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+# EVENTRPG: EVENT DATA AUGMENTATION WITH RELEVANCE PROPAGATION GUIDANCE
+
+Mingyuan Sun $^{1,2}$ , Donghao Zhang $^{3}$ , Zongyuan Ge $^{4}$ , Jiaxu Wang $^{1}$ , Jia Li $^{5}$ , Zheng Fang $^{2,*}$  & Renjing Xu $^{1,*}$
+
+<sup>1</sup>The Hong Kong University of Science and Technology (Guangzhou) <sup>2</sup>Northeastern University  
+<sup>3</sup>Seeing Machines <sup>4</sup>Monash University <sup>5</sup>Peking University  
+mingyuansun20@gmail.com, peter.zhang1@seeingmachines.com, zongyuan.ge@monash.edu, jwang457@connect.hkust-gz.edu.cn, j.gaga.lee@gmail.com, fangzheng@mail.neu.edu.cn, renjingxu@hkust-gz.edu.cn
+
+# ABSTRACT
+
+Event camera, a novel bio-inspired vision sensor, has drawn a lot of attention for its low latency, low power consumption, and high dynamic range. Currently, overfitting remains a critical problem in event-based classification tasks for Spiking Neural Network (SNN) due to its relatively weak spatial representation capability. Data augmentation is a simple but efficient method to alleviate overfitting and improve the generalization ability of neural networks, and saliency-based augmentation methods are proven to be effective in the image processing field. However, there is no approach available for extracting saliency maps from SNNs. Therefore, for the first time, we present Spiking Layer-Time-wise Relevance Propagation rule (SLTRP) and Spiking Layer-wise Relevance Propagation rule (SLRP) in order for SNN to generate stable and accurate CAMs and saliency maps. Based on this, we propose EventRPG, which leverages relevance propagation on the spiking neural network for more efficient augmentation. Our proposed method has been evaluated on several SNN structures, achieving state-of-the-art performance in object recognition tasks including N-Caltech101, CIFAR10-DVS, with accuracies of  $85.62\%$  and  $85.55\%$ , as well as action recognition task SL-Animals with an accuracy of  $91.59\%$ . Our code is available at https://github.com/myuansun/EventRPG.
+
+# 1 INTRODUCTION
+
+With the advent of event cameras, researchers have focused on applying the brain-inspired technique to achieve a variety of tasks, as the asynchronous nature of event cameras mimics the way the biological visual system works (Gallego et al., 2020). Event cameras record the change in brightness of each pixel, and once the change in brightness of a pixel exceeds a predetermined threshold, an event is triggered. The intrinsic properties of event cameras give them several advantages over RGB cameras, including low power consumption, high dynamic range, low latency, and high temporal resolution. These benefits highlight the potential of event cameras in challenging scenarios, such as low-light and high-speed conditions, which has led to some research emphasizing the use of event cameras for robotic sensing in challenging situations (Zuo et al., 2022; Chen et al., 2023). Spiking Neural Network (SNN) (Maass, 1997) is a type of neural network that is inspired by the way biological neurons communicate with each other. By integrating the biological neuronal dynamics into individual neurons, SNN becomes capable of representing intricate spatio-temporal information and dealing with asynchronous data naturally, typically event-based data.
+
+In terms of classification tasks, a number of event-based datasets, such as N-MNIST, N-Caltech101 (Orchard et al., 2015), and CIFAR10-DVS (Li et al., 2017), have been used to evaluate the performance of artificial neural networks (ANNs) and SNNs. However, the issue of overfitting still poses a significant challenge for event-based datasets. Data augmentation is an efficient method for improving the generalization and performance of a model. Lots of methods have been proposed to augment event-based data, for example, transferring classic geometric augmentations from image
+
+field to event-based field (Li et al., 2022), randomly dropping events (Gu et al., 2021), and mixing two event streams with a randomly sampled mask (Shen et al., 2023). Nevertheless, current mixing augmentation strategies in event-based field do not consider the size and location information of label-related objects, and thus may produce events with incorrect labels and disrupt the training process. To address this problem in image processing field, Uddin et al. (2020); Kim et al. (2020) mix the label-related objects together based on the saliency information obtained from neural networks. This paradigm achieves better results compared with conventional non-saliency augmentations. Kim & Panda (2021b) was the first to explore acquiring saliency information from SNNs. In this work, Spiking Activation Map (SAM) was presented to reveal the model's attention by weighted adding of the intermediate feature maps. Since the CAMs obtained from this method are not related to the network's predictions, their precision regarding the shape and position of label-related objects is still not satisfactory.
+
+In this paper, we present two novel methods extended from layer-wise relevance propagation (LRP) (Bach et al., 2015) to visualize the label-related saliency information of SNNs, each offering distinct advantages in terms of temporal precision and computational time. Moreover, guided by this saliency information, we develop two data augmentation approaches targeted at event-based data, demonstrating significant improvements in the performance and generalization capability of SNNs across multiple classification tasks.
+
+Our contributions are summarized as follows:
+
+- We propose Spiking Layer-Time-wise Relevance Propagation (SLTRP) and Spiking Layerwise Relevance Propagation (SLRP) to accurately reveal the saliency information of SNNs. The former reveals information across time, while the latter is less time-consuming.  
+- We present RPGDrop and RPGMix. By dropping and mixing events with the guidance of relevance propagation obtained from SLRP or STLRP, the augmented samples exhibit increased diversity and tight correlation with the labels. Combined with several geometric data augmentations, we formulate our data augmentation strategy namely EventRPG.  
+- We evaluate our proposed saliency visualizing method and data augmentation method using various SNNs on event-based object and action recognition datasets. Experiments demonstrate that our SLRP and SLTRP can generate high quality CAMs and saliency maps with sub-optimal computing time. EventRPG achieves state-of-the-art performance on both object recognition and action recognition tasks with limited time consumption.
+
+# 2 PRELIMINARY
+
+Layer-wise Relevance propagation (LRP) was first introduced in (Bach et al., 2015) as a visualization tool for generating saliency maps that show the contribution of individual pixels in the input data to the model prediction or a specific class, facilitating the interpretability of neural networks. According to the LRP rule, we assign a value to each neuron to represent the neuron's contribution to the prediction or target class, called Relevance Score. The idea of LRP is to find a propagation rule satisfying the following definition.
+
+Definition 1 (Conservation Property). On a neural network with  $L$  layers, the relevance score of each layer satisfies
+
+$$
+c = \sum_ {i} R _ {i} ^ {(l)}, \quad \forall l \in [ 0, L ], \tag {1}
+$$
+
+where  $c$  is a constant value and  $R_{i}^{(l)}$  is the relevance score of  $i^{th}$  neuron on layer  $l$ .  $R^{(0)}$  is the relevance score before input layer.
+
+We calculate the relevance score of  $i^{th}$  input neuron on layer  $l$ $R_{i}^{(l - 1)}$  by the sum of all relevance scores propagated from all the connections in this layer:
+
+$$
+R _ {i} ^ {(l - 1)} = \sum_ {j} R _ {i \leftarrow j} ^ {(l - 1, l)}. \tag {2}
+$$
+
+Utilizing the  $\alpha \beta$ -rule (Montavon et al., 2017) and abandoning the bias term  $b$ , we obtain the relevance score that should be propagated from the  $j^{th}$  neuron in layer  $l$  to the  $i^{th}$  neuron in layer  $l-1$ :
+
+$$
+R _ {i \leftarrow j} ^ {(l - 1, l)} = R _ {j} ^ {(l)} \cdot \left(\alpha \cdot \frac {z _ {i j} ^ {+}}{\sum_ {i} z _ {i j} ^ {+}} + \beta \cdot \frac {z _ {i j} ^ {-}}{\sum_ {i} z _ {i j} ^ {-}}\right), \tag {3}
+$$
+
+![](images/3e9bdfd8780abb9454a024ef65e83c90417b1e5f4c1455a462c808353c436948.jpg)  
+(a) Layer-wise Relevance Propagation
+
+![](images/83b52e7dd036ce5940f4eef5f9cd6b70066ebc1519765b7da8544ee9f28ea34c.jpg)  
+(b) Spiking Layer-Time-wise Relevance Propagation  
+Figure 1: Forward propagation flow and relevance propagation flow of ANNs (a) and SNNs (b, c).
+
+![](images/237e2fcf19e53eae886594b0b8ef4fcf60e91c90aafa45de00ac504a09154b69.jpg)  
+(c) Spiking Layer-wise Relevance Propagation
+
+where  $\alpha +\beta = 1$ .  $z_{ij}^{+} = w_{ij}^{+}\cdot x_{i}^{+} + w_{ij}^{-}\cdot x_{i}^{-}$  and  $z_{ij}^{-} = w_{ij}^{+}\cdot x_{i}^{-} + w_{ij}^{-}\cdot x_{i}^{+}$  are respectively the positive and negative contribution of the  $i^{th}$  neuron in layer  $l - 1$  to the  $j^{th}$  neuron in layer  $l$ . This relevance propagation rule satisfies definition 1, and the proof can be seen in appendix D.1.
+
+# 3 SPIKING RELEVANCE PROPAGATION RULE
+
+In an SNN, information is represented by the timing and frequency of spikes. Neurons in an SNN generate spikes when their membrane voltage reaches a certain threshold. These spikes propagate through the network, influencing the activity of other neurons layer by layer. Two popular and fundamental models used to represent basic spiking neurons are the Leaky Integrate-and-Fire (LIF) neuron model and the Integrate-and-Fire (IF) neuron model:
+
+$$
+f _ {L I F} (V, I) = e ^ {- \frac {\Delta t}{\tau}} V [ t - 1 ] + \left(1 - e ^ {- \frac {\Delta t}{\tau}}\right) I [ t ], \tag {4}
+$$
+
+$$
+f _ {I F} (V, I) = V [ t - 1 ] + I [ t ], \tag {5}
+$$
+
+where  $\tau$  is the attenuation factor and  $\Delta t$  is the interval between time steps satisfying  $\Delta t < \tau$ . Membrane voltage  $V$  can be interpreted as the information reserved by a neuron from the previous time step  $t - 1$ .  $\bar{I}[t]$  indicates the input of the neuron at the current time step.
+
+In practice, researchers would replace the activation layer in an ANN with spiking layers represented by eq. (4) or eq. (5) to construct an SNN. As shown in fig. 1a, an ANN consists of linear layers and activation layers and passes information layer by layer. These two layer types do not include variables related to time and as such we propagate the relevance scores through them layer by layer.
+
+# 3.1 RELEVANCE PROPAGATION FOR LINEAR LAYERS IN SNNS
+
+Information in SNNs is propagated through time in spiking layers, while the linear layers do not transmit information across different time steps. Any linear layer's output  $y[t]$  at time step  $t$  only correlates to its input  $x[t]$  at time step  $t$ , and we conduct relevance propagation using eq. (3) on this layer separately for each time step. Specifically, for linear layer  $l$ , we extend the eq. (3) with suffix "[t]" representing the time step  $t$ , and we have
+
+$$
+R _ {i} ^ {(l - 1)} [ t ] = \sum_ {j} R _ {i \leftarrow j} ^ {(l - 1, l)} [ t ], \tag {6}
+$$
+
+$$
+R _ {i \leftarrow j} ^ {(l - 1, l)} [ t ] = R _ {j} ^ {(l)} [ t ] \cdot \left(\alpha \cdot \frac {z _ {i j} ^ {+} [ t ]}{\sum_ {i} z _ {i j} ^ {+} [ t ]} + \beta \cdot \frac {z _ {i j} ^ {-} [ t ]}{\sum_ {i} z _ {i j} ^ {-} [ t ]}\right), \tag {7}
+$$
+
+in which  $z_{ij}^{+}[t] = w_{ij}^{+} \cdot x_{i}^{+}[t] + w_{ij}^{-} \cdot x_{i}^{-}[t]$  and  $z_{ij}^{-}[t] = w_{ij}^{+} \cdot x_{i}^{-}[t] + w_{ij}^{-} \cdot x_{i}^{+}[t]$ . Next, we focus on the derivation of the relevance propagation rule on spiking layers.
+
+# 3.2 SPIKING LAYER-TIME-WISE RELEVANCE PROPAGATION
+
+From eq. (4) we know that in the forward propagation, the output of a LIF neuron at time step  $t$  ( $t > 0$ ) depends on this layer's input current  $I[t]$  at the current time step and the membrane voltage  $V[t - 1]$  from the previous time step. Therefore, we should propagate the relevance score to the neuron's internal voltage at the previous time step and to the input current at the current time step, as shown in fig. 1b. Assume there are overall T time steps. Similar to  $\alpha \beta$ -rule, we propagate
+
+the relevance score based on its positive contribution and negative contribution at each time step  $t \in [1, T]$ . Consider a neuron in spiking layer  $l$  at time step  $t$ . We decompose and represent it in a general manner:
+
+$$
+f (V, I) = c \cdot V [ t - 1 ] + d \cdot I [ t ] \tag {8}
+$$
+
+where c and d are coefficients depending on the neuron_type:
+
+$$
+c = \left\{ \begin{array}{l l} e ^ {- \frac {\Delta t}{\tau}} & \text {L I F N e u r o n ,} \\ 1 & \text {I F N e u r o n ,} \end{array} \right. \quad d = \left\{ \begin{array}{l l} 1 - e ^ {- \frac {\Delta t}{\tau}} & \text {L I F N e u r o n ,} \\ 1 & \text {I F N e u r o n .} \end{array} \right.
