](images/ef6fcbcd48fee87ec30dda344809d30858bff9f6fbe0b25410fc358ef3050137.jpg)
+
+case the episode was prematurely truncated due to the agent not covering new ground in  $\tau$  consecutive steps. However, neither of these rewards affected the end result, likely due to the fact that they are sparse compared to the other rewards, and do not provide a strong reward signal. The coverage reward already encourages complete coverage, and the constant negative reward heavily penalizes not reaching full coverage. Since both of these are dense, adding sparse rewards for the same purpose does not provide any benefit.
+
+# A.6. Generalization of the Lidar Feature Representation
+
+Since we represent the lidar distance measurements as a vector of a fixed size and normalize the distances to  $[0,1]$ , the sensor observation is tied to a specific sensor setup. Due to the normalization, the sensor observation is tied to a specific sensor range. This is important to keep in mind when transferring to another sensor setup. If the maximum range differs, the normalization constant needs to be considered, such that the same distance corresponds to the same observed value, and the maximum observable value should be limited to 1. If the number of lidar rays differ, the representation might not generalize as easily, as the dimensionality of the sensor vector would change. One option is to interpolate the observed distances at the ray angles that were used during training. Although, inaccuracies may occur if the angles are not well aligned. However, the map representation is more flexible, and does in theory also contain the information of the raw lidar data. So excluding the sensor feature extractor would in theory generalize to any sensor setup.
+
+# B. Additional Implementation Details
+
+# B.1. Physical Dimensions
+
+Table 6 shows the physical dimensions in the three different variations of the CPP problem that were considered. We match the dimensions of the methods under comparison to get comparable performance values.
+
+# B.2. Agent Architecture Details
+
+As proposed for soft actor-critic (SAC) (Haarnoja et al., 2018b), we use two Q-functions that are trained independently, together with accompanying target Q-functions whose weights are exponentially moving averages. No separate state-value network is used. All networks, including actor network, Q-functions, and target Q-functions share the same network architecture, which is either a multilayer perceptron (MLP), a naive convolutional neural network (CNN), or our proposed scale-grouped convolutional neural network (SGCNN). These are described in the following paragraphs. The MLP contains a total of 3.2M parameters, where most are part of the initial fully connected layer, which takes the entire observation vector as input. Meanwhile, the CNN-based architectures only contain 0.8M parameters, as the convolution layers process the maps with fewer parameters.
+
+MLP. As mentioned in the main paper, the map and sensor feature extractors for MLP simply consist of flattening layers
+
+Table 6. Physical dimensions for the three different CPP variations.  
+
+<table><tr><td></td><td>Omnidirectional exploration</td><td>Non-omnidirectional exploration</td><td>Lawn mowing</td></tr><tr><td>Coverage radius</td><td>7 m</td><td>3.5 m</td><td>0.15 m</td></tr><tr><td>Agent radius</td><td>0.08 m</td><td>0.15 m</td><td>0.15 m</td></tr><tr><td>Maximum linear velocity</td><td>0.5 m/s</td><td>0.26 m/s</td><td>0.26 m/s</td></tr><tr><td>Maximum angular velocity</td><td>1 rad/s</td><td>1 rad/s</td><td>1 rad/s</td></tr><tr><td>Simulation step size</td><td>0.5 s</td><td>0.5 s</td><td>0.5 s</td></tr><tr><td>Lidar rays</td><td>20</td><td>24</td><td>24</td></tr><tr><td>Lidar range</td><td>7 m</td><td>3.5 m</td><td>3.5 m</td></tr><tr><td>Lidar field-of-view</td><td>360°</td><td>180°</td><td>180°</td></tr></table>
+
+that restructure the inputs into vectors. The flattened multi-scale coverage maps, obstacle maps, frontier maps, and lidar detections are concatenated and fed to the fusion module. For the Q-functions, the predicted action is appended to the input of the fusion module. The fusion module consists of two fully connected layers with ReLU and 256 units each. A final linear prediction head is appended, which predicts the mean and standard deviation for sampling the action in the actor network, or the Q-value in the Q-functions.
+
+Naive CNN. This architecture is identical to SGCNN, see below, except that it convolves all maps simultaneously in one go, without utilizing grouped convolutions.
+
+SGCNN. Our proposed SGCNN architecture uses the same fusion module as MLP, but uses additional layers for the feature extractors, including convolutional layers for the image-like maps. Due to the simple nature of the distance readings for the lidar sensor, the lidar feature extractor only uses a single fully connected layer with ReLU. It has the same number of hidden neurons as there are lidar rays. The map feature extractor consists of a  $2 \times 2$  convolution with stride 2 to reduce the spatial resolution, followed by three  $3 \times 3$  convolutions with stride 1 and without padding. We use 24 total channels in the convolution layers. Note that the maps are grouped by scale, and convolved separately by their own set of filters. The map features are flattened into a vector and fed to a fully connected layer of 256 units, which is the final output of the map feature extractor. ReLU is used as the activation function throughout.
+
+# B.3. Choice of Hyperparameters
+
+Multi-scale map parameters. We started by considering the size and resolution in the finest scale, such that sufficiently small details could be represented in the nearest vicinity. We concluded that a resolution of 0.0375 meters per pixel was a good choice as it corresponds to 8 pixels for a robot with a diameter of  $30~\mathrm{cm}$ . Next, we chose a size of  $32 \times 32$  pixels, as this results in the finest scale spanning  $1.2 \times 1.2$  meters. Next, we chose the scale factor  $s = 4$ , as this allows only a few maps to represent a relatively large region, while maintaining sufficient detail in the second finest scale. Finally, the training needs to be done with a fixed number of scales. The most practical way is to simply train with sufficiently many scales to cover the maximum size for a particular use case. Thus, we chose  $m = 4$  scales, which in total spans a  $76.8 \times 76.8$  m region, and can contain any of the considered training and evaluation environments. If the model is deployed in a small area, the larger scales would not contain any frontier points in the far distance, but the agent can still cover the area by utilizing the smaller scales. For a larger use case, increasing the size of the represented area by adding more scales is fairly cheap, as the computational cost is  $\mathcal{O}(\log n)$ .
+
+Reward parameters. Due to the normalization of the area and TV rewards, they are given approximately the same weight for  $\lambda_{\mathrm{area}} = 1$  and  $\lambda_{\mathrm{TV}} = 1$ . Thus, we used this as a starting point, which seemed to work well for the lawn mowing problem, while a lower weight for the TV rewards was advantageous for exploration.
+
+Episode truncation. For large environments, we found it important not to truncate the episodes too early, as this would hinder learning. If the truncation parameter  $\tau$  was set too low, the agent would not be forced to learn to cover an area completely, as the episode would simply truncate whenever the agent could not progress, without a large penalty. With the chosen value of  $\tau = 1000$ , the agent would be greatly penalized by the constant reward  $R_{\mathrm{const}}$  for not reaching the goal coverage, and would be forced to learn to cover the complete area in order to maximize the reward.
+
+# B.4. Random Map Generation
+
+Inspired by procedural environment generation for reinforcement learning (Justesen et al., 2018), we use randomly generated maps during training to increase the variation in the training data and to improve generalization. We consider random floor plans and randomly placed circular obstacles. First, the side length of a square area is chosen uniformly at random, from the interval [2.4, 7.5] meters for mowing and [9.6, 15] meters for exploration. Subsequently, a random floor plan is created with a  $70\%$  probability. Finally, with  $70\%$  probability, a number of obstacles are placed such that they are far enough apart to guarantee that the agent can visit the entire free space, i.e. that they are not blocking the path between different parts of the free space. An empty map can be generated if neither a floor plan is created nor obstacles placed. In the following paragraphs, we describe the floor plan generation and obstacle placement in more detail.
+
+Random floor plans. The random floor plans contain square rooms of equal size in a grid-like configuration, where neighboring rooms can be accessed through door openings. On occasion, a wall is removed completely or some openings are closed off to increase the variation further. First, floor plan parameters are chosen uniformly at random, such as the side length of the rooms from [1.5, 4.8] meters, wall thickness from [0.075, 0.3] meters, and door opening size from [0.6, 1.2] meters. Subsequently, each vertical and horizontal wall spanning the whole map either top-to-bottom or left-to-right is placed with a  $90\%$  probability. After that, door openings are created at random positions between each neighboring room. Finally, one opening is closed off at random for either each top-to-bottom spanning vertical wall or each left-to-right spanning horizontal wall, not both. This is to ensure that each part of the free space can be reached by the agent.
+
+Random circular obstacles. Circular obstacles with radius 0.25 meters are randomly scattered across the map, where one obstacle is placed for every four square meters. If the closest distance between a new obstacle and any previous obstacle or wall is less than 0.6 meters, the new obstacle is removed to ensure that large enough gaps are left for the agent to navigate through and that it can reach every part of the free space.
+
+# B.5. Progressive Training
+
+To improve convergence in the early parts of training, we apply curriculum learning (Bengio et al., 2009), which has shown to be effective in reinforcement learning (Narvekar et al., 2020). We use simple maps and increase their difficulty progressively. To this end, we rank the fixed training maps by difficulty, and assign them into tiers, see Figure 11. The maps in the lower tiers have smaller sizes and simpler obstacles. For the higher tiers, the map size is increased together with the complexity of the obstacles.
+
+Furthermore, we define levels containing certain sets of maps and specific goal coverage percentages, see Table 7. The agent starts at level 1, and progresses through the levels during training. To progress to the next level, the agent has to complete each map in the current level, by reaching the specified goal coverage. Note that randomly generated maps are also used in the higher levels, in which case the agent has to additionally complete a map with random floor plans, and a map with randomly placed circular obstacles to progress. Whenever random maps are used in a level, either a fixed map or a random map is chosen with a  $50\%$  probability each at the start of every episode.
+
+Table 7. The progressive training levels contain maps of increasingly higher tiers and goal coverage percentages. At the highest levels, generated maps with random floor plans and obstacles are also used.  
+
+<table><tr><td rowspan="2">Level</td><td colspan="3">Exploration</td><td colspan="3">Lawn mowing</td></tr><tr><td>Map tiers</td><td>Random maps</td><td>Goal coverage</td><td>Map tiers</td><td>Random maps</td><td>Goal coverage</td></tr><tr><td>1</td><td>1,2</td><td></td><td>90%</td><td>0</td><td></td><td>90%</td></tr><tr><td>2</td><td>1,2,4</td><td></td><td>90%</td><td>0,1</td><td></td><td>90%</td></tr><tr><td>3</td><td>1,2,4</td><td></td><td>95%</td><td>0,1</td><td></td><td>95%</td></tr><tr><td>4</td><td>1,2,4</td><td></td><td>97%</td><td>0-2</td><td></td><td>95%</td></tr><tr><td>5</td><td>1,2,4</td><td></td><td>99%</td><td>0-2</td><td></td><td>97%</td></tr><tr><td>6</td><td>1-4</td><td></td><td>99%</td><td>0-2</td><td></td><td>99%</td></tr><tr><td>7</td><td>1-4</td><td>✓</td><td>99%</td><td>0-3</td><td></td><td>99%</td></tr><tr><td>8</td><td>1-5</td><td>✓</td><td>99%</td><td>0-3</td><td>✓</td><td>99%</td></tr></table>
+
+# B.6. Details on the Compared Methods
+
+TSP-based solution with  $\mathbf{A}^*$ . For the TSP baseline we subdivide the environment into square grid cells of comparable size to the agent, and apply a TSP solver to find the shortest path to visit the center of each cell, which are the nodes. In order to guarantee that each cell is fully covered by a circular agent, the side length for each grid cell is set to  $\sqrt{2} r$ , where  $r$  is the agent radius. Note that this leads to overlap and thus increases the time to reach full coverage. In the offline case, we compute the weight between each pair of nodes using the shortest path algorithm,  $\mathbf{A}^*$  (Hart et al., 1968). However, due to the size of our environments, the computation time was infeasible. Thus, we implemented a supremum heuristic for distant nodes, where they would be assigned the largest possible path length instead of running  $\mathbf{A}^*$ . This improved the runtime considerably, while not affecting the path length to a noticeable extent. For the online case we applied the TSP solver on all visible nodes, executed the path while observing new nodes, and repeated until all nodes were covered. In this case, the TSP execution time on an Epyc 7742 64-core CPU was included in the coverage time, as the path would need to be replanned online in a realistic setting. However, the execution time was relatively small compared to the time to navigate the path. We also tried replanning in shorter intervals, but the gain in time due to a decreased path length was smaller than the increase in computation time.
+
+Code repositories. Table 8 lists the code implementations used for the methods under comparison. Note that for omnidirectional exploration, we evaluate our method under the same settings as in Explore-Bench, for which Xu et al. (2022) report the performance of the compared methods. We report these results in Table 1.
+
+Table 8. Links to code implementations used for the methods under comparison.  
+
+<table><tr><td>Omnidirectional exploration
+Official Explore-Bench implementation (Xu et al., 2022):
+https://github.com/efc-robot/Explore-Bench</td></tr><tr><td>Non-omnidirectional exploration
+Official Frontier RL implementation (Hu et al., 2020):
+https://github.com/hanlinniu/turtlebot3_dpg_release_avoidance
+Frontier RL implementation in simulation:
+https://github.com/Peace1997/Voronoi-Based_Multi_Robot_Collaborate_Exploration_Uknow_Enviroment</td></tr><tr><td>Lawn mowing
+Implementation of BSA (Gonzalez et al., 2005):
+https://github.com/nobleo/full_coverage_path_planner</td></tr></table>
+
+# B.7. Evaluation Maps
+
+Figure 12 shows the maps used for evaluating our method on the different CPP variations, as well as additional maps for the ablation study. Figures 12(a) and 12(c) show the maps in order from left to right as they appear in Tables 1 and 2 respectively.
+
+![](images/4f31c1295e755c428443043fb5ea2bed61ddfc9d240c518f672244c94463c1c6.jpg)
+
+![](images/1b303508c55413d87ce74be300e9fa2616a29f0711635b63ca4432316d946314.jpg)
+
+![](images/870b06b2f3182b488e7cfdae7d84bf4b284a13cac58e32339b3f1f958244dbac.jpg)
+
+![](images/0024e03331c9cbfd476bb5e94749e8c8a258dbd539afc53595f88f8b87445e5a.jpg)
+
+![](images/a2421169fff6c1383dfce671aebc002b38b573d1d2b14a5009502da2d6e13ba3.jpg)
+
+![](images/5755b6040871f36d5b6b398aed1f7a72bb07f2567120194226b80bb3ce034896.jpg)
+
+![](images/136e6e9a41170f3e5546696912b7b8ebfa985550b2e3f2ae2a9a7f6c29c7e5bb.jpg)
+
+![](images/69edf886bf5976a80c516435ece7d51c0a4a1852f71d9d86f54ae4fa956d6497.jpg)
+
+![](images/41d4f5acc4ff88b619a2197092b67c397a29e6344d16e07b9e128b7500aeac5e.jpg)  
+Figure 11. The fixed training maps are grouped into tiers by difficulty, depending on their size and complexity of the obstacles.
+
+![](images/293c8b2a42a07db2d75ee74217244ef8a5a8568699646a4a2e42f12202a77534.jpg)
+
+![](images/d0697e0a35e41082e4b54c99d7e483d78c742b77c262e869a92b5b0c4b425c47.jpg)
+
+![](images/c11c75549963aa34b1e917d8b52c1c7845ddd6fd1c9413bcc0a210a126523a73.jpg)
+
+![](images/14872962d84a42e6acbb895db45f462c0f24b09cced8f1effd678d96f7da1b8c.jpg)
+
+![](images/d89452782eae3013a757787acefe65d84e1935ebe029b669c093018b21887afe.jpg)  
+Figure 12. Evaluation maps used for (a) omnidirectional exploration, (b) non-omnidirectional exploration, and (c) lawn mowing. The maps in the last two rows were used additionally for ablations in (d) exploration and (e) lawn mowing.

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+# Learning Divergence Fields for Shift-Robust Graph Representations
+
+Qitian Wu<sup>1</sup> Fan Nie<sup>1</sup> Chenxiao Yang<sup>1</sup> Junchi Yan<sup>1</sup>
+
+# Abstract
+
+Real-world data generation often involves certain geometries (e.g., graphs) that induce instance-level interdependence. This characteristic makes the generalization of learning models more difficult due to the intricate interdependent patterns that impact data-generative distributions and can vary from training to testing. In this work, we propose a geometric diffusion model with learnable divergence fields for the challenging generalization problem with interdependent data. We generalize the diffusion equation with stochastic diffusivity at each time step, which aims to capture the multi-faceted information flows among interdependent data. Furthermore, we derive a new learning objective through causal inference, which can guide the model to learn generalizable patterns of interdependence that are insensitive across domains. Regarding practical implementation, we introduce three model instantiations that can be considered as the generalized versions of GCN, GAT, and Transformers, respectively, which possess advanced robustness against distribution shifts. We demonstrate their promising efficacy for out-of-distribution generalization on diverse real-world datasets. Source codes are available at https://github.com/fannie1208/GLIND.
+
+# 1. Introduction
+
+Learning from data involving certain geometries is a fundamental challenge in machine learning (Tenenbaum et al., 2000; Belkin & Niyogi, 2003; Belkin et al., 2006). One common scenario entails explicit graph structures, where the observed edges create interdependence among data points; another more challenging scenario involves implicit struc
+
+$^{1}$ School of Artificial Intelligence & Department of Computer Science and Engineering & MoE Lab of AI, Shanghai Jiao Tong University, Shanghai, China. Correspondence to: Junchi Yan <yanjunchi@sjtu.edu.cn>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+![](images/019d73858862c914ceed8c5153c7d29efbd8ece0395cc5980071864504405bb6.jpg)  
+Figure 1: The challenge of generalization with interdependent data involves distribution shifts regarding the underlying manifolds that define the proximity among data samples.
+
+tures, where interdependence also exists but is not directly observable from data. In both scenarios, due to the interdependence of data points, the commonly used i.i.d. assumption for modeling becomes invalid. Moreover, given the dynamic environment where the model interacts in the open-world setting, the training and testing data are often generated from different distributions, necessitating generalization under distribution shifts (Mansour et al., 2009; Blanchard et al., 2011; Muandet et al., 2013; Gong et al., 2016; Arjovsky et al., 2019).
+
+However, unlike standard learning settings (e.g., for image and text data) where each data sample can be treated as independent (Koh et al., 2021; Ye et al., 2021), the interdependence among data points (e.g., nodes in an observed graph or other geometries with implicit structures) significantly increases the difficulty of generalization (Ma et al., 2021; Wu et al., 2022; Zhao et al., 2020; Stadler et al., 2021). Particularly, as illustrated in Fig. 1, since the training and testing data lie on different underlying manifolds that induce proximity among data points, the model solely optimized on training data with specific geometry may not generalize well to testing data generated from a different manifold. In contrast, an ideal model should be able to capture the generalizable patterns of data interdependence that can transfer across domains (or interchangeably, environment).
+
+While generalization across potentially different environments is desirable in practice, the challenge lies in how to model the generalizable patterns of the interdependence among data points, which can be abstract and convoluted in nature. Given the data interdependence, the label of each instance depends on both the instance itself and other
+
+instances, and such input-label dependency, which is important for prediction, would vary accordingly when distribution shifts exist (between training and testing data). Thereby, for generalization, the model needs to learn causal predictive relations from the inputs of interdependent data with certain geometries to their labels, which stays insensitive to distribution shifts.
+
+In this paper, we commence with diffusion equations on manifolds (Weickert et al., 1998; Romeny, 2013) as the foundation for learning generalizable predictive relations with interdependent data. We propose a geometric diffusion model with learnable divergence fields as an effective means to model the complex interdependent patterns among data points. In particular, we generalize the diffusivity function, which measures the rate of information flows between two points on the manifold, as random samples from a variational distribution over a set of diffusivity hypotheses, inducing stochastic evolutionary directions at each step. In this way, the model can accommodate and capture the multi-faceted information flows at each diffusion step.
+
+To facilitate the generalization, we harness deconfounded learning, a technique from causal inference, and derive a step-wise re-weighting regularization approach for the inferred latent diffusivity. This gives rise to a new optimization objective that guides the model to learn the causal relation induced by the diffusion dynamics trajectory, which captures the generalizable relations from initial states  $(x)$  to output states  $(y)$  that are insensitive to data distribution shifts. In terms of practical implementation, we present three instantiations based on our model formulation, which can be treated as generalized versions of GCN (Kipf & Welling, 2017), GAT (Velickovic et al., 2018), and DIFFormer (Wu et al., 2023c), respectively, for out-of-distribution generalization in situations where the data geometries are observed (e.g., as graphs) or unobserved.
+
+We evaluate the practical efficacy of our models on diverse experimental datasets. The results show that the proposed models can effectively handle various distribution shifts. The contributions are summarized below.
+
+- We propose a geometric diffusion model with learnable divergence fields that accommodates multi-faceted information flows among data points. The diffusivity at each step is estimated via a variational distribution conditioned on the diffusion dynamics trajectories.  
+- We design a causal regularization approach for learning the generalizable patterns of interdependence. Our scheme serves as the first attempt to unlock the potential of diffusion models for generalization with interdependent data.  
+- We introduce three practical versions of model implementation and demonstrate their superiority through extensive experiments, particularly in handling various distribution
+
+shifts with observed and unobserved data geometries.
+
+# 2. Background
+
+In this section, we review some technical background as building blocks of our proposed model.
+
+Diffusion on Manifolds. The diffusion process over some abstract domain  $\Omega$  composed of  $N$  points describes the evolution of particular signal  $z(u,t)$ , a scalar-valued function on  $\Omega \times [0,\infty)$ , for each point  $u\in \Omega$  at arbitrary time  $t$ . In specific, the evolution of  $z(u,t)$  by a (heat) diffusion process can be described via a partial differential equation (PDE) with boundary conditions (Freidlin & Wentzell, 1993; Medvedev, 2014; Romeny, 2013):
+
+$$
+\frac {\partial z (u , t)}{\partial t} = \nabla^ {*} \left(D (u, t) \odot \nabla z (u, t)\right), \tag {1}
+$$
+
+with initial conditions  $z(u,0) = z_0(u), t \geq 0, u \in \Omega$ . Here  $\nabla, \nabla^*$  and  $D(u,t)$  reflect some spatial characteristics and are associated with the structure of  $\Omega$ . Without loss of generality, we assume  $\Omega$  as a Riemannian manifold and define  $\mathcal{Z}(\Omega)$  and  $\mathcal{Z}(T\Omega)$  as the scalar and (tangent) vector fields on  $\Omega$ , respectively. Then the gradient operator  $\nabla: \mathcal{Z}(\Omega) \to \mathcal{Z}(T\Omega)$  returns a vector field  $\nabla z(u,t)$  that provides the direction of the steepest change of  $z$  at the point  $u$ . The divergence operator  $\nabla^*: \mathcal{Z}(T\Omega) \to \mathcal{Z}(\Omega)$  is a scalar field that reflects the flow of  $\nabla z(u,t)$  through an infinitesimal volume around  $u$ . The  $D(u,t)$  in Eqn. 1 is the diffusivity that describes some thermal conductance properties of  $\Omega$ , i.e., the measure of the rate at which heat can spread over the space (Rosenberg & Steven, 1997). For homogeneous system, the diffusivity  $D(u,t)$  remains a constant for arbitrary  $u$ , while  $D(u,t)$  is position-dependent for inhomogeneous system. In the latter case,  $D(u,t)$  can be scalar-valued (isotropic diffusion) or matrix-valued (anisotropic diffusion) (Weickert et al., 1998).
+
+Causal Deconfounded Learning. The principle of deconfounded learning is originally rooted in the study of Causality (Pearl et al., 2016; Rubin, 2019), which pursuits the causal (a.k.a. stable) mappings between input  $x$  (cause) and output  $y$  (effect) and eliminates the influence from a latent confounder  $c$ , i.e., the common cause of  $x$  and  $y$ . In machine learning tasks, the domain context in a dataset can be a common confounder between input features  $x$  and labels  $y$ . For example, in image classification, the background often associates with both  $x$  and  $y$ , e.g., a horse often appears on the grass. The domain context would mislead the model to learn non-stable correlation between inputs and labels, e.g., a shortcut between the 'green' background and the 'horse' (Beery et al., 2018; Geirhos et al., 2018). Such correlation however does not hold on other datasets collected with different contexts and may impair the model generalization. Deconfounded learning aims to remove the
+
+confounding bias from domain context by intervention on  $x$  and maximize  $p(y|do(x))$  where the do-operation cuts off the dependence from  $c$  and  $x$ . The probability with the do operator can be computed by randomized controlled trial using data recollection or statistically estimated by leveraging backdoor adjustment (Pearl et al., 2016; Christakopoulou et al., 2020; Roberts et al., 2020; Yang et al., 2022a).
+
+# 3. Model Formulation
+
+We consider  $N$  data samples (a.k.a. instances)  $\{\mathbf{x}_u, y_u\}_{u=1}^N$  that are partially labeled and the labeled portion is  $\{\mathbf{x}_u, y_u\}_{u=1}^M$  where  $M < N$ . There exist interconnecting structures among the  $N$  data points, reflected by  $\mathcal{G} = \{a_{uv}\}_{N \times N}$  where  $a_{uv} = 1$  indicates the connectivity between instance  $u$  and  $v$  and 0 for non-connectivity. In this situation, each instance  $u$  can be considered as the node in the graph  $\mathcal{G}$ . Furthermore, without loss of generality, the structures can also be unobserved, in which case we can assume  $a_{uv} = 1$  for  $\forall u, v$  suggesting that there can exist potential interactions between every instance pair. We introduce  $\mathbf{Z} = [\mathbf{z}_u]_{u=1}^N$  where  $\mathbf{z}_u \in \mathbb{R}^d$  denotes the representation (a.k.a. embedding) of instance  $u$ .