+$$
+
+Based on eq. (8), at time step  $t$ , we define the proportion of relevance score that should be propagated to the previous time step as
+
+$$
+\gamma [ t ] = \alpha \cdot \frac {c \cdot V ^ {+} [ t - 1 ]}{c \cdot V ^ {+} [ t - 1 ] + d \cdot I ^ {+} [ t ]} + \beta \cdot \frac {c \cdot V ^ {-} [ t - 1 ]}{c \cdot V ^ {-} [ t - 1 ] + d \cdot I ^ {-} [ t ]}, \tag {9}
+$$
+
+where superscripts  $^+$  and  $-$  denote the positive and negative values, respectively. For each neuron in spiking layer  $l$ , By initializing the relevance score at the final time step as  $R^{(l - 1)}[T] = R^{(l)}[T]$  and subsequently updating relevance scores iteratively as time step  $t$  from  $T$  to 1 using
+
+$$
+R ^ {(l - 1)} [ t - 1 ] \leftarrow \gamma [ t ] \cdot R ^ {(l - 1)} [ t ] + R ^ {(l)} [ t - 1 ], \tag {10}
+$$
+
+$$
+R ^ {(l - 1)} [ t ] \leftarrow (1 - \gamma [ t ]) \cdot R ^ {(l - 1)} [ t ], \tag {11}
+$$
+
+we propagate the relevance score of every neuron in spiking layers to all time steps. With iteration formulas (10) and (11), we could derive the relevance score at each time step
+
+$$
+R ^ {(l - 1)} [ t ] = (1 - \gamma [ t ]) \left(\sum_ {i = t + 1} ^ {T} R ^ {(l)} [ i ] \prod_ {j = t + 1} ^ {i} \gamma [ j ] + R ^ {(l)} [ t ]\right). \tag {12}
+$$
+
+Proposition 1. The sum of relevance scores propagated after spiking layer  $l$  from time step 1 to  $k(k < T)$  is
+
+$$
+\sum_ {t = 1} ^ {k} R ^ {(l - 1)} [ t ] = \sum_ {t = 1} ^ {k} R ^ {(l)} [ t ] + \sum_ {i = k + 1} ^ {T} R ^ {(l)} [ i ] \prod_ {j = k + 1} ^ {i} \gamma [ j ]. \tag {13}
+$$
+
+The proof is provided in appendix D.2. With proposition 1, we have
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} R ^ {(l - 1)} [ t ] = \sum_ {t = 1} ^ {T - 1} R ^ {(l - 1)} [ t ] + R ^ {(l - 1)} [ T ] = \sum_ {t = 1} ^ {T - 1} R ^ {(l)} [ t ] + \gamma [ T ] R ^ {(l)} [ T ] + (1 - \gamma [ T ]) R ^ {(l)} [ T ] \\ = \sum_ {t = 1} ^ {T - 1} R ^ {(l)} [ t ] + R ^ {(l)} [ T ] = \sum_ {t = 1} ^ {T} R ^ {(l)} [ t ]. \tag {14} \\ \end{array}
+$$
+
+This is a stronger Conservation Property since the relevance score stays unchanged for every neuron in a spiking layer. This indicates that leveraging formulas (10) and (11), we are able to propagate relevance scores through any spiking layer while satisfying the Conservation Property. Combining with relevance propagation rules introduced in section 3.1, we could propagate relevance scores to any layer in an SNN, thereby revealing the saliency information across time, namely Spiking Layer-Time-wise Relevance Propagation (SLTRP).
+
+# 3.3 SPIKING LAYER-WISE RELEVANCE PROPAGATION
+
+SLTRP could reveal the saliency information at any time step, and would thus cost more time to conduct the whole relevance propagation process compared with ANNs. Under some circumstances, e.g., on datasets transformed from static images where coordinates of events tend to be fixed w.r.t. the time, we don't need to obtain the saliency score of some specific time step and only require the saliency information stacked from all time steps  $R^{(l)} \equiv \frac{1}{T} \sum_{t}^{T} R^{(l)}[t]$ . For any spiking layer  $l$ , from eq. (14) we have  $R^{(l-1)} = R^{(l)}$ .
+
+In terms of linear layers, we sum the positive and negative contribution values of the  $i^{th}$  neuron to  $j^{th}$  neuron through time dimension as
+
+$$
+\begin{array}{l} z _ {i j} ^ {+} \equiv \frac {1}{T} \sum_ {t} z _ {i j} ^ {+} [ t ] = \frac {1}{T} \sum_ {t} (w _ {i j} ^ {+} \cdot x _ {i} ^ {+} [ t ] + w _ {i j} ^ {-} \cdot x _ {i} ^ {-} [ t ]) = \frac {1}{T} (w _ {i j} ^ {+} \cdot \sum_ {t} x _ {i} ^ {+} [ t ] + w _ {i j} ^ {-} \cdot \sum_ {t} x _ {i} ^ {-} [ t ]), \\ z _ {i j} ^ {-} \equiv \frac {1}{T} \sum_ {t} z _ {i j} ^ {-} [ t ] = \frac {1}{T} \sum_ {t} (w _ {i j} ^ {+} \cdot x _ {i} ^ {-} [ t ] + w _ {i j} ^ {-} \cdot x _ {i} ^ {+} [ t ]) = \frac {1}{T} (w _ {i j} ^ {+} \cdot \sum_ {t} x _ {i} ^ {-} [ t ] + w _ {i j} ^ {-} \cdot \sum_ {t} x _ {i} ^ {+} [ t ]). \\ \end{array}
+$$
+
+![](images/501865dd4e1d5ebb32ed231b0ddebdd400a9480c7ac4399fa770db19d91ebb75.jpg)  
+(a)
+
+![](images/7c957b1aee777c7808c356b2f4b07f50877f0eba798ed72cbb71f6d939a9c96e.jpg)  
+(b)  
+Figure 2: Illustration of Relevance Propagation Guided Event Mix and Event Drop. (a) RPGDrop. Where the saliency map offers a higher value, events are more likely to be dropped. (b) RPGMix.
+
+Then we propagate the relevance scores of linear layer using eq. (2) and eq. (3) with different definitions of  $R^{(l-1)}$ ,  $R^{(l)}$ ,  $z_{ij}^{+}$  and  $z_{ij}^{-}$ . This enables a spiking relevance propagation process without consideration of the time dimension, saving time costs and being more practical for datasets transformed from static images, namely Spiking Layer-wise Relevance Propagation (SLRP) (see fig. 1c).
+
+# 4 RELEVANCE PROPAGATION GUIDED EVENT DATA AUGMENTATION
+
+# 4.1 SALIENCY MAP AND CLASS ACTIVATION MAP
+
+In an SNN, we first leverage Contrastive Layer-wise Relevance Propagation (CLRP) (Gu et al., 2018) to initialize the relevance scores of the output layer for each time step. Then we propagate the relevance scores backward using SLRP or SLTRP depending on the dataset. Relevance scores are propagated throughout all layers to create saliency maps, while class activation maps (CAMs) can be formed in two ways. One method involves summing the relevance scores from a specific intermediate layer across the channel dimension, resulting in SLRP-CAM and SLTRP-CAM. Alternatively, CAMs can be generated by calculating a weighted sum between relevance scores and feature maps, a method referred to as SLRP-RelCAM and SLTRP-RelCAM (Lee et al., 2021).
+
+# 4.2 RELEVANCE PROPAGATION GUIDED EVENT DROP
+
+Gu et al. (2021) has proven randomly dropping events to be an effective augmentation strategy. Furthermore, we expect to drop events more frequently in regions with label-related objects, motivated by the fact that disturbing regions with no label-related information (namely background) would have a negligible impact on the classifier's prediction. The label-related information can be provided by CAM and saliency map, where higher values imply higher model attention and label relevance. Since saliency map accurately reveals the relevance score of each pixel to the target in the input data, we leverage saliency map to guide dropping, dually illustrated in fig. 2a. The higher the value of a pixel, the higher the probability that we will drop events on that pixel.  $\theta$  is the parameter controlling the magnitude of augmentation.
+
+# 4.3 RELEVANCE PROPAGATION GUIDED EVENT MIX
+
+Event-based data, in contrast to image-based data, does not include color details, with the most crucial aspect being the texture information it contains. The overlapping of label-related objects will impair the texture details of these objects, which in turn further degrades the quality of features extracted in SNNs. Building upon this motivation, we propose Relevance Propagation Guided Event Mix (RPGMix). The whole mixing strategy is illustrated in fig. 2b. For two event-based data candidates, we utilize relevance propagation to localize the label-related regions and obtain two bounding boxes. To mix two objects with clear texture features, we randomly select two positions ensuring minimal overlap of their bounding boxes. This involves initially positioning one box at a corner to maximize the nonoverlapping area for the other box's placement, then selecting positions for both boxes in order, maintaining minimal overlap and maximizing sampling options. Finally, the two event streams are moved to the sampled positions. Although this linear translation part prevents the overlapping of label-related objects, the background of one object would still overlap with the other object. Moreover, in one single time step, the representation ability of the spiking neurons
+
+(which only output binary information) is much worse than that of the activation layer (usually ReLU) of ANNs, making them less capable of spatial resolution and more likely to fall into local optima. Therefore, to promise the presence of only events from a single event stream candidate per pixel, avoiding regions with mixed information from interfering with the SNN, we adopt a CutMix strategy to mask the two event streams based on the bounding box of the second event stream, as demonstrated in the left part in fig. 2b. Kim et al. (2020) takes the sum of each sample's mask as the ratio of their corresponding labels. This ensures that the proportion of labels in the mixed label matches the proportion of pixels belonging to each sample. In our approach, we further aim to align the proportion of labels with the proportion of label-related pixels, which can be estimated using the bounding boxes. As a result, the labels of the two event streams are mixed as
+
+$$
+L _ {m i x} = \frac {L _ {1} \left(w _ {1} h _ {1} - S _ {\text {o v e r l a p}}\right) + L _ {2} w _ {2} h _ {2}}{w _ {1} h _ {1} + w _ {2} h _ {2} - S _ {\text {o v e r l a p}}}, \tag {15}
+$$
+
+where  $w_{i}$  and  $h_i$  denote the width and height of the bounding box in the event stream  $i$ .  $L_{1}$  and  $L_{2}$  are the one-hot labels of the two event streams and  $S_{\text{overlap}}$  is the area of the overlapping region of the two bounding boxes.
+
+# 4.4 EVENTRPG
+
+Combined with a few geometric data augmentation methods in NDA (Li et al., 2022), we formulate our data augmentation strategy namely EventRPG as shown in fig. 3. Specifically, for a batch of input event streams, each of the event streams would be augmented with randomly sampled policy and magnitude. They then have a probability of 0.5 to be augmented by RPGMix.
+
+![](images/5c90497da406a738157744e19aefd34aa02ad34a0b35c70aad7f61c2917d0e07.jpg)  
+Figure 3: Augmentation process of EventRPG.
+
+# 5 EXPERIMENTS
+
+# 5.1 EFFECTIVENESS OF SLRP AND SLTRP
+
+In this subsection, we evaluate the effectiveness of our approach for generating CAMs and saliency maps. Current event-based datasets for classification can be divided into two tasks: object recognition task and action recognition task. Generally, the former mainly involves event streams generated from jittering of static images, while the latter mainly involves event streams recorded in real environments, containing more dynamic information. We perform experiments on both types of datasets to showcase the effectiveness of our approach. We visualize the feature before the last fully connected layer as CAM.
+
+<table><tr><td rowspan="3">Method</td><td colspan="6">Object Recognition</td><td colspan="4">Action Recognition</td></tr><tr><td colspan="2">N-Caltech101</td><td colspan="2">CIFAR10-DVS</td><td colspan="2">N-Cars</td><td colspan="2">DVSGesture</td><td colspan="2">SL-Animals</td></tr><tr><td>A.I.↑</td><td>A.D.↓</td><td>A.I.↑</td><td>A.D.↓</td><td>A.I.↑</td><td>A.D.↓</td><td>A.I.↑</td><td>A.D.↓</td><td>A.I.↑</td><td>A.D.↓</td></tr><tr><td>SAM (Kim &amp; Panda, 2021b)</td><td>0.86</td><td>22.33</td><td>1.23</td><td>46.83</td><td>5.24</td><td>5.89</td><td>8.27</td><td>10.83</td><td>20.60</td><td>8.23</td></tr><tr><td>Grad-CAM (Selvaraju et al., 2017)</td><td>12.03</td><td>41.71</td><td>8.31</td><td>22.15</td><td>21.77</td><td>17.74</td><td>0.41</td><td>67.09</td><td>0.80</td><td>81.44</td></tr><tr><td>Grad-CAM++ (Chattopadhyay et al., 2018)</td><td>24.82</td><td>10.32</td><td>6.23</td><td>26.33</td><td>26.04</td><td>5.41</td><td>7.99</td><td>10.41</td><td>11.78</td><td>22.21</td></tr><tr><td>SLRP-RelCAM (Lee et al., 2021)</td><td>17.51</td><td>10.44</td><td>7.60</td><td>23.41</td><td>22.05</td><td>19.00</td><td>13.11</td><td>6.41</td><td>26.49</td><td>10.67</td></tr><tr><td>SLTRP-RelCAM (Lee et al., 2021)</td><td>17.54</td><td>10.44</td><td>7.60</td><td>23.41</td><td>22.05</td><td>19.00</td><td>13.05</td><td>6.40</td><td>26.49</td><td>10.67</td></tr><tr><td>SLRP-CAM</td><td>34.24</td><td>5.75</td><td>8.41</td><td>23.50</td><td>15.72</td><td>23.80</td><td>7.79</td><td>29.99</td><td>10.88</td><td>40.95</td></tr><tr><td>SLTRP-CAM</td><td>34.24</td><td>5.75</td><td>8.41</td><td>23.50</td><td>15.72</td><td>23.80</td><td>7.79</td><td>30.02</td><td>10.88</td><td>40.95</td></tr><tr><td>SLRP-Saliency Map</td><td>34.12</td><td>4.18</td><td>9.44</td><td>19.99</td><td>22.98</td><td>6.77</td><td>22.20</td><td>13.86</td><td>22.30</td><td>20.57</td></tr><tr><td>SLTRP-Saliency Map</td><td>34.17</td><td>4.19</td><td>9.51</td><td>19.98</td><td>22.84</td><td>6.75</td><td>21.17</td><td>13.95</td><td>22.30</td><td>20.79</td></tr></table>
+
+Table 1: Comparison of A.I. and A.D. on event-based object recognition and action recognition datasets. We highlight the best results in bold and the second best results with underlining.
+
+# 5.1.1 OBJECTIVE FAITHFULNESS
+
+We adopt two widely used metrics, Average Drop (A.D.) and Average Increase (A.I.), to measure the objective faithfulness of our method compared with other typical visualization tools. These two metrics explain how well an attention map explains a model's attention by measuring the average change in the model's output when the attention map is applied as a mask to the input. The higher the A.I. and the lower the A.D., the better the attention map explains the model's attention. We
+
+compare our method with Grad-CAM (Selvaraju et al., 2017), Grad-CAM++ (Chattopadhyay et al., 2018), and SAM (Kim & Panda, 2021b) on both object recognition datasets and action recognition datasets.
+
+On object recognition task, our method achieves the best performance in terms of A.I. and A.D. on large datasets including N-Caltech101 and CIFAR10DVS, demonstrating the effectiveness of our method in generating CAMs and saliency maps. On the N-Cars dataset, Grad-CAM++ outperforms our method. This disparity may be due to N-Cars being a binary classification task that solely focuses on detecting the presence of a car in the event stream, in contrast to other datasets requiring multi-class object localization.
+
+In terms of action recognition task, methods with spiking relevance propagation outperform other methods significantly, with an A.I. metric almost thrice as good as the best method without spiking relevance propagation — Grad-CAM++, in the DVSGesture dataset. In the SL-Animals dataset, RelCAM achieves the best results in terms of A.I. and also has a low A.D. value. Note that RelCAM is also obtained based on the relevance scores from SLRP and SLTRP. Therefore, its good results also helps to demonstrate the effectiveness of our spiking relevance propagation rules.
+
+![](images/ba3a57cf668777282943359996dfa919abbcdcbaadb63d0cdf933aaa95ae9086.jpg)
+
+![](images/2eaed45214bd42064871388b13ae9f867aeddefab7df1cfaccb77a2f0409efb0.jpg)  
+(a)  
+(c)  
+Figure 4: CAM and saliency map by different methods on (a) N-Caltech101 and (b) CIFAR10-DVS. (c) Saliency maps generated from SLTRP on DVS-Gesture and SL-Animals.
+
+# 5.1.2 EVALUATION OF SELECTIVITY
+
+We visualized the saliency map and CAM results of ours and other methods from Spiking-VGG11 and SEW Resnet18. As shown in fig. 4a, our method is more selective than other methods, with a higher value on the label-related objects and a lower value on the background. In contrast, Grad-CAM and SAM are more likely to be affected by the background, and Grad-CAM++ failed to locate the "dragonfly" on the top row, even though there are no other label-unrelated events in this sample. On CIFAR10-DVS, SLRP-CAM and SLRP-RelCAM successfully localize the dog's head, whereas the attention of the other methods drifted to other regions.
+
+The saliency maps generated from SLTRP are able to track the exact moving object in the dataset (see fig. 4c). In the demonstration of SL-Animals dataset, the saliency map first focuses on the hand part to recognize the gesture. In the later time steps, it transfers its attention to the arm to capture the moving information. This proves its capability of temporal saliency information capturing, also yielding high selectivity.
+
+<table><tr><td>Model</td><td>Resolution</td><td>Grad-CAM</td><td>Grad-CAM++</td><td>SLRP-RelCAM</td><td>SLTRP-RelCAM</td><td>SAM</td><td>SLRP-CAM</td><td>SLRP-Saliency Map</td><td>SLTRP-CAM</td><td>SLTRP-Saliency Map</td></tr><tr><td rowspan="2">Spiking VGG-11</td><td>(48, 48)</td><td>0.0776</td><td>0.0884</td><td>0.0225</td><td>0.0233</td><td>0.0267</td><td>0.0246</td><td>0.0645</td><td>0.0279</td><td>0.1157</td></tr><tr><td>(128, 128)</td><td>0.0842</td><td>0.0996</td><td>0.0335</td><td>0.0347</td><td>0.0303</td><td>0.0323</td><td>0.0716</td><td>0.0345</td><td>0.1886</td></tr><tr><td>SEW Resnet18</td><td>(128, 128)</td><td>0.2629</td><td>0.2846</td><td>0.0902</td><td>0.1072</td><td>0.0909</td><td>0.0869</td><td>0.1960</td><td>0.0785</td><td>0.2881</td></tr></table>
+
+Table 2: Average time cost (s) of generating CAM and saliency map on N-Caltech101 (SpikingVGG11) and DVSGesture (SEW Resnet18). Results within the quickest tier are highlighted in bold.
+
+# 5.1.3 COMPUTATION TIME
+
+As shown in table 2, SLRP-CAM, SLTRP-CAM are on the same level as SLTRP-RelCAM, SLRP-RelCAM, and SAM, all among the fastest methods. SLRP-Saliency Map and SLTRP-Saliency Map are slower than other methods, while still being competitive on SEW Resnet18 compared with Grad-CAM and Grad-CAM++. The time cost of SLTRP-CAM does not increase a lot compared to SLRP-CAM, since it only requires the relevance computation of the last fully connected layer. In contrast, the time cost of SLTRP-Saliency Map is much higher than SLRP-Saliency Map, since it requires the relevance computation of all layers.
+
+# 5.2 RESULTS OF EVENTRPG
+
+<table><tr><td>Dataset</td><td>Data Augmentation</td><td>Training Method</td><td>Neural Network</td><td>Neuron</td><td>Timesteps</td><td>Resolution</td><td>Accuracy</td></tr><tr><td rowspan="8">N-Catech101</td><td>Flip</td><td>SALT (Kim &amp; Panda, 2021a)</td><td>Spike-VGG16</td><td>LIF</td><td>20</td><td>(80,80)</td><td>55.00</td></tr><tr><td>NDA (Li et al., 2022)</td><td>STBP-tdBN (Zheng et al., 2021)</td><td>Spike-VGG11</td><td>LIF</td><td>10</td><td>(48,48)</td><td>78.20</td></tr><tr><td>NDA (Li et al., 2022)</td><td>STBP-tdBN (Zheng et al., 2021)</td><td>Spike-VGG11</td><td>LIF</td><td>10</td><td>(128,128)</td><td>83.70</td></tr><tr><td>Eventmix (Shen et al., 2023)</td><td>STBP</td><td>Pre-Act Resnet18</td><td>PLIF</td><td>10</td><td>(48,48)</td><td>79.47</td></tr><tr><td>Identity</td><td></td><td></td><td></td><td></td><td></td><td>75.70</td></tr><tr><td>EventDrop (Gu et al., 2021)</td><td>TET (Deng et al., 2022)</td><td>Spike-VGG11</td><td>LIF</td><td>10</td><td>(128,128)</td><td>74.04</td></tr><tr><td>EventRPG (CAM)</td><td></td><td></td><td></td><td></td><td></td><td>85.00</td></tr><tr><td>EventRPG (Saliency Map)</td><td></td><td></td><td></td><td></td><td></td><td>85.62</td></tr><tr><td rowspan="12">CIFAR10-DVS</td><td>Drop by time</td><td>STBP</td><td>SEW Wide-7B-Net</td><td>PLIF</td><td>16</td><td>(128,128)</td><td>74.40</td></tr><tr><td>Flip</td><td>SALT (Kim &amp; Panda, 2021a)</td><td>Spike-VGG16</td><td>LIF</td><td>20</td><td>(64,64)</td><td>67.10</td></tr><tr><td>Random Crop</td><td>DSR (Meng et al., 2022)</td><td>Spike-VGG11</td><td>IF</td><td>20</td><td>(48,48)</td><td>75.03 ± 0.39</td></tr><tr><td>Random Crop</td><td>DSR (Meng et al., 2022)</td><td>Spike-VGG11</td><td>LIF</td><td>20</td><td>(48,48)</td><td>77.27±0.24</td></tr><tr><td>FlipTranslation</td><td>TET (Deng et al., 2022)</td><td>Spike-VGG11</td><td>LIF</td><td>10</td><td>(48,48)</td><td>83.17±0.15</td></tr><tr><td>NDA (Li et al., 2022)</td><td>STBP-tdBN (Zheng et al., 2021)</td><td>Spike-VGG11</td><td>LIF</td><td>10</td><td>(48,48)</td><td>79.60</td></tr><tr><td>NDA (Li et al., 2022)</td><td>STBP-tdBN (Zheng et al., 2021)</td><td>Spike-VGG11</td><td>LIF</td><td>10</td><td>(128,128)</td><td>81.70</td></tr><tr><td>Eventmix (Shen et al., 2023)</td><td>STBP</td><td>Pre-Act Resnet18</td><td>PLIF</td><td>10</td><td>(48,48)</td><td>81.45</td></tr><tr><td>Identity</td><td></td><td></td><td></td><td></td><td></td><td>78.85</td></tr><tr><td>EventDrop (Gu et al., 2021)</td><td>TET (Deng et al., 2022)</td><td>Spike-VGG11</td><td>LIF</td><td>10</td><td>(48,48)</td><td>77.73</td></tr><tr><td>EventRPG (CAM)</td><td></td><td></td><td></td><td></td><td></td><td>85.55</td></tr><tr><td>EventRPG (Saliency Map)</td><td></td><td></td><td></td><td></td><td></td><td>84.96</td></tr><tr><td rowspan="7">N-Cars</td><td>NDA (Li et al., 2022)</td><td>STBP-tdBN (Zheng et al., 2021)</td><td>Spike-VGG11</td><td>LIF</td><td>10</td><td>(48,48)</td><td>90.10</td></tr><tr><td>NDA (Li et al., 2022)</td><td>STBP-tdBN (Zheng et al., 2021)</td><td>Spike-VGG11</td><td>LIF</td><td>10</td><td>(128,128)</td><td>91.90</td></tr><tr><td>Eventmix (Shen et al., 2023)</td><td>STBP</td><td>Pre-Act Resnet18</td><td>PLIF</td><td>10</td><td>(48,48)</td><td>96.29</td></tr><tr><td>Identity</td><td></td><td></td><td></td><td></td><td></td><td>94.92</td></tr><tr><td>EventDrop (Gu et al., 2021)</td><td>TET (Deng et al., 2022)</td><td>Spike-VGG11</td><td>LIF</td><td>10</td><td>(48,48)</td><td>95.46</td></tr><tr><td>EventRPG (CAM)</td><td></td><td></td><td></td><td></td><td></td><td>95.76</td></tr><tr><td>EventRPG (Saliency Map)</td><td></td><td></td><td></td><td></td><td></td><td>96.00</td></tr></table>
+
+Table 3: Accuracy of various data augmentation methods on event-based object recognition datasets. All of the datasets used are created using event cameras. Specifically, N-Caltech101 and CIFAR10DVS are derived from static images, while N-Cars is recorded in real-world environments.
+
+In this section, we evaluate our proposed EventRPG with other augmentation methods including EventDrop (Gu et al., 2021), NDA (Li et al., 2022), and EventMix (Shen et al., 2023) across several object recognition and action recognition datasets. They could illustrate the performance of our methods in terms of static objects and moving objects, respectively. The datasets used and training settings are introduced in appendix C in details. Since SLRTP and SLRP yield nearly identical results for object recognition tasks, we only leverage SLRP in object recognition experiments since it costs fewer time. Eventdrop (Gu et al., 2021) did not conduct experiments on SNNs, so we reproduce it using its public code and conduct experiments under the same training setting of ours. We report the best accuracy for each experiment using same random seed.
+
+# 5.2.1 OBJECT RECOGNITION TASKS
+
+From table 3 we see that EventRPG achieves state-of-the-art performance on N-Caltech101 and CIFAR10-DVS datasets, bringing  $9.92\%$  and  $6.7\%$  improvements compared with identity (no augmentation), respectively. On N-Cars, EventRPG achieves the second-best performance, only  $0.29\%$  lower than EventMix. This might be attributed to the fact that N-Cars is a binary classification dataset, which only contains label "car" and "background". Most samples belonging to "background" do not have a specific label-related object to locate, making it difficult for our method to generate saliency map and CAM with high quality, thus decreasing the performance.
+
+# 5.2.2 ACTION RECOGNITION TASKS
+
+We implement EventRPG on SEW Resnet18 for action recognition tasks. On DVSGesture dataset, our method achieves best results compared with other reproduced augmentation methods under the same training settings, though slightly lower than EventMix on Pre-Act Resnet18. On SL-Animals dataset, our method achieves state-of-the-art performance among all data augmentation approaches
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Spike</td><td rowspan="2">Method</td><td rowspan="2">DVSGesture</td><td colspan="2">SL-Animals</td></tr><tr><td>4 sets</td><td>3 sets</td></tr><tr><td>7-Layer Spiking CNN</td><td>Hybrid</td><td>SCTFA (Cai et al., 2023)</td><td>98.96</td><td>90.04</td><td>-</td></tr><tr><td>GoogLeNet</td><td>X</td><td>TORE (Baldwin et al., 2022)</td><td>96.20</td><td>85.10</td><td>-</td></tr><tr><td>Event Transformer</td><td>X</td><td>EvT (Sabater et al., 2022)</td><td>96.20</td><td>88.12</td><td>87.45</td></tr><tr><td>Pre-Act Resnet18</td><td>✓</td><td>EventMix (Shen et al., 2023)</td><td>96.75</td><td>-</td><td>-</td></tr><tr><td rowspan="7">SEW Resnet18</td><td rowspan="7">✓</td><td>Identity</td><td>94.33</td><td>85.42</td><td>89.09</td></tr><tr><td>EventDrop (Gu et al., 2021)</td><td>92.33</td><td>86.33</td><td>88.99</td></tr><tr><td>NDA (Li et al., 2022)</td><td>93.67</td><td>87.77</td><td>89.55</td></tr><tr><td>EventRPG (CAM, SLRP)</td><td>95.83</td><td>90.97</td><td>91.96</td></tr><tr><td>EventRPG (Saliency Map, SLRP)</td><td>95.49</td><td>91.59</td><td>93.30</td></tr><tr><td>EventRPG (CAM, SLTRP)</td><td>96.18</td><td>90.54</td><td>90.63</td></tr><tr><td>EventRPG (Saliency Map, SLTRP)</td><td>96.53</td><td>90.04</td><td>93.75</td></tr></table>
+
+Table 4: Accuracy of various data augmentation methods on event-based action recognition datasets.
+
+and all neural networks, with  $3.82\%$  and  $4.2\%$  improvements compared with the second-best augmentation method on 4 sets and 3 sets, respectively. Action recognition tasks include more dynamic information compared to object recognition tasks. Thus, the success of our method on action recognition tasks demonstrates its great potential for other dynamic event-based datasets, which is a future direction worth exploring.
+
+# 5.2.3 TIME CONSUMPTION ANALYSIS
+
+![](images/146cae39e64f34afa04d00eb2b67a98e1517af221ebe183b2f309f092cd0a8f9.jpg)  
+(a)
+
+![](images/5cd405787214dcd6195449fcef12c9a9b6b7a65547f45beb162a8ccd1cef65f9.jpg)  
+(b)  
+Figure 5: Average time cost of augmentation methods with SEW Resnet18 on two datasets.