+
+# 3.1. Geometric Diffusion Model
+
+Diffusion Equation over Discrete Space. We first characterize a neural message passing model induced by generic diffusion process Eqn. 1 defined over a discrete space consisting of  $N$  points as locations on the manifold. Through treating the instance representations as signals of locations in the discrete space, we can define the gradient and divergence operators according to the discretization of the continuous notions. The gradient operator  $\nabla$  measures the difference between source and target locations, i.e.,  $(\nabla \mathbf{Z}(t))_{uv} = \mathbf{z}_u(t) - \mathbf{z}_v(t)$ . The divergence operator  $\nabla^*$  aggregates the information flows at a point, i.e.,  $(\nabla^*)_u = \sum_{v,a_{uv} = 1}\mathbf{d}_v(\mathbf{Z}(t),u,t)(\nabla \mathbf{Z}(t))_{uv}$ , where  $\mathbf{d}(\mathbf{Z}(t),u,t)$  determines the diffusivity of the location  $u$  at time  $t$ . We can then obtain the diffusion equation that describes the evolution of instance representations over the observed discrete structures among  $N$  data points:
+
+$$
+\frac {\partial \mathbf {z} _ {u} (t)}{\partial t} = \sum_ {v, a _ {u v} = 1} \mathbf {d} _ {v} (\mathbf {Z} (t), u, t) \left(\mathbf {z} _ {v} (t) - \mathbf {z} _ {u} (t)\right), \tag {2}
+$$
+
+with initial conditions  $\mathbf{Z}(0) = [\mathbf{x}_u]_{u=1}^N$  and  $t \geq 0$ . The above equation can be essentially considered as a continuous version of neural message passing, commonly used by graph neural networks (Chamberlain et al., 2021a) and Transformers (Wu et al., 2023c), while the latter models use discrete layers as the approximation of time. On top of this connection, recent works harness diffusion equations on graphs as a principled perspective for justifying architectural choices (Chamberlain et al., 2021b; Thorpe et al., 2022),
+
+analyzing GNNs' behaviors (Wu et al., 2023b) and guiding model designs for difficult problems, e.g., geometric knowledge distillation (Yang et al., 2022b).
+
+Branching-Structured Divergence Fields. The above diffusion system assumes that at each time step the information flows between connected locations are determined by a deterministic diffusivity function  $\mathbf{d}(\mathbf{Z}(t), u, t)$ . The diffusivity essentially measures the rate of information flows between any pair of locations on the manifold. Based on the analogy between diffusion on manifolds and message passing among interconnected data, the diffusivity can be treated as quantification of pairwise influence among data points. The latter, however, could be driven by multiple criteria along with uncertainty, instead of remaining to be deterministic. In light of this observation, we extend Eqn. 2 to accommodate the heterogeneity and stochasticity of the diffusivity, which is assumed to be sampled from a generative distribution  $p(\mathbf{d}^{(t)}|\mathbf{Z}(t), u, t)$ , over a set of hypothesis, dependent on the current states  $\mathbf{Z}(t)$ , position  $u$  and time  $t$ :
+
+$$
+\frac {\partial \mathbf {z} _ {u} (t)}{\partial t} = \sum_ {v, a _ {u v} = 1} d _ {u v} ^ {(t)} \cdot \left(\mathbf {z} _ {v} (t) - \mathbf {z} _ {u} (t)\right), \tag {3}
+$$
+
+where  $[d_{uv}^{(t)}]_{v=1}^N = \mathbf{d}_u^{(t)} \sim p(\mathbf{d}^{(t)}|\mathbf{Z}(t), u, t)$ . Through the diffusion process defined by Eqn. 3, the change of instance representation at infinitesimal time is equal to the divergence at each point  $u$  induced by the current diffusivity  $\mathbf{d}_u^{(t)}$ . The evolutionary direction of instance representation at time  $t$  is stochastic according to  $p(\mathbf{d}^{(t)}|\mathbf{Z}(t), u, t)$  that yields multiple branches of subsequent diffusion dynamics from the current time (as illustrated in Fig. 2(d)). This formulation further paves the way for generalization by enabling the model to learn the stable diffusion dynamics as we will discuss in later subsections.
+
+Diffusion Trajectories with Stochastic Diffusivity. The differential equation of Eqn. 3 can be solved through numerical methods by using discrete time steps  $\{l\}$  to approximate the continuous time  $t$ . In particular, with step size  $\alpha$ , Eqn. 3 induces a trajectory of instance embeddings:
+
+$$
+\mathbf {z} _ {u} ^ {(l + 1)} = \mathbf {z} _ {u} ^ {(l)} + \alpha \sum_ {v, a _ {u v} = 1} d _ {u v} ^ {(l)} \cdot \left(\mathbf {z} _ {v} ^ {(l)} - \mathbf {z} _ {u} ^ {(l)}\right), \tag {4}
+$$
+
+where  $[d_{uv}^{(t)}]_{v=1}^N = \mathbf{d}_u^{(l)} \sim p(\mathbf{d}^{(l)}|\mathbf{z}_u^{(l)})$ . The diffusion model Eqn. 4 with  $L$  layers yields a trajectory of embeddings for each instance  $u$ :  $\mathbf{x}_u = \mathbf{z}_u^{(0)} \to \mathbf{z}_u^{(1)} \to \dots \to \mathbf{z}_u^{(L)} = \hat{\mathbf{y}}_u$ , where  $\hat{\mathbf{y}}_u$  denotes the predicted label for  $u$ .
+
+# 3.2. Optimization with Causal Regularization
+
+For generalization, we next discuss on how to guide the geometric diffusion model to learn causal relations from inputs to outputs that are insensitive to distribution shifts. Consider
+
+![](images/001b410a8e272b1cdc5e407020b23a4649d29c4b3dd154afd5f63e7354eedd72.jpg)  
+(a) Data generation
+
+![](images/9f42bcea37fddcab6d9aca7ff7ae9c85ee5de4f2931766e9a3d1c2a229fef3bd.jpg)  
+(b) Model deconfounded learning
+
+![](images/d24302f0d5cb3eb9b15c7d48d15fe92d97a44b344d6971def17d051ccff1cae7.jpg)  
+(c) Diffusion dynamics with inference for diffusivity
+
+![](images/248389382300733327f8e681767fa7990e59c59b0f45f014ccbd170f180378e9.jpg)  
+(d) Proposed model and learning objective  
+Figure 2: (a)~ (c) Dependence among random variables of interest (here we omit  $\mathcal{G}$  for brevity since everything can be treated as conditioned on  $\mathcal{G}$ ). (a) Causal dependence for data generation where the diffusivity  $\mathbf{d}$  is the common cause of  $\mathbf{x}$  and  $\mathbf{y}$ . (b) Deconfounded learning  $p_{\theta}(\mathbf{y}|do(\mathbf{x}),\mathcal{G})$  which aims to cut off the dependence path from  $\mathbf{d}$  to  $\mathbf{x}$  in order to learn causal (a.k.a. stable) relation between  $\mathbf{x}$  and  $\mathbf{y}$  for generalization. (c) Our diffusion model whose feed-forward dynamics  $\mathbf{x}_u = \mathbf{z}_u^{(0)}\rightarrow \mathbf{z}_u^{(1)}\rightarrow \dots \rightarrow \mathbf{z}_u^{(L)} = \hat{\mathbf{y}}_u$  is given by a predictive distribution  $p_{\theta}(\mathbf{z}^{(l + 1)}|\mathbf{z}^{(l)},\mathbf{d}^{(l)},\mathcal{G})$  and a variational distribution  $q_{\phi}(\mathbf{d}^{(l)}|\mathbf{z}^{(l)})$ . (d) The geometric diffusion model that is optimized by a new learning objective (comprised of a supervised term and a regularization term) which can achieve the goal of deconfounded learning  $p_{\theta}(\mathbf{y}|do(\mathbf{x}),\mathcal{G})$ .
+
+the instance embedding  $\mathbf{z}_u^{(l)}$  of an arbitrary given layer as a random variable, we define by  $p_{\theta}(\mathbf{z}^{(l + 1)}|\mathbf{z}^{(l)},\mathbf{d}^{(l)},\mathcal{G})$  the (delta) predictive distribution induced by the  $l$ -th layer's updating of Eqn. 4, where the condition on  $\mathcal{G}$  stems from the message passing among instances. Then the generative distribution at the  $l$ -th layer can be expressed as
+
+$$
+p _ {\theta} (\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathcal {G}) = \mathbb {E} _ {p (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} [ p _ {\theta} (\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}) ],
+$$
+
+where the expectation marginalizes over all the possible results of  $\mathbf{d}^{(l)}$ . The log-likelihood  $\log p_{\theta}(\mathbf{y}|\mathbf{x},\mathcal{G})$  for each data sample can be computed by  $\log \prod_{l = 0}^{L - 1}p_{\theta}(\mathbf{z}^{(l + 1)}|\mathbf{z}^{(l)},\mathcal{G}) =$
+
+$$
+\sum_ {l = 0} ^ {L - 1} \log \mathbb {E} _ {p (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} [ p _ {\theta} (\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}) ]. \tag {5}
+$$
+
+Inference for Diffusivity. Due to the intractable integration over  $\mathbf{d}^{(l)}$  in the above objective, we turn to the evidence lower bound of Eqn. 5 by introducing a variational distribution  $q_{\phi}(\mathbf{d}^{(l)}|\mathbf{z}^{(l)})$  (where  $\phi$  denotes its parameterization):
+
+$$
+\sum_ {l = 0} ^ {L - 1} \mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} \left[ \log p _ {\theta} (\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}) \frac {p (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})}{q _ {\phi} (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} \right]. \tag {6}
+$$
+
+With some re-arranging, the evidence lower bound of the log-likelihood  $\log p_{\theta}(\mathbf{y}|\mathbf{x},\mathcal{G})$  induces the following objective (see derivation in Appendix A.1)
+
+$$
+\mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(0)} | \mathbf {z} ^ {(0)}), \dots , q _ {\phi} (\mathbf {d} ^ {(L - 1)} | \mathbf {z} ^ {(L - 1)})} \left[ \log p _ {\theta} (\mathbf {y} | \mathbf {x}, \mathbf {d} ^ {(0)}, \dots , \right.
+$$
+
+$$
+\left. \mathbf {d} ^ {(L - 1)}, \mathcal {G}\right) \Big ] - \sum_ {l = 0} ^ {L - 1} \mathrm {K L} \left(q _ {\phi} \left(\mathbf {d} ^ {(l)} \mid \mathbf {z} ^ {(l)}\right), p \left(\mathbf {d} ^ {(l)} \mid \mathbf {z} ^ {(l)}\right)\right), \tag {7}
+$$
+
+where the first term corresponds to the supervised loss by computing the difference between the ground-truth label  $\mathbf{y}_u$  and the prediction  $\hat{\mathbf{y}}_u$ , and the second term has a regularization effect on the inferred latent diffusivity. Eqn. 7 is a tractable objective that approximates the log-likelihood.
+
+However, the generative distribution  $p(\mathbf{d}^{(l)}|\mathbf{z}^{(l)})$  is a model-based prior for diffusivity over a hypothesis space that expresses how plausible the model thought the ideal diffusivity were in light of the previous diffusion dynamics. Though it incorporates the information of  $\mathbf{z}^{(l)}$  for estimating the adaptive diffusivity at the current step, the model learning would make the estimation of diffusivity biased towards training data. In this way, the predictive relations from  $\mathbf{z}^{(l)}$  (resp.  $\mathbf{x}$ ) to  $\mathbf{z}^{(l+1)}$  (resp.  $\mathbf{y}$ ) learned by the model would be spuriously associated by the latent confounder, the diffusivity  $\mathbf{d}^{(l)}$  (resp.  $\{\mathbf{d}^{(l)}\}_{l=0}^{L-1}$ ), as illustrated by Fig. 2(a). The confounding effect of the diffusivity, which corresponds to the context information that differs across domains, would hamper the generalization of the model when transferring from the training distribution to a testing one where the data lies on another manifold with disparate proximity context.
+
+Step-wise Re-weighting Regularization To facilitate generalization, the ideal solution is to achieve deconfounded learning, i.e.,  $p_{\theta}(\mathbf{y}|do(\mathbf{x}), \mathcal{G})$  that cuts off the dependence path from  $\mathbf{d}$  to  $\mathbf{x}$  (as illustrated in Fig. 2(b)), in which situation the model can learn the causal (a.k.a. stable) relation between  $\mathbf{x}$  and  $\mathbf{y}$  that is generalizable across domains. While the technical difficulty lies in how to compute  $p_{\theta}(\mathbf{y}|do(\mathbf{x}), \mathcal{G})$  in a feasible manner, the next theorem presents a tractable surrogate that approximates the target.
+
+Theorem 3.1. For any given diffusion model  $p_{\theta}(\mathbf{z}^{(l + 1)}|\mathbf{z}^{(l)},\mathbf{d}^{(l)},\mathcal{G})$  we have a lower bound of the deconfounded learning objective:  $\log p_{\theta}(\mathbf{y}|do(\mathbf{x}),\mathcal{G})\geq$
+
+$$
+\sum_ {l = 0} ^ {L - 1} \mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} \left[ \log p _ {\theta} \left(\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}\right) \frac {p _ {0} \left(\mathbf {d} ^ {(l)}\right)}{q _ {\phi} \left(\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)}\right)} \right], \tag {8}
+$$
+
+where  $p_0(\mathbf{d}^{(l)})$  is a model-free prior distribution. In particular, the equality holds for Eqn. 8 iff  $q_{\phi}(\mathbf{d}^{(l)}|\mathbf{z}^{(l)}) = p_{\theta}(\mathbf{d}^{(l)}|\mathbf{z}^{(l)},\mathbf{z}^{(l + 1)},\mathcal{G})\cdot \frac{p_0(\mathbf{d}^{(l)})}{p(\mathbf{d}^{(l)}|\mathbf{z}^{(l)})}$
+
+The proof is deferred to Appendix A.2. We can optimize Eqn. 8 as a surrogate objective by joint learning of  $q_{\phi}$  and  $p_{\theta}$  that has the effect of optimizing the deconfounded learning objective. As another perspective for understanding the effect of Eqn. 8, it can be seen as multiplying a re-weighting term  $\frac{p_0(\mathbf{d}^{(l)})}{p(\mathbf{d}^{(l)}|\mathbf{z}^{(l)})}$  with Eqn. 6, which can down-weight the frequent diffusivity components and up-weight the infrequent ones to eliminate the observation bias caused by limited training data. This can facilitate learning the stable relation from  $\mathbf{z}^{(l)}$  (resp.  $\mathbf{x}$ ) to  $\mathbf{z}^{(l+1)}$  (resp.  $\mathbf{y}$ ) that holds across domains, instead of the non-stable correlation pertaining to specific context in training data.
+
+Based on Eqn. 8, we can obtain the final learning objective (see derivation in Appendix A.3 which is similar to Eqn. 7):
+
+$$
+\mathbb {E} _ {q _ {\phi} \left(\mathbf {d} ^ {(0)} \mid \mathbf {z} ^ {(0)}\right), \dots , q _ {\phi} \left(\mathbf {d} ^ {(L - 1)} \mid \mathbf {z} ^ {(L - 1)}\right)} \left[ \log p _ {\theta} (\mathbf {y} \mid \mathbf {x}, \mathbf {d} ^ {(0)}, \dots , \right.
+$$
+
+$$
+\left. \mathbf {d} ^ {(L - 1)}, \mathcal {G}\right) \Big ] - \sum_ {l = 0} ^ {L - 1} \mathrm {K L} \left(q _ {\phi} \left(\mathbf {d} ^ {(l)} \mid \mathbf {z} ^ {(l)}\right), p _ {0} \left(\mathbf {d} ^ {(l)}\right)\right). \tag {9}
+$$
+
+Fig. 2(d) provides an illustration for the model. The computation of the regularization term depends on specific form of  $q_{\phi}(\mathbf{d}^{(l)}|\mathbf{z}^{(l)})$ , which we will discuss in the next section.
+
+# 4. Model Instantiations
+
+In this section, we delve into the model instantiations on the basis of the formulation of our diffusion model in Sec. 3. We call our model GLIND, short for diffusion-induced Generalizable Learning with INterdependent Data. For simplicity, we assume the inference distribution for diffusivity  $q_{\phi}(\mathbf{d}^{(l)}|\mathbf{z}^{(l)})$  as a multinomial distribution: at each time step  $l$  for arbitrary instance  $u$ , the diffusivity  $\mathbf{d}_u^{(l)}$  is sampled from a limited hypothesis set  $\{\mathbf{d}_u^{(l,k)}\}_{k=1}^K$  with probabilities  $\{\pi_u^{(l,k)}\}_{k=1}^K$  for each component, where  $\sum_{k=1}^K \pi_u^{(l,k)} = 1$ . We can further assume  $\alpha = 1$ , and then the diffusion trajectory of Eqn. 4 induces a feed-forward propagation layer:
+
+$$
+\mathbf {z} _ {u} ^ {(l + 1)} = \mathbf {z} _ {u} ^ {(l)} + \sum_ {k = 1} ^ {K} h _ {u, k} ^ {(l)} \sum_ {v, a _ {u v} = 1} d _ {u v} ^ {(l, k)} (\mathbf {z} _ {v} ^ {(l)} - \mathbf {z} _ {u} ^ {(l)}),
+$$
+
+where  $\mathbf{h}_u^{(l)}\sim \mathcal{M}(\pmb{\pi}_u^{(l)})$  is a one-hot vector where the entry '1' indicates the sampled result from the multinomial
+
+distribution  $\mathcal{M}(\pi_u^{(l)})$ . Here  $\pi_u^{(l)}$  and  $\mathbf{d}_u^{(l,k)}$  determine the divergence (the updating signals) at the current time step  $l$  and control the evolutionary direction of the diffusion process, and both of them can be parameterized.
+
+Parameterization. We adopt a feed-forward layer with Softmax to model the probability of  $K$  diffusivity components and leverage the Gumbel trick (Jang et al., 2017) to approximate the sampling process:
+
+$$
+h _ {u, k} ^ {(l)} = \frac {\exp \left(\left(\pi_ {u} ^ {(l , k)} + g _ {k}\right) / \tau\right)}{\sum_ {k ^ {\prime}} \exp \left(\left(\pi_ {u} ^ {(l , k ^ {\prime})} + g _ {k ^ {\prime}}\right) / \tau\right)}, g _ {k} \sim \operatorname {G u m b e l} (0, 1),
+$$
+
+where  $[\pi_u^{(l,k)}]_{k=1}^K = \pi_u^{(l)} = \operatorname{Softmax}(\mathbf{W}_L^{(l)}\mathbf{z}_u^{(l)})$  and  $\mathbf{W}_L^{(l)} \in \mathbb{R}^{K \times d}$  is a trainable weight matrix. The message passing is driven by the diffusivity hypothesis set  $\{\mathbf{d}_u^{(l,k)}\}_{k=1}^K$ . We next introduce three model instantiations that can be considered as the generalized implementations of GCN, GAT and Transformers, respectively.
+
+GLIND-GCN. We first consider a model version that uses the aggregated embeddings of connected instances to update the representation of the target instance. Inspired by the anisotropic diffusion which is position- and direction-dependent, we assume each branch  $k$  converts the embeddings for the target instance (resp. other connected instances) through  $\mathbf{W}_S^{(l,k)}\in \mathbb{R}^{d\times d}$  (resp.  $\mathbf{W}_D^{(l,k)}\in \mathbb{R}^{d\times d}$ ) and the gradient of signals is scaled by the number of connected instances  $\tilde{d}_u = \sum_v a_{uv}$ :
+
+$$
+\mathbf {z} _ {u} ^ {(l + 1)} = \mathbf {z} _ {u} ^ {(l)} + \sum_ {k = 1} ^ {K} h _ {u, k} ^ {(l)} \left(\sum_ {v, a _ {u} v = 1} \frac {1}{\tilde {d} _ {u}} \mathbf {W} _ {D} ^ {(l, k)} \mathbf {z} _ {v} ^ {(l)} + \mathbf {W} _ {S} ^ {(l, k)} \mathbf {z} _ {u} ^ {(l)}\right),
+$$
+
+where  $\{\mathbf{W}_S^{(l,k)}\}_{k=1}^K$  and  $\{\mathbf{W}_D^{(l,k)}\}_{k=1}^K$  are trainable weights at the  $l$ -th layer. This model can be seen as a generalized implementation of the GCN architecture (Kipf & Welling, 2017), where the approximate one-hot vector  $\mathbf{h}_u^{(l)}$  dynamically selects the convolution filter in each layer.
+
+GLIND-GAT. We can further harness an attention network for each branch to model the intensity of the pairwise influence between connected nodes, by extending the spirit of GAT (Velickovic et al., 2018):
+
+$$
+\mathbf {z} _ {u} ^ {(l + 1)} = \mathbf {z} _ {u} ^ {(l)} + \sum_ {k = 1} ^ {K} h _ {u, k} ^ {(l)} \left(\sum_ {v, a _ {u v} = 1} w _ {u v} ^ {(l, k)} \mathbf {W} _ {D} ^ {(l, k)} \mathbf {z} _ {v} ^ {(l)} + \mathbf {W} _ {S} ^ {(l, k)} \mathbf {z} _ {u} ^ {(l)}\right),
+$$
+
+$$
+w _ {u v} ^ {(l, k)} = \frac {\delta ((\mathbf {c} ^ {(l , k)}) ^ {\top} [ \mathbf {W} _ {A} ^ {(l , k)} \mathbf {z} _ {u} ^ {(l)} \| \mathbf {W} _ {A} ^ {(l , k)} \mathbf {z} _ {v} ^ {(l)} ])}{\sum_ {w , a u w = 1} \delta (\mathbf {c} ^ {(l , k)}) ^ {\top} [ \mathbf {W} _ {A} ^ {(l , k)} \mathbf {z} _ {u} ^ {(l)} \| \mathbf {W} _ {A} ^ {(l , k)} \mathbf {z} _ {w} ^ {(l)} ])},
+$$
+
+where  $\delta$  is instantiated as LeakyReLU and  $\mathbf{W}_A^{(l,k)}\in \mathbb{R}^{d\times d}$  and  $\mathbf{c}^{(l,k)}\in \mathbb{R}^{2d}$  are both trainable parameters. Here,  $\mathbf{h}_u^{(l)}$  adaptively selects the path for attentive propagation.
+
+GLIND-TRANS. The preceding two models resort to message passing over observed structures. There also exist
+
+scenarios where the data geometries are partially observed or unobserved. To accommodate the potential interactions among arbitrary instance pairs, we extend the attention-based propagation to a latent topology as is done by Transformer (Vaswani et al., 2017):
+
+$$
+\mathbf {z} _ {u} ^ {(l + 1)} = \mathbf {z} _ {u} ^ {(l)} + \sum_ {k = 1} ^ {K} h _ {u, k} ^ {(l)} \left(\mathbf {W} _ {D} ^ {(l, k)} \mathbf {b} _ {u} ^ {(l, k)} + \mathbf {W} _ {S} ^ {(l, k)} \mathbf {z} _ {u} ^ {(l)}\right),
+$$
+
+$$
+\mathbf {b} _ {u} ^ {(l, k)} = \sum_ {v} \frac {\eta (\mathbf {W} _ {K} ^ {(l , k)} \mathbf {z} _ {v} ^ {(l)} , \mathbf {W} _ {Q} ^ {(l , k)} \mathbf {k} _ {u} ^ {(l)})}{\sum_ {w = 1} ^ {N} \eta (\mathbf {W} _ {K} ^ {(l , k)} \mathbf {z} _ {w} ^ {(l)} , \mathbf {W} _ {Q} ^ {(l , k)} \mathbf {k} _ {u} ^ {(l)})} \cdot \mathbf {z} _ {v} ^ {(l)},
+$$
+
+where  $\mathbf{W}_K^{(l,k)}$  and  $\mathbf{W}_Q^{(l,k)}$  are trainable weights at the  $l$ -th layer, and  $\eta$  denotes a certain similarity function. However, due to the all-pair global attention, the above computation requires quadratic complexity w.r.t.  $N$ , which hinders the scalability to large numbers of instances. We thus turn to a linearly scalable and numerically stable version of all-pair attention proposed by DIFFormer (Wu et al., 2023c), which specifically instantiates  $\eta$  as  $\eta(\mathbf{a},\mathbf{b}) = 1 + (\frac{\mathbf{a}}{\|\mathbf{a}\|_2})^\top \frac{\mathbf{b}}{\|\mathbf{b}\|_2}$ . Then assuming  $\mathbf{k}_u^{(l)} = \frac{\mathbf{W}_K^{(l,k)}\mathbf{z}_u^{(l)}}{\|\mathbf{W}_K^{(l,k)}\mathbf{z}_u^{(l)}\|_2}$  and  $\mathbf{q}_u^{(l)} = \frac{\mathbf{W}_Q^{(l,k)}\mathbf{z}_u^{(l)}}{\|\mathbf{W}_Q^{(l,k)}\mathbf{z}_u^{(l)}\|_2}$ , we can efficiently compute  $\mathbf{b}_{u,v}^{(l,k)}$  by
+
+$$
+\mathbf {b} _ {u} ^ {(l, k)} = \frac {\sum_ {v = 1} ^ {N} \mathbf {z} _ {v} ^ {(l)} + \left(\sum_ {v = 1} ^ {N} (\mathbf {k} _ {v} ^ {(l)}) (\mathbf {z} _ {v} ^ {(l)}) ^ {\top}\right) (\mathbf {q} _ {u} ^ {(l)})}{N + (\mathbf {q} _ {u} ^ {(l)}) ^ {\top} (\sum_ {v = 1} ^ {N} \mathbf {k} _ {v} ^ {(l)})}.