+
+In order to assess the computational efficiency of our data augmentation method, we perform two experiments to measure the time required for executing EventRPG with SLRP and SLTRP in object recognition and action recognition tasks. As depicted in fig. 5, the computation time of other data augmentation approaches remains constant regardless of the batch size, whereas our method exhibits a nearly linear decrease in computation time as the batch size increases, since the relevance propagation process, which is the main contributor to the time consumption, can be speeded up by parallizing computing across samples in a batch, similar to the gradient backpropagation process. When the batch size for each GPU exceeds 4, both SLRP and SLTRP achieve comparable speeds to NDA and EventDrop, confirming the time efficiency of EventRPG.
+
+# 6 CONCLUSION AND LIMITATION
+
+Conclusion In this paper, for the first time, we propose SLTRP and SLRP, two efficient and practical methods for generating CAMs and saliency maps for SNNs. Building upon this, we propose EventRPG, i.e., dropping events and mixing events with Relevance Propagation Guidance. Since EventRPG only disturbs and mixes regions on which model concerns most, it is more efficient compared to vanilla dropping and mixing, and also alleviates the likely misalignment problem between data and label. In our experiments, SLRP and SLTRP not only both yield best results compared with other feature visualization tools, but also consume very little time to compute. EventRPG achieves state-of-the-art performance on N-Caltech101, CIFAR10-DVS and SL-Animals datasets, proving its strong generalization ability across different models and datasets.
+
+Limitation Currently, EventRPG can only be implemented on classification tasks, remaining as a limitation. However, we could still leverage multi-task training paradigm which has been proven to be effective to implement it into other downstream tasks, and we will also explore more possibilities of using EventRPG in self-supervised learning tasks.
+
+# 7 ACKNOWLEDGEMENT
+
+We would like to thank all anonymous reviewers for their committed work and insightful feedback. This work was supported in part by the National Natural Science Foundation of China under Grants 62073066 and U20A20197, in part by the Fundamental Research Funds for the Central Universities under Grant N2226001, in part by 111 Project under Grant B16009, in part by the Intel Neuromorphic Research Community (INRC) Grant Award (RV2.137.Fang), and in part by Guangzhou-HKUST(GZ) Joint Funding Program under Grant 2023A03J0682.
+
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+
+# APPENDIX
+
+# A ADDITIONAL EXPERIMENTAL RESULTS
+
+Here we present more experiments to further prove the effectiveness of our approach.
+
+# A.1 CLASSIFICATION ON MINI N-IMAGENET DATASET
+
+N-ImageNet, as proposed by Kim et al. (2021), represents the neuromorphic adaptation of the well-known ImageNet dataset. It encompasses a thousand object categories, making it the most challenging task in event-based classification to date. We conduct experiments to evaluate the performances of different data augmentation methods on mini N-ImageNet with SEW Resnet-18.
+
+<table><tr><td>Data Augmentation</td><td>Identity</td><td>EventDrop</td><td>NDA</td><td>EventRPG</td></tr><tr><td>Top-1 Accuracy</td><td>28.16</td><td>34.18</td><td>35.84</td><td>40.90</td></tr><tr><td>Top-5 Accuracy</td><td>52.14</td><td>60.94</td><td>63.64</td><td>67.74</td></tr></table>
+
+Table 5: Top-1 and Top-5 Accuracies (%) on mini N-ImageNet.
+
+Table 5 demonstrates that data augmentations significantly enhance the performance of the model, as shown by the notable positive effects. In this comparison, our approach outperforms the nearest competitor NDA, achieving a higher Top-1 accuracy by  $5.06\%$  and Top-5 accuracy by  $4.1\%$ .
+
+![](images/60ef0027e78e9d12fad869e4c2c531d420eb00b9fe6204b699ec7022839675d9.jpg)  
+(a)
+
+![](images/6ff49296cbb53a4ba96a90de0f06ff86c68496b58b788e36309d92e6470ea516.jpg)  
+(b)  
+Figure 6: The loss and Accuracy curves of different augmentations on mini N-ImageNet.
+
+As shown in fig. 6a, the performance disparity between EventRPG and NDA is as significant as the one observed between EventDrop and Identity. Additionally, late epochs of EventDrop and Identity exhibit overfitting, a problem that is effectively mitigated in NDA and EventRPG, as illustrated in fig. 6b. These outcomes highlight again the efficiency of our approach in boosting the model's performance and mitigating overfitting.
+
+# A.2 COMPARISON WITH OTHER SALIENCY-BASED MIX METHODS
+
+We also compare the performance of various saliency-based mix methods on two typical datasets: N-Caltech101 and SL-Animals. In the experiments, there is a  $50\%$  chance for each event stream to be augmented by corresponding mix approach with no other augmentations applied. Notably, it's observed in table 6 that both Saliency Mix (Uddin et al., 2020) and Puzzle Mix (Kim et al., 2020) occasionally result in lower performance than the identity approach. This underscores RPGMix's robust generalization capability, as it consistently excels across all datasets.
+
+To clearly showcase the differences among these mix methods, we present visualizations of the augmented samples they produce (see fig. 7). In most cases, our RPGMix aims to preserve as many label-related pixels as possible.
+
+![](images/ca1282c97e3c18728c7eda4c63dcefa2a2a5623d447c139bed14eece9cf0539a.jpg)  
+Figure 7: Mixed samples generated by different methods on N-Caltech101.
+
+<table><tr><td>Dataset</td><td>Model</td><td>Identity</td><td>Saliency Mix</td><td>Puzzle Mix</td><td>Puzzle Mix (mask only)</td><td>RPGMix</td></tr><tr><td>N-Caltech101</td><td>Spike-VGG11</td><td>75.70</td><td>72.63</td><td>79.38</td><td>78.13</td><td>81.75</td></tr><tr><td>SL-Animals-4Sets</td><td>SEW Resnet18</td><td>85.42</td><td>84.00</td><td>85.34</td><td>85.68</td><td>88.67</td></tr><tr><td>SL-Animals-3Sets</td><td>SEW Resnet18</td><td>89.09</td><td>89.29</td><td>88.39</td><td>90.18</td><td>90.45</td></tr></table>
+
+Table 6: Accuracy of Puzzle Mix, Saliency Mix, and RPGMix on object and action recognition tasks.
+
+# A.3 ADDITIONAL QUAILITATIVE RESULTS
+
+We present additional qualitative results of our method applied to the N-Caltech101, SL-Animals, and DVS-Gesture datasets. The saliency maps created using SLTRP, as shown in fig. 8, exhibit strong selectivity. They assign high values to label-related objects and low values to label-unrelated objects. In action recognition datasets, saliency maps primarily focus on label-related moving objects to capture the dynamic aspects of actions, as illustrated in fig. 9. In SL-Animals datasets (second row), the saliency maps focus on the person's hand raised to his/her head, assigning minimal attention to the lower area of the person, despite its high event density. Similarly, in the fourth row, the saliency maps show little interest in the person's right hand (from our perspective), focusing instead on the left hand, even though both hands have similar event densities. These results illustrate that it's the label-related actions, rather than event density, that truly draw the focus of the saliency maps. This observation further confirms the selectivity of SLTRP-generated saliency maps in both action recognition and object recognition datasets.
+
+# B RELATED WORK
+
+# B.1 EVENT-BASED DATA
+
+Event-based data is generated by asynchronous sensors, usually referred to as event cameras (Mahowald, 1994; Son et al., 2017). Similar to point cloud, event-based data consists of four-dimensional points, denoted as  $(x,y,t,p)$ , where  $x$  and  $y$  are the spatial coordinates,  $t$  is the timestamp, and  $p$  is the polarity. Due to the inherent advantages of event cameras, event-based data is widely used in the field of perception, e.g., optical flow estimation (Gehrig et al., 2021), depth estimation (Wang et al., 2021), 3d reconstruction (Baudron et al., 2020; Rudnev et al., 2022), motion segmentation (Stoffregen et al., 2019; Zhou et al., 2021), semantic segmentation (Kim et al., 2022), etc.
+
+There are two mainstream approaches to processing event-based data, frame-based approaches and event-by-event-based approaches. Frame-based approaches (Kogler et al., 2009; Bardow et al., 2016; Lagorce et al., 2016; Gehrig et al., 2019) are similar to image processing, where events are first converted into frames with a fixed shape  $(C, H, W)$ , and then the frames are fed into an ANN for downstream classification or regression tasks. Event-by-event-based approaches handle events on a one-by-one basis, which is natural for SNNs (Lee et al., 2020; Fang et al., 2021a; Deng et al., 2022).
+
+# B.2 DATA AUGMENTATION
+
+Vanilla data augmentations include simple transformations such as scale, rotation, flip, etc. Mixup (Zhang et al., 2018) proposes mixing two randomly selected samples and their corresponding labels as new data and label for training, based on which several works that modify the mixing details were proposed to further improve the robustness and performance (Yun et al., 2019; Hendrycks et al., 2019). Kim et al. (2020); Uddin et al. (2020) are similar to our proposed RPGMix that leverages the model's saliency information to augment the data, while our motivation and implementation details are quite different from them.
+
+# B.3 RELEVANCE PROPAGATION
+
+Relevance propagation was initially presented in (Montavon et al., 2017) as a method for creating saliency maps to visualize the impact of each pixel in the input data on the prediction of the model
+
+![](images/c5eaea19fa5d0f5b09e5096241ad45fa6f755febd9ba401034ef25bca1ebc49a.jpg)  
+Figure 8: SLTRP-Saliency Maps from Spike-VGG11 on N-Caltech101.
+
+![](images/a00e602ffc771d0ededd83403012eb662527cbf1dc7a6ac5f01ec705ddc36a4f.jpg)  
+SL-Animals
+
+![](images/cb6c61c1b26999828ab2599b78c03457d8cbca2413ad040c56b8cb1267b8b79d.jpg)  
+DVS-Gesture  
+Figure 9: SLTRP-Saliency Maps from SEW Resnet-18 on SL-Animals and DVS-Gesture.
+
+or a particular class, thereby enhancing the interpretability of neural networks. The specificity of saliency maps towards the target class can be improved by employing Contrastive Layer-wise Televance Propagation (CLRP) as described by Gu et al. (2018), which diminishes the relevance scores for non-target classes.
+
+# B.4 CLASS ACTIVATION MAP
+
+As another visualization tool describing the area of most interest to the model, Class Activation Map (CAM) is widely used to interpret the model's attention and to find objects belonging to a certain class in the input data. The original CAM (Zhou et al., 2016) could only visualize the activation map of the last global average pooling (GAP) layer of the model, requiring the model to have a special structure. Grad-CAM (Selvaraju et al., 2017) overcomes this limitation by replacing the weights of GAP with the sum of gradients, enabling the acquisition of CAM from any CNN model. On this basis, Grad-CAM++ (Chattopadhyay et al., 2018), Score-CAM (Wang et al., 2020), and Relevance-CAM (Lee et al., 2021) were proposed to improve the quality of CAMs.
+
+# C DATASETS AND TRAINING DETAILS
+
+# C.1 OBJECT RECOGNITION TASK
+
+N-Caltech101 The neuromorphic version (Orchard et al., 2015) of Caltech101 (Fei-Fei et al., 2004). It is artificially created by moving an asynchronous time-based image sensor (ATIS) mounted on a pan tilt unit in front of an LCD screen that presents the image data in Caltech101. We use the same dataset as Li et al. (2022) that is split into the training set and test set by  $9:1$ . The resolution of N-Caltech101 is  $180 \times 240$  which, in our implementation, is padded to  $240 \times 240$  and rescaled to  $128 \times 128$ .
+
+CIFAR10-DVS The DVS version (Li et al., 2017) of CIFAR10 (Krizhevsky et al., 2009). The generation of CIFAR10-DVS is similar to N-Caltech101. Following NDA (Li et al., 2022), we divide it by  $9:1$  as the training set and test set, and scale the resolution from  $128 \times 128$  to  $48 \times 48$ .
+
+N-Cars A binary classification dataset (Sironi et al., 2018). Unlike N-Caltech101 and CIFAR10-DVS transformed from image datasets, N-Cars is obtained from the recording of an ATIS in real driving scenarios. Similar to the preprocessing of N-Caltech101, we first pad the resolution from  $100 \times 120$  to  $120 \times 120$  and then scale it to  $48 \times 48$  on SNN.
+
+Mini N-ImageNet A subset of N-ImageNet dataset (Kim et al., 2021). Despite being a smaller segment, it includes over 100,000 samples spanning 100 classes, making it the largest dataset used in our experiments. In line with the original authors' setup, we pad the resolution from  $640 \times 480$  to  $640 \times 640$  and then scale it down to  $224 \times 224$  as the input resolution of the SNN.
+
+# C.2 ACTION RECOGNITION TASK
+
+For object recognition tasks, shapes and textures are the most important information to recognize an object, and thus we do not care how an object moves in the event streams. While in action recognition tasks, the movements of the objects should be considered. They represent two different recognition strategies and thus are both essential to validate our proposed method.
+
+DVS128 Gesture A hand gesture dataset (Amir et al., 2017) recorded from a DVS128 camera. It comprises 1464 samples with 11 classes, split into the training set and test set by 8:2. The resolution is set as  $128 \times 128$  on SNN.
+
+SL-Animals-DVS A sign language dataset (Vasudevan et al., 2021) including 19 Spanish Sign Language signs corresponding to animals. Following the raw paper, we split it by 7.5:2.5 into the training set and test set. Also, we separate the datasets into “4 sets” and “3 sets”, which excludes the samples disturbed by the indoor lighting conditions reflecting on the patterned clothing of the user. The resolution processing of SL-Animals-DVS is identical to that of DVS 128 Gesture.
+
+<table><tr><td>Neural Networks</td><td>Neuron Model</td><td>Datasets</td><td>Epoch</td><td>Batch Size</td><td>T</td><td>Learning Rate</td></tr><tr><td rowspan="3">Spike-VGG11</td><td rowspan="3">LIF</td><td>N-Caltech101</td><td rowspan="3">100</td><td>16</td><td rowspan="3">10</td><td rowspan="3">\( 1 \times 10^{-3} \)</td></tr><tr><td>CIFAR10-DVS</td><td>64</td></tr><tr><td>N-Cars</td><td>64</td></tr><tr><td rowspan="3">SEW Resnet-18</td><td rowspan="3">PLIF</td><td>SL-Animals</td><td rowspan="2">200</td><td>20</td><td rowspan="2">16</td><td rowspan="2">\( 5 \times 10^{-4} \)</td></tr><tr><td>DVS-Gesture</td><td rowspan="2">64</td></tr><tr><td>Mini N-Imagenet</td><td>100</td><td>10</td><td>\( 1 \times 10^{-3} \)</td></tr></table>
+
+Table 7: Hyper-parameters for different models and datasets.