+$$
+
+The above computation for updating  $N$  instances' embeddings in each layer can be achieved within  $O(N)$  complexity since the three summation terms are shared by all instances and only requires once computation in practice. Moreover, if the observed structures are available, we can incorporate the aggregated embeddings of connected instances to the updating signals:  $\mathbf{b}_u^{(l,k)}\gets \frac{1}{2} (\mathbf{b}_u^{(l,k)} + \sum_{v,a_{uv} = 1}\frac{1}{d_u}\mathbf{z}_v^{(l)})$
+
+Data-Driven Prior via Mixture of Posteriors. A proper setting of the prior distribution  $p_0(\mathbf{d}^{(l)})$  is important but non-trivial. One simple solution is to assume the prior as some pre-defined trivial forms (Dinh et al., 2015), which however may potentially lead to over-regularization (Burda et al., 2015). Alternatively, one can estimate the prior by the average of the variational posterior (Hoffman & Johnson, 2016), i.e.,  $p_0(\mathbf{d}^{(l)}) \approx 1 / N\sum_{u = 1}^N q(\mathbf{d}^{(l)}|\mathbf{z}^{(l)} = \mathbf{z}_u^{(l)})$ . Nevertheless, such an approach can be computationally expensive (Tomczak & Welling, 2018) and would lead to biased estimation given limited training data. In our case, inspired by (Tomczak & Welling, 2018) using mixture of Gaussian as a learnable prior, we propose to use a mixture of pseudo variational posteriors as a flexible prior estimation:
+
+$$
+p _ {0} (\mathbf {d} ^ {(l)}) = \frac {1}{T} \sum_ {t = 1} ^ {T} q \left(\mathbf {d} ^ {(l)} \mid \mathbf {z} ^ {(l)} = \tilde {\mathbf {z}} _ {t} ^ {(l)}\right). \tag {10}
+$$
+
+where  $\tilde{\mathbf{z}}_t^{(l)}$  is the embedding of instance  $t$  in the generated pseudo dataset  $\{\tilde{\mathbf{x}}_t,\tilde{y}_t\}_{t = 1}^T$ . The latter is constructed by randomly sampling  $T$  instances from the  $N$  instances in the observed dataset, and the structures of the pseudo dataset are randomly generated with a pre-defined probability for each potential edge. We set  $T\ll N$  to reduce the computational cost. In this way, the prior estimation by Eqn. (10) plays as a general reflection of how the model recognizes each diffusivity hypothesis given uninformative inputs and is learned with the model in a fully data-driven manner.
+
+# 5. Experiments
+
+The goal of our experiments is to evaluate the generalization ability of the model under distribution shifts with interdependent data. We consider various real-world datasets that involve observed and unobserved data geometries, and the distribution shifts between training and testing sets are led by distinct contexts. Following common practice, we use the prediction accuracy on the out-of-distribution testing data for measuring the generalization performance.
+
+# 5.1. Experiment Setup
+
+Datasets. We adopt five datasets Twitch, Arxiv, DPPIN, STL and CIFAR for evaluation. Detailed information about these datasets are deferred to Appendix B. Since the properties of these datasets vary case by case, we consider different ways to split the observed data into multiple domains or construct out-of-distribution data with new unseen domains (where the data from different domains are considered as samples from different distributions). We next describe the evaluation protocol in terms of how we split the training and testing sets with distribution shifts in each case.
+
+- Twitch is a multi-graph dataset (Rozemberczki & Sarkar, 2021) where nodes of graphs are instances whose interconnectivity is reflected by observed structures. Each subgraph is comprised of a social network of users from a particular region, so one can use the nodes in different subgraphs as samples from different domains, since these subgraphs have obviously different topological features (Wu et al., 2022). In particular, we use the nodes from three subgraphs as training data (where we hold out  $25\%$  for validation), and the nodes from the other three subgraphs as testing data. The task is a binary classification for users' genders, and the performance is measured by ROC-AUC.  
+- Arxiv is a temporal network (Hu et al., 2020b) which evolves with time. Each node (i.e., a paper) as an instance has a time label indicating its publication year. Since the citation relationships among papers can significantly vary with different time-sensitive contexts, the distribution shifts naturally exist for the network data collected within different time windows. We thus use the papers published
+
+Table 1: Testing (mean±standard deviation) Accuracy (%) for Arxiv and ROC-AUC (%) for Twitch on different subsets of out-of-distribution data (determined by publication times and subgraphs, respectively). The missing results of EERM on Arxiv stem from the out-of-memory issue. We mark the first/second/third methods with the top performance.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Arxiv</td><td colspan="3">Twitch</td></tr><tr><td>2014-2016</td><td>2016-2018</td><td>2018-2020</td><td>ES</td><td>FR</td><td>EN</td></tr><tr><td>ERM-GCN</td><td>56.33 ± 0.17</td><td>53.53 ± 0.44</td><td>45.83 ± 0.47</td><td>66.07 ± 0.14</td><td>52.62 ± 0.01</td><td>63.15 ± 0.08</td></tr><tr><td>IRM-GCN</td><td>55.92 ± 0.24</td><td>53.25 ± 0.49</td><td>45.66 ± 0.83</td><td>66.95 ± 0.27</td><td>52.53 ± 0.02</td><td>62.91 ± 0.08</td></tr><tr><td>GroupDRO-GCN</td><td>56.52 ± 0.27</td><td>53.40 ± 0.29</td><td>45.76 ± 0.59</td><td>66.82 ± 0.26</td><td>52.69 ± 0.02</td><td>62.95 ± 0.11</td></tr><tr><td>DANN-GCN</td><td>56.35 ± 0.11</td><td>53.81 ± 0.33</td><td>45.89 ± 0.37</td><td>66.15 ± 0.13</td><td>52.66 ± 0.02</td><td>63.20 ± 0.06</td></tr><tr><td>Mixup-GCN</td><td>56.67 ± 0.46</td><td>54.02 ± 0.51</td><td>46.09 ± 0.58</td><td>65.76 ± 0.30</td><td>52.78 ± 0.04</td><td>63.15 ± 0.08</td></tr><tr><td>EERM-GCN</td><td>-</td><td>-</td><td>-</td><td>67.50 ± 0.74</td><td>51.88 ± 0.07</td><td>62.56 ± 0.02</td></tr><tr><td>GLIND-GCN</td><td>59.42 ± 0.33</td><td>56.84 ± 0.54</td><td>57.06 ± 1.21</td><td>67.72 ± 0.10</td><td>53.16 ± 0.08</td><td>64.18 ± 0.03</td></tr><tr><td>ERM-GAT</td><td>57.15 ± 0.25</td><td>55.07 ± 0.58</td><td>46.22 ± 0.82</td><td>65.67 ± 0.02</td><td>52.00 ± 0.10</td><td>61.85 ± 0.05</td></tr><tr><td>IRM-GAT</td><td>56.55 ± 0.18</td><td>54.53 ± 0.32</td><td>46.01 ± 0.33</td><td>67.27 ± 0.19</td><td>52.85 ± 0.15</td><td>62.40 ± 0.24</td></tr><tr><td>GroupDRO-GAT</td><td>56.69 ± 0.27</td><td>54.51 ± 0.49</td><td>46.00 ± 0.59</td><td>67.41 ± 0.04</td><td>52.99 ± 0.08</td><td>62.29 ± 0.03</td></tr><tr><td>DANN-GAT</td><td>57.23 ± 0.18</td><td>55.13 ± 0.46</td><td>46.61 ± 0.57</td><td>66.59 ± 0.38</td><td>52.88 ± 0.12</td><td>62.47 ± 0.32</td></tr><tr><td>Mixup-GAT</td><td>57.17 ± 0.33</td><td>55.33 ± 0.37</td><td>47.17 ± 0.84</td><td>65.58 ± 0.13</td><td>52.04 ± 0.04</td><td>61.75 ± 0.13</td></tr><tr><td>EERM-GAT</td><td>-</td><td>-</td><td>-</td><td>66.80 ± 0.46</td><td>52.39 ± 0.20</td><td>62.07 ± 0.68</td></tr><tr><td>GLIND-GAT</td><td>60.36 ± 0.36</td><td>58.98 ± 0.43</td><td>59.71 ± 0.53</td><td>67.82 ± 0.10</td><td>54.50 ± 0.12</td><td>64.32 ± 0.12</td></tr></table>
+
+Table 2: Testing RMSE for DPPIN on different domains (determined by protein identification methods).  
+
+<table><tr><td>Method</td><td>Hazbun</td><td>Krogan (LCMS)</td><td>Krogan (MALDI)</td><td>Lambert</td><td>Tarassov</td><td>Uetz</td><td>Yu</td></tr><tr><td>ERM-Trans</td><td>1.82 ± 0.17</td><td>1.63 ± 0.04</td><td>1.57 ± 0.03</td><td>1.49 ± 0.07</td><td>1.62 ± 0.03</td><td>1.52 ± 0.04</td><td>1.51 ± 0.04</td></tr><tr><td>IRM-Trans</td><td>1.66 ± 0.14</td><td>1.86 ± 0.04</td><td>1.84 ± 0.04</td><td>1.52 ± 0.07</td><td>1.76 ± 0.03</td><td>1.66 ± 0.05</td><td>1.66 ± 0.04</td></tr><tr><td>DANN-Trans</td><td>1.69 ± 0.11</td><td>1.66 ± 0.02</td><td>1.62 ± 0.03</td><td>1.39 ± 0.05</td><td>1.63 ± 0.01</td><td>1.49 ± 0.01</td><td>1.50 ± 0.01</td></tr><tr><td>GroupDRO-Trans</td><td>1.65 ± 0.13</td><td>1.68 ± 0.02</td><td>1.65 ± 0.02</td><td>1.48 ± 0.03</td><td>1.72 ± 0.01</td><td>1.53 ± 0.04</td><td>1.53 ± 0.01</td></tr><tr><td>Mixup-Trans</td><td>1.46 ± 0.13</td><td>1.79 ± 0.05</td><td>1.76 ± 0.04</td><td>1.50 ± 0.06</td><td>1.70 ± 0.05</td><td>1.56 ± 0.06</td><td>1.59 ± 0.06</td></tr><tr><td>EERM-Trans</td><td>1.68 ± 0.47</td><td>1.91 ± 0.23</td><td>1.92 ± 0.09</td><td>1.47 ± 0.05</td><td>1.79 ± 0.11</td><td>1.67 ± 0.07</td><td>1.65 ± 0.08</td></tr><tr><td>GLIND-TRANS</td><td>1.02 ± 0.07</td><td>1.38 ± 0.07</td><td>1.33 ± 0.05</td><td>1.08 ± 0.04</td><td>1.40 ± 0.04</td><td>1.20 ± 0.04</td><td>1.20 ± 0.04</td></tr></table>
+
+before 2014 as training data (where we hold out  $25\%$  for validation), and the papers published after 2014 as testing data. The task is to predict the subareas of papers, and the performance is measured by Accuracy.
+
+○ DPPIN consists of multiple datasets of biological protein interactions (Fu & He, 2022). Each dataset is comprised of protein instances corresponding to a particular protein identification method (that can be treated as the domain). Each protein has a time-evolving scalar representing the gene expression value and the proteins within each domain have certain interactions exposed by the co-expressed activities. The interaction networks at one time, reflecting the partially observed data geometries, can be noisy and incomplete given the dynamical evolution of the co-expression levels. We use the proteins of four datasets for training, one dataset for validation and the other seven datasets for testing. We consider the regression task for protein's gene expression values, and the performance is measured by RMSE.
+
+- STL-10 is an image dataset where each instance is an image. Since there are no observed structures interconnecting the instances, we create the inter-instance relations via  $k$ -nearest-neighbor ( $k$ NN) with the Euclidean distance between two instances' input features. We consider using dif
+
+ferent  $k$ 's for constructing the data from different domains. To be specific, the value of  $k$  is set as  $\{2,3,4,8,9,10\}$  which results in six domains where each domain corresponds to a particular kNN graph interconnecting the fix set of instances. We split all the image instances into training and testing sets with the ratio 1:1. Then we use the training instances with the structures of the first three domains as training data (where we hold out  $25\%$  for validation), and the testing instances with the structures of the other three domains as testing data. In this way, the training and testing data can be seen as samples with disparate underlying geometries. The task is to predict the image labels, and the performance is measured by Accuracy.
+
+$\circ$  CIFAR-10 is another image dataset. Similarly, we use  $k\mathbf{NN}$  to construct the inter-instance relations. Differently, we fix  $k = 5$  and consider cosine similarity as the distance function. To introduce distribution shifts, we add angle bias to the cosine similarity to construct data from different domains. To be specific, we denote by  $s_i$  the distance function used for  $k\mathbf{NN}$  in the  $i$ -th domain, and particularly we consider  $s_i(\mathbf{x}_1,\mathbf{x}_2) = 1 - \cos (\angle (\mathbf{x}_1,\mathbf{x}_2) + \theta_i)$  where  $\angle$  denotes the angle of input feature vectors  $\mathbf{x}_1$  and  $\mathbf{x}_2$ . We set  $\theta_{i}\in \{0^{o},30^{o},90^{o},150^{o},160^{o},170^{o}\}$ , resulting in six
+
+domains. We adopt the same way for data splitting as STL, resulting in training data from three domains and testing data from the other three domains. The angle bias causes the interconnecting structures of training and testing data to have different proximity patterns.
+
+Competitors. We compare with a set of learning algorithms including empirical risk minimization (ERM), which is the most commonly used supervised learning objective, and state-of-the-art approaches for improving generalization such as the adversarial learning approach DANN (Liu et al., 2020), the distributionally robust optimization Group-DRO (Sagawa et al., 2019), the data augmentation approach Mixup (Zhang et al., 2017), and invariant risk minimization approaches IRM (Arjovsky et al., 2019) and EERM (Wu et al., 2022). These models are agnostic to encoder architectures. For fair comparison, we use GCN (Kipf & Welling, 2017), GAT (Velickovic et al., 2018) and DIFFormer (Wu et al., 2023c) as their encoder backbones to compare with our models GLIND-GCN, GLIND-GAT and GLIND-TRANS, respectively. Due to space limit, we defer implementation details and hyper-parameter searching space to Appendix C.
+
+# 5.2. Comparative Results
+
+Generalization with Observed Geometries. We first test the models on data with observed structures where we compare GLIND-GCN and GLIND-GAT with the competitors using GCN and GAT as the encoder backbones, respectively. Table 1 reports the performance on testing data of Arxiv and Twitch. In particular, for Arxiv, we further divide the testing data into three subsets according to the publication years of the papers to closely check the generalization performance on testing data with different levels of distribution shifts. As we can see, the accuracy produced by all methods exhibits an overall decrease as the time gap (between training and testing data) enlarges, while GLIND-GCN and GLIND-GAT significantly alleviate the performance drop. Notably, for testing data within 2018-2020, GLIND-GCN and GLIND-GAT achieve  $23.8\%$  and  $26.6\%$  improvement of the accuracy over the runner-ups, respectively. Furthermore, for Twitch, we separately report the ROC-AUC on three testing subgraphs to compare the generalization performance on specific cases. The results show that GLIND-GCN and GLIND-GAT achieve overall superior ROC-AUCs over these strong competitors, respectively.
+
+Generalization with Partially Observed Geometries. We next consider the case where the data geometries are partially observed. This situation can be commonly encountered in practice, as the observed relational structures can be noisy and incomplete. To accommodate the potential interactions, we use our model GLIND-TRANS and compare it with the competitors using DIFFformer as the encoder. Table 2 reports the performance on testing data of DPPIN. We
+
+found that our model GLIND-TRANS consistently achieves the first-ranking results (the lowest RMSE) in seven testing domains, with an averaged reduction of  $21.6\%$  on RMSE over the runner-up. This demonstrates the practical efficacy of the proposed model for handling generalization with partially observed data geometries.
+
+Generalization with Unobserved Geometries. The last scenario we study involves data geometries that are unobserved. We use the image datasets for demonstration where the inter-instance structures are synthetically constructed through the  $k$ -nearest-neighbor. We adopt the three model versions for comparison, where GLIND-GCN and GLIND-GAT purely rely on the input structures and GLIND-TRANS estimates the potential interactions among arbitrary instance pairs. Table 3 reports the performance on testing data of CIFAR and STL. In contrast with the competitors, our model yields the highest averaged accuracy on two datasets, demonstrating the superiority of the proposed model. Comparing the models using different backbones, we found that the Transformer-style architecture contributes to more stable performance (with lower variance across different testing sets). This is probably because the Transformer-style architecture can capture the latent interactions that are more insensitive to distribution shifts of structures than the GNNs that purely rely on observed structures.
+
+# 5.3. Further Discussions
+
+![](images/deff2a2e42161e3a7aa14ca1e8fb705f604ee46495e6556e513121bd0e01db3a.jpg)  
+(a) Accuracy on testing sets
+
+![](images/dca139c7c198ef90e22904af0fe278c42fe67dcc85963394e2636162379e4585.jpg)  
+(b) Accuracy on testing sets  
+Figure 3: (a) Ablation studies for GLIND on STL. (b) Performance of GLIND on  $\mathsf{Arxiv}$  with different  $K$ 's.
+
+Ablation Studies. To further verify the effectiveness of GLIND, we next dissect the efficacy of the specific proposed components in our model. We conduct ablation studies w.r.t. model architectures and learning objectives, respectively, and report the comparison results on STL in Fig. 5(a) (more results are deferred to Appendix D). For the model architecture, we compare with three variants: 1)  $w / o$  Res removes the residual link  $(\mathbf{z}_u^{(l)}$  added to each layer); 2)  $w / o$  Feat removes the self-loop feature aggregation  $(\mathbf{W}_S^{(l,k)}\mathbf{z}_u^{(l)})$ ; 3)  $w / o$  Multi sets  $K = 1$  that removes the multiple branches at each layer. We found that all of these components contribute to consistent performance improvements in the three testing domains, which verifies the efficacy of these designs and
+
+Table 3: Testing Accuracy (\%) for CIFAR and STL on out-of-distribution data with different domains. We use the  $k$ -nearest-neighbor to construct the inter-instance relations in each domain. For STL (resp. CIFAR), different  $k$ 's (resp. angle bias for computing the cosine similarity) are used for creating distinct domains.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">CIFAR</td><td colspan="3">STL</td></tr><tr><td>150°</td><td>160°</td><td>170°</td><td>k=8</td><td>k=9</td><td>k=10</td></tr><tr><td>ERM-GCN</td><td>72.30 ± 0.22</td><td>73.16 ± 0.05</td><td>72.55 ± 0.30</td><td>72.89 ± 0.34</td><td>73.27 ± 0.40</td><td>74.18 ± 0.23</td></tr><tr><td>IRM-GCN</td><td>72.96 ± 0.32</td><td>73.97 ± 0.44</td><td>73.33 ± 0.08</td><td>72.96 ± 0.38</td><td>73.67 ± 0.22</td><td>74.28 ± 0.40</td></tr><tr><td>GroupDRO-GCN</td><td>72.73 ± 0.45</td><td>73.40 ± 0.37</td><td>72.85 ± 0.24</td><td>72.79 ± 0.39</td><td>73.31 ± 0.19</td><td>74.44 ± 0.67</td></tr><tr><td>DANN-GCN</td><td>72.19 ± 0.34</td><td>73.12 ± 0.19</td><td>72.34 ± 0.37</td><td>72.77 ± 0.22</td><td>73.48 ± 0.56</td><td>74.38 ± 0.73</td></tr><tr><td>Mixup-GCN</td><td>72.66 ± 0.51</td><td>73.49 ± 0.38</td><td>72.60 ± 0.49</td><td>73.04 ± 0.09</td><td>73.99 ± 0.19</td><td>74.68 ± 0.57</td></tr><tr><td>EERM-GCN</td><td>71.01 ± 0.95</td><td>72.29 ± 1.35</td><td>71.16 ± 1.36</td><td>72.14 ± 1.30</td><td>72.11 ± 1.73</td><td>72.07 ± 1.41</td></tr><tr><td>GLIND-GCN</td><td>79.24 ± 0.53</td><td>80.26 ± 0.60</td><td>79.39 ± 0.89</td><td>78.00 ± 0.34</td><td>78.63 ± 0.29</td><td>78.17 ± 0.29</td></tr><tr><td>ERM-GAT</td><td>72.92 ± 0.22</td><td>74.39 ± 0.18</td><td>73.22 ± 0.14</td><td>73.29 ± 0.36</td><td>73.38 ± 0.58</td><td>74.15 ± 0.74</td></tr><tr><td>IRM-GAT</td><td>72.96 ± 0.17</td><td>73.97 ± 0.38</td><td>73.33 ± 0.10</td><td>72.30 ± 1.42</td><td>73.16 ± 0.63</td><td>74.22 ± 1.24</td></tr><tr><td>GroupDRO-GAT</td><td>72.87 ± 0.18</td><td>74.07 ± 0.09</td><td>73.21 ± 0.11</td><td>73.38 ± 0.19</td><td>73.70 ± 0.12</td><td>74.61 ± 0.51</td></tr><tr><td>DANN-GAT</td><td>72.81 ± 0.38</td><td>74.16 ± 0.15</td><td>73.23 ± 0.45</td><td>73.45 ± 0.40</td><td>73.70 ± 0.19</td><td>74.90 ± 0.54</td></tr><tr><td>Mixup-GAT</td><td>72.98 ± 0.17</td><td>74.41 ± 0.19</td><td>73.65 ± 0.13</td><td>73.36 ± 0.21</td><td>74.28 ± 0.17</td><td>74.75 ± 0.01</td></tr><tr><td>EERM-GAT</td><td>71.03 ± 1.69</td><td>72.82 ± 1.87</td><td>71.84 ± 1.14</td><td>71.31 ± 1.01</td><td>72.65 ± 1.89</td><td>71.96 ± 1.55</td></tr><tr><td>GLIND-GAT</td><td>78.23 ± 0.80</td><td>79.01 ± 0.63</td><td>77.98 ± 0.88</td><td>77.91 ± 0.84</td><td>79.17 ± 0.23</td><td>78.81 ± 0.66</td></tr><tr><td>ERM-Trans</td><td>76.88 ± 0.11</td><td>77.51 ± 0.25</td><td>76.35 ± 0.28</td><td>76.53 ± 0.25</td><td>77.10 ± 0.65</td><td>77.90 ± 0.22</td></tr><tr><td>IRM-Trans</td><td>76.53 ± 0.03</td><td>77.11 ± 0.05</td><td>76.42 ± 0.31</td><td>76.95 ± 0.14</td><td>77.49 ± 0.25</td><td>78.02 ± 0.35</td></tr><tr><td>GroupDRO-Trans</td><td>76.94 ± 0.65</td><td>76.99 ± 0.31</td><td>76.37 ± 0.53</td><td>77.81 ± 0.59</td><td>78.01 ± 0.54</td><td>78.10 ± 0.27</td></tr><tr><td>DANN-Trans</td><td>76.91 ± 0.17</td><td>77.13 ± 0.37</td><td>76.61 ± 0.30</td><td>77.64 ±0.13</td><td>78.29 ± 0.54</td><td>78.19 ± 0.35</td></tr><tr><td>Mixup-Trans</td><td>77.49 ± 0.39</td><td>77.91 ± 0.14</td><td>77.45 ± 0.34</td><td>77.76 ± 0.30</td><td>78.32 ± 0.57</td><td>78.73 ± 0.76</td></tr><tr><td>EERM-Trans</td><td>79.68 ± 0.51</td><td>79.89 ± 0.32</td><td>78.82 ± 0.54</td><td>77.92 ± 0.93</td><td>78.58 ± 0.20</td><td>78.18 ± 0.38</td></tr><tr><td>GLIND-TRANS</td><td>80.72 ± 0.39</td><td>81.06 ± 0.32</td><td>80.24 ± 0.38</td><td>78.06 ± 0.46</td><td>79.39 ± 0.28</td><td>78.41 ± 0.57</td></tr></table>
+
+our model architecture induced by the principled diffusion framework. For the learning objective, we compare with two variants: 1)  $w / o$  Reg removes the regularization term (the KL loss in Eqn. 9); 2)  $w / o$  Mix replaces the prior distribution in Eqn. 10 with the average of variational posterior  $p_0(\mathbf{d}^{(l)}) \approx 1 / N \sum_{u=1}^N q(\mathbf{d}^{(l)} | \mathbf{z}^{(l)} = \mathbf{z}_u^{(l)})$ . We found that these two variants lead to some performance drops, which verifies the effectiveness of our learning objective.
+
+Impact of Numbers of Diffusivity Hypothesis. We proceed to study the impact of the hyper-parameter  $K$ , i.e., the number of diffusivity hypotheses at each message passing layer, on the model performance. Fig. 5(b) presents the testing accuracy of GLIND-GCN on Arxiv with  $K$  ranging from 2 to 10 (more results are deferred to Appendix D). We found that setting  $K$  as a moderate value can bring up overall the optimal performance. If  $K$  is too small, the model would have insufficient capacity for learning multi-faceted data interactions from complex underlying generative mechanisms; if  $K$  is too large, the model would become overly complicated which might impair generalization.
+
+# 6. Conclusion
+
+This paper has explored a geometric diffusion framework for generalization with interdependent data. The model is generalized from the diffusion process on manifolds with
+
+stochastic diffusivity function generated from a variational distribution, conditioned on the diffusion dynamics trajectories. We further develop a re-weighting regularization approach to guide the diffusion model to learn causal relations from inputs to outputs throughout the diffusion process that facilitates generalization. Experiments on diverse real-world datasets with different kinds of distribution shifts verify the effectiveness of the proposed models.
+
+Future Work. There can be several promising directions for future investigation. For example, apart from the experimental scenarios and datasets considered in this paper, the proposed methodology and general ideas can be potentially applied and extended to other tasks where out-of-distribution generalization matters, e.g., molecule representations (Yang et al., 2022c). Another problem closely related to out-of-distribution generalization is out-of-distribution detection (Wu et al., 2023a; Li et al., 2022), whose goal is to enable the model to identify testing data from different distributions than training. It also remains under-explored how the diffusion framework explored in this paper can be used for guiding the methodology in the context of out-of-distribution detection with interdependent data.
+
+# Impact Statement
+
+This work presents an attempt to improve the generalization for handling interdependent data. Researching the generalization of machine learning models holds profound implications for advancing the reliability and applicability of artificial intelligence systems. Understanding and enhancing the generalization capabilities of models is paramount for deploying them across diverse real-world scenarios. Improved generalization not only enhances model performance on unseen data but also addresses ethical concerns related to bias and fairness. Furthermore, research in this domain contributes to the creation of models that are transferable across different domains, fostering broader adoption of machine learning solutions. As the impact of AI continues to grow across various sectors, unraveling the intricacies of generalization becomes instrumental in ensuring the responsible and equitable deployment of machine learning technologies.
+
+# Acknowledgement
+
+The work was in part supported by NSFC (92370201, 622222607, 72342023).
+
+# References
+
+Arjovsky, M., Bottou, L., Gulrajani, I., and Lopez-Paz, D. Invariant risk minimization. arXiv preprint arXiv:1907.02893, 2019.  
+Beery, S., Van Horn, G., and Perona, P. Recognition in terra incognita. In European conference on computer vision (ECCV), pp. 456-473, 2018.  
+Belkin, M. and Niyogi, P. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput., 15(6):1373-1396, 2003.  
+Belkin, M., Niyogi, P., and Sindhwani, V. Manifold regularization: A geometric framework for learning from labeled and unlabeled examples. Journal of machine learning research, 7(11), 2006.  
+Blanchard, G., Lee, G., and Scott, C. Generalizing from several related classification tasks to a new unlabeled sample. In Advances in Neural Information Processing Systems (NeurIPS), pp. 2178-2186, 2011.  
+Burda, Y., Grosse, R., and Salakhutdinov, R. Importance weighted autoencoders. arXiv:1509.00519, 2015.  
+Chamberlain, B., Rowbottom, J., Gorinova, M. I., Bronstein, M. M., Webb, S., and Rossi, E. GRAND: graph neural diffusion. In International Conference on Machine Learning (ICML), pp. 1407-1418, 2021a.
+
+Chamberlain, B. P., Rowbottom, J., Eynard, D., Giovanni, F. D., Dong, X., and Bronstein, M. M. Beltrami flow and neural diffusion on graphs. In Advances in Neural Information Processing Systems (NeurIPS), 2021b.  
+Chen, T., Kornblith, S., Norouzi, M., and Hinton, G. A simple framework for contrastive learning of visual representations, 2020.  
+Christakopoulou, K., Traverse, M., Potter, T., Marriott, E., Li, D., Haulk, C., Chi, E. H., and Chen, M. Deconfounding user satisfaction estimation from response rate bias. In RecSys, pp. 450-455, 2020.  
+Dinh, L., Krueger, D., and Bengio, Y. Nice: Non-linear independent components estimation. ICLR, 2015.  
+Freidlin, M. I. and Wentzell, A. D. Diffusion processes on graphs and the averaging principle. The Annals of probability, pp. 2215-2245, 1993.  
+Fu, D. and He, J. Dppin: A biological repository of dynamic protein-protein interaction network data. In 2022 IEEE International Conference on Big Data (Big Data), pp. 5269-5277. IEEE, 2022.  
+Geirhos, R., Rubisch, P., Michaelis, C., Bethge, M., Wichmann, F. A., and Brendel, W. Imagenet-trained cnns are biased towards texture; increasing shape bias improves accuracy and robustness. arXiv preprint arXiv:1811.12231, 2018.  
+Gong, M., Zhang, K., Liu, T., Tao, D., Glymour, C., and Schölkopf, B. Domain adaptation with conditional transferable components. In International Conference on Machine Learning (ICML), pp. 2839-2848, 2016.  
+Hoffman, M. D. and Johnson, M. J. Elbo surgery: yet another way to carve up the variational evidence lower bound. In Workshop in Advances in Approximate Bayesian Inference, NeurIPS, 2016.  
+Hu, W., Fey, M., Zitnik, M., Dong, Y., Ren, H., Liu, B., Catasta, M., and Leskovec, J. Open graph benchmark: Datasets for machine learning on graphs. In Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., and Lin, H. (eds.), NeurIPS 2020, 2020a.  
+Hu, W., Fey, M., Zitnik, M., Dong, Y., Ren, H., Liu, B., Catasta, M., and Leskovec, J. Open graph benchmark: Datasets for machine learning on graphs. In Advances in Neural Information Processing Systems, 2020b.  
+Jang, E., Gu, S., and Poole, B. Categorical reparameterization with gumbel-softmax. In International Conference on Learning Representations, 2017.
+
+Kipf, T. N. and Welling, M. Semi-supervised classification with graph convolutional networks. In International Conference on Learning Representations (ICLR), 2017.  
+Koh, P. W., Sagawa, S., Marklund, H., Xie, S. M., Zhang, M., Balsubramani, A., Hu, W., Yasunaga, M., Phillips, R. L., Gao, I., Lee, T., David, E., Stavness, I., Guo, W., Earnshaw, B., Haque, I., Beery, S. M., Leskovec, J., Kundaje, A., Pierson, E., Levine, S., Finn, C., and Liang, P. WILDS: A benchmark of in-the-wild distribution shifts. In International Conference on Machine Learning (ICML), pp. 5637-5664, 2021.  
+Li, Z., Wu, Q., Nie, F., and Yan, J. Graphde: A generative framework for debiased learning and out-of-distribution detection on graphs. In Advances in Neural Information Processing Systems, 2022.  
+Liu, M., Gao, H., and Ji, S. Towards deeper graph neural networks. In Gupta, R., Liu, Y., Tang, J., and Prakash, B. A. (eds.), KDD, pp. 338-348. ACM, 2020.  
+Ma, J., Deng, J., and Mei, Q. Subgroup generalization and fairness of graph neural networks. Advances in Neural Information Processing Systems, 34:1048-1061, 2021.  
+Mansour, Y., Mohri, M., and Rostamizadeh, A. Domain adaptation: Learning bounds and algorithms. In Conference on Learning Theory (COLT), 2009.  
+Medvedev, G. S. The nonlinear heat equation on dense graphs and graph limits. SIAM Journal on Mathematical Analysis, 46(4):2743-2766, 2014.  
+Muandet, K., Balduzzi, D., and Scholkopf, B. Domain generalization via invariant feature representation. In International Conference on Machine Learning (ICML), pp. 10-18, 2013.  
+Pearl, J., Glymour, M., and Jewell, N. P. Causal inference in statistics: A primer. John Wiley & Sons, 2016.  
+Roberts, M. E., Stewart, B. M., and Nielsen, R. A. Adjusting for confounding with text matching. American Journal of Political Science, 64(4):887-903, 2020.  
+Romeny, B. M. H. Geometry-driven diffusion in computer vision, volume 1. Springer Science & Business Media, 2013.  
+Rosenberg, S. and Steven, R. The Laplacian on a Riemannian manifold: an introduction to analysis on manifolds. Number 31. Cambridge University Press, 1997.  
+Rozemberczki, B. and Sarkar, R. Twitch gamers: a dataset for evaluating proximity preserving and structural role-based node embeddings, 2021.
+
+Rubin, D. B. Essential concepts of causal inference: a remarkable history and an intriguing future. *Biostatistics & Epidemiology*, 3:140–155, 2019.  
+Sagawa, S., Koh, P. W., Hashimoto, T. B., and Liang, P. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization. arXiv preprint arXiv:1911.08731, 2019.  
+Stadler, M., Charpentier, B., Geisler, S., Zügner, D., and Gunnemann, S. Graph posterior network: Bayesian predictive uncertainty for node classification. Advances in Neural Information Processing Systems, 34:18033-18048, 2021.  
+Tenenbaum, J. B., Silva, V. d., and Langford, J. C. A global geometric framework for nonlinear dimensionality reduction. science, 290(5500):2319-2323, 2000.  
+Thorpe, M., Xia, H., Nguyen, T., Strohmer, T., Bertozzi, A. L., Osher, S. J., and Wang, B. GRAND++: graph neural diffusion with a source term. In International Conference on Learning Representations (ICLR), 2022.  
+Tomczak, J. and Welling, M. Vae with a vampprior. In AISTATS, 2018.  
+Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., and Polosukhin, I. Attention is all you need. Advances in neural information processing systems, 30, 2017.  
+Velickovic, P., Cucurull, G., Casanova, A., Romero, A., Lio, P., and Bengio, Y. Graph attention networks. In International Conference on Learning Representations (ICLR), 2018.  
+Weickert, J. et al. Anisotropic diffusion in image processing, volume 1. Teubner Stuttgart, 1998.  
+Wu, Q., Zhang, H., Yan, J., and Wipf, D. Handling distribution shifts on graphs: An invariance perspective. In International Conference on Learning Representations, 2022.  
+Wu, Q., Chen, Y., Yang, C., and Yan, J. Energy-based out-of-distribution detection for graph neural networks. In International Conference on Learning Representations, 2023a.  
+Wu, Q., Yang, C., Zeng, K., Nie, F., Bronstein, M., and Yan, J. Advective diffusion transformers for topological generalization in graph learning. CoRR, abs/2310.06417, 2023b.  
+Wu, Q., Yang, C., Zhao, W., He, Y., Wipf, D., and Yan, J. DIFFormer: Scalable (graph) transformers induced by energy constrained diffusion. In International Conference on Learning Representations, 2023c.
+
+Yang, C., Wu, Q., Wen, Q., Zhou, Z., Sun, L., and Yan, J. Towards out-of-distribution sequential event prediction: A causal treatment. In Advances in Neural Information Processing Systems, 2022a.  
+Yang, C., Wu, Q., and Yan, J. Geometric knowledge distillation: Topology compression for graph neural networks. In Advances in Neural Information Processing Systems, 2022b.  
+Yang, N., Zeng, K., Wu, Q., Jia, X., and Yan, J. Learning substructure invariance for out-of-distribution molecular representations. In Advances in Neural Information Processing Systems, 2022c.  
+Ye, N., Li, K., Hong, L., Bai, H., Chen, Y., Zhou, F., and Li, Z. Ood-bench: Benchmarking and understanding out-of-distribution generalization datasets and algorithms. CoRR, abs/2106.03721, 2021.  
+Zhang, H., Cisse, M., Dauphin, Y. N., and Lopez-Paz, D. mixup: Beyond empirical risk minimization. arXiv preprint arXiv:1710.09412, 2017.  
+Zhao, X., Chen, F., Hu, S., and Cho, J.-H. Uncertainty aware semi-supervised learning on graph data. Advances in Neural Information Processing Systems, 33:12827-12836, 2020.
+
+# Appendix
+
+# A. Derivation and Proof
+
+# A.1. Derivation for Eqn. 7
+
+The log-likelihood for observed data  $\log p_{\theta}(\mathbf{y}|\mathbf{x},\mathcal{G})$  can be computed by
+
+$$
+\begin{array}{l} \log p _ {\theta} (\mathbf {y} | \mathbf {x}, \mathcal {G}) = \log \sum_ {\mathbf {z} ^ {(1)}, \dots , \mathbf {z} ^ {(L - 1)}} \prod_ {l = 0} ^ {L - 1} p _ {\theta} (\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathcal {G}) \\ = \sum_ {l = 0} ^ {L - 2} \log \sum_ {\mathbf {z} ^ {(l + 1)}} p _ {\theta} \left(\mathbf {z} ^ {(l + 1)} \mid \mathbf {z} ^ {(l)}, \mathcal {G}\right) + \log p _ {\theta} \left(\mathbf {z} ^ {(L)} \mid \mathbf {z} ^ {(L - 1)}, \mathcal {G}\right) \tag {11} \\ = \sum_ {l = 0} ^ {L - 2} \log \sum_ {\mathbf {z} ^ {(l + 1)}} \mathbb {E} _ {p (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} [ p _ {\theta} (\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}) ] \\ + \log \mathbb {E} _ {p (\mathbf {d} ^ {(L - 1)} | \mathbf {z} ^ {(L - 1)})} [ p _ {\theta} (\mathbf {z} ^ {(L)} | \mathbf {z} ^ {(L - 1)}, \mathbf {d} ^ {(L - 1)}, \mathcal {G}) ], \\ \end{array}
+$$
+
+where the marginalization over  $\mathbf{z}^{(1)},\dots ,\mathbf{z}^{(L - 1)}$  (which is omitted in Eqn. 5 for brevity) is through the delta predictive distributions  $p_{\theta}(\mathbf{z}^{(l + 1)}|\mathbf{z}^{(l)},\mathbf{d}^{(l)},\mathcal{G})$ . Due to the intractable integration over  $\mathbf{d}^{(l)}$ , we consider the evidence lower bound of Eqn. 11 by introducing  $q_{\phi}(\mathbf{d}^{(l)}|\mathbf{z}^{(l)})$ :
+
+$$
+\begin{array}{l} \sum_ {l = 0} ^ {L - 2} \mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} \left[ \log \sum_ {\mathbf {z} ^ {(l + 1)}} p _ {\theta} (\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}) \frac {p (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})}{q _ {\phi} (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} \right] \\ \left. + \mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(L - 1)} | \mathbf {z} ^ {(L - 1)})} \left[ \log p _ {\theta} (\mathbf {z} ^ {(L)} | \mathbf {z} ^ {(L - 1)}, \mathbf {d} ^ {(L - 1)}, \mathcal {G}) \frac {p (\mathbf {d} ^ {(L - 1)} | \mathbf {z} ^ {(L - 1)})}{q _ {\phi} (\mathbf {d} ^ {(L - 1)} | \mathbf {z} ^ {(L - 1)})} \right] \right] \\ = \sum_ {l = 0} ^ {L - 2} \mathbb {E} _ {q _ {\phi} \left(\mathbf {d} ^ {(0)} \mid \mathbf {z} ^ {(0)}\right), \dots , q _ {\phi} \left(\mathbf {d} ^ {(L - 2)} \mid \mathbf {z} ^ {(L - 2)}\right)} \left[ \log \sum_ {\mathbf {z} ^ {(l + 1)}} p _ {\theta} \left(\mathbf {z} ^ {(l + 1)} \mid \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}\right) \right] \tag {12} \\ + \mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(L - 1)} | \mathbf {z} ^ {(L - 1)})} [ \log p _ {\theta} (\mathbf {z} ^ {(L)} | \mathbf {z} ^ {(L - 1)}, \mathbf {d} ^ {(L - 1)}, \mathcal {G}) ] + \sum_ {l = 0} ^ {L - 1} \mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} \left[ \log \frac {p (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})}{q _ {\phi} (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} \right] \\ = \mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(0)} | \mathbf {z} ^ {(0)}), \dots , q _ {\phi} (\mathbf {d} ^ {(L - 1)} | \mathbf {z} ^ {(L - 1)})} [ \log p _ {\theta} (\mathbf {z} ^ {(L)} | \mathbf {z} ^ {(0)}, \mathbf {d} ^ {(0)}, \dots , \mathbf {d} ^ {(L - 1)}, \mathcal {G}) ] \\ - \sum_ {l = 0} ^ {L - 1} \mathrm {K L} \left(q _ {\phi} \left(\mathbf {d} ^ {(l)} \mid \mathbf {z} ^ {(l)}\right), p \left(\mathbf {d} ^ {(l)} \mid \mathbf {z} ^ {(l)}\right)\right). \\ \end{array}
+$$
+
+This gives rise to the objective of Eqn. 7.
+
+# A.2. Proof for Theorem 3.1
+
+Before the proof, we first introduce two fundamental rules of do-calculus (Pearl et al., 2016) which will be used as the building blocks later. Consider a causal directed acyclic graph  $\mathcal{A}$  with three nodes:  $B$ ,  $D$  and  $E$ . We denote  $\mathcal{A}_{\overline{B}}$  as the intervened causal graph by cutting off all arrows coming into  $B$ , and  $\mathcal{A}_{\underline{B}}$  as the graph by cutting off all arrows going out from  $B$ . For any interventional distribution compatible with  $\mathcal{A}$ , the do-calculus induces the following two fundamental rules.
+
+i) Action/observation exchange:
+
+$$
+P (d | d o (b), d o (e)) = P (d | d o (b), e), \text {i f} (D \perp E | B) _ {\mathcal {A} _ {\overline {{B}} \underline {{E}}}}.
+$$
+
+ii) Insertion/deletion of actions:
+
+$$
+P (d | d o (b), d o (e)) = P (d | d o (b)), \text {i f} (D \perp E | B) _ {\mathcal {A} _ {\overline {{B E}}}}.
+$$
+
+Back to our case where we study the causal graph comprised of  $\mathbf{x}$ ,  $\mathbf{y}$  and  $\mathbf{d}$ . We have
+
+$$
+\begin{array}{l} p _ {\theta} (\mathbf {y} | d o (\mathbf {x}), \mathcal {G}) = \sum_ {\mathbf {d}} p _ {\theta} (\mathbf {y} | d o (\mathbf {x}), \mathbf {d}, \mathcal {G}) p (\mathbf {d} | d o (\mathbf {x})) \\ = \sum_ {\mathbf {d}} p _ {\theta} (\mathbf {y} | \mathbf {x}, \mathbf {d}, \mathcal {G}) p (\mathbf {d} | d o (\mathbf {x})) \tag {13} \\ = \sum_ {\mathbf {d}} p _ {\theta} (\mathbf {y} | \mathbf {x}, \mathbf {d}, \mathcal {G}) p _ {0} (\mathbf {d}), \\ \end{array}
+$$
+
+where the first step is given by the law of total probability, the second step is according to the first rule (since  $\mathbf{y} \perp \mathbf{x}|\mathbf{d}$  in  $\mathcal{A}_{\mathbf{x}}$ ), and the third step is due to the second rule (since we have  $\mathbf{d} \perp \mathbf{x}$  in  $\mathcal{A}_{\overline{\mathbf{x}}}$ ). The above derivation shows that  $p_{\theta}(\mathbf{y}|do(\mathbf{x}), \mathcal{G}) = \mathbb{E}_{p_{0}(\mathbf{d})}[p_{\theta}(\mathbf{y}|\mathbf{x}, \mathbf{d}, \mathcal{G})]$  where  $p_{0}$  is the prior distribution of the diffusivity.
+
+Now consider the diffusion model that induces a trajectory  $\mathbf{x} = \mathbf{z}^{(0)}\rightarrow \mathbf{z}^{(1)}\rightarrow \dots \rightarrow \mathbf{z}^{(L)} = \mathbf{y}$ . By treating  $\{\mathbf{d}^{(l)}\}_{l = 0}^{L - 1}$  as a whole random variable, we can extend the result of Eqn. 13 and obtain
+
+$$
+p _ {\theta} (\mathbf {y} \mid d o (\mathbf {x}), \mathcal {G}) = \sum_ {\mathbf {d} ^ {(0)}, \dots , \mathbf {d} ^ {(L - 1)}} p _ {\theta} (\mathbf {y} \mid \mathbf {x}, \mathbf {d} ^ {(0)}, \dots , \mathbf {d} ^ {(L - 1)}, \mathcal {G}) p _ {0} \left(\mathbf {d} ^ {(0)}, \dots , \mathbf {d} ^ {(L - 1)}\right). \tag {14}
+$$
+
+Notice that  $p_0(\mathbf{d}^{(0)},\dots ,\mathbf{d}^{(L - 1)}) = \prod_{l = 0}^{L - 1}p_0(\mathbf{d}^{(l)})$ . By inserting the delta predictive distribution  $p_{\theta}(\mathbf{z}^{(l + 1)}|\mathbf{z}^{(l)},\mathbf{d}^{(l)},\mathcal{G})$  induced by the diffusion model, we can derive the log-likelihood:
+
+$$
+\begin{array}{l} \log p _ {\theta} (\mathbf {y} \mid d o (\mathbf {x}), \mathcal {G}) \\ = \log \sum_ {\mathbf {z} ^ {(1)}, \dots , \mathbf {z} ^ {(L - 1)}} \sum_ {\mathbf {d} ^ {(0)}, \dots , \mathbf {d} ^ {(L - 1)}} \prod_ {l = 0} ^ {L - 1} p _ {\theta} (\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}) p _ {0} (\mathbf {d} ^ {(l)}) \\ = \sum_ {l = 0} ^ {L - 2} \log \sum_ {\mathbf {z} ^ {(l + 1)}} \sum_ {\mathbf {d} ^ {(l)}} p _ {\theta} \left(\mathbf {z} ^ {(l + 1)} \mid \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}\right) p _ {0} \left(\mathbf {d} ^ {(l)}\right) + \log \sum_ {\mathbf {d} ^ {(L - 1)}} p _ {\theta} \left(\mathbf {z} ^ {(L)} \mid \mathbf {z} ^ {(L - 1)}, \mathbf {d} ^ {(L - 1)}, \mathcal {G}\right) p _ {0} \left(\mathbf {d} ^ {(L - 1)}\right) \tag {15} \\ = \sum_ {l = 0} ^ {L - 2} \log \sum_ {\mathbf {z} ^ {(l + 1)}} \sum_ {\mathbf {d} ^ {(l)}} p _ {\theta} (\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}) p _ {0} (\mathbf {d} ^ {(l)}) + \log \sum_ {\mathbf {d} ^ {(L - 1)}} p _ {\theta} (\mathbf {z} ^ {(L)} | \mathbf {z} ^ {(L - 1)}, \mathbf {d} ^ {(L - 1)}, \mathcal {G}) p _ {0} (\mathbf {d} ^ {(L - 1)}) \\ \end{array}
+$$
+
+We next separately consider the derivation for the two terms in the above equation. For the first term in Eqn. 15, using Jensen Inequality we have
+
+$$
+\begin{array}{l} \sum_ {l = 0} ^ {L - 2} \log \sum_ {\mathbf {z} ^ {(l + 1)}} \sum_ {\mathbf {d} ^ {(l)}} p _ {\theta} (\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}) p _ {0} (\mathbf {d} ^ {(l)}) \\ = \sum_ {l = 0} ^ {L - 2} \log \sum_ {\mathbf {z} ^ {(l + 1)}} \sum_ {\mathbf {d} ^ {(l)}} p _ {\theta} \left(\mathbf {z} ^ {(l + 1)} \mid \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}\right) p _ {0} \left(\mathbf {d} ^ {(l)}\right) \frac {q _ {\phi} \left(\mathbf {d} ^ {(l)} \mid \mathbf {z} ^ {(l)}\right)}{q _ {\phi} \left(\mathbf {d} ^ {(l)} \mid \mathbf {z} ^ {(l)}\right)} \tag {16} \\ \geq \sum_ {l = 0} ^ {L - 2} \sum_ {\mathbf {d} ^ {(l)}} q _ {\phi} (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)}) \cdot \left[ \log \sum_ {\mathbf {z} ^ {(l + 1)}} p _ {\theta} (\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}) \frac {p _ {0} (\mathbf {d} ^ {(l)})}{q _ {\phi} (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} \right]. \\ \end{array}
+$$
+
+For the second term in Eqn. 15, similarly we have
+
+$$
+\begin{array}{l} \log \sum_ {\mathbf {d} ^ {(L - 1)}} p _ {\theta} (\mathbf {z} ^ {(L)} | \mathbf {z} ^ {(L - 1)}, \mathbf {d} ^ {(L - 1)}, \mathcal {G}) p _ {0} (\mathbf {d} ^ {(L - 1)}) \\ \geq \sum_ {\mathbf {d} ^ {(L - 1)}} q _ {\phi} \left(\mathbf {d} ^ {(L - 1)} \mid \mathbf {z} ^ {(L - 1)}\right) \cdot \left[ \log p _ {\theta} \left(\mathbf {z} ^ {(L)} \mid \mathbf {z} ^ {(L - 1)}, \mathbf {d} ^ {(L - 1)}, \mathcal {G}\right) \frac {p _ {0} \left(\mathbf {d} ^ {(L - 1)}\right)}{q _ {\phi} \left(\mathbf {d} ^ {(L - 1)} \mid \mathbf {z} ^ {(L - 1)}\right)} \right]. \tag {17} \\ \end{array}
+$$
+
+By combing the results of Eqn. 16 and Eqn. 17 with Eqn. 15, we have the variational lower bound of  $\log p_{\theta}(\mathbf{y}|do(\mathbf{x}),\mathcal{G})$
+
+$$
+\begin{array}{l} \sum_ {l = 0} ^ {L - 2} \mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} \left[ \log \sum_ {\mathbf {z} ^ {(l + 1)}} p _ {\theta} \left(\mathbf {z} ^ {(l + 1)} \mid \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}\right) \frac {p _ {0} \left(\mathbf {d} ^ {(l)}\right)}{q _ {\phi} \left(\mathbf {d} ^ {(l)} \mid \mathbf {z} ^ {(l)}\right)} \right] \tag {18} \\ + \mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(L - 1)} | \mathbf {z} ^ {(L - 1)})} \left[ \log p _ {\theta} (\mathbf {z} ^ {(L)} | \mathbf {z} ^ {(L - 1)}, \mathbf {d} ^ {(L - 1)}, \mathcal {G}) \frac {p _ {0} (\mathbf {d} ^ {(L - 1)})}{q _ {\phi} (\mathbf {d} ^ {(L - 1)} | \mathbf {z} ^ {(L - 1)})} \right], \\ \end{array}
+$$
+
+where the marginalization over  $\mathbf{z}^{(1)},\dots ,\mathbf{z}^{(L - 1)}$  (which is omitted in Eqn. 8 for brevity) is through the delta predictive distribution  $p_{\theta}(\mathbf{z}^{(l + 1)}|\mathbf{z}^{(l)},\mathbf{d}^{(l)},\mathcal{G})$
+
+Furthermore, in Eqn. 16 the equality holds if and only if  $q_{\phi}(\mathbf{d}^{(l)}|\mathbf{z}^{(l)}) = p(\mathbf{d}^{(l)}|\mathbf{z}^{(l + 1)},\mathbf{z}^{(l)},\mathcal{G}) \cdot \frac{p_0(\mathbf{d}^{(l)})}{p(\mathbf{d}^{(l)}|\mathbf{z}^{(l)})}$  in which case the function inside the expectation stays as a constant w.r.t.  $\mathbf{d}^{(l)}$ . Similarly, for Eqn. 17, the equality holds if and only if  $q_{\phi}(\mathbf{d}^{(L - 1)}|\mathbf{z}^{(L - 1)}) = p(\mathbf{d}^{(L - 1)}|\mathbf{z}^{(L)},\mathbf{z}^{(L - 1)},\mathcal{G}) \cdot \frac{p_0(\mathbf{d}^{(L - 1)})}{p(\mathbf{d}^{(L - 1)}|\mathbf{z}^{(L - 1)})}$ .