+
+# C.3 TRAINING DETAILS
+
+In all experiments, we use Adam optimizer with the default setting  $(\beta_{1},\beta_{2}) = (0.9,0.999)$  to update the parameters and Cosine Annealing Scheduler for the decrease of learning rate. Other hyper-parameters are shown in table 7. For fair comparisons, an identical seed is leveraged for all experiments. We utilize the public code of TET (Deng et al., 2022) to build Spike-VGG11 and Spikingjelly (Fang et al., 2020) to build SEW Resnet-18 with PLIF (Fang et al., 2021b).
+
+# D PROOFS
+
+In this section, we provide proofs of the propositions.
+
+# D.1 PROOF FOR CONSERVATION PROPERTY ON  $\alpha \beta$ -RULE
+
+Proof. The sum of relevance scores in layer  $l - 1$  is
+
+$$
+\begin{array}{l} \sum_ {i} R _ {i} ^ {(l - 1)} = \sum_ {i} \sum_ {j} R _ {i \leftarrow j} ^ {(l - 1, l)} = \sum_ {i} \sum_ {j} R _ {j} ^ {(l)} \cdot \left(\alpha \cdot \frac {z _ {i j} ^ {+}}{\sum_ {i} z _ {i j} ^ {+}} + \beta \cdot \frac {z _ {i j} ^ {-}}{\sum_ {i} z _ {i j} ^ {-}}\right) \\ = \sum_ {j} R _ {j} ^ {(l)} \sum_ {i} \left(\alpha \cdot \frac {z _ {i j} ^ {+}}{\sum_ {i} z _ {i j} ^ {+}} + \beta \cdot \frac {z _ {i j} ^ {-}}{\sum_ {i} z _ {i j} ^ {-}}\right) \\ = \sum_ {j} R _ {j} ^ {(l)} \left(\alpha \cdot \frac {\sum_ {i} z _ {i j} ^ {+}}{\sum_ {i} z _ {i j} ^ {+}} + \beta \cdot \frac {\sum_ {i} z _ {i j} ^ {-}}{\sum_ {i} z _ {i j} ^ {-}}\right) \\ = \sum_ {j} R _ {j} ^ {(l)} (\alpha + \beta) \\ = \sum_ {j} R _ {j} ^ {(l)}. \\ \end{array}
+$$
+
+![](images/14f58bd81f0fc7aa9f4c09099bc76128fb2e2220b5eff11f36f086895da4771d.jpg)
+
+# D.2 PROOF FOR PROPOSITION 1
+
+Proof. We prove proposition 1 by induction.
+
+For case  $t = 1$ , according to eq. (12) we have
+
+$$
+R ^ {(l - 1)} [ 1 ] = (1 - \gamma [ 1 ]) \left(\sum_ {i = 2} ^ {T} R ^ {(l)} [ i ] \prod_ {j = 2} ^ {i} \gamma [ j ] + R ^ {(l)} [ 1 ]\right),
+$$
+
+where  $\gamma [1] = 0$ , since at the first time step, the membrane voltage is initialized to be 0. Clearly, proposition 1 holds for case  $t = 1$ .
+
+Next, suppose proposition 1 holds for case  $t = k - 1$ , we have
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {k - 1} R ^ {(l - 1)} [ t ] = \sum_ {t = 1} ^ {k - 1} R ^ {(l)} [ t ] + \sum_ {i = k} ^ {T} R ^ {(l)} [ i ] \prod_ {j = k} ^ {i} \gamma [ j ] \\ = \sum_ {t = 1} ^ {k - 1} R ^ {(l)} [ t ] + \gamma [ k ] \left(R ^ {(l)} [ k ] + \sum_ {i = k + 1} ^ {T} R ^ {(l)} [ i ] \prod_ {j = k + 1} ^ {i} \gamma [ j ]\right) \\ = \sum_ {t = 1} ^ {k - 1} R ^ {(l)} [ t ] + \frac {\gamma [ k ]}{1 - \gamma [ k ]} R ^ {(l - 1)} [ k ]. \\ \end{array}
+$$
+
+Then
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {k} R ^ {(l - 1)} [ t ] = \sum_ {t = 1} ^ {k - 1} R ^ {(l - 1)} [ t ] + R ^ {(l - 1)} [ k ] = \sum_ {t = 1} ^ {k - 1} R ^ {(l)} [ t ] + \frac {\gamma [ k ]}{1 - \gamma [ k ]} R ^ {(l - 1)} [ k ] + R ^ {(l - 1)} [ k ] \\ = \sum_ {t = 1} ^ {k - 1} R ^ {(l)} [ t ] + \frac {R ^ {(l - 1)} [ k ]}{1 - \gamma [ k ]} \\ = \sum_ {t = 1} ^ {k - 1} R ^ {(l)} [ t ] + R ^ {(l)} [ k ] + \sum_ {i = k + 1} ^ {T} R ^ {(l)} [ i ] \prod_ {j = k + 1} ^ {i} \gamma [ j ] \\ = \sum_ {t = 1} ^ {k} R ^ {(l)} [ t ] + \sum_ {i = k + 1} ^ {T} R ^ {(l)} [ i ] \prod_ {j = k + 1} ^ {i} \gamma [ j ]. \\ \end{array}
+$$
+
+![](images/f770b50201b36206a2f6c69219c60919f7e9837dea2781f9da3a8ff8a1382206.jpg)

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+# EVOKE: EVOKING CRITICAL THINKING ABILITIES IN LLMS VIA REVIEWER-AUTHOR PROMPT EDITING
+
+Xinyu Hu\* Pengfei Tang\* Simiao Zuo\* Zihan Wang\* Bowen Song\* Qiang Lou\* Jian Jiao\* Denis Charles
+
+Microsoft University of Washington University of Michigan
+
+# ABSTRACT
+
+Large language models (LLMs) have made impressive progress in natural language processing. These models rely on proper human instructions (or prompts) to generate suitable responses. However, the potential of LLMs are not fully harnessed by commonly-used prompting methods: many human-in-the-loop algorithms employ ad-hoc procedures for prompt selection; while auto prompt generation approaches are essentially searching all possible prompts randomly and inefficiently. We propose Evoke, an automatic prompt refinement framework. In Evoke, there are two instances of a same LLM: one as a reviewer (LLM-Reviewer), it scores the current prompt; the other as an author (LLM-Author), it edits the prompt by considering the edit history and the reviewer's feedback. Such an author-reviewer feedback loop ensures that the prompt is refined in each iteration. We further aggregate a data selection approach to Evoke, where only the hard samples are exposed to the LLM. The hard samples are more important because the LLM can develop deeper understanding of the tasks out of them, while the model may already know how to solve the easier cases. Experimental results show that Evoke significantly outperforms existing methods. For instance, in the challenging task of logical fallacy detection, Evoke scores above 80, while all other baseline methods struggle to reach 20.
+
+# 1 INTRODUCTION
+
+Consider an intriguing trio that at first glance seems unrelated: bumble bees, cell phones, and exciting news. At a superficial level, their commonality might note their plural forms; however, a more profound analysis reveals a shared essence: they all "create a buzz." This comparison sheds light on the depth and intricacy of human cognitive processes. At the heart of such processes is critical thinking, the ability to conceptualize, analyze, question, and evaluate ideas and beliefs. As we transition to the domain of artificial intelligence, it is observed that large language models (LLMs) have remarkably evolved as general problem solvers, urging us to ponder:
+
+# Can LLMs think on their own?
+
+In practice, we observe that existing prompting methods are inadequate in evoking the critical thinking abilities of LLMs. For example, in Figure 1, we show two prompts for solving a common concept task. From Figure 1 (left), we see that for the input trio “bumble bees, cell phones, and exciting news”, the LLM outputs a superficial common concept “plural form” using the hand-crafted prompt. On the other hand, with the prompt generated by the proposed method, the LLM demonstrates much deeper understanding about the task, i.e., it generates the correct answer “can cause a buzz” (see Figure 1, right). These results indicate that the quality of prompts are directly related to the performance of LLMs. In this work, we focus on prompting methods that enables LLMs to think on their own.
+
+The current prompting methodologies exhibit significant drawbacks. Many prompting methods are ad hoc because of their human-in-the loop development paradigm. In such a process, given a target task, we first draft an initial prompt. Then, we refine the prompt using techniques such as chain-of-thought, few-shot demonstrations, and coding-style problem descriptions (Wei et al., 2022c;a;
+
+![](images/e38785e7b028f862cd184958cc13d8a3f410059929d3fd2f225bada0b5a37bc7.jpg)  
+Figure 1: Comparison between hand-crafted and Evoke prompts.
+
+![](images/8ec39f983e8bd527e97e378790d7281e0d03c60b4c690cb6dda8924fcb2daf21.jpg)  
+Figure 2: Simplified workflow of Evoke.
+
+Gao et al., 2023) based on the model's performance on the target task. We note that in practice, a hand-crafted prompt optimized for one task rarely translates to satisfactory performance in another task (Zhang et al., 2023). Therefore, each task becomes a new expedition, with its own set of trials, errors, and validations. Such an ad hoc human-in-the-loop development procedure introduces extensive human labor requirements, which significantly hinder the applicability of LLMs in real-world applications.
+
+Existing works develop algorithms to automatically generate prompts instead of relying on ad hoc human optimization (Shin et al., 2020; Honovich et al., 2022; Zhou et al., 2022). However, these methods often lack feedback loops, such that the refinement procedure essentially performs a random search. For example, in each refinement iteration, Zhou et al. (2022) simply rephrases the prompt into multiple candidates, and then select the candidate that yields the best performance as the refined prompt. Note that such a procedure fails to learn from past successes and failures, such that refined prompt does not enrich the original prompt with additional context.
+
+We propose Evoke, which addresses the aforementioned drawbacks by leveraging an author-reviewer paradigm. In this paradigm, there are two distinct purposes: one instance as an author (LLM-Author) tasked with editing prompts, and another instance as a reviewer (LLM-Reviewer) tasked with evaluating the quality of the prompts generated by the LLM-Author. Each role is played independently by separate instances of the same LLM.
+
+Critical thinking is not something you do once with an issue and then drop it. It requires that we update our knowledge as new information comes in. Daniel Levitin
+
+The essence of this quote resonates with the feedback loop in the workflow of Evoke, as depicted in Figure 2. The workflow comprises three steps: First, the LLM-Author edits prompts from previous iterations, taking into account the past edits and the feedback from the LLM-Reviewer. Second, the LLM-Reviewer scores the revised prompts from the LLM-Author, and the top-n candidates with the highest scores are selected for subsequent procedures. The LLM-Reviewer employs a memory module that stores history edits, prompts and task accuracy of history prompts. Finally, the task accuracy for each instruction is computed.
+
+To further enhance the efficacy of Evoke, we propose a data selection strategy. In this strategy, only the hard samples selected by a selector are exposed to the LLM. The intuition is that the LLM can develop deeper understanding of the tasks out of the hard samples, while it already knows how to solve the easier cases. Through extensive experiments (see Figure 10 in the experiments), we see that retaining the hard samples indeed improves efficacy of Evoke.
+
+We conduct extensive experiments to demonstrate the effectiveness of Evoke. Specifically, on eight tasks from the Instruction Induction (Honovich et al., 2022) dataset and the Big Bench Instruction Induction (Zhou et al., 2022) dataset, we show that Evoke significantly outperforms existing automatic prompt engineering approaches. For example, on the challenging logical fallacy detection task, Evoke achieves a score of over 80, while all the baseline methods struggle to reach 20. We also show that Evoke can improve LLMs' robustness against adversarial attacks, and can also handle fine-grained named entity recognition tasks with exceptional performance. As an example, Evoke
+
+achieves significant performance gain on an adversarially constructed dataset, indicating that the proposed method can improve robustness of LLMs. Additionally, we provide detailed analysis on the effectiveness of each component of Evoke.
+
+# 2 RELATED WORK
+
+Large Language Models Recently, LLMs have shown emergent abilities—capabilities to perform tasks they weren't explicitly trained for (Wei et al., 2022a;b; Bubeck et al., 2023). This includes common sense question answering, code generation, and cross-domain problem solving, enriching their utility across unforeseen domains (Chen et al., 2021; Sarsa et al., 2022; Thirunavukarasu et al., 2023; Huang & Chang, 2022; Du et al., 2023). Subsequently, adapting LLMs to specific problems has drawn attention, and several methods have been proposed: Reinforcement Learning from Human Feedback (RLHF Ouyang et al. 2022), efficient fine-tuning (Hu et al., 2022; Dettmers et al., 2023), and prompt engineering (White et al., 2023), among others. Each method has its pros and cons. For instance, RLHF can significantly improve performance but may require extensive human annotations. Efficient fine-tuning, on the other hand, can be less resource-intensive but might fall short in achieving the desired level of task-specific optimization. Prompt engineering, while innovative, may require a well-crafted prompt to effectively guide the model towards accurate outputs.
+
+In-Context Learning and Prompt Engineering In-Context Learning (ICL) refers to the ability of LLMs to learn a new task from a small set of examples presented within the context (the prompt) at inference time, without updating any parameters (Wei et al., 2022a). This paradigm has significantly improved the capabilities of LLMs across various tasks. Many studies have explored the reasons behind such improvements, examining aspects like Bayesian optimization and the difficulty of demonstrations (Xie et al., 2022; Min et al., 2022; Liu et al., 2022; Yoo et al., 2022).
+
+Prompt engineering plays a pivotal role in facilitating ICL. It entails the design of prompts that arm the LLM with the essential information needed to learn and adeptly perform the new task. Each prompt essentially sets the stage for the LLM, enclosing the task's requirements and guiding the model towards producing the desired output. By carefully crafting prompts, it is possible to leverage the inherent capability of LLMs, enabling them to tackle a wide range of tasks even with limited or no prior explicit training on those tasks. Recently, methods such as Chain-of-Thought (CoT), Zero-CoT, Self-Consistency, Program-Aided, and Few-Shot Prompting have been demonstrated to be effective (Wei et al., 2022c; Kojima et al., 2022; Wang et al., 2022; Gao et al., 2023; Reynolds & McDonell, 2021).