+
+# A.3. Derivation for Eqn. 9
+
+The objective Eqn. 9 can be derived in a similar way as Eqn. 12 on the basis of Eqn. 18. In specific, we leverage the model-free prior  $p_0(\mathbf{d}^{(l)})$  to replace  $p(\mathbf{d}^{(l)}|\mathbf{z}^{(l)})$  in the first line of Eqn. 12:
+
+$$
+\begin{array}{l} \sum_ {l = 0} ^ {L - 2} \mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} \left[ \log \sum_ {\mathbf {z} ^ {(l + 1)}} p _ {\theta} (\mathbf {z} ^ {(l + 1)} | \mathbf {z} ^ {(l)}, \mathbf {d} ^ {(l)}, \mathcal {G}) \frac {p _ {0} (\mathbf {d} ^ {(l)})}{q _ {\phi} (\mathbf {d} ^ {(l)} | \mathbf {z} ^ {(l)})} \right] \\ + \mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(L - 1)} | \mathbf {z} ^ {(L - 1)})} \left[ \log p _ {\theta} \left(\mathbf {z} ^ {(L)} \mid \mathbf {z} ^ {(L - 1)}, \mathbf {d} ^ {(L - 1)}, \mathcal {G}\right) \frac {p _ {0} \left(\mathbf {d} ^ {(L - 1)}\right)}{q _ {\phi} \left(\mathbf {d} ^ {(L - 1)} \mid \mathbf {z} ^ {(L - 1)}\right)} \right] \tag {19} \\ = \mathbb {E} _ {q _ {\phi} (\mathbf {d} ^ {(0)} | \mathbf {z} ^ {(0)}), \dots , q _ {\phi} (\mathbf {d} ^ {(L - 1)} | \mathbf {z} ^ {(L - 1)})} [ \log p _ {\theta} (\mathbf {z} ^ {(L)} | \mathbf {z} ^ {(0)}, \mathbf {d} ^ {(0)}, \dots , \mathbf {d} ^ {(L - 1)}, \mathcal {G}) ] \\ - \sum_ {l = 0} ^ {L - 1} \operatorname {K L} \left(q _ {\phi} \left(\mathbf {d} ^ {(l)} \mid \mathbf {z} ^ {(l)}\right), p _ {0} \left(\mathbf {d} ^ {(l)}\right)\right). \\ \end{array}
+$$
+
+# B. Dataset Introduction
+
+Twitch (Rozemberczki & Sarkar, 2021) is a social network dataset which comprises multiple disconnected subgraphs. Each subgraph is a social network from a particular region and contains thousands of node instances that are interconnected. Each node is a use on Twitch and the node label is the gender of the user. Edges denote the friendship among users. Since in this dataset, nodes from different subgraphs can be seen as samples from distinct distributions (Table 4 shows different statistics of these subgraphs), we split the training and testing data according to subgraphs. Specifically, we use the nodes from subgraphs DE, PT and RU as training data, and the nodes from subgraphs ES, FR and EN as testing data.
+
+Arxiv is a citation network dataset provided by (Hu et al., 2020a). It records the citation relationship between papers and the goal is to predict the paper's subarea based on the word embeddings of paper keywords as the node features. The dataset also provides the time information about when the paper is published, and we use the publication year to split the training and testing sets. Since the research subareas and citation relationships of papers are sensitive to time, there natural exist distribution shifts between papers published in different time windows. In our experiments, we use papers published before 2014 for training, and papers published after 2014 for testing. We further divide the testing data into three subsets comprised of the papers published within 2014-2016, 2016-2018 and 2018-2020, respectively. As verification for the distribution shifts, Table 4 visualizes the different distributions of training and testing instances.
+
+CIFAR-10 and STL-10 are two image datasets where each image is an instance and there are no observed structures interconnecting the instances. Following the recent work (Wu et al., 2023c), we use all the 13000 images, each of which belongs to one of ten classes, for STL-10, and choose 1500 images from each of 10 classes for CIFAR-10. The input features of image instances are pre-computed by using the self-supervised approach SimCLR (Chen et al., 2020) (that does not use labels for training) and training a ResNet-18 that extracts the feature map as the input features of each instance. We use the  $k$ -nearest-neighbor to construct the inter-instance relational structures for these datasets, and consider different  $k$ 's and distance functions to introduce the distribution shifts. More details about how we choose the values of  $k$  and distance functions are presented in Sec. 5.1.
+
+Table 4: Statistics for different subgraphs of Twitch.  
+
+<table><tr><td>Datasets</td><td>DE</td><td>PT</td><td>RU</td><td>ES</td><td>FR</td><td>EN</td></tr><tr><td>#Instances</td><td>9498</td><td>1912</td><td>4385</td><td>4648</td><td>6549</td><td>7126</td></tr><tr><td>Density</td><td>0.0033</td><td>0.0171</td><td>0.0038</td><td>0.0054</td><td>0.0052</td><td>0.0013</td></tr><tr><td>Max Node Degree</td><td>3475</td><td>455</td><td>575</td><td>809</td><td>1517</td><td>465</td></tr><tr><td>Mean Node Degree</td><td>16</td><td>16</td><td>8</td><td>12</td><td>17</td><td>4</td></tr></table>
+
+DPPIN (Fu & He, 2022) is comprised of 12 individual dynamic network datasets at different scales, and each network records the protein-protein interactions. Each dynamic network is obtained by one protein identification method, which can be seen as one domain, and consists of 36 snapshots. Each protein has a sequence of continuous scalar features with 36 time stamps, which records the evolution of gene expression values within metabolic cycles of yeast cells. The interconnecting structures among protein instances are determined by co-expressed protein pairs at one time, and can change as the co-expression activities evolve with time. We consider the regression task for the gene expression value of each protein instance within one snapshot, and ignore the temporal dependence between different snapshots. In specific, the model input contains the proteins' gene expression value in the previous ten snapshot (as input features) and the interaction structures at the current snapshot, based on which the prediction target is the gene expression value of proteins at the current snapshot. As observed by (Fu & He, 2022), the protein instances from different datasets have distinct topological patterns (e.g., the distributions of cliques), so we split the training and testing data based on different datasets.
+
+![](images/b5b6eb0fb306fa613a8d807c57bfd6c9c8924536466b2a51af792e0d14393573.jpg)  
+(a) 2014-2016 test instances
+
+![](images/5e0fe27ac88a5118c9db119ecec6b900471cb2716468ef5b69d1da774cc74c61.jpg)  
+(b) 2016-2018 test instances  
+Figure 4: T-SNE visualization of input features for training instances (blue) and testing instances (red) on Arxiv.
+
+![](images/86794abcc8eaba43d630179214d393f7064df541f9970b448dce2eaebcb102a3.jpg)  
+(c) 2018-2020 test instances
+
+# C. Implementation Details
+
+Our implementation is based on PyTorch 1.9.0 and PyTorch Geometric 2.0.3. All of our experiments are run on a Tesla V100 with 16 GB memory. We adopt Adam with weight decay for training. We set a fixed training budget with 1000 epochs for DPPIN and 500 epochs for other datasets. The testing performance achieved by the epoch where the model gives the best performance on validation data is reported.
+
+The detailed architecture of GLIND is described as follows. The model architecture consists of the following modules in sequential order:
+
+- A fully-connected layer with hidden size  $D \times d$  (transforming  $D$ -dim input raw features into  $d$ -dim embeddings).  
+-  $L$ -layer message passing network with hidden size  $d$  (each layer contains  $K$  branches that have independent parameterization), based on the three instantiations in Sec. 4.  
+- A fully-connected layer with hidden size  $d \times C$  (mapping  $d$ -dim embeddings to  $C$  classes).
+
+In each layer, we use ReLU activation, dropout and the residual link where the weight  $\alpha$  is set as 0.5 across all datasets.
+
+The probability estimation for each branch is modeled by a feed-forward layer with hidden size  $d \times K$  in each layer as introduced in Sec. 4. The sampling process for the selected branch is approximated by Gumbel Softmax. For computing the prior distribution in the objective, we introduce a pseudo dataset of size  $T$  as described in Eqn. 10. We set  $T = \operatorname{int}(0.01 * N)$ , where  $N$  denotes the number of instances in the dataset, for all the datasets.
+
+For other hyper-parameters, we search them for each dataset with grid search on the validation set. The searching spaces for all the hyper-parameters are as follows.
+
+- Number of message passing layers  $L$  : [2, 3, 4, 5].  
+- Hidden dimension  $d$ : [32, 64, 128].  
+- Dropout ratio: [0.0, 0.1, 0.2, 0.5].  
+- Learning rate: [0.001, 0.005, 0.01, 0.02].  
+Weight decay: [0, 5e-5, 5e-4, 1e-3].  
+- Number of diffusivity hypothesis  $K$  : [3, 4, 5, 10].  
+Gumbel Softmax temperature  $\tau$ : [1, 2, 3, 5].
+
+![](images/794a629a9aaf786a92b1d221c1df17e0dc6e4dc40ce452cad33c97f21c732646.jpg)  
+D. Additional Experiment Results  
+(a) Accuracy on CIFAR
+
+![](images/dca61ffae1b3d401b32506cab708188c82feb4e0e62e9c152d9baf0310180f0e.jpg)  
+(b) Accuracy on Twitch  
+Figure 5: (a) Ablation studies for GLIND on CIFAR. (b) Performance of GLIND on three testing sets of Twitch with different numbers of diffusivity  $K$ 's.

2024/Learning Divergence Fields for Shift-Robust Graph Representations/images.zip

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2024/Learning Exceptional Subgroups by End-to-End Maximizing KL-Divergence/c2dc423f-c89d-4bb0-9b36-b4bf8f60b75e_origin.pdf

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2024/Learning Exceptional Subgroups by End-to-End Maximizing KL-Divergence/full.md

@@ -0,0 +1,614 @@
+# Learning Exceptional Subgroups by End-to-End Maximizing KL-divergence
+
+Sascha Xu  $^{*1}$  Nils Philipp Walter  $^{*1}$  Janis Kalofolias  $^{1}$  Jilles Vreeken  $^{1}$
+
+# Abstract
+
+Finding and describing sub-populations that are exceptional in terms of a target property has important applications in many scientific disciplines, from identifying disadvantaged demographic groups in census data to finding conducive molecules within gold nanoparticles. Current approaches to finding such subgroups require pre-discretized predictive variables, do not permit non-trivial target distributions, do not scale to large datasets, and struggle to find diverse results. To address these limitations, we propose SYFLOW, an end-to-end customizable approach in which we leverage normalizing flows to model arbitrary target distributions and introduce a novel neural layer that results in easily interpretable subgroup descriptions. We demonstrate on synthetic data, real-world data, and via a case study, that SYFLOW reliably finds highly exceptional subgroups accompanied by insightful descriptions.
+
+# 1. Introduction
+
+The majority of modern machine learning focuses on finding global models that perform well on predictive tasks such as classification. In this domain, deep neural networks often achieve state-of-the-art performance, at the expense of human-interpretable insight.
+
+Orthogonal to the advances in predictive modeling, many scientific applications require descriptive modeling such as finding sub-populations that are somehow exceptional, and providing a human-interpretable description for these. Applications of finding such subgroups range from identifying disadvantaged demographic groups in census data (Boll & Lagemann, 2019; Ortiz & Cummins, 2011) to learning those combinations of properties that single out materials with desirable properties (Sutton et al., 2020).
+
+*Equal contribution  ${}^{1}$  CISPA Helmholtz Center for Information Security, Saarbrücken, Germany. Correspondence to: Sascha Xu <sascha.xu@cispa.de>, Nils P. Walter <nils.walter@cispa.de>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+The common denominator in such applications is to present the relevant subgroups to a domain expert. In other words, not only do we require to find subsets with exceptional behavior, but also, that these can clearly be interpreted by the respective audience. That is, we have a joint optimization task of learning simple descriptions of sub-populations for which the property of interest is (locally) exceptionally distributed compared to the rest of the dataset. Typically, such a description is a conjunction of predicates, each based on the features of the dataset. For example on census data, where wage is the target property, a subgroup could be "Women without higher education" (Fig. 1a) have an exceptionally low salary compared to the overall population (Fig. 1b).
+
+Since the introduction of subgroup discovery by Klössgen, many approaches have been proposed (Atzmueller, 2015). However, these have not kept up with the recent advances in machine learning and suffer from three main limitations. First, due to combinatorial optimization, these methods are limited to small datasets. Second, most methods assume that the target follows a simple distribution, e.g. normal or binomial distribution. Although there are proposals to learn a proxy of the target distribution, their results are less interpretable. Third, existing methods require a pre-quantization of the continuous features, which is independent of the optimization procedure. As we show in our experiments, this greatly influences the quality of the results.
+
+To overcome these limitations, we propose SYFLOW. Our key contributions are as follows:
+
+(i) We formulate subgroup discovery as a continuous optimization problem based on KL-divergence. This enables first-order optimization, which significantly improves runtime and performance.  
+(ii) We leverage Normalizing Flows (Rezende & Mohamed, 2015a) to accurately learn the target distribution from data, enabling us to deal with intricate real-world distributions.  
+(iii) We propose a neuro-symbolic rule layer to learn interpretable subgroup descriptions and the corresponding discretization in an end-to-end fashion.
+
+We extensively evaluate SYFLOW on synthetic and real-world data. We show that SYFLOW accurately and reliably learns and characterizes exceptional subgroups, even for complex target distributions. We demonstrate that in con
+
+![](images/81586690864fbe802c79ac08f8a10b5610e0b2a59f5a313abbf27ea55438efba.jpg)  
+(a) Wages Subgroup  $S$  "Women without higher education" earn significantly less.
+
+![](images/3c41b8918844089f7597d1e90db5e43d991ad147adaca58973fd4018af81ac75.jpg)  
+(b) Exceptional distribution in subgroup (blue) vs. overall (green).  
+(c) SYFLOW learns subgroups of gold molecules with exceptional electron configurations.  
+Figure 1. Subgroups. SYFLOW learns subgroups, named subpopulations of which the distribution of the target variable is exceptional. In (a) SYFLOW precisely describes the subgroup of "Women without higher education", whose distribution of the target quantity wage is significantly lower (b). In general, SYFLOW is applicable on any data with non-trivial target distributions, e.g. material science (c).
+
+![](images/535a356dd7e400686f3da621316e6fb8d439538a73946a1a6f7e8d6d57000559.jpg)
+
+trast to the state-of-the-art baselines, SYFLOW identifies diverse sets of subgroups. To showcase SYFLOW's strength on real-world tasks, we perform a case study in the domain of materials science, where we search for characteristics of gold clusters with exceptional conductivity (Fig. 1c) and reactivity. In both cases, SYFLOW identifies physically meaningful subgroups and their respective descriptions.
+
+# 2. Preliminaries
+
+We consider a dataset of  $n$  pairs  $(\mathbf{x},y)$ , where  $y \in \mathcal{V}$  represents a property of interest, the target property, and  $\mathbf{x} \in \mathbb{R}^m$  is a feature vector. From a statistical perspective, we assume  $(\mathbf{x},y)$  is a realization of a pair of random variables  $(\mathbf{X},Y) \sim \mathbb{P}(\mathbf{X},Y)$ . We denote random variables by capitals, write  $p$  for their density, and  $P$  for their law.
+
+We are interested in learning subgroups for which the conditional distribution of the target attribute  $P_{Y|S = 1}$  is exceptional compared to  $P_{Y}$ . A subgroup membership function, or rule,  $\sigma (\mathbf{x})\in \{0,1\}$  determines whether a sample  $x$  belongs to the subgroup (1) or not (0). Formally, it is a conjunction  $\sigma :\mathbf{x}\mapsto \wedge_{i = 1}^{m}\pi (x_{i})$  of Boolean-valued predicates  $\pi :\mathbb{R}\to \{0,1\}$ , where each predicate defines an interval over which its output is 1, e.g. "  $18 < \mathrm{age} < 65$  ".
+
+To permit continuous optimization, we consider soft predicates  $\hat{\pi} \in [0,1]$ . These are smooth functions that model the probability of a sample  $x$  to be inside an interval  $\alpha < x < \beta$ . We can control the steepness of the transition via a temperature parameter  $t$ . We write  $s(x) \in [0,1]$  to denote a soft rule based on soft predicates, and  $S \in \{0,1\}$  for the indicator (random) variable that  $s$  defines, i.e.  $\mathbb{P}(S = 1 \mid \mathbf{X} = x) = s(x)$ . Note that by reducing  $t$  to 0 we again obtain binary predicates and rules that are easy to interpret.
+
+# 3. Method
+
+In this section, we introduce SYFLOW for learning exceptional subgroups by end-to-end maximization of KL-divergence. We first give an overview and then the details.
+
+# 3.1. Overview
+
+A subgroup is characterized by a membership function  $\sigma$ , which is commonly constrained to be a directly interpretable rule  $\sigma$ , i.e. a logical conjunction over predicates  $\pi$ . Finding the rule that identifies the most exceptional conditional distribution of the target is an NP-hard combinatorial problem (Boley & Grosskreutz, 2009).
+
+We propose to take a different, end-to-end optimization approach to learning subgroups. To this end, we propose a continuous relaxation of the binary-valued rule function that is designed to be differentiable, avoiding the need for pre-discretization, yet giving directly interpretable results. We propose to optimize the exceptionality of a subgroup in terms of KL-divergence between the conditional distribution  $P_{Y|S = 1}$  and the marginal distribution  $P_{Y}$  of the target, where we model these distributions non-parametrically using Normalizing Flows - with the added benefit that we have a single solution for univariate or multivariate targets.
+
+As our entire framework is differentiable, we can optimize all components with gradient descent, which as we will see is often both faster and more performant than combinatorial approaches. Finally, our framework naturally enables iteratively learning multiple non-redundant subgroups by regularizing with the similarity of already learned subgroups.
+
+# 3.2. Differentiable Rule Induction
+
+Subgroup membership  $\sigma : \mathbf{x} \mapsto \bigwedge_{i=1}^{m} \pi(x_i; \alpha_i, \beta_i)$  is defined by a logical conjunction over binary predicates  $\pi$ ,
+
+![](images/c021ebfa16b110bd9841b5621f460d91ad68ed3aa90b3b47748be19759fcd55e.jpg)  
+(a) Soft predicate  $\hat{\pi} (x;t)$  for different temperatures  $t$
+
+![](images/5474e2f524351df1ad0084e773c92a2a756c06d0284ec751d3a658c509c31b49.jpg)  
+(b) Subgroup membership probability for soft rule  $s(x)$ .  
+Figure 2. Example soft predicate (a) and soft rule (b).
+
+where we use one predicate per feature. Predicates for features that are not relevant for a subgroup are always true.
+
+Our key idea is to redefine the subgroup membership function in probabilistic terms, such that we obtain a function that acts akin to logical conjunctions while at the same time it is differentiable and therewith can be learned using a gradient-descent-based method. In particular, we propose to associate each feature  $X_{i}$  with a soft predicate  $\pi_{i}$
+
+$$
+\hat {\pi} \left(x _ {i}; \alpha_ {i}, \beta_ {i}, t\right) = \frac {e ^ {\frac {1}{t} \left(2 x _ {i} - \alpha_ {i}\right)}}{e ^ {\frac {1}{t} x _ {i}} + e ^ {\frac {1}{t} \left(2 x _ {i} - \alpha_ {i}\right)} + e ^ {\frac {1}{t} \left(3 x _ {i} - \alpha_ {i} - \beta_ {i}\right)}} (1)
+$$
+
+where we adapt the idea of approximate and differentiable splits from deep decision trees (Yang et al., 2018), we use a lower and upper bound  $\alpha, \beta \in \mathbb{R}$ , and introduce a temperature parameter  $t > 0$  that controls the steepness of the function at these bounds. The lower the temperature, the less soft the predicate, and in the limit of  $t \to 0$  a soft predicate converges to a strict predicate. In Fig. 2a we show an example soft predicate for different temperatures. For smaller  $t$ , the soft predicate closely approximates the strict predicate, except for a negligible region around the bounds.
+
+Theorem 1 Given its lower and upper bounds  $\alpha_{i},\beta_{i}\in \mathbb{R}$  the soft predicate of Eq. (1) applied on  $x\in R$  converges to the crisp predicate that decides whether  $x\in (\alpha ,\beta)$
+
+$$
+\lim  _ {t \to 0} \hat {\pi} (x _ {i}; \alpha_ {i}, \beta_ {i}, t) = \left\{ \begin{array}{l l} 1 & i f \alpha_ {i} <   x _ {i} <   \beta_ {i} \\ 0. 5 & i f x _ {i} = \alpha_ {i} \lor x _ {i} = \beta_ {i} \\ 0 & o t h e r w i s e \end{array} \right..
+$$
+
+We provide the proof for the general case in Appendix A.
+
+The soft predicate  $\hat{\pi}$  provides a differentiable, adaptable binning function. Next, we propose to combine the predicates  $\hat{\pi}$  for each feature  $x_{i}$  into a soft rule  $s$  that acts akin to logical conjunctions but remains differentiable. It is possible to model a logical conjunction using multiplication, but this leads to vanishing gradients (Hochreiter, 1998), which is problematic especially for non-trivial amounts of soft predicates (features). The harmonic mean
+
+$$
+\mathcal {M} (x) = \frac {p}{\sum_ {i = 1} ^ {m} \hat {\pi} (x _ {i} ; \alpha_ {i} , \beta_ {i} , t) ^ {- 1}},
+$$
+
+behaves as desired for strictly binary predicates, i.e.  $\exists \hat{\pi}(x_i; \alpha_i, \beta_i, t) = 0 \Rightarrow \mathcal{M}(x) = 0$  and  $\forall \hat{\pi}(x_i; \alpha_i, \beta_i, t) = 1 \Rightarrow \mathcal{M}(x) = 1$ , but tends to break down when given many soft predicates, e.g. for a very high dimensional feature space. To avoid this, we propose to instead use the weighted harmonic mean to model logical conjunctions, and construct the soft rule function  $s$  as
+
+$$
+s (x; \alpha , \beta , a, t) = \frac {\sum_ {i = 1} ^ {m} a _ {i}}{\sum_ {i = 1} ^ {m} a _ {i} \hat {\pi} \left(x _ {i} ; \alpha_ {i} , \beta_ {i} , t\right) ^ {- 1}}. \tag {2}
+$$
+
+The weights  $a \in \mathbb{R}^m$ , which are constrained to be positive through a ReLU function, allow the conjunction layer to disable unnecessary predicates. Wherever  $a_i > 0$ , the weights do not affect behavior for strictly binary predicates.
+
+In Fig. 2b, we show an example subgroup membership function for a soft rule  $s$ . The subgroup here is characterized by a conjunction of two predicates on  $X_{1}$  and  $X_{2}$ , i.e. a box, with a gradual, smooth transition from subgroup inclusion to exclusion at the boundaries.
+
+In general, our formulation of a soft rule is completely flexible in regards to the thresholds of the binned features, and for  $t \to 0$  is asymptotically equivalent to a strict rule.
+
+# 3.3. Differentiable Density Estimation
+
+Besides a differential rule function, we require accurate estimations of the conditional resp. marginal distributions of the target. We deviate from existing work by taking a differentiable non-parametric approach in the form of Normalizing Flows. These are an increasingly popular class of density estimators (Rezende & Mohamed, 2015b; Dinh et al., 2017; Papamakarios et al., 2021). The fundamental idea behind a normalizing flow is to start with a distribution with a known density function, e.g. a Gaussian distribution with  $p_{\mathcal{N}}$ , and fit an invertible function  $f$  to transform it onto the target density.
+
+Our method, SYFLOW, allows to seamlessly use any normalizing flow architecture. In this work, our architecture of choice are Neural Spline Flows (Durkan et al., 2019), which use expressive yet invertible piece-wise, polynomial spline functions. In general, the idea is to train the function  $f$  so that  $p_{y} \approx f(p_{\mathcal{N}})$ . Given a sample  $y$ , we can compute the likelihood of that point under the current function  $f$  as  $p_{f(\mathcal{N})}(y) = p_{\mathcal{N}}(f^{-1}(y))|\det \left(\frac{\delta f^{-1}(y)}{\delta y}\right)|$ . Thus, given a sample of  $\mathbb{P}(Y)$ , we can maximize the likelihood of  $p_{f(\mathcal{N})}(y)$  and hence fit  $p_{f(\mathcal{N})} \approx p_{y}$ .
+
+# 3.4. Differentiable Exceptionality Measure
+
+We now have a differentiable rule function  $s$  and versatile density estimate  $p_{Y}$  resp.  $p_{Y|S=1}$  of the target distribution. Next, we propose a differentiable measure of exceptionality between the conditional target distribution and the marginal
+
+target distribution. We adopt the Kullback-Leibler (KL) divergence and show how we can reformulate it towards this goal. We start with the standard definition,
+
+$$
+D _ {\mathrm {K L}} \left(P _ {Y \mid S = 1} \| P _ {Y}\right) = \int_ {y \in \mathcal {Y}} p _ {Y \mid S = 1} (y) \log \left(\frac {p _ {Y \mid S = 1} (y)}{p _ {Y} (y)}\right) d y. \tag {3}
+$$
+
+Here, the soft rule  $s$  does not explicitly appear, although we need it to take gradients wrt. the parameters. Towards this, we rewrite the first occurrence of  $p_{Y|S = 1}$  in Eq. (3) as
+
+$$
+\begin{array}{l} p _ {Y \mid S = 1} (y) = \int_ {\mathbf {x} \in \mathbb {R} ^ {m}} p _ {Y \mid S = 1, \mathbf {x}} (y, \mathbf {x}) p _ {\mathbf {x} \mid S = 1} (\mathbf {x}) d \mathbf {x} \\ = \int_ {\mathbf {x} \in \mathbb {R} ^ {m}} p _ {Y \mid S = 1, \mathbf {x}} (y, \mathbf {x}) \frac {p _ {S = 1 \mid \mathbf {x}} (\mathbf {x}) p _ {\mathbf {x}} (\mathbf {x})}{\mathbb {P} (S = 1)} d \mathbf {x}, \tag {4} \\ \end{array}
+$$
+
+by using the rules of marginal probability resp. Bayes' rule. We will first approximate Eq. (4), and then show how to estimate the KL divergence of Eq. (3) for its optimization.
+
+To this end, we first note that the subgroup indicator  $S$  takes two discrete values, indicating whether  $\mathbf{x}$  belongs to the subgroup. The rule function  $s(\mathbf{x})$  is deterministic in the limit of  $t\to 0$  as per Theorem 1. We use this to partition the domain of integration  $\mathbb{R}^m$  into  $\mathbb{R}_{\in}^{m}:= \{\mathbf{x}\in$ $\mathbb{R}^m |s_{t\to 0}(\mathbf{x}) = 1\}$  and  $\mathbb{R}_{\notin}^{m}:= \{\mathbf{x}\in \mathbb{R}^{m}|s_{t\to 0}(\mathbf{x}) = 0\}$ . Under the following four assumptions, we can bound the approximation of density  $p_{Y|S = 1}$  in Eq. (4). First, we assume that both  $p_{\mathbf{x}}$  and  $p_{Y|S = 1,\mathbf{x}}$  are upper bounded by the finite constants  $C_{\mathbf{X}} > 0$  and  $C_Y > 0$  respectively. Secondly, we assume that in a subset  $\mathbb{R}_{\notin C}^{m}\subset \mathbb{R}_{\notin}^{m}$ , where  $s$  is neither zero or one, i.e. the yellow region in Figure 2b, is negligible. Lastly, we assume that  $\mathbb{R}_{\notin C}^{m}$  covers almost all of the non-membership domain  $\mathbb{R}_{\notin}^{m}$  and the probability mass in this area is also negligible. Formally, we assume
+
+$$
+p _ {\mathbf {x}} (\mathbf {x}) \leq C _ {\mathbf {X}}, \tag {5}
+$$
+
+$$
+p _ {Y \mid S = 1, \mathbf {X}} \leq C _ {Y}, \tag {6}
+$$
+
+$$
+\int_ {\mathbf {x} \in \mathbb {R} _ {\neq C} ^ {m}} p _ {S = 1 | \mathbf {X}} (\mathbf {x}) d \mathbf {x} \leq \epsilon_ {1}, \tag {7}
+$$
+
+$$
+\int_ {\mathbf {x} \in \mathbb {R} _ {\notin \varnothing} ^ {m} \backslash \mathbb {R} _ {\notin \subset} ^ {m}} p _ {\mathbf {x}} (\mathbf {x}) d \mathbf {x} <   \epsilon_ {2}. \tag {8}
+$$
+
+Theorem 2 When Eqs. (5), (6), (7), and (8) hold, it is
+
+$$
+p _ {Y \mid S = 1} (y) - \int_ {\mathbf {x} \in \mathbb {R} _ {\epsilon} ^ {m}} p _ {Y \mid S = 1, \mathbf {x}} (y, \mathbf {x}) d \mathbf {x} \leq \frac {C _ {Y} \left(\epsilon_ {2} + C _ {\mathbf {X}} \epsilon_ {1}\right)}{\mathbb {P} (S = 1)}.
+$$
+
+Further, during learning, this bound becomes tighter until it asymptotically vanishes, assuming a decreasing annealing schedule for the temperature parameter.
+
+We postpone the proof to Appendix B.
+
+For an element of the subgroup  $\mathbf{x} \in \mathbb{R}_{\in}^{m} p_{S=1|x}$  is either zero or one, up to a negligible region (Ass. 7). We further approximate the target property conditional for  $\mathbf{x} \in \mathbb{R}_{\in}^{m}$
+
+$$
+\begin{array}{l} p _ {Y | \mathbf {x}} (y, \mathbf {x}) = p _ {Y | S = 1, \mathbf {x}} (y, \mathbf {x}) p _ {S = 1 | \mathbf {x}} (\mathbf {x}) \\ + p _ {Y \mid S = 0, \mathbf {x}} (y, \mathbf {x}) p _ {S = 0 \mid \mathbf {x}} (\mathbf {x}) \\ \approx p _ {Y \mid S = 1, \mathbf {x}} (y, \mathbf {x}), \\ \end{array}
+$$
+
+The approximation follows from a case distinction on  $p_{S = 1|x}$ . This allows us to approximate the subgroup-conditional target distribution from Eq. (4) as
+
+$$
+\begin{array}{l} p _ {Y \mid S = 1} (y) = \int_ {\mathbf {x} \in \mathbb {R} _ {\epsilon} ^ {m}} p _ {Y \mid S = 1, \mathbf {x}} (y, \mathbf {x}) \frac {p _ {S = 1 \mid \mathbf {x}} (\mathbf {x}) p _ {\mathbf {x}} (\mathbf {x})}{\mathbb {P} (S = 1)} d x \\ \approx \int_ {\mathbf {x} \in \mathbb {R} _ {\in} ^ {m}} p _ {Y, \mathbf {x}} (y, \mathbf {x}) \frac {p _ {S = 1 | \mathbf {x}} (\mathbf {x})}{\mathbb {P} (S = 1)} d x. \tag {9} \\ \end{array}
+$$
+
+Finally, we replace Eq. (9) into Eq. (3) to obtain our final approximation
+
+$$
+D _ {\mathrm {K L}} \left(P _ {Y \mid S = 1} \| P _ {Y \mid S = 1}\right) \approx
+$$
+
+$$
+\int_ {y \in \mathcal {Y}} \int_ {\mathbf {x} \in \mathbb {R} _ {\epsilon} ^ {m}} p _ {Y, \mathbf {x}} (y, \mathbf {x}) \frac {p _ {S = 1 | \mathbf {x}} (\mathbf {x})}{\mathbb {P} (S = 1)} d x \log \left(\frac {p _ {Y | S = 1} (y)}{p _ {Y} (y)}\right) d y.
+$$
+
+From this point onward, we can use the standard Monte Carlo estimation of this integral. This gives
+
+$$
+D _ {\mathrm {K L}} \left(P _ {Y | S = 1} \| P _ {Y}\right) \approx \frac {1}{n _ {s}} \sum_ {k = 1} ^ {n} s \left(\mathbf {x} ^ {(k)}\right) \log \left(\frac {p _ {Y | S = 1} \left(y ^ {(k)}\right)}{p _ {Y} \left(y ^ {(k)}\right)}\right),
+$$
+
+where  $p_{Y}$  and  $p_{Y|S=1}$  stand for the models trained from the normalizing flows,  $s$  is our subgroup membership model (see Sec. 3.2) and  $n_{s}$  is estimated as  $\frac{1}{n}\sum_{i=1}^{n}s(\mathbf{x}^{(i)})$ . Our approximation is directly computable given the density estimates  $p_{Y}$  and  $p_{Y|S=1}$  from the normalizing flows. Crucially this allows us to update the subgroups rule  $s$  as to maximize its exceptionality/KL-Divergence. Finally, we can now deal with some fine tuning of the objective to discover both representative and diverse subgroups.
+
+# 3.5. Rule Generality and Diversity
+
+The KL-divergence measures dissimilarity between two distributions. Naively maximizing it, however, has a drawback as we could easily craft a small subgroup consisting of the single most deviating sample on its own, defined by a rule with a very narrow scope and relatively low value. Thus, we employ a common technique in subgroup discovery (Boley et al., 2017) in order to steer the results towards larger subgroups: we multiply the statistic of dissimilarity with the size of the subgroup  $n_s^\gamma$ . The power  $\gamma$  tunes the trade-off in the importance of subgroup exceptionality and size.
+
+As we will show in our experiments, traditional subgroup discovery approaches often find sets of nearly identical subgroups. In the typical top- $k$  scheme, the best-scored subgroups are often slight variations of the same rule. To encourage SYFLOW to learn subgroups with diverse distributions, we introduce a regularizer, based on the KL-divergence of the current subgroup  $S$  to the previously found subgroups  $S_{k}$  to our objective. Hence, summarizing all the above, we obtain as our final objective our variant of the size-corrected KL (van Leeuwen & Knobbe, 2011):
+
+$$
+\begin{array}{l} D _ {\mathrm {W K L}} \left(P _ {Y | S = 1} \| P _ {Y}\right) = \\ n _ {s} ^ {\gamma} \hat {D} _ {\mathrm {K L}} \left(P _ {Y \mid S = 1} \| P _ {Y}\right) + \lambda \sum_ {j = 1} ^ {k} \hat {D} _ {\mathrm {K L}} \left(P _ {Y \mid S = 1} \| P _ {Y \mid S _ {j} = 1}\right). \tag {10} \\ \end{array}
+$$
+
+The parameter  $\lambda$  controls the strength of the regularizer.
+
+# 3.6. Full Model
+
+In the previous sections, we detailed our rule learning architecture with differentiable thresholding and aggregation (Sec. 3.2). We described how to use Neural Spline Flows to obtain non-parametric density estimates (Sec. 3.3), and finally derived Objective (10), a size aware Kullback-Leibler Divergence that allows us to optimize our rule function  $s(\mathbf{x})$  with gradient descent. Together these make up the components of our SYFLOW architecture for subgroup discovery with normalizing flows. Given a dataset  $\{(\mathbf{x}^{(k)},y^{(k)})\}_{k = 1}^{N}\sim \mathbb{P}(\mathbf{X},Y)$ , SYFLOW undergoes the following three steps for each sample  $(\mathbf{x}^{(k)},y^{(k)}):$
+
+1. Feature Thresholding: All continuous features  $\mathbf{x}_i^{(k)}$  are thresholded with learned parameters  $\alpha_{i}$  and  $\beta_{i}$  using the soft-binning from Eq. (2). Thereby we obtain a predicate vector  $\hat{\pi} (\mathbf{x}^{(k)};\alpha ,\beta ,t)\in [0,1]^p$  
+2. Subgroup Rule: We employ weights  $a_{i}$  to combine the individual predicates  $\hat{\pi} (\mathbf{x}_i^{(k)})$  into a conjunction  $s(\mathbf{x};\alpha ,\beta ,a,t)$ . This rule represents the probability of subgroup membership  $\mathbb{P}(S = 1|X = \mathbf{x}^{(k)})$ .  
+3. Distribution Exceptionality: We estimate the likelihood of  $p_{Y}(y^{(k)})$  and  $p_{Y|S=1}(y^{(k)})$  with two separately fitted normalizing flow models. Then, according to Objective (10), we can estimate the KL-Divergence between the current subgroup and the general distribution.
+
+By repeating the aforementioned steps over all samples  $(\mathbf{x}^{(k)},y^{(k)})$  and summing up the results, Objective (10) gives us a differentiable estimate of the KL-Divergence in regards to the subgroup rule  $s(\mathbf{x})$ . We optimize  $s(\mathbf{x})$  using standard gradient descent techniques with the Adam optimizer (Kingma & Ba, 2015). After the subgroup rule
+
+has been updated, we again update the normalizing flow of the subgroups density as described in Sec. 3.3, and repeat this process for a user-specified amount of epochs. During the training, we gradually decrease the temperature  $t$  by a pre-determined schedule to obtain increasingly crisp subgroup assignments. Finally, at the last epoch, the discovered subgroup is then the output of the subgroup rule  $s(\mathbf{x})$ . We provide a diagram overviewing and the pseudo-code for SYFLOW in the Appendix C.
+
+# 4. Related Work
+
+Subgroup Discovery. Traditional approaches for subgroup discovery (Klösgen, 1996) can be split based on type of search and on exceptionality measures. Subgroup discovery is NP-hard (Boley & Grosskreutz, 2009) and hence most proposals resort to greedy heuristics (Duivesteijn et al., 2016; Atzmueller, 2015) without guarantees. Branch-and-bound based algorithms (Boley et al., 2017; Kalofolias et al., 2019) permit results with guarantees for some exceptionality measures, but generally do not scale beyond tens of features (Atzmueller & Puppe, 2006).
+
+Most proposals for subgroup discovery assess exceptionality by comparing the conditional and marginal distributions of a single univariate target (Song et al., 2016; Helal, 2016). Basic measures of exceptionality include mean-shift (Grosskreutz & Rüping, 2009) for continuous, and weighted relative accuracy (Song et al., 2016) for discrete targets, but there exists a wide range of proposals for many data types (Kalofolias & Vreeken, 2022). Most, however, make strong assumptions about the distribution of the target such as normal (Friedman & Fisher, 1999; Lavrac et al., 2004), binomial or  $\chi^2$  (Grosskreutz & Rüping, 2009).
+
+Duivesteijn et al. (2016) generalize subgroup discovery to multivariate targets by measuring exceptionality via the difference between models just trained on the subgroup versus on all of the data. Due to computational costs, only simple models can be used, leading to a compromise in performance. Proença et al. (2022) instead measure exceptionality using a proxy of KL-divergence based on the Minimum Description Length (MDL). In contrast, SYFLOW optimizes KL-divergence directly, can employ any type of normalizing flow, and is equally suited for uni-/multi-variate targets.
+
+Neural Set Functions (NSFs) (Zaheer et al., 2017; Ou et al., 2022) are similar in spirit to subgroup discovery, as they also seek to find exceptional points. However, they significantly differ in two aspects. First, NSFs are learned using (partial) supervision, while subgroup discovery is unsupervised. Second, subgroup discovery finds interpretable descriptions, while NSFs come in a black-box.
+
+Subgroup discovery is also related to slice discovery (Chung et al., 2019; d'Eon et al., 2022; Eyuboglu et al., 2022) which
+
+![](images/16b32e1421026ca32d6eff82898b6c68e60665a65012c5a285e408aa5d2d2cd4.jpg)  
+(a) F1-scores of recovering a planted subgroup for different target distributions, higher is better.
+
+![](images/627319fecd211d60fe6c22c95eff38c810d2b0c1cf963d713aa827328a728ce6.jpg)  
+(b) F1-scores for SYFLOW and  $\mathrm{SD - }\mu$  higher is better.
+
+![](images/174032ed39cd05ede81c1d64562deacba5ca2e06769378b5a6b6437f5740bfe7.jpg)  
+(c) Runtime of SYFLOW and  $\mathrm{SD - }\mu$  ,lower is better.  
+Figure 3. Subgroup Predictive Accuracy. (a) Method comparison in terms of F1-score recovering subgroups in synthetic data. (b) Across different distributions: SYFLOW outperforms the competition on distributions with higher order moments. (c) With increasing number of cutpoints, SD-  $\mu$  matches SYFLOW accuracy around 40 bins, but needs  $10\times$  more time.
+
+aims to find subsets of the data where a deep learning model has high error rates. Slice discovery methods return membership functions defined on the latent space of the network. The focus lies on discovering concept-aligned slices rather than explicit rules on latent features. This makes slice discovery methods not applicable for generic subgroup discovery tasks. In contrast, methods such as SYFLOW can be useful for slice discovery with appropriate modifications.
+
+Differentiable Rule Induction Classical rule induction methods aim to find rules of the form “if  $X_{1} = 1 \land X_{5} = 1$  then  $Y = 0$ ” through expensive combinatorial optimization. Recently, highly scalable differentiable rule induction methods were proposed based on differentiable analogues of logical connectives, e.g. conjunctions and disjunctions (Fischer & Vreeken, 2021; Wang et al., 2020) that permit extracting crisp logical rules after training. Most work in this direction focuses on learning a global classifier (Yang et al., 2018; Qiao et al., 2021; Wang et al., 2021; Dierckx et al., 2023) as opposed to our goal of learning concise rules that identify exceptional subgroups. In this sense most related is Walter et al. (2024), who propose a neural architecture to find conjunctions of binary features that are over resp. under-expressed for a particular label. In contrast, SYFLOW considers continuous features, and is not constrained to a type or number of target variables.
+
+# 5. Experiments
+
+We evaluate SYFLOW against four state-of-the-art methods on synthetic and real-world data. We compare against Bump Hunting (BH, Friedman & Fisher, 1999), subgroup discovery using mean-shift  $(\mathrm{SD} - \mu$ , Lemmerich & Becker, 2018), subgroup discovery using KL-divergence (SD-KL), and Robust Subgroup Discovery (RSD, Proença et al., 2022). We give the hyperparameters in Appendix D and provide the data generators as well as the code online.
+
+# 5.1. Synthetic Data
+
+To evaluate on datasets with known ground truth we consider synthetic data. We begin by generating  $m$  feature variables  $\mathbf{X}_i$  and the target variable  $Y$  from a uniform distribution  $\mathcal{U}(0,1)$  to create a dataset of  $n = 20000$  samples. Within this dataset, we plant a rule  $\sigma(x) = \wedge_i^c \pi(x_i; \alpha_i, \beta_i)$  of  $c$  conditions. The hypercube described by the rule is set to have a volume of 0.1, i.e.,  $10\%$  of the population. For the samples within the planted subgroup, we re-sample the target variable  $Y$  using a separate distribution  $\mathbb{P}(Y \mid S = 1)$ . We run each experiment five times and report the average.
+
+Target Distribution First, we assess for all methods their accuracy in recovering the planted subgroup for different distributions of the target property  $Y$ . To this end, we vary  $\mathbb{P}(Y \mid S = 1)$  to be respectively a normal distribution  $\mathcal{N}(1.5, 0.5)$ , uniform distribution  $\mathcal{U}(0.5, 1.5)$ , exponential distribution  $\mathrm{Exp}(0.5)$ , Rayleigh distribution  $\mathcal{R}(2)$ , Cauchy distribution  $\mathcal{C}(0, 1)$ , beta distribution  $\mathcal{B}(0.2, 0.2)$ , and a balanced mixture distribution of two Gaussians (Bi-N)  $\mathcal{N}(-1.5, 0.5)$  and  $\mathcal{N}(1.5, 0.5)$ . The distribution are shown in Figure 8 in Appendix G. The number of features  $m = 10$  and complexity  $c = 4$  remains fixed.
+
+For each method, we compute the F1-score between the ground truth and discovered subgroup labels (i.e.  $S = 1$ ). We present the results in Figure 3a; We see that for distri
+
+![](images/7eb2e035aa2d7b8a10335b501cd00781a51371bda475017b7071b422175e1d74.jpg)  
+(a) F1 under increasingly more features  $m$ , higher is better.  
+(b) Runtime under increasingly more features  $m$ , lower is better.  
+Figure 4. Scalability of SYFLOW and baselines.
+
+![](images/da44b9d8f7a3986a348ecc5386cf0b4d5dbcc377ed920baba3a3d078f83f3131.jpg)  
+(a) Insurance: SYFLOW rules
+
+![](images/00e4cbb35eb3d4917569b9a61dc518271d4332b843077f64ab2ec7ad456d5d1d.jpg)  
+Figure 5. Subgroups learned on the Insurance and Wages datasets. Only SYFLOW learns diverse and exceptional subgroups.
+
+![](images/1247575db7f37af8504ecb829eb2c1abfbcf8afd90b1c03a338bd19855e072b1.jpg)  
+(b) Insurance: Sd-KL rules  
+(c) Wages: SYFLOW rules
+
+![](images/11017f142238d622769147dec91353c22b63d7401079938e5ef6832900fdb846.jpg)  
+(d) Wages:  $\mathbf{SD} - \mu$  rules
+
+butions that are well characterized by their first moment, i.e. the uniform and normal distribution, SYFLOW,  $\mathrm{SD - }\mu$  and BH are all able to recover the planted subgroup. On the exponential, Rayleigh and bi-modal distributions, only SYFLOW is able to recover the majority of the planted subgroup. In general, SYFLOW reliably recovers subgroups independent of the underlying target distribution.
+
+**Thresholding** Next, we study the efficacy of the differentiable feature thresholding. We generate data as before, considering only Normal distributions for the target variable  $Y$ , and setting  $m = 50$ . We compare the accuracy of SYFLOW against SD-µ, whilst gradually increasing the amount of bins per feature in the pre-processing for SD-µ.
+
+As we can see in Figure 3b, as the number of cutpoints increases, the F1-score of  $\mathrm{SD} - \mu$  improves. Under 20 cutpoints  $\mathrm{SD} - \mu$  performs much worse than SYFLOW, while for more cutpoints it slightly outperforms it in terms of accuracy. At the same time, as the number of cutpoints increases,  $\mathrm{SD} - \mu$  runtime increases rapidly (Figure 3c), requiring an order of magnitude more runtime to perform on par with SYFLOW.
+
+Scalability In the final experiment on synthetic data, we assess performance and runtime when varying the number of features. We allow each method up to 24 hours. We present our results in Figure 4. We observe that all methods perform remarkably stable in terms of F1-score, as well as that SYFLOW is significantly faster than all of its competitors. As a continuous optimization based method, SYFLOW avoids the typical combinatorial explosion in runtime, and additionally takes advantage of GPU acceleration. In comparison, SD-  $\mu$  SYFLOWS closest competitor, takes 50 times longer for 1000 features, whereas RSD does not finish within 24 hours for more than 100 features.
+
+Hyperparameter Sensitivity Lastly, we study the sensitivity of SYFLOW to its hyperparameters. We report the average F1-score over all target distributions, varying each hyperparameter individually whilst keeping the others fixed as in the previous experiments.
+
+For  $\gamma$ , which controls the trade-off between size and excep
+
+tionality, SYFLOW works well in a range of  $\gamma \in [0.4,1]$  with an average F1 score of 0.85 upwards. For the regularizer  $\lambda$ , we observe a 0.05 improvement when it is active, whilst varying its strength between [0.5,3] does not have a noticeable impact. Finally, for the predicate temperature  $t$  that controls the softness of the binning, the ideal range by empirical observation lies in between [0.01,0.2]. In both synthetic and real-world experiments we used a temperature of  $t = 0.2$  which provides good performance across the board. We provide a detailed analysis in Appendix D.
+
+# 5.2. Real World Data
+
+We now turn to real-world data, where we evaluate on regression datasets from the UCI-Machine Learning Repository. They provide a variety of target distributions and feature spaces to search and allow us to qualitatively inspect the learned rules. On each dataset, we let each method return 5 subgroups. For SYFLOW we report the average over 100 runs and report the standard deviation in Tab. 3. As the ground truth is unknown, we assess the subgroup quality in terms of KL-divergence  $(D_{KL})$ , the distribution overlap or Bhattacharyya Coefficient  $(BC)$ , and by absolute difference in mean (AMD). As  $D_{KL}$  and AMD are strongly influenced by the size  $n_s$  of the subgroup, we correct for this. For a formal definition we refer to Appendix E. In Table 1 we report, per metric, the scores of the best subgroup each method found.
+
+Across the board, SYFLOW reliably finds exceptional subgroups, and is either the best or close to the best method when it comes to the distribution based exceptionality measures  $(D_{KL}$  and  $BC)$ . SD-  $\mu$  optimizes for mean difference, and hence it is not surprising that it outperforms SYFLOW on this metric, but that SYFLOW nevertheless comes so close in terms of scores is a very positive result indeed.
+
+For the Insurance dataset it is inconclusive which method learns the most exceptional subgroups; SYFLOW returns the best subgroup in terms of  $D_{KL}$ , SD-KL finds the subgroup with the largest mean difference. For further analysis, we
+
+Table 1. Quantitative results of exceptionality of the subgroups discovered by resp. SYFLOW (ours), SD-KL, SD-μ, RSD, and BH, as measured KL-Divergence ( $D_{KL}$ , higher is better), Bhattacharyya coefficient ( $BC$ , lower is better), and absolute mean distance ( $AMD$ , higher is better).  