+
+Automatic Prompt Engineering The existing methodologies for automating discrete prompt optimization have their roots in instruction induction, as discussed by Honovich et al. 2022. It was discovered that LLMs can generate natural language instructions based on a small number of input-output pair examples. Building on this, Zhou et al. (2022) proposed a new algorithm for the automatic generation and selection of instructions for LLMs. The algorithm, named Automatic Prompt Engineer (APE), is capable of generating prompts that achieve human-level performance across a diverse range of NLP tasks. Work has also been done on automating prompt generation for specific domains like code generation, as discussed in Shrivastava et al. 2023.
+
+# 3 ITERATIVE REVIEWER-AUTHOR PROMPT EDITING
+
+# 3.1 OVERVIEW
+
+In Evoke, the same LLM plays two different roles: an author (LLM-Author) that is in charge of editing and refine prompts, and a reviewer (LLM-Reviewer) that is in charge of scoring the refined prompts. We use two different prompts for the author's and the reviewer's task.
+
+$\diamond$  LLM-Author edits and generates new prompts based on feedback from LLM-Reviewer. The prompt for LLM-Author consists of several components:
+
+a Input for editing: Current task instruction to be refined and training data;  
+b Instruction for editing: "We've provided pairs consisting of inputs, the teacher's correct answers, and the students' responses. Please review the incorrect responses from the stu
+
+Algorithm 1: Evoke  
+Return: The prompt with the highest task accuracy.  
+Require: Training set; Initial prompt for the target task (i.e., the one we want to refine).  
+// Initialization  
+LLM-Selector: Initialize data scoring instruction.  
+LLM-Author: Initialize prompt editing instruction.  
+LLM-Review: Initialize prompt reviewing instruction.  
+while  $t \leq T$  do  
+// LLM-Selector  
+Assign difficulty scores for each data point in the training set.  
+Select a training subset based on the difficulty level.  
+// LLM-Author  
+LLM-Author generates multiple prompts based on the training data and its own memory.  
+// LLM-Reviewer  
+LLM-Reviewer scores the quality of each generated prompt from LLM-Author based on its own memory.  
+Select top-n prompts based on the generated scores from LLM-Reviewer.  
+Get task accuracy for all prompts.  
+// Memory update  
+Memory of LLM-Author appends ( edits, scores).  
+Memory of LLM-Reviewer appends ( edits, prompts, task accuracy).
+
+dents and summarize key points that could be adjusted in the instruction to enhance student accuracy. Highlight major edits and present the updated task instruction."
+
+c Memory: prior history ( edits, scores).
+
+LLM-Author refines the instructions (prompts for the given task) by utilizing the training data and a memory component. We note that the memory consists of all prior (edit, score) pairs, where the score comes from LLM-Reviewer. This memory component enables LLM-Author to execute increasingly effective edits, drawing upon feedback from previous edits.
+
+$\diamond$  LLM-Reviewer scores the quality of prompts generated by LLM-Author. The input prompt for LLM-Reviewer consists of several components:
+
+a Input for scoring: problem description and current instruction from LLM-Author;  
+b Instruction for scoring: "Please rate the following instruction on a scale of 1 to 10, where 10 represents the highest level of clarity in problem description, execution steps, and a comprehensive explanation of the problem."  
+c Memory: prior ( edits, instructions, task accuracy).
+
+The instructions generated by LLM-Author are forwarded to LLM-Reviewer for evaluation. Based on the scores generated by LLM-Reviewer, only a subset of high-scoring candidates is selected to move on to the subsequent iteration. Through this iterative editing process between LLM-Author and LLM-Reviewer, LLM-Author can refine instructions in each iteration. Details of the algorithm can be found in Algorithm 1.
+
+To illustrate the effectiveness of Evoke, first three edits from the Movie Recommendation task in Big Bench are presented in Figure 3. To start with, the prompt contains the basic task instruction. Next, it extracts key factors considered in movie recommendation, such as the genre of each movie, the distance between the given movies and the movies the user has watched before, and the popularity of the movies. In the final step, a well-explained example is presented with a detailed explanation following aforementioned factors. In summary, Evoke successfully concludes the key components of movie recommendation, and curates a demonstration with detailed explanation.
+
+# 3.2 DATA SELECTION VIA LLM-SELECTOR
+
+In practice, we find that not all samples are equally important to model performance (see Figure 10). In particular, we find that even without prompt refinement, the LLM already knows how to solve
+
+# Step 1:
+
+Given user's interest in movies he watched previously: \{'watched\} Now given four different movies from A to D, please recommend one that might be the most interest of the user. Options: \{'list\}
+
+# Step 2:
+
+# {Prompt from Step 1}
+
+To help you make a good recommendation, consider the following factors:
+
+- The genre, theme, and tone of the movies.  
+- The similarity or difference between the movies and the ones the user watched before.  
+- The popularity, ratings, and reviews of the movies.
+
+# Step 3:
+
+# {Prompt from Step 2}
+
+Example: If the user watched The Godfather, The Godfather Part II, and Goodfellas, and the options are A) The Departed, B) Scarface, C) The Irishman, and D) Casino, a possible answer is:
+
+<Answer>A<Answer>
+
+{detailed explanations}
+
+Figure 3: Illustration of Prompt Editing for the first three steps in the Task of Movie Recommendation within Big Bench.
+
+some "easier" cases. Therefore, we only use "hard" samples in each refinement iteration. Specifically, we assign a third role besides an author a reviewer to the LLM: a data selector. The LLM-Selector evaluates the difficulty level (on a scale of 1 to 10) of each data point by assessing, based on the current task instruction, how challenging it is to derive the correct answer from the input. The input prompt for LLM-Selector consists of several components:
+
+a Input for evaluating difficulty level: current instruction and input-output pair;  
+b Instruction for evaluating difficulty level: "As an experienced teacher with insight into the various levels of difficulty of exam questions, please rate the following question on a scale of 1 to 10, considering factors such as conceptual understanding, application of knowledge, problem-solving skills, time required, clarity of language, and accessibility, where 1 denotes extremely easy and 10 denotes extremely difficult."
+
+Empirically, we can further improve effectiveness of Evoke by using such a data selection strategy.
+
+# 4 EXPERIMENTS
+
+We conduct extensive experiments to demonstrate the effectiveness of Evoke. We show that for any given task, the prompts generated by Evoke include clear definitions and well-structured task execution steps. Moreover, these prompts feature demonstrations accompanied by detailed explanations. In all experiments, we utilize the Azure OpenAI API service (GPT-4) for the involved LLMs.
+
+# 4.1 MAIN RESULTS
+
+Datasets. We perform a comprehensive evaluation on eight tasks from Instruction Induction (Honovich et al., 2022) and Big Bench Instruction Induction (BBII) (Zhou et al., 2022), including
+
+orthography starts with: Extract the words starting with a given letter from the input sentence.
+
+common concept: Find a common characteristic for the given objects.
+
+rhymes: Write a word that rhymes with the input word.
+
+movie recommendation: Recommend movies similar to the given list of movies.
+
+logical fallacy detection: Detect informal and formal logical fallacies.
+
+presuppositions as nli: Determine whether the first sentence entails or contradicts the second.
+
+winowhy: Evaluate the reasoning in answering Winograd Schema Challenge questions.
+
+epistemic reasoning: Determine whether one sentence entails the next.
+
+These tasks covers a wide range of natural language understanding, reasoning and inference tasks. For each task, we divide the dataset randomly into two sets,  $60\%$  of the data is allocated for training (prompt refinement) and the remaining  $40\%$  is for testing (prompt evaluation).
+
+Baselines. We compare our methods against two baselines: human curated prompts (Human) from Honovich et al. (2022); Suzgun et al. (2022) and automatic prompt engineer (APE) proposed in (Zhou et al., 2022). APE first deduces an initial prompt from input-output pairs, and subsequently employs LLMs to refine and generate new prompt candidates. However, prompts are simply paraphrased during the refinement process of APE, which largely resembles random searching in the space of all possible prompts.
+
+Main Results. Figure 4 demonstrates experimental results. We observe that Evoke outperforms all the baselines in all eight tasks. For example, on the challenging logical fallacy detection task from BBII, performance of Evoke is more than 80, while performance of both APE and Human are below 20. This is because Evoke is adept at conceptualizing the core definition of a task, decomposing a complex task into smaller subtasks, and curating relevant demonstrations accompanied by detailed explanations. To demonstrate the power of Evoke, we show the generated prompt for logical fallacy detection in Table 1. We see that the prompt begins with a clear task introduction and objective, followed by a fine-grained definition of logical fallacy. It then articulates the criteria for evaluation and the task steps to follow. Lastly, it provides a list of common logical fallacies, each accompanied by a detailed description. Additionally, a well-structured prompt for epistemic reasoning is presented in Table 2.
+
+![](images/253360105f7bcaa26b2ab63526c04622c083de31dbbd7c9a39dbf82deede1c3f.jpg)  
+Figure 4: Results on eight tasks from the Instruction Induction and the Big Bench Instruction Induction datasets. We report the average accuracy over three runs.
+
+![](images/5d04d8197df62b8e12b5812633611b2ebe804917fa9d0b76eadd9f8260392a57.jpg)
+
+# 4.2 TOWARDS ADVERSARIAL ROBUSTNESS
+
+Despite their superior performance, LLMs are not robust to adversarial attacks (Wang et al., 2023). For example, when asking GPT-4 whether "pretty" is a positive word, the model can output the correct answer. However, if we ask whether "pretty", a clear typo of "pretty", is a positive word, the LLM outputs an opposite answer. We show that Evoke can generate prompts which alert the LLM in paying attention to potential typos, and thus can improve model robustness.
+
+Datasets. We adopt two datasets: SST-2 (Socher et al., 2013) is a sentiment classification task, where we need to decide whether a movie review is positive or negative; and QQP (Wang et al., 2019) is a task where we need to determine whether two sentences are paraphrases of each other.
+
+To evaluate whether Evoke can improve LLMs' robustness, we add typos to the datasets. Specifically, we perform character-level adversarial attacks for each sam
+
+ple. In the attack, we change at most one character in each word, and we change at most 4 words in each sentence (Jones et al., 2020). In this way, the constructed adversarial texts are human-
+
+![](images/1706d7a791d3d54ba9b858e38ca18745017ce602e3d2fb7b2c691167ec1cf09c.jpg)  
+Figure 5: Results on clean and adversarially attacked SST2 and QQP datasets. We report the average accuracy over three runs. We note that RobEnc is only applied to the attacked data.
+
+Table 1: Prompt generated by Evoke for logical fallacy detection.  
+
+<table><tr><td>Prompt from Evoke</td><td>Breakdown</td></tr><tr><td>In this task, you are required to identify both informal and formal logical fallacies in the provided input statements. Your response should be a binary value: return 1 if the query is logically valid (i.e., free from any logical fallacies), and return 0 if the query is logically invalid (i.e., contains at least one logical fallacy).</td><td>Introduction and objective</td></tr><tr><td>A logical fallacy refers to an error in reasoning. Informal fallacies are often content-dependent, such as appealing to irrelevant authority or making hasty generalizations. Formal fallacies, on the other hand, are structural errors in reasoning that occur regardless of the content.</td><td>Definition of logical fallacy</td></tr><tr><td>It is crucial to consider the structure and the substance of the argument, evaluating whether the conclusions follow logically from the premises, and whether the premises and assumptions are factual and valid. Be cautious not to let personal beliefs interfere with your analysis.</td><td>Evaluation criteria</td></tr><tr><td>For each given pair, compare the input statement against the principles of logical reasoning, to determine whether it contains a logical fallacy or not. Ensure your answer reflects the presence or absence of logical fallacies, thus determining the logical validity or invalidity of the statement.</td><td>Task steps</td></tr><tr><td>Here are some common examples of logical fallacies: 
+- Ad Hominem: {details} 
+- Appeal to Nature: {details} 
+- Hasty Generalization: {details} 
+- Post Hoc: {details} 
+- False Cause: {details}</td><td>Common examples of logical fallacy</td></tr></table>
+
+Table 2: Prompt generated by Evoke for epistemic reasoning.  
+
+<table><tr><td>Prompt from Evoke</td><td>Breakdown</td></tr><tr><td>In this task, your goal is to determine whether the statement in the “Hypothesis” logically follows from the statement in the “Premise.” This is known as entailment. If the “Hypothesis” statement is a logical consequence of the “Premise” statement, then it is an entailment. If it is not, then it is a non-entailment.</td><td>Introduction and objective</td></tr><tr><td>-Make sure to carefully consider the relations and assumptions mentioned in both the “Premise” and the “Hypothesis” statements.-The entailment does not depend on the truth of the statements, but rather whether the logic in the “Hypothesis” follows from the “Premise”.-Pay close attention to the wording and structure of the sentences to analyze whether one entails the other.</td><td>Guidelines</td></tr><tr><td>Examples:EntailmentPremise: The sun rises in the east.Hypothesis: The sun rises.Explanation: The Hypothesis is a simplified version of the Premise and does not introduce any new information or contradictions, hence it&#x27;s an entailment.Non-entailmentPremise: Sarah believes that all cats are black.Hypothesis: All cats are black. Explanation: Even though the Hypothesis is expressed in the Premise, it&#x27;s tied to Sarah&#x27;s belief and not presented as a fact, hence it&#x27;s a non-entailment.</td><td>Examples</td></tr><tr><td>Now, review the provided pairs of statements. Determine if the Hypothesis logically follows from the Premise and respond with either entailment or non-entailment.</td><td>Task Execution</td></tr></table>
+
+interpretable and simulate real typos. As an example, one sample from SST-2 is "that's pure pr hype", and its corresponding adversarial (corrupted) sample after the attack is "tha'cs pure pr hyp". We evaluate performance of different prompting methods on the corrupted samples.
+
+Baselines. Besides APE and Evoke, we evaluate another model: RobEnc (Jones et al., 2020), which is a widely-used rule-based defense approach. RobEnc works as a clustering denoiser to cluster and
+
+# APE
+
+Provided with a user query, please conduct a sentiment analysis. In this analysis, a score of 0 represents negative sentiment, while a score of 1 represents positive sentiment. Ensure to carefully check for any typos before providing a response.