+
+<table><tr><td rowspan="2"></td><td colspan="5">DKL</td><td colspan="5">BC</td><td colspan="5">AMD</td></tr><tr><td>ours</td><td>SD-KL</td><td>SD-μ</td><td>RSD</td><td>BH</td><td>ours</td><td>SD-KL</td><td>SD-μ</td><td>RSD</td><td>BH</td><td>ours</td><td>SD-KL</td><td>SD-μ</td><td>RSD</td><td>BH</td></tr><tr><td>Abalone</td><td>0.15</td><td>0.02</td><td>0.12</td><td>0</td><td>0.05</td><td>0.69</td><td>0.99</td><td>0.93</td><td>1</td><td>0.87</td><td>0.73</td><td>0.25</td><td>0.84</td><td>0</td><td>0.16</td></tr><tr><td>Airquality</td><td>0.22</td><td>0.22</td><td>0.24</td><td>0</td><td>0.0</td><td>0.65</td><td>0.86</td><td>0.79</td><td>1</td><td>1.0</td><td>0.37</td><td>0.53</td><td>0.49</td><td>0</td><td>0.0</td></tr><tr><td>Automobile</td><td>0.25</td><td>0.24</td><td>0.23</td><td>0.26</td><td>0.21</td><td>0.63</td><td>0.85</td><td>0.79</td><td>0.64</td><td>0.6</td><td>1838</td><td>2807</td><td>2683</td><td>2218</td><td>2476</td></tr><tr><td>Bike</td><td>0.17</td><td>0.1</td><td>0.15</td><td>0.17</td><td>0.13</td><td>0.64</td><td>0.95</td><td>0.9</td><td>0.67</td><td>0.73</td><td>584</td><td>570</td><td>630</td><td>432</td><td>622</td></tr><tr><td>California</td><td>0.13</td><td>0.06</td><td>0.11</td><td>0</td><td>0.0</td><td>0.74</td><td>0.97</td><td>0.93</td><td>1</td><td>1.0</td><td>0.25</td><td>0.3</td><td>0.32</td><td>0</td><td>0.0</td></tr><tr><td>Insurance</td><td>0.26</td><td>0.13</td><td>0.26</td><td>0</td><td>0.19</td><td>0.41</td><td>0.93</td><td>0.52</td><td>1</td><td>0.84</td><td>3846</td><td>3973</td><td>3845</td><td>0</td><td>1518</td></tr><tr><td>Mpg</td><td>0.27</td><td>0.26</td><td>0.24</td><td>0.21</td><td>0.24</td><td>0.57</td><td>0.76</td><td>0.8</td><td>0.47</td><td>0.61</td><td>2.99</td><td>2.85</td><td>2.96</td><td>1.66</td><td>2.79</td></tr><tr><td>Student</td><td>0.08</td><td>0.03</td><td>0.08</td><td>0.09</td><td>0.04</td><td>0.85</td><td>0.99</td><td>0.94</td><td>0.71</td><td>0.97</td><td>0.46</td><td>0.52</td><td>0</td><td>0.47</td><td>0.45</td></tr><tr><td>Wages</td><td>0.1</td><td>0.02</td><td>0.1</td><td>0</td><td>0.03</td><td>0.88</td><td>0.99</td><td>0.9</td><td>1</td><td>0.99</td><td>6043</td><td>2994</td><td>5916</td><td>0</td><td>5149</td></tr><tr><td>Wine</td><td>0.06</td><td>0.0</td><td>0.06</td><td>0</td><td>0.01</td><td>0.9</td><td>1.0</td><td>0.97</td><td>1</td><td>0.97</td><td>0.17</td><td>0.04</td><td>0.19</td><td>0</td><td>0.04</td></tr><tr><td>Avg. rank</td><td>1.3</td><td>3.6</td><td>2.0</td><td>3.5</td><td>3.6</td><td>1.3</td><td>4.0</td><td>2.9</td><td>3.4</td><td>2.9</td><td>2.6</td><td>2.4</td><td>1.5</td><td>4.5</td><td>3.6</td></tr></table>
+
+plot the best three subgroups found by SYFLOW and SD-KL in Figure 5a and 5b. The specific rules that SYFLOW finds are succinct and informative ( $\neg$ Smoker  $\land$  Age  $< 36$ ), and represent a diverse set of subgroups (low, medium and high premiums). In contrast, SD-KL finds highly redundant rules, e.g. "Smoker" and "Smoker  $\land$  Age  $>18$ ", that all describe the same subpopulation.
+
+Generally, on all datasets where the target variable is exponentially distributed (Wages, Insurance, California), SYFLOW performs well in any measured metric, which matches the results seen on the exponential synthetic data (Sec. 5.1). Finally, in Fig. 5c and 5d, we plot the learned subgroups on the Wages dataset mentioned in the introduction. Again, SYFLOW finds diverse subgroups of disadvantaged demographics (Female  $\wedge$  Low Education) and advantaged groups (Male  $\wedge$  White  $\wedge$  Age  $>38$ ), whilst its competitor find either much less exceptional subgroups (RSD,BH) or variants of the same subgroup ( $\mathrm{SD}-\mu$ , SD-KL).
+
+Overall, SYFLOW shows versatility in finding exceptional subgroups across a variety of datasets and metrics. The discovered rules are succinct and diverse, can scale to large datasets, and are robust to the underlying target distribution. In the following, we will use SYFLOW on a specific application in materials science.
+
+# 5.3. Case Study: Materials Science
+
+Next, we consider a case study on materials science data (Goldsmith et al., 2017), an application where learning diverse exceptional subgroups has a clear scientific benefit. In particular, we consider a dataset of properties of gold-nano-clusters. These are key components in photo-voltaics, as well as in medical applications (Giljohann et al., 2020). The key goal is to better understand which molecular configurations lead to materials with better properties in terms
+
+of absorbing photons or with which cells they interact.
+
+We first focus on characterizing the HOMO-LUMO gap, the difference between the highest occupied and lowest unoccupied molecular orbit, which corresponds to the efficiency in energy absorption of the material for photons of specific frequencies. Gold nano-clusters are also increasingly used in medical applications, in which the HOMO-LUMO gap is crucial to determine their reactivity with other molecules.
+
+We show the subgroups that SYFLOW learns for the HOMO-LUMO gap in Figure 1c. The discovered subgroups relate to known ground-truth factors (Goldsmith et al., 2017), i.e. odd number of atoms leads to low gaps, but also ones with more complex descriptors such as clusters with an even amount of atoms that are mostly made up of double bond connections. SYFLOW provides out of the box interpretable, exceptional and diverse subgroups on this complex target quantity.
+
+Next, we additionally consider the relative intramolecular Van der Waals energies as the target quantity. These quantities are important to determine the strength of molecular binding, e.g. allow a compound to selectively interact with the intended target. We show the results in Figure 6. It is
+
+![](images/4094eef6ee03c9975c0788da84a76052db6f687f12526beef458aa03a7da7f13.jpg)  
+Figure 6. Gold Nano-Clusters. SYFLOW discovers a subgroup of molecules that have an outstanding joint distribution of Van-derWaals energy and HOMO-LUMO gap.
+
+easy to see that samples fitting subgroup rule  $s$  are indeed exceptional with regard to the overall distribution. Moreover, when we analyze the rule in detail, it makes physical sense. Molecules of between 8 to 14 atoms with a low (close to 1) gyration tend to have a significantly higher Van der Waals energy than those that are more strongly gyrated (i.e. are less 'flat') (Goldsmith et al., 2017).
+
+# 6. Conclusion
+
+We explored the problem of discovering a diverse set of exceptional subgroups, where the distribution of the target within each subgroup differs significantly from the overall population. Existing works suffer from combinatorial explosion, can not deal with intricate real world distributions, and heavily depend on the pre-quantization of the features. To overcome these limitations, we propose SYFLOW, where we formulate subgroup discovery as a continuous optimization problem. We use normalizing flows to accurately learn the distribution of the target property, enabling us to deal with arbitrary distributions. We propose a differentiable rule learner, which simultaneously learns the subgroup description and the corresponding discretization. We show on synthetic and real-world data, including a case study on gold nano-clusters, that SYFLOW reliably discovers diverting subgroups, especially when the target distribution is complex. In a case study on gold nano-clusters, we find subgroups that correspond to physically plausible processes.
+
+# Limitations
+
+A current limitation of all subgroup discovery methods, including SYFLOW, is that the description language of conjunctions of Boolean predicates may be too simple to describe true subgroups for physical data. We are specifically interested in exploring how we can extend SYFLOW to perform symbolic regression. Furthermore, we are interested in adapting SYFLOW to deal with structured data, such as images. We provide first experiments in Appendix I, where we find that SYFLOW is capable of discovering subgroups corresponding to the digits 0 and 1 in the MNIST dataset. Whilst this at the moment requires extensive pre-processing, we will pursue future work to allow SYFLOW to directly work with structured data such as images.
+
+# Impact Statement
+
+We propose a method, which can assist practitioners in making new scientific discoveries. For example, new insights into gold-cluster can have a potential impact on biomedical applications. However, when applied to sensitive census data, it is important to stress that SYFLOW is based on correlations and is not designed to make a definite causal statement. Thus, the results must be handled responsibly.
+
+# References
+
+Atzmueller, M. Subgroup discovery. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 5(1): 35-49, 2015.  
+Atzmueller, M. and Puppe, F. SD-Map - A Fast Algorithm for Exhaustive Subgroup Discovery. In Knowledge Discovery in Databases: PKDD 2006, pp. 6-17. Springer Berlin Heidelberg, 2006.  
+Boley, M. and Grosskreutz, H. Non-redundant subgroup discovery using a closure system. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 179-194. Springer, Springer, 2009.  
+Boley, M., Goldsmith, B. R., Ghiringhelli, L. M., and Vreeken, J. Identifying Consistent Statements about Numerical Data with Dispersion-Corrected Subgroup Discovery. Data Mining and Knowledge Discovery, pp. 1391-1418, September 2017.  
+Boll, C. and Lagemann, A. The gender pay gap in eu countries—new evidence based on eu-ses 2014 data. *Intereconomics*, 54:101–105, 2019.  
+Chung, Y., Kraska, T., Polyzotis, N., Tae, K. H., and Whang, S. E. Slice finder: Automated data slicing for model validation. In 2019 IEEE 35th International Conference on Data Engineering (ICDE), pp. 1550-1553. IEEE, 2019.  
+d'Eon, G., d'Eon, J., Wright, J. R., and Leyton-Brown, K. The spotlight: A general method for discovering systematic errors in deep learning models. In Proceedings of the 2022 ACM Conference on Fairness, Accountability, and Transparency, pp. 1962-1981, 2022.  
+Dierckx, L., Veroneze, R., and Nijssen, S. Rl-net: Interpretable rule learning with neural networks. In Pacific-Asia Conference on Knowledge Discovery and Data Mining, pp. 95-107. Springer, 2023.  
+Dinh, L., Sohl-Dickstein, J., and Bengio, S. Density estimation using real nvp. In International Conference on Learning Representations, 2017.  
+Duivesteijn, W., Feelders, A. J., and Knobbe, A. Exceptional Model Mining: Supervised descriptive local pattern mining with complex target concepts. Data Mining and Knowledge Discovery, pp. 47-98, January 2016.  
+Durkan, C., Bekasov, A., Murray, I., and Papamakarios, G. Neural spline flows. Advances in Neural Information Processing Systems, 32, 2019.  
+Eyuboglu, S., Varma, M., Saab, K. K., Delbrouck, J.-B., Lee-Messer, C., Dunnmon, J., Zou, J., and Re, C. Domino: Discovering systematic errors with cross-modal embeddings. In International Conference on Learning Representations, 2022.
+
+Fischer, J. and Vreeken, J. Differentiable pattern set mining. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pp. 383-392, 2021.  
+Friedman, J. H. and Fisher, N. I. Bump hunting in high-dimensional data. Statistics and computing, 9(2):123-143, 1999.  
+Giljohann, D. A., Seferos, D. S., Daniel, W. L., Massich, M. D., Patel, P. C., and Mirkin, C. A. Gold nanoparticles for biology and medicine. *Spherical Nucleic Acids*, pp. 55-90, 2020.  
+Goldsmith, B. R., Boley, M., Vreeken, J., Scheffler, M., and Ghiringhelli, L. M. Uncovering structure-property relationships of materials by subgroup discovery. New Journal of Physics, 19(1):013031, 2017.  
+Grosskreutz, H. and Rüping, S. On subgroup discovery in numerical domains. Data Mining and Knowledge Discovery, pp. 210-226, October 2009.  
+Helal, S. Subgroup discovery algorithms: a survey and empirical evaluation. Journal of computer science and technology, 31:561-576, 2016.  
+Hochreiter, S. The vanishing gradient problem during learning recurrent neural nets and problem solutions. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 6(02):107-116, 1998.  
+Kalofolias, J. and Vreeken, J. Naming the most anomalous cluster in hilbert space for structures with attribute information. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, 2022.  
+Kalofolias, J., Boley, M., and Vreeken, J. Discovering robustly connected subgraphs with simple descriptions. In 2019 IEEE International Conference on Data Mining (ICDM), pp. 1150-1155, November 2019.  
+Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization. In Bengio, Y. and LeCun, Y. (eds.), 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015.  
+Klösgen, W. Explora: A multipattern and multistrategy discovery assistant. In Advances in knowledge discovery and data mining, pp. 249-271. 1996.  
+Lavrac, N., Kavšek, B., Flach, P., and Todorovski, L. Subgroup discovery with CN2-SD. Journal of Machine Learning Research, pp. 153–188, 2004.  
+LeCun, Y., Bottou, L., Bengio, Y., and Haffner, P. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278-2324, 1998.
+
+Lemmerich, F. and Becker, M. pysubgroup: Easy-to-use subgroup discovery in python. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 658-662, 2018.  
+Ortiz, I. and Cummins, M. Global inequality: Beyond the bottom billion-a rapid review of income distribution in 141 countries. UNICEF, Division of Policy and Strategy Working papers, (1102), 2011.  
+Ou, Z., Xu, T., Su, Q., Li, Y., Zhao, P., and Bian, Y. Learning neural set functions under the optimal subset oracle. Advances in Neural Information Processing Systems, 35, 2022.  
+Papamakarios, G., Nalisnick, E., Rezende, D. J., Mohamed, S., and Lakshminarayanan, B. Normalizing flows for probabilistic modeling and inference. The Journal of Machine Learning Research, 22(1):2617-2680, 2021.  
+Proença, H. M., Grünwald, P., Bäck, T., and van Leeuwen, M. Robust subgroup discovery: Discovering subgroup lists using mdI. Data Mining and Knowledge Discovery, 36(5):1885-1970, 2022.  
+Qiao, L., Wang, W., and Lin, B. Learning accurate and interpretable decision rule sets from neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 4303-4311, 2021.  
+Rezende, D. and Mohamed, S. Variational inference with normalizing flows. In International Conference on Machine Learning, pp. 1530-1538. PMLR, 2015a.  
+Rezende, D. and Mohamed, S. Variational inference with normalizing flows. In International conference on machine learning, pp. 1530-1538. PMLR, 2015b.  
+Song, H., Kull, M., Flach, P., and Kalogridis, G. Subgroup Discovery with Proper Scoring Rules. In Machine Learning and Knowledge Discovery in Databases, pp. 492-510. Springer, Cham, September 2016.  
+Sutton, C., Boley, M., Ghiringhelli, L. M., Rupp, M., Vreeken, J., and Scheffler, M. Identifying domains of applicability of machine learning models for materials science. Nature communications, 11(1):4428, 2020.  
+van Leeuwen, M. and Knobbe, A. Non-redundant subgroup discovery in large and complex data. In Machine Learning and Knowledge Discovery in Databases, pp. 459-474. Springer, 2011.  
+Walter, N. P., Fischer, J., and Vreeken, J. Finding interpretable class-specific patterns through efficient neural search. In Proceedings of the AAAI Conference on Artificial Intelligence, 2024.
+
+Wang, Z., Zhang, W., Liu, N., and Wang, J. Transparent Classification with Multilayer Logical Perceptrons and Random Binarization. Proceedings of the AAAI Conference on Artificial Intelligence, pp. 6331-6339, April 2020.  
+Wang, Z., Zhang, W., Liu, N., and Wang, J. Scalable rule-based representation learning for interpretable classification. Advances in Neural Information Processing Systems, 34:30479-30491, 2021.  
+Yang, Y., Morillo, I. G., and Hospedales, T. M. Deep neural decision trees. arXiv preprint arXiv:1806.06988, 2018.  
+Zaheer, M., Kottur, S., Ravanbakhsh, S., Poczos, B., Salakhutdinov, R. R., and Smola, A. J. Deep sets. Advances in neural information processing systems, 30, 2017.
+
+# A. Proof of Asymptotic Correctness of Soft-Binning
+
+Proof: Consider a real value  $x_{i} \in \mathbb{R}$  and  $M$  sorted bin thresholds  $\beta_{i,j} \in \mathbb{R}$ , i.e.  $\beta_{i,j} < \beta_{i,j+1}$ . From the thresholds  $\beta_{i,j}$  we construct the bias vector  $b_{i} \in \mathbb{R}^{M+1}$  defined as
+
+$$
+b _ {i} = \left(0, \sum_ {j = 1} ^ {1} - \beta_ {i, j}, \dots , \sum_ {j = 1} ^ {M} - \beta_ {i, j}\right) ^ {T}.
+$$
+
+Additionally, we define a weight vector  $w \in \mathbb{R}^{M + 1}$  with  $w_{j} = j$ , so that
+
+$$
+w = (1, 2, \dots , M + 1) ^ {T}.
+$$
+
+The soft-binning result  $z\in [0,1]^{\mathcal{M} + 1}$  is defined as
+
+$$
+z = \operatorname {s o f t m a x} \left(\left(w x _ {i} + b _ {i}\right) / t\right).
+$$
+
+Now, let  $x_{i}$  be in the  $l$ -th bin, i.e.  $\beta_{i,l-1} < x_{i} < \beta_{i,l}$ , then we now firstly prove that  $\forall j \neq l: z_{l} > z_{j}$ . We do this by showing that the  $l$ -th logit  $\bar{z}_{l} = w_{l}x_{i} + b_{i,l}$  is the largest and hence also has the highest softmax activation.
+
+Firstly, note that the bin thresholds are sorted in order, so that for  $j < l$  it also holds that  $\beta_{i,j} < \beta_{i,l}$ .  $\bar{z}_l$  is defined as
+
+$$
+\bar {z} _ {l} = w _ {l} x _ {i} + b _ {i, l} = w _ {l} x _ {i} - \sum_ {k = 1} ^ {l - 1} \beta_ {i, k}.
+$$
+
+We can simply transform  $\bar{z}_l$  into  $\bar{z}_{l - 1}$  by subtracting  $x_{i} - \beta_{i,l - 1}$ , so that
+
+$$
+\bar {z} _ {l} - x _ {i} + \beta_ {i, l} = w _ {l - 1} x _ {i} - \sum_ {k = 1} ^ {l - 2} \beta_ {i, k} = \bar {z} _ {l - 1}.
+$$
+
+Now, as  $x_{i}$  is in the  $l$ -th bin, we know that  $\beta_{i,l-1} < x_{i}$  and hence  $x_{i} - \beta_{i,l} < 0$ . For all other  $j < l$ $\beta_{i,j} < x_{i}$  holds, and hence also  $\bar{z}_{l} > \bar{z}_{j}$ .
+
+Now consider the case where  $j > l$ . Here, it holds that
+
+$$
+\bar {z} _ {l} + x _ {i} - \beta_ {i, l + 1} = w _ {l + 1} x _ {i} - \sum_ {k = 1} ^ {l + 1} \beta_ {i, k} = \bar {z} _ {l + 1}.
+$$
+
+In general, we may transform  $\bar{z}_l$  into  $\bar{z}_j$  by repeatedly adding  $x_i - \beta_{i,k}$  for  $k\in [l + 1,\dots ,j]$ . For all thresholds  $x_{i} < \beta_{i,k}$  holds. Thus, each time we add a strictly negative number to the logit  $\bar{z}_l$ , which proves that also here  $\forall j > l:\bar{z}_l > \bar{z}_j$ . Thus, it holds that  $\forall j\neq l:\bar{z}_l > \bar{z}_j$
+
+Lastly, it remains to prove that with temperature  $t \to 0$ ,  $z$  is a one-hot bin encoding, i.e.  $z_{l} = 1$  and  $\forall j \neq l: z_{j} = 0$ . The soft-binning of  $z_{l}$  is defined as
+
+$$
+\lim _ {t \to 0} z _ {l} = \lim _ {t \to 0} \frac {\exp (\bar {z} _ {l} / t)}{\sum_ {j = 1} ^ {M + 1} \exp (\bar {z} _ {j} / t)} = \lim _ {t \to 0} \frac {1}{\sum_ {j = 1} ^ {M + 1} \exp ((\bar {z} _ {j} - \bar {z} _ {l}) / t)}.
+$$
+
+For  $j = l$ , the sum term evaluates to  $\exp (\bar{z}_l - \bar{z}_l) / t = \exp (0) = 1$ . For  $j \neq l$ , it holds that  $\bar{z}_l > \bar{z}_j$  as show previously, and hence in the limit
+
+$$
+\lim  _ {t \rightarrow 0} \exp (\bar {z} _ {j} - \bar {z} _ {j}) / t = \exp (- \infty) = 0.
+$$
+
+Thus, in the limit  $t \to 0$ , the denominator sums up to 1 and hence  $z_{l} = 1$ , and as the softmax is positive and sums up to zero, it follows that  $\forall j \neq l: z_{j} = 0$ .
+
+# B. Proof of Theorem 2
+
+Theorem 2 When Eqs. (5), (6), (7), and (8) hold, it is
+
+$$
+p _ {Y \mid S = 1} (y) - \int_ {\mathbf {x} \in \mathbb {R} _ {\epsilon} ^ {m}} p _ {Y \mid S = 1, \mathbf {x}} (y, \mathbf {x}) d \mathbf {x} \leq \frac {C _ {Y} (\epsilon_ {2} + C _ {\mathbf {X}} \epsilon_ {1})}{\mathbb {P} (S = 1)}.
+$$
+
+Further, during learning, this bound becomes tighter until it asymptotically vanishes, assuming a decreasing annealing schedule for the temperature parameter.
+
+Proof: We first recall that, under our model,  $p_{S = 1|\mathbf{x}}(\mathbf{x}) = s_{t\to 0}(\mathbf{x};\alpha ,\beta)$  for some  $\alpha ,\beta \in \mathbb{R}^n$  , and is therefore a smooth function of  $\mathbf{x}$  . Intuitively, there are two regions of interest within  $\mathbb{R}_{\neq}^{m}$  : one within which it transitions from the value of almost 1 to that of almost O, which is the region  $\mathbb{R}_c^m\setminus \mathbb{R}_{\neq C}^m$  , and a saturation region, where  $p_{S = 1|\mathbf{x}}\rightarrow 0$  super-exponentially, which is the region  $\mathbb{R}_{\neq C}^{m}$  . The particular thresholds that define these regions are not important, and any reasonable scheme leads to vanishing bounds  $\epsilon_1,\epsilon_2$
+
+More formally, using the partitioning of  $\mathbb{R}^n$ , we can split the integral of Eq. (4) into
+
+$$
+p _ {Y | S = 1} (y) = \int_ {\mathbf {x} \in \mathbb {R} _ {\epsilon} ^ {m}} p _ {Y | S = 1, \mathbf {x}} (y, \mathbf {x}) \frac {p _ {S = 1 | \mathbf {x}} (\mathbf {x}) p _ {\mathbf {x}} (\mathbf {x})}{\mathbb {P} (S = 1)} d x + \int_ {\mathbf {x} \in \mathbb {R} _ {\epsilon} ^ {m}} p _ {Y | S = 1, \mathbf {x}} (y, \mathbf {x}) \frac {p _ {S = 1 | \mathbf {x}} (\mathbf {x}) p _ {\mathbf {x}} (\mathbf {x})}{\mathbb {P} (S = 1)},
+$$
+
+with the goal to upper bound (and hence ignore) the second term, which we consider as an error. This second term can be now bounded as
+
+$$
+\begin{array}{l} \int_ {\mathbf {x} \in \mathbb {R} _ {\neq} ^ {m}} p _ {Y | S = 1, \mathbf {x}} (y, \mathbf {x}) \frac {p _ {S = 1 | \mathbf {x}} (\mathbf {x}) p _ {\mathbf {x}} (\mathbf {x})}{\mathbb {P} (S = 1)} d \mathbf {x} \leq \\ \frac{1}{\mathbb{P}(S = 1)}\int_{\mathbf{x}\in \mathbb{R}^{m}_{\not\leq}}C_{Y}p_{S = 1|\mathbf{x}}(\mathbf{x})p_{\mathbf{x}}(\mathbf{x})d\mathbf{x}\leq \\ \frac {C _ {Y}}{\mathbb {P} (S = 1)} \left[ \int_ {\mathbf {x} \in \mathbb {R} _ {\notin S} ^ {m} \backslash \mathbb {R} _ {\notin C} ^ {m}} \underbrace {p _ {S = 1 | \mathbf {x}} (\mathbf {x})} _ {\leq 1} p _ {\mathbf {x}} (\mathbf {x}) d \mathbf {x} + \int_ {\mathbf {x} \in \mathbb {R} _ {\notin C} ^ {m}} p _ {S = 1 | \mathbf {x}} (\mathbf {x}) \underbrace {p _ {\mathbf {x}} (\mathbf {x})} _ {\leq C _ {\mathbf {x}}} d \mathbf {x} \right] \leq \\ \frac {C _ {Y}}{\mathbb {P} (S = 1)} \left[ \int_ {\mathbf {x} \in \mathbb {R} _ {\not \in} ^ {m} \backslash \mathbb {R} _ {\not \in \subset} ^ {m}} p _ {\mathbf {x}} (\mathbf {x}) d \mathbf {x} + C _ {\mathbf {X}} \int_ {\mathbf {x} \in \mathbb {R} _ {\not \in \subset} ^ {m}} p _ {S = 1 | \mathbf {x}} (\mathbf {x}) d \mathbf {x} \right] \leq \\ \begin{array}{c} \frac {C _ {Y} (\epsilon_ {2} + C _ {\mathbf {X}} \epsilon_ {1})}{\mathbb {P} (S = 1)}  , \end{array} \\ \end{array}
+$$
+
+where  $p_{S = 1|\mathbf{X}}\leq 1$  since  $S$  is a discrete random variable.