+
+# Evoke
+
+Now given a user query, please do a sentiment analysis where 0 represents negative sentiment and 1 represents positive sentiment. Please note that the input may contain typos. These are not intentional and do not reflect the user's actual sentiment. You should try to correct the typos or infer the intended meaning from the context before assigning a sentiment score.
+
+For example, `to merely badv trather than painfully awfl` could be corrected to `to merely bad rather than painfully awful' and scored as 0. Similarly, `l comelling mobion` could be corrected to `a compelling motion` and scored as 1.
+
+denoise potentially corrupted inputs into an encoding, and then the denoised encoding is fed to the subsequent model (e.g., GPT-4) for inference. RobEnc learns rule-based word cluster for denoising: for example, if the word "hallo" is clustered around the word "hello", then all the "hallo" in the input will be converted to "hello".
+
+Results. Figure 5 summarizes experimental results. We observe that Evoke significantly outperforms all the baselines in all the tasks. The performance gain is more significant for adversarially constructed datasets, e.g., Adversarial-SST2 and Adversarial-QQP. To understand this, we show the prompts generated by APE and Evoke in Figure 6. We see that although the prompt from APE provides a clear instruction regarding the given task and acknowledges the existence of typos, it does not provide clear guidelines on how to address the typo. On the other hand, the prompt from Evoke provides detailed explanations and actionable suggestions about defending against typos.
+
+# 4.3 TOWARDS FINE-GRAINED TASKS: NAMED ENTITY RECOGNITION
+
+Tasks in Figure 4 and Figure 5 are all sentence-level classification tasks, e.g., deciding whether a sentence is of positive or negative sentiment. In this section, we investigate whether Evoke can handle more fine-grained tasks, such as token-level named entity recognition (Schneider et al., 2020; Zuo et al., 2023).
+
+![](images/143c0000c6908177754375ba60d911724aa764505ba0b18e12acbd4d12215483.jpg)  
+Figure 7: Results of APE and Evoke on an in-house multi-lingual NER dataset.
+
+![](images/085c27da15087eee86540e3834d4c7f4c76c2699a94564a4409d961e58a6c1aa.jpg)  
+Figure 6: Prompts from APE and Evoke on adversarial attacked SST-2 task  
+Figure 8: Prompt from Evoke on the NER task.
+
+We collect multi-lingual in-house query data from a search engine, and for each token in the query, our goal is to assign the token to a pre-defined class (e.g., brand, location).
+
+We illustrate results of Evoke on the fine-grained NER task in Figure 7. We see that Evoke significantly outperforms APE on all the languages. We further show the prompt generated by Evoke in Figure 8. From the prompt, we see that Evoke is able to automatically generate examples and explanations about the task.
+
+![](images/469370551a259f473d04cb754cf76768f514cf7f98bb5ff4bc349b22282b76a6.jpg)
+
+![](images/f9be997215f0133ccb4de98d03c0d84149befe5e0547b814bf82b847b41371f3.jpg)  
+Figure 9: Correlation between scores generated by LLM-Reviewer and task accuracy.
+
+![](images/f53056f18fe49bfe19af6ce36e34b2e8b80f158df935910b21afaf5b2dd1a937.jpg)
+
+![](images/934db7ae2b25b9fc4896a56535317a9dacf54d7a17202672d28a12d6ca26270e.jpg)  
+Figure 10: Task accuracy over the number of iteration steps.
+
+# 4.4 ANALYSIS
+
+LLM-Reviewer can judge the quality of prompts. Recall that in Evoke, LLM-Reviewer scores all the prompts generated by LLM-Author. We empirically show that the scores can reflect the quality of the generated prompts. To examine the effectiveness of these scores, we illustrate the relationship between the scores and the task accuracy in Figure 9. In the experiments, we consider two tasks: Adversarial-SST2 and Common-Concept. From the results, we see that the scores can indeed reflect the final task accuracy. For example, for Common-Concept, we see that the task accuracy is about  $5\%$  when the prompt score is 6, and the task accuracy increases to about  $17\%$  when the prompt score increases to 7. A similar trend is also revealed on the Adversarial-SST2 task. We see that when the score is 7.5, the final task accuracy barely reaches  $75\%$ . And when the score increases to 8, the task accuracy increases to  $85\%$ .
+
+LLM-Author iteratively improves prompt generation. In Evoke, because LLM-Author takes the feedback from LLM-Reviewer into consideration, it can iteratively improve the generated prompt. We demonstrate this in Figure 10 (the left-most orange bars). From the results, we see that indeed the final task accuracy continues to increase when we increase the number of iteration steps. For example, on Adversarial-SST2, with one refinement iteration, the final task accuracy is about  $75\%$ . When we increase the number of refinement iterations to 3, we see that task accuracy significantly increases to above  $90\%$ .
+
+Effectiveness of LLM-Selector. Recall that in Evoke, we only consider the "hard" samples in each iteration. We demonstrate the effectiveness of such a strategy in Figure 10. We consider three settings: Hard is the strategy that we adopt in Evoke; Random is when we randomly select samples instead of selecting based on a score; and Easy is when we select the easy samples instead of the hard ones. From the results, we see that on both Common-Concept and Adversarial SST-2, Easy yields the worst performance, indicating that the hard samples are more helpful than the easy ones. Moreover, we observe that performance of Random is worse than Hard (i.e., Evoke), further implying the effectiveness of the proposed data selection strategy.
+
+# 5 CONCLUSION
+
+We propose Evoke, an author-reviewer framework for automatic prompt engineering. In Evoke, the same LLM serves two roles: as a reviewer it scores the quality of the prompt; and as an author it refines the prompt, taking the feedback of the reviewer into account. We further propose a data selection strategy, where we only expose the hard samples to the model. Extensive experiments show that Evoke outperforms existing automatic prompt engineering approaches.
+
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+
+# A INSTRUCTION INDUCTION
+
+In Evoke, we use the author-reviewer framework to modify a task-specific prompt. In the experiments, we use an off-the-shelf algorithm to generate the initial task-specific prompt. Table 3 demonstrates examples of using instruction induction (Honovich et al., 2022) for prompt initialization.
+
+Table 3: Three examples of instruction inferred from input-output pairs  
+
+<table><tr><td>Input</td><td>Output</td><td>Inferred Instruction</td></tr><tr><td>Departure</td><td>Arrival</td><td>Get antonym</td></tr><tr><td>I am Mike</td><td>Ich ben Mike</td><td>Translate to German</td></tr><tr><td>Build</td><td>Built</td><td>Get passive voice of the given verb</td></tr></table>
+
+# B PROMPTS OF LLM ROLES IN EVOKE
+
+# B.1 LLM-REVIEWER
+
+# Prompt for LLM-Reviewer
+
+As an experienced teacher, you are well-versed in discerning effective instruction that guides students toward correct answers. Please rate the following instruction on a scale of 1 to 10, where 10 represents the highest level of clarity in problem description, execution steps, and a comprehensive explanation of the problem.
+
+The task at hand is titled: {description}
+
+History that may help you: {memory}
+
+The instruction to be rated is as follows: {instruction}
+
+Kindly provide your rating below.
+
+# B.2 LLM-AUTHOR
+
+# Prompt for LLM-Author
+
+Task Instruction: {instruction}
+
+We've provided pairs consisting of inputs, the teacher's correct answers, and the students' responses. Please review the incorrect responses from the students and summarize key points that could be adjusted in the instruction to enhance student accuracy.
+
+Pairs: {pairs}
+
+History that may help you: {memory}
+
+To improve the outcome, please revise the task instruction. Highlight major edits and present the updated task instruction.
+
+# B.3 LLM-SELECTOR
+
+# Prompt for LLM-Selector
+
+As an experienced teacher with insight into the various levels of difficulty of exam questions, please rate the following question on a scale of 1 to 10, considering factors such as conceptual understanding, application of knowledge, problem-solving skills, time required, clarity of language, and accessibility, where 1 denotes extremely easy and 10 denotes extremely difficult.
+
+Task instruction: {instruction}
+
+Input: {input}
+
+Correct answer: {answer}
+
+# C GENERATED INSTRUCTIONS
+
+We include generated instructions from all tasks below.
+
+# C.1 ORTHOGRAPHY STARTS WITH
+
+# Prompt from Evoke
+
+Given an input sentence and a specified letter, identify the word or words starting with the given letter. If there are two or more words in a sequence starting with the specified letter, include all of them as a single answer. Ensure to present the word or group of words.
+
+Here are the steps to follow:  
+-Read the provided input sentence carefully.  
+-Identify the word or words that start with the specified letter.  
+-If there are consecutive words starting with the specified letter, group them together as one entity.  
+-For example, if the input is "I prefer eating apples." and the specified letter is [e], your answer should be eating.
+
+# C.2 COMMON CONCEPT
+
+# Prompt from Evoke
+
+Given a list, find the commonality between the inputs. The commonality should be a meaningful characteristic, property, or relation that applies to all the inputs, not just a superficial or coincidental feature.
+
+For example, can be used for repairs is a valid commonality for ['sewing', 'wrenches', 'glue', 'surgery'], but tools or skills for joining is too broad and vague, and contain the letter e is too trivial and irrelevant.
+
+# C.3 RHYMES
+
+# Prompt from Evoke
+
+For this task, you are required to find a word that rhymes with the given word. The word you provide should not be the same as the given word, and should be a real, correctly spelled word from the English language. A rhyming word is defined as a word that has the last syllable sounding identical to the last syllable of the given word. For example, if the given word is "hat", a word that rhymes with it is "cat".
+
+Here are the steps to complete this task:
+
+1. Read the given word carefully.  
+2. Think of a word that has the same ending sound as the given word.  
+3. Ensure that the word you thought of is a real word, is spelled correctly, and is not the same as the given word.  
+4. Write down the rhyming word next to the given word.  
+Now, please proceed with finding a word that rhymes with each of the following words.
+
+# C.4 MOVIE RECOMMENDATION
+
+# Prompt from Evoke
+
+Given user's interest in movies he watched previously: 'watched'. Now given four different movies from A to D, please recommend one that might be the most interest of the user.
+
+To help you make a good recommendation, consider the following factors:
+
+- The genre, theme, and tone of the movies. For example, if the user likes comedy, action, or drama.  
+- The similarity or difference between the movies and the ones the user watched before. For example, if the movies are part of a series, a remake, or a spin-off.  
+- The popularity, ratings, and reviews of the movies. For example, if the movies are critically acclaimed, award-winning, or have a large fan base.
+
+Use these factors to compare and contrast the movies and explain why you think one of them is the best choice for the user. Do not just pick a movie based on your personal preference or guesswork. Example: If the user watched The Godfather, The Godfather Part II, and Goodfellas, and the options are A) The Departed, B) Scarface, C) The Irishman, and D) Casino, a possible answer is: A
+
+The Departed is a crime thriller that has a similar genre, theme, and tone to the movies the user watched before. It is also a remake of a Hong Kong film called Infernal Affairs, which adds a twist to the familiar story of undercover agents and mobsters. The Departed is a highly popular and acclaimed movie that won four Oscars, including Best Picture and Best Director. It has a star-studded cast that includes Leonardo DiCaprio, Matt Damon, Jack Nicholson, and Mark Wahlberg. The user might enjoy the suspense, the plot twists, and the performances of the actors in this movie. Therefore, I recommend The Departed as the best option for the user.
+
+# C.5 LOGICAL FALLACY DETECTION
+
+# Prompt from Evoke
+
+In this task, you are required to identify both informal and formal logical fallacies in the provided input statements. Your response should be a binary value: return 1 if the query is logically valid (i.e., free from any logical fallacies), and return 0 if the query is logically invalid (i.e., contains at least one logical fallacy). A logical fallacy refers to an error in reasoning. Informal fallacies are often content-dependent, such as appealing to irrelevant authority or making hasty generalizations. Formal fallacies, on the other hand, are structural errors in reasoning that occur regardless of the content. It is crucial to consider the structure and the substance of the argument, evaluating whether the conclusions follow logically from the premises, and whether the premises and assumptions are factual and valid. Be cautious not to let personal beliefs interfere with your analysis. For each given pair, compare the input statement against the principles of logical reasoning, to determine whether it contains a logical fallacy or not. Ensure your answer reflects the presence or absence of logical fallacies, thus determining the logical validity or invalidity of the statement. Here are some common examples of logical fallacies:
+
+- Ad Hominem: Attacking the character of a person making an argument rather than the argument itself.  
+- Appeal to Nature: Claiming something is good because it's natural, or bad because it's unnatural.  
+- Hasty Generalization: Making a broad claim based on a small or unrepresentative sample size.  
+- Post Hoc: Assuming that because one event followed another, the first event caused the second event.  
+- False Cause: Assuming a false or misleading cause-and-effect relationship.
+
+# C.6 PRESUPPOSITIONS AS NLI
+
+# Prompt from Evoke
+
+Determine whether the first sentence entails, contradicts, or is neutral to the second sentence. The term "entailment" means that the information in the first sentence logically supports or leads to the conclusion presented in the second sentence. The term "contradiction" means that the information in the first sentence logically opposes or dis proves the information in the second sentence. The term "neutral" implies that the information in the first sentence neither supports nor opposes the information in the second sentence; they are unrelated or the relation between them is ambiguous.
+
+It's important to focus on the factual information provided rather than assumptions or external knowledge. Make sure to carefully read both sentences and analyze their logical relation based only on the given text.
+
+Entailment: The information in the first sentence supports the conclusion in the second sentence.
+
+Contradiction: The information in the first sentence opposes or disproves the information in the second sentence.
+
+Neutral: The information in the first sentence neither supports nor opposes the information in the second sentence, or the relation between them is ambiguous.
+
+For each pair, please provide the correct judgment between entailment, contradiction, and neutral, based only on the provided text. Please avoid assumptions and focus solely on the text provided.
+
+# C.7 WINOWHY
+
+# Prompt from Evoke
+
+In the given text, you are required to evaluate the reasoning provided concerning the identification of the antecedent of a pronoun in a sentence. The antecedent is the noun that the pronoun is referring to. Carefully examine the reasoning to determine if it accurately identifies the antecedent based solely on the information presented within the sentence itself. Here are the steps you should follow: Read the sentence and the reasoning provided thoroughly.
+
+-Assess whether the reasoning accurately identifies the antecedent of the pronoun based solely on the provided text. Avoid making assumptions or using external knowledge.