+
+We argue about the second part by claiming that both bounds  $\epsilon_{1}$  and  $\epsilon_{2}$  vanish during learning. Indeed, the form  $s(\mathbf{x})\to p_{S = 1|\mathbf{x}}(\mathbf{x})$  satisfies the assumption of Eq. (7) for a steep enough temperature parameter  $t$ , while it is also learning the correct domain  $\mathbb{R}_{\neq}^{m}$ , so that indeed the assumption of Eq. (8) is satisfied, both with inexorably diminishing bounds  $\epsilon_{1}$  and  $\epsilon_{2}$ , respectively.
+
+# C. Algorithm and Hyperparameters
+
+In this section, we provide pseudocode for SYFLOW.
+
+Algorithm 1: fit_flow  $\left(\{x^{(1)},\ldots ,x^{(n)}\} ,\{y^{(1)},\ldots ,y^{(n)}\} ,s,p,\bar{p}\right)$  
+$\begin{array}{rl} & {\log \mathcal{L}\gets \frac{1}{n}\sum_{i = 1}^{n}\log [p(y^{(i)})\cdot s(x^{(i)}) + \bar{p} (y^{(i)})\cdot (1 - s(x^{(i)}))];}\\ & {\mathrm{loss}\gets -\log \mathcal{L};}\\ & {\mathrm{loss backwards();}}\\ & {\mathrm{Update~}p;} \end{array}$
+
+Algorithm 2: SYFLOW(X, Y, epochsFlowY, epochsFlowy, IrFlow, Irs, priors,  $\gamma$ ,  $\lambda$ , t)  
+1 \(p_{Y}\gets\) Neural Spline Flow;   
+2 for \(i\gets 1\) to epochsFlowy do   
+3 \(\log \mathcal{L}\leftarrow \frac{1}{n}\sum_{i = 1}^{n}\log [p_{Y}(y^{(i)})];\)   
+4 loss \(\leftarrow -\log \mathcal{L}\)   
+5 loss backwards();   
+6 Update \(p_Y\) .   
+7 end   
+8 \(\alpha_{i}\gets \min X_{i};\)   
+9 \(\beta_{i}\gets \max X_{i};\)   
+10 \(a_i\gets 1\)   
+11 Rule(x) \(\leftarrow s(x;\alpha ,\beta ,a,t)\)   
+12 \(p_{Y|S = 1}\gets\) Neural Spline Flow;   
+13 \(p_{Y|S = 0}\gets\) Neural Spline Flow;   
+14 for \(i\gets 1\) to epochsFlowy do   
+15 Compute subgroup membership probabilities \(s(x^{(i)})\) ..   
+16 KL \(\leftarrow \frac{1}{n}\sum_{i = 1}^{n}s(x^{(i)})\cdot (\log [p_{Y|S = 1}(y^{(i)})] - \log [p_Y(y^{(i)})])\) .   
+17 \(n_s\gets \frac{1}{n}\sum_{i = 1}^{n}s(x^{(i)})\)   
+18 Weighted- \(KL\gets KL\cdot \bar{s}^{\gamma}\) Regularizer \(\leftarrow 0\)   
+20 for \(p_{SG_k}\) in priors do   
+21 Regularizer \(\leftarrow\) Regularizer + \(\sum_{i = 1}^{n}s(x^{(i)})\cdot (\log [p_{Y|S = 1}(y^{(i)})] - \log [p_{Y|SG_k = 1}(y^{(i)})])\)   
+end   
+23 Regularizer \(\leftarrow \frac{\lambda}{[primes]}\cdot\) Regularizer ;   
+24 loss \(\leftarrow\) Weighted-KL -Regularizer;   
+25 loss/backwards();   
+26 Update rule \(s\) to minimize objective/maximize weighted, regularized KL;   
+27 fit_flow({x(1),..,x(n)},{y(1),..,y(n)},s,p_{Y|S=1},p_{Y|S=0});\)   
+28 if (t = epochsFlowys/2) V (t = epochsFlowys·3/4) then   
+29 t \(\leftarrow t / 2\)   
+30 end   
+31 end   
+32 return Rule, \(p_{Y|S = 1}\)
+
+# D. Hyperparameters for experimental evaluation
+
+For all methods we optimize their respective hyperparameters such to maximize the measures used to evaluate the experiments. Since, BH has no hyperparameters no fine-tuning is required.
+
+# D.1. Synthetic experiments
+
+We used one hyperparameter setting for all synthetic experiments For SYFLOW the hyperparameter setting is:  $t = 0.2$ ,  $\gamma = 0.5$ ,  $\lambda = 0.5$ ,  $\mathrm{Ir}_{\mathrm{Flow}} = 5 \times 10^{-2}$ ,  $\mathrm{Ir}_{\mathrm{s}} = 2 \times 10^{-2}$ ,  $\mathrm{epoch}_{\mathrm{Flow}_{\mathrm{Y}}} = 2000$  and  $\mathrm{epoch}_{\mathrm{Flow}_{\mathrm{Y}_{\mathrm{s}}}} = 1500$ . For SD-  $\mu$ , SD-KL and RSD, we used 20 cutpoints and a beamwidth of 100, while  $\gamma$  is set to 1.0. Although  $\gamma$  has for all approaches the same functionality i.e. control the size of the subgroup, the absolute value of a  $\gamma$  has drastically different meanings. As SYFLOW outputs a soft assignment, we require a smaller alpha to achieve the same effect. For example, if SYFLOW assigns 0.9, then contribution to the size correction for  $\gamma = 0.5$  is  $\sqrt{0.9} \approx 0.95$ , thus we require a smaller  $\gamma$ .
+
+# D.2. Real world data
+
+For the experiments on Kaggle and UCI data (i.e. Section 5.2), we used for SYFLOW:  $t = 0.2$ ,  $\gamma = 0.3$ ,  $\lambda = 2.0$ ,  $\mathrm{lr}_{\mathrm{Flow}} = 5 \times 10^{-2}$ ,  $\mathrm{lr}_{\mathrm{s}} = 2 \times 10^{-2}$ ,  $\mathrm{epoch}_{\mathrm{Flow}_{\mathrm{Y}}} = 1000$  and  $\mathrm{epoch}_{\mathrm{Flow}_{\mathrm{Y}_{\mathrm{s}}}} = 1000$ . For SD-  $\mu$ , SD-KL and RSD, we used 20 cutpoints, a beamwidth of 100, and while  $\gamma = 1.0$ . For the experiments on gold cluster data present in Figure 1c, we used  $\mathrm{epoch}_{\mathrm{Flow}_{\mathrm{Y}}} = 7000$ ,  $\mathrm{epoch}_{\mathrm{Flow}_{\mathrm{Y}_{\mathrm{s}}}} = 3000$ ,  $\lambda = 10.0$  and  $\gamma = 0.2$ . For the case study, we used  $\mathrm{epoch}_{\mathrm{Flow}_{\mathrm{Y}}} = 3000$ ,  $\mathrm{epoch}_{\mathrm{Flow}_{\mathrm{Y}_{\mathrm{s}}}} = 2000$ ,  $\lambda = 5$  and  $\gamma = 0.2$ .
+
+# E. Evaluation metric
+
+To objectively evaluate the discovered subgroups on real-world data, we use Bhattacharyya coefficient (BC), KL-divergence (KL) and absolute mean distance (AMD) between the distribution of the subgroup  $P_{Y|S=1}$  and overall distribution of  $P_{Y}$ . KL and AMD are size corrected, since both metrics are heavily influenced by the size of the subgroup. We use histograms to estimate the probability distributions. The edges of the bins are computed using the Freedman Diaconis Estimator, which is robust to outliers and less sensitive towards distribution shapes. For the subgroup distribution  $P_{Y|S=1}$  and overall distribution  $P_{Y}$ , the metrics are formally defined as
+
+$$
+B C (P _ {Y | S = 1}, P _ {Y}) = \sum_ {y \in \mathcal {Y}} \sqrt {p _ {Y | S = 1} (y) p _ {Y} (y)}
+$$
+
+$$
+D _ {K L} \left(P _ {Y \mid S = 1}, P _ {Y}\right) = \sum_ {y \in \mathcal {Y}} p _ {Y \mid S = 1} (y) \log \left(\frac {p _ {Y \mid S = 1} (y)}{p _ {Y} (y)}\right)
+$$
+
+$$
+A M D (\mathcal {Y} _ {s}, \mathcal {Y}) = \left| \left(\frac {1}{| \mathcal {Y} _ {s} |} \sum_ {y \in \mathcal {Y} _ {s}} y\right) - \left(\frac {1}{| \mathcal {Y} |} \sum_ {y \in \mathcal {Y}} y\right) \right|.
+$$
+
+In the last definition, with slight abuse of notation, we used  $\mathcal{V}_s$  to denote all points in the subgroup. For the Table 1, we size corrected KL and AMD using  $\gamma = 1$ . Note, this is exactly the metric that SD-  $\mu$  and SD-KL optimize.
+
+# F. Rule Complexity Experiment
+
+We study how SYFLOW learns subgroups of increasing complexity. This is achieved by increasing the number of predicates in a generated rule up to ten. We keep the target distribution fixed to a Normal distribution.
+
+We present our results in Figure 7; Here, as the complexity of the rule increases, the accuracy of all methods generally decreases as the task becomes progressively harder. Amongst all methods, SYFLOW recovers the planted subgroup with the highest accuracy. In particular, SYFLOW improves the most over its competitors on the more complex subgroups.
+
+# G. Target distributions for synthetic experiment
+
+In Figure 8, we show the different target distributions for the first synthetic experiment.
+
+![](images/f7d09e2126eb43d976eb9925c94cadc68b31f5eb98e45ad600a4fa0fedc6c4b7.jpg)
+
+![](images/e637aa599614c43c0dd6ed71b37af4ed0aa01f2bf46b096a18f1dc009b6f3c06.jpg)  
+Figure 7. F1 under increasingly complex rules, higher is better
+
+![](images/7a6e353bcf399809b1198880a2ee31101192885b691217716a24daea89f67788.jpg)  
+(a)  $\mathcal{N}(1.5,0.5)$
+
+![](images/9e3905e5005745e5e34313f035b0de7770536b9e6779b093121b63e82ae13d8c.jpg)
+
+![](images/a1184f658fb65389684232f16cd26f74100a5ab4d5e49923a08d61ba2df7f8e2.jpg)
+
+![](images/80d50d1eb32010e12d47d464ccfd7bc5c983aa222661116b522fe778dd64e675.jpg)
+
+![](images/82ed6b0f50ff3225b5a1597d8526d48538f28c3dd77162f46a582eb653b9d5c1.jpg)  
+(b)  $\mathcal{B}(0.2,0.2)$  
+(e)  $\mathcal{R}(2)$  
+Figure 8. Here, we show the different target distributions for second synthetic experiment in Section 5.
+
+![](images/d7388771b5730356e3db16b36427f47c63f81d22acbcd761fdb809561ba9c78d.jpg)  
+(c)  $\mathcal{C}(0,1)$  
+(f)  $\mathcal{N}(-1.5,0.5) + \mathcal{N}(1.5,0.5)$
+
+![](images/d38ae93ed7e2e68949d75fc3f68f1fc7fab40739d37e6cbb10d7f6eaeb264dcf.jpg)  
+(d)  $Exp(0.5)$  
+(g)  $\mathcal{U}(0.5,1.5)$
+
+# H. Subgroup descriptions
+
+We show in Table 2 further examples of subgroups found on the real life datasets. For each method we select the first subgroup that the respective method found. Since BH did not find relevant subgroups for most datasets, we do not consider it in the table.
+
+# I. Subgroup Discovery on Image Data
+
+One of the defining characteristics of SYFLOW is its ability to give an interpretable rule for each subgroup. Hence, we focus our experiments on tabular data where we can directly interpret the learned rules. Still, SYFLOW also has potential applications with structured data such as images. We conduct a preliminary experiment on the MNIST dataset (LeCun et al., 1998) using the digits 0 and 1 only. As the target variable  $Y$ , we use pixel values of the downsampled, normalized image in  $14 \times 14$  resolution as a vector  $Y \in \mathbb{R}^{196}$ . For the feature variable  $X$ , preprocessing is required for the rule learner because of the high variability between individual pixel values. Instead, we use the two-dimensional t-SNE embedding of the pixel values as features  $X \in \mathbb{R}^2$ .
+
+We show in Figure 9 the discovered subgroups for the digits 0 and 1. SYFLOW is able to learn a subgroup of resp. the digit 0 and the digit 1, and the sampled images from the subgroup distributions closely resemble the digits. The corresponding descriptors are boxes around the regions of the t-SNE embedding where the digit is located. This shows that in principle, SYFLOW can be applied to image data. Future work could focus on extending SYFLOW to directly work with image data, for example by using convolutional neural networks to learn the subgroups and giving up the interpretability of the rules, or
+
+<table><tr><td>Dataset</td><td>Method</td><td>Rule</td></tr><tr><td rowspan="4">Abalone</td><td>SYFLOW</td><td>0.20 &lt; whole-weight &lt; 2.83 ∧ 0.13 &lt; viscera-weight &lt; 0.60 ∧ 0.26 &lt; shell-weight &lt; 1.00 ∧ sex-I =0</td></tr><tr><td>RSD</td><td>shell-weight ≥ 0.43 ∧ 0.46 ≤ diameter &lt; 0.49 ∧ shucked-weight &lt; 0.55 ∧ whole-weight ≥ 1.03</td></tr><tr><td>SD-KL</td><td>shell-weight &lt; 0.48</td></tr><tr><td>SD-μ</td><td>height ≥ 0.10 ∧ shucked-weight ≥ 0.16 ∧ shell-weight ≥ 0.11 ∧ shell-weight ≥ 0.19</td></tr><tr><td rowspan="4">Airquality</td><td>SYFLOW</td><td>404.67 &lt; NOx(GT) &lt; 1479.00 ∧ -44.90 &lt; NO2(GT) &lt; 340.00</td></tr><tr><td>RSD</td><td>PT08.S4(NO2) ≥ 2026.0 ∧ PT08.S3(NOx) &lt; 373.0 ∧ month ≥ 11.0</td></tr><tr><td>SD-KL</td><td>PT08.S1(CO) &lt; 1285.0 ∧ NMHC(GT) &lt; 185.0 ∧ PT08.S2(NMHC) &lt; 1068.0 ∧ NOx(GT) &lt; 326.0 ∧ NOx(GT) ≥ -200.0 ∧ PT08.S5(O3) &lt; 1780.0</td></tr><tr><td>SD-μ</td><td>PT08.S1(CO) ≥ 956.0 ∧ PT08.S2(NMHC) ≥ 979.0 ∧ PT08.S5(O3) ≥ 704.0 ∧ PT08.S5(O3) ≥ 917.0</td></tr><tr><td rowspan="4">Automobile</td><td>SYFLOW</td><td>If 159.42 &lt; length &lt; 208.10 ∧ 60.64 &lt; width &lt; 72.00 ∧ 2684.19 &lt; curb-weight &lt; 4066.00 ∧ 64.84 &lt; engine-size &lt; 270.46 ∧ 7.00 &lt; compression-ratio &lt; 19.11 ∧ 16.00 &lt; highway-mpg &lt; 31.72 ∧ jaguar=0 ∧ mercedes-benz=0 ∧ plymouth=0 ∧ subaru=0 ∧ fuel-type-gas=1 ∧ engine-location-front=1 ∧ engine-location-rear=0 ∧ engine-type-ohcf=0 ∧ fuel-system-2bbl=0 ∧ fuel-system-mpf=1</td></tr><tr><td>RSD</td><td>engine-size ≥ 201.5</td></tr><tr><td>SD-KL</td><td>highway-mpg ≥ 29.0 ∧ audi=0 ∧ bmw=0 ∧ wheel-base &lt; 104.90 ∧ width &lt; 66.90 ∧ body-style-convertible=0 ∧ engine-size &lt; 161.0</td></tr><tr><td>SD-μ</td><td>highway-mpg &lt; 30.0 ∧ honda=0 ∧ isuzu=0 ∧ plymouth=0 ∧ subaru=0 ∧ curb-weight ≥ 2385.0 ∧ fuel-system-mfi=0</td></tr><tr><td rowspan="4">Bike</td><td>SYFLOW</td><td>holiday=0 ∧ 1.00 &lt; weathersit &lt; 2.35 ∧ 0.47 &lt; atemp &lt; 0.78 ∧ 0.18 &lt; hum &lt; 0.83 ∧ season-1=0</td></tr><tr><td>RSD</td><td>temp &lt; 0.2</td></tr><tr><td>SD-KL</td><td>temp ≥ 0.26 ∧ atemp ≥ 0.28 ∧ hum &lt; 0.87 ∧ windspeed &lt; 0.34 ∧ season-1=0</td></tr><tr><td>SD-μ</td><td>temp ≥ 0.40 ∧ temp ≥ 0.43 ∧ hum &lt; 0.82</td></tr><tr><td rowspan="4">California</td><td>SYFLOW</td><td>0.50 &lt; MedInc &lt; 2.4</td></tr><tr><td>RSD</td><td>MedInc ≥ 7.4 ∧ 35.0 ≤ HouseAge &lt; 38.0 ∧ -121.19 ≤ Longitude &lt; -118.34 ∧ AveBedrms ≥ 0.96</td></tr><tr><td>SD-KL</td><td>MedInc &lt; 5.11 ∧ MedInc &lt; 6.16 ∧ HouseAge &lt; 52.0 ∧ AveOccup ≥ 2.08</td></tr><tr><td>SD-μ</td><td>MedInc ≥ 3.32 ∧ AveOccup &lt; 3.89 ∧ AveOccup &lt; 4.33 ∧ Latitude &lt; 37.99 ∧ Longitude &lt; -117.08</td></tr><tr><td rowspan="4">Insurance</td><td>SYFLOW</td><td>44.00 &lt; age &lt; 64.00 ∧ smoker=0</td></tr><tr><td>RSD</td><td>smoker=1 ∧ bmi ≥ 30.0 ∧ age ≥ 59.0</td></tr><tr><td>SD-KL</td><td>smoker=0</td></tr><tr><td>SD-μ</td><td>smoker=1</td></tr><tr><td rowspan="4">Mpg</td><td>SYFLOW</td><td>3.35 &lt; cylinders &lt; 5.25</td></tr><tr><td>RSD</td><td>weight ≥ 3845.0 ∧ 71.0 ≤ model-year &lt; 74.5</td></tr><tr><td>SD-KL</td><td>displacement ≥ 151.0 ∧ weight ≥ 2671.0 ∧ weight ≥ 3085.0</td></tr><tr><td>SD-μ</td><td>cylinders=4.0 ∧ weight &lt; 2807.0 ∧ weight &lt; 4278.0</td></tr><tr><td rowspan="4">Student</td><td>SYFLOW</td><td>school=1 ∧ address=0 ∧ failures=0 ∧ schoolsup=1 ∧ nursery=0 ∧ higher=0 ∧ internet=1 ∧ 1.00 &lt; Dalc &lt; 3.34 ∧ Medu=1/0 ∧ Fedu=1=0</td></tr><tr><td>RSD</td><td>failures=1 ∧ famsize=1 ∧ absences &lt; 1 ∧ famsup=0</td></tr><tr><td>SD-KL</td><td>failures=0</td></tr><tr><td>SD-μ</td><td>school=0 ∧ higher=0 ∧ absences &lt; 16 ∧ absences &lt; 18 ∧ failures=0 ∧ schoolsup=1</td></tr><tr><td rowspan="4">Wages</td><td>SYFLOW</td><td>sex=0 ∧ 10.68 &lt; ed &lt; 18.00 ∧ 29.86 &lt; age &lt; 95.00</td></tr><tr><td>RSD</td><td>ed ≥ 18.0 ∧ 37.0 ≤ age &lt; 43.0 ∧ sex=0 ∧ height &lt; 70.25</td></tr><tr><td>SD-KL</td><td>race-other=0.0</td></tr><tr><td>SD-μ</td><td>height ≥ 65.05 ∧ height ≥ 65.79 ∧ sex=0 ∧ age ≥ 30.0</td></tr><tr><td rowspan="4">Wine</td><td>SYFLOW</td><td>0.23 &lt; volatile acidity &lt; 1.10 ∧ 0.99 &lt; density &lt; 1.04 ∧ 8.00 &lt; alcohol &lt; 10.12</td></tr><tr><td>RSD</td><td>alcohol ≥ 12.75 ∧ 29.0 &lt; free sulfur dioxide &lt; 41.0 ∧ pH &lt; 3.24 ∧ 0.26 ≤ citric acid;0.34 ∧ residual sugar ≥ 1.4</td></tr><tr><td>SD-KL</td><td>free sulfur dioxide ≥ 11.0</td></tr><tr><td>SD-μ</td><td>fixed acidity &lt; 8.30 ∧ alcohol ≥ 10.40 ∧ free sulfur dioxide ≥ 11.0 ∧ total sulfur dioxide &lt; 195.0 ∧ density &lt; 1.00</td></tr></table>
+
+Table 2. Symbolic subgroup descriptions for real life datasets in Section 5.2
+
+Table 3. We show the standard deviation for the 100 reruns of SYFLOW on real world data.  
+
+<table><tr><td>Dataset</td><td>DKL</td><td>BC</td><td>AMD</td></tr><tr><td>abalone</td><td>0.01</td><td>0.02</td><td>0.04</td></tr><tr><td>airquality</td><td>0.01</td><td>0.02</td><td>0.02</td></tr><tr><td>automobile</td><td>0.02</td><td>0.04</td><td>239.14</td></tr><tr><td>bike</td><td>0.0</td><td>0.01</td><td>73.98</td></tr><tr><td>california</td><td>0.01</td><td>0.04</td><td>0.02</td></tr><tr><td>insurance</td><td>0.01</td><td>0.1</td><td>320.22</td></tr><tr><td>mpg</td><td>0.0</td><td>0.0</td><td>0.08</td></tr><tr><td>student</td><td>0.02</td><td>0.05</td><td>0.1</td></tr><tr><td>wages</td><td>0.0</td><td>0.02</td><td>172.48</td></tr><tr><td>wine</td><td>0.01</td><td>0.03</td><td>0.03</td></tr></table>
+
+![](images/c8dda82457abd3f182852cb1b6978cdad9065b708a8135af99c7038164524f3a.jpg)  
+Figure 9. Subgroup discovery on the MNIST dataset with features  $X$  obtained by 2D t-SNE embedding. The discovered subgroups resemble the digits 0 and 1 resp.
+
+![](images/4d59a9b095e5ad678433d21d4601a49cb2a12aa3406524d367b1bf4a9e224f81.jpg)
+
+![](images/15f24d7ec3e020aa91e4a58620dbc158a83de2702b941daaf44b68453d47fd62.jpg)
+
+by using an image-targeted normalizing flow architecture to model the subgroup distribution.
+
+# J. Sensitivity of SYFLOW
+
+We here provide preliminary results for the average F1 score across the different target distributions, shown in Figure D. Overall, SYFLOW performs well with a variety of different hyperparameters, and gives the end user room for customization in regards to the desired size of the subgroups through  $\gamma$  and the diversity between subgroups through  $\lambda$ . When it comes to discovering underlying subgroup structures, as demonstrated in the experiments, SYFLOW does so reliably for a wide range of parameters.
+
+In Table 3, we show the standard deviations of SYFLOW for the 100 runs on real world data.
+
+![](images/8e5778c2a700186a692fdcc6d33ef3c44ae44dd563dfa98b3a85e943c23fbf91.jpg)  
+(a) Subgroup size weighting  $\gamma$
+
+![](images/1068b25e619117119726d447435dab0816d794a039801a744ecd3761bc7c60ba.jpg)  
+(b) Regularization strength  $\lambda$
+
+![](images/9d2653cb3cbcb6b6a27a79b8c283c27317b4c84284f5bf8fddccad3c1d951695.jpg)  
+(c) Predicate temperature/softness  $t$  
+Figure 10. Here we conduct a sensitivity analysis of the hyperparameters of SYFLOW. Subgroup size weighting  $\gamma$ : SYFLOW works well in a range of values for  $\gamma \in [0.4, 1]$ . Regularization strength  $\lambda$ : Using the regularizer improves the F1 score by 0.05, whilst varying its strength between [0.5, 3] does not have a noticeable impact on accuracy. Predicate temperature/softness  $t$ : the starting temperature controls the softness of the binning function. Its ideal range by empirical observation lies in between [0.01, 0.2]. Setting it too high can result in un-crisp binning functions, which in turn results in an inaccurate subgroup assignment yielding a worse F1 score. In both synthetic and real-world experiments we used a temperature of  $t = 0.2$  which provides good performance across the board

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