+
+-If the reasoning correctly identifies the antecedent of the pronoun, based on the information given in the sentence.
+
+-If the reasoning fails to accurately identify the antecedent of the pronoun or relies on assumptions or external information.
+
+Remember,
+
+Your evaluation should strictly be based on the information provided in the text.
+
+Your goal is to assess the accuracy of the reasoning in identifying the antecedent of the pronoun.
+
+# C.8 EPISTEMIC REASONING
+
+# Prompt from Evoke
+
+In this task, your goal is to determine whether the statement in the "Hypothesis" logically follows from the statement in the "Premise." This is known as entailment. If the "Hypothesis" statement is a logical consequence of the "Premise" statement, then it is an entailment. If it is not, then it is a non-entailment.
+
+-Make sure to carefully consider the relations and assumptions mentioned in both the "Premise" and the "Hypothesis" statements.
+
+- The entailment does not depend on the truth of the statements, but rather whether the logic in the "Hypothesis" follows from the "Premise".
+
+-Pay close attention to the wording and structure of the sentences to analyze whether one entails the other.
+
+Examples:
+
+Entailment
+
+Premise: The sun rises in the east. Hypothesis: The sun rises.
+
+Explanation: The Hypothesis is a simplified version of the Premise and does not introduce any new information or contradictions, hence it's an entailment.
+
+Non-entailment
+
+Premise: Sarah believes that all cats are black.
+
+Hypothesis: All cats are black. Explanation: Even though the Hypothesis is expressed in the Premise, it's tied to Sarah's belief and not presented as a fact, hence it's a non-entailment.
+
+Now, review the provided pairs of statements. Determine if the Hypothesis logically follows from the Premise and respond with either entailment or non-entailment.
+
+# C.9 ADVERSARIAL SST2
+
+# Prompt from Evoke
+
+Now given a user query, please do a sentiment analysis where 0 represents negative sentiment and 1 represents positive sentiment. Please note that the input may contain typos. These are not intentional and do not reflect the user's actual sentiment. You should try to correct the typos or infer the intended meaning from the context before assigning a sentiment score. For example, 'to merely badly trather than painfully awfl' could be corrected to 'to merely bad rather than painfully awful' and scored as 0. Similarly, '1 comelling mobion' could be corrected to 'a compelling motion' and scored as 1. Sentiment analysis.
+
+# C.10 ADVERSARIAL QQP
+
+# Prompt from Evoke
+
+You will be given a pair of questions and asked to determine whether they are paraphrases of each other. Paraphrases are questions that have the same meaning or ask about the same information, even if they use different words or structures. Please answer with a binary value of 1 if the questions are paraphrases, or 0 if they are not. Please pay close attention to typos, spelling, grammar, and punctuation before answering, as they may affect the meaning of the questions. If you are not sure whether the questions are paraphrases or not, you can use some strategies to help you decide, such as:
+
+- Compare the keywords and topics of the questions. Do they match or relate to each other? - Rewrite one question in a different way and see if it still conveys the same message as the other question. - Think about the context and purpose of the questions. Are they asking for the same type of information or response?
+
+For example, the questions What is the capital of France? and Which city is the seat of the French government? are paraphrases, because they both ask about the same fact and can be answered with the same word (Paris). However, the questions How do you play the guitar? and What are some guitar chords? are not paraphrases, because they ask for different kinds of information and have different levels of specificity.
+
+# D ADDITIONAL EXPERIMENTS
+
+# D.1 INITIAL PROMPT
+
+The influence of initial prompt quality on the efficacy of Evoke was an important aspect of our investigation. To thoroughly examine this, we conducted additional experiments using two alternative initial prompts: a task description and an empty string. As illustrated in Table below, Evoke's performance showed relative insensitivity to the quality of the initial prompt. These results, further underscore the robustness of our approach.
+
+![](images/5ab9c5b44854bab0eca0a4c804a104806306df306fdc681518a18adaf3c9c918.jpg)  
+Figure 11: Comparison of different prompt initialization methods: The blue bar indicates that, on average, using a task description as the initialization prompt is more effective, while the orange bar suggests that an empty string performs on par with instruction induction on average.
+
+# D.2 OPEN SOURCE MODELS
+
+We expanded our experiment scope to include open source models Mistral-7B-Openorca and Llama2-70B-Instruct. Our findings reveal notable performance enhancements with these models, affirming the versatility and robustness of our Evoke framework beyond GPT-4.
+
+![](images/ceef2693da5c34b2fbb4d82f10816950bde6fc8c2d0683a020a3aaac3e7d200f.jpg)  
+Figure 12: Comparison of test accuracy on APE and Evoke for Mistral-7B-Openorca and Llama2-70B-Instruct: the Evoke prompt outperforms the baseline for both open source models.

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+# EXPECTED FLOW NETWORKS IN STOCHASTIC ENVIRONMENTS AND TWO-PLAYER ZERO-SUM GAMES
+
+Marco Jiralerspong*, Bilun Sun*, Danilo Vucetic*, Tianyu Zhang,
+
+Yoshua Bengio $^{\diamond}$ , Gauthier Gidel $^{\dagger}$ , Nikolay Malkin
+
+Mila - Quebec AI Institute, Université de Montréal
+
+marco.jiralerspong,bilun.sun,danilo.vucetic,tianyu.zhang, @mila.quebec
+
+# ABSTRACT
+
+Generative flow networks (GFlowNets) are sequential sampling models trained to match a given distribution. GFlowNets have been successfully applied to various structured object generation tasks, sampling a diverse set of high-reward objects quickly. We propose expected flow networks (EFlowNets), which extend GFlowNets to stochastic environments. We show that EFlowNets outperform other GFlowNet formulations in stochastic tasks such as protein design. We then extend the concept of EFlowNets to adversarial environments, proposing adversarial flow networks (AFlowNets) for two-player zero-sum games. We show that AFlowNets learn to find above  $80\%$  of optimal moves in Connect-4 via self-play and outperform AlphaZero in tournaments.
+
+Code: https://github.com/GFNOrg/AdversarialFlowNetworks.
+
+# 1 INTRODUCTION
+
+Generative flow networks (GFlowNets; Bengio et al., 2021; 2023; Lahlou et al., 2023) are a unifying algorithmic framework for training stochastic policies in Markov decision processes (MDPs; Sutton & Barto, 2018) to sample from a given distribution over terminal states. GFlowNets have been used as an efficient alternative to Monte Carlo methods for amortized sampling in applications that require finding diverse high-reward samples (Jain et al., 2022; Zhang et al., 2022; 2023a; Li et al., 2023, see §5). This paper revisits and extends the view of GFlowNets as diversity-seeking learners in MDPs, enabling the training of robust agents in stochastic environments and two-player adversarial games.
+
+The main application of GFlowNets to date has been generation of structured objects  $x \in \mathcal{X}$  – where  $\mathcal{X}$  is the set of terminal states of an episodic MDP – given a reward function  $R: \mathcal{X} \to \mathbb{R}_{>0}$  interpreted as an unnormalized density. The generation of  $x$  follows a trajectory  $s_0 \to s_1 \to \dots \to s_n = x$ , representing iterative construction or refinement (e.g., building a graph by adding one edge at a time). These settings assume that applying an action to a partially constructed object  $s_i$  deterministically yields the object  $s_{i+1}$ . It is natural to attempt to generalize the GFlowNet framework to stochastic environments, in which a given sequence of actions does not always produce the same final state.
+
+However, the common notions of GFlowNets must be modified to recover a useful stochastic generalization. An existing attempt (Pan et al., 2023) proposes to treat stochastic transitions in the environment as actions of the GFlowNet drawn from a fixed policy. One of the starting points for this paper is that this previous formulation sacrifices several desirable theoretical properties, inducing poor sampling performance in many practical settings. We propose an alternative notion of expected flow networks (EFlowNets), which provably do not suffer from the previous limitations (Fig. 1a).
+
+As EFlowNets can perform inference via stochastic control in a fixed environment, they can be used to learn robust strategies against a stochastic opponent in a two-player game. We further define an adversarial flow network (AFlowNet) as a collection of EFlowNet players, each with its own reward function, taking actions in a shared environment. We show the existence and uniqueness of a joint optimum of the players' respective objectives. This stable point can be characterized via a probabilistic notion of game-theoretic equilibrium. We perform additional theoretical analysis and develop efficient training objectives for two-player zero-sum games.
+
+Figure 1: We extend GFlowNets to stochastic environments (a) and games (b).  
+![](images/8a11a2eb744e419dac636870388dd653438ab5f3707b7d55750ecda4cdb0fcde.jpg)  
+(a) Left: A stochastic MDP with four terminal states. The first (black) transition is chosen by the agent from a learnable policy with parameter  $p$ , and the second (red) action is sampled from the environment's fixed transition distribution. Middle: Our proposed EFlowNets (§3.1) integrate over the uncertainty of the environment's transitions. We derive expected detailed balance training objectives to amortize this integration. Right: A prior approach to GFlowNets in stochastic environments (Pan et al., 2023) proposes to train the agent to sample from the reward distribution by treating the environment transitions as actions sampled from an immutable policy, leading to unsatisfiable constraints.  
+(b) Optimizing two EFlowNets that play against each other yields robust game-playing agents. Here, a branch-adjusted AFlowNet (§3.3) for a two-player zero-sum game is shown: each player receives a reward of  $e^{\lambda}$  for a win, 1 for a draw, and  $e^{-\lambda}$  for a loss, adjusted appropriately by branching factors.
+
+The contributions of this work are as follows:
+
+(1) We propose expected flow networks (EFlowNets, §3.1), a class of sequential sampling models and learning objectives generalizing GFlowNets on tree-structured state spaces. We demonstrate theoretically and experimentally the advantages of the EFlowNet formulation over past attempts to generalize GFlowNets to stochastic environments (§3.1, §4.1).  
+(2) We define adversarial flow networks (AFlowNets, §3.2) for two-player games and prove the existence and uniqueness of equilibrium solutions. In Proposition 4 we exploit the zero-sum structure of two-player games to get a novel trajectory balance (TB) loss for AFlowNets. We believe this new loss is a major algorithmic novelty. We conduct extensive experiments on two-player games showing that AFlowNets are able to learn robust policies via self-play (§4.2).  
+(3) We connect GFlowNets, EFlowNets, and AFlowNets to models of imperfect agents in psychology and behavioral economics (§A).
+
+# 2 BACKGROUND
+
+# 2.1 GFLOWNETS IN TREE-SHAPED ENVIRONMENTS
+
+We review GFlowNets in deterministic environments, mainly following the conventions from Malkin et al. (2022). (Notably, we assume that terminating states have no children and rewards are strictly positive.) In keeping with past work, we use the language of directed acyclic graphs (DAGs), rather than the equivalent language of deterministic MDPs. We state all results only for tree-structured state spaces, but note that they are special cases of results for general DAGs.
+
+Setting. Let  $G = (S, \mathcal{A})$  be a directed tree, with finite sets of vertices (states)  $S$  and edges (actions)  $\mathcal{A} \subset S \times S$ , oriented away from the root (initial state)  $s_0 \in S$ . Denote by  $\mathrm{Ch}(s)$  the set of children of a state  $s$  and by  $\mathrm{Pa}(s)$  the parent of  $s$ , which exists unless  $s = s_0$ . The set of childless (terminal) states is denoted  $X$ . A complete trajectory is a sequence  $\tau = (s_0 \to s_1 \to \dots \to s_n)$ , where  $s_n \in X$  and each  $s_i \to s_{i+1}$  is an action. The set of complete trajectories is denoted  $\mathcal{T}$ .
+
+A (forward) policy is a collection of distributions  $P_F(\cdot \mid s)$  over  $\mathrm{Ch}(s)$  for every  $s \in S \setminus X$ . A policy induces a distribution over  $\mathcal{T}$ , with  $P_F(s_0 \to s_1 \to \dots \to s_n) = \prod_{i=1}^{n} P_F(s_i \mid s_{i-1})$ . This in turn induces a terminating distribution  $P_F^\top$  over  $X$ , defined as the marginal distribution over the final state of a trajectory  $\tau \sim P_F(\tau)$ . One can sample  $x \sim P_F^\top(x)$  by running a chain starting at  $s_0$  and transitioning according to  $P_F$  until a terminal state  $x$  is reached.
+
+A reward function is a function  $R: \mathcal{X} \to \mathbb{R}_{>0}$ . A policy  $P_F$  is said to sample proportionally to the reward  $R$  if  $P_F^\top(x) \propto R(x)$ , i.e.,  $P_F^\top(x) = R(x)/Z$  for all  $x \in \mathcal{X}$ , where  $Z = \sum_{x \in \mathcal{X}} R(x)$ . Given a
+
+reward function  $R$ , GFlowNet algorithms aim to produce a policy  $P_F$  that samples proportionally to  $R$ . This is, in essence, a generative modeling problem, but the setting is close to maximum-entropy reinforcement learning (RL; Haarnoja et al., 2017): one is not given samples from the target density, as in typical generative modeling settings, but must rather explore the reward landscape through sequential sampling.
+
+FM and DB objectives. We review the two GFlowNet objectives of flow matching (FM; Bengio et al., 2021) and detailed balance (DB; Bengio et al., 2023) in tree-structured state spaces.
+
+The FM objective optimizes a function  $F: S \to \mathbb{R}_{>0}$ , called the state flow. The objective enforces a pair of constraints, which in tree-structured DAGs are
+
+$$
+F (s) = \sum_ {s ^ {\prime} \in \operatorname {C h} (s)} F \left(s ^ {\prime}\right) \forall s \in \mathcal {S} \backslash \mathcal {X} \quad \text {a n d} \quad F (x) = R (x) \forall x \in \mathcal {X}. \tag {1}
+$$
+
+Any edge flow  $F$  induces a policy  $P_F$ , defined by  $P_F(s' \mid s) = \frac{F(s')}{F(s)}$  for all  $(s, s') \in \mathcal{A}$ . If the flow satisfies the FM constraints (1), then it holds that  $P_F$  samples proportionally to the reward  $R$ .
+
+The DB objective avoids the explicit summation over children in (1) and jointly optimizes both  $F$  and the policy  $P_F$ , replacing the first constraint by
+
+$$
+F (s) P _ {F} \left(s ^ {\prime} \mid s\right) = F \left(s ^ {\prime}\right) \forall (s, s ^ {\prime}) \in \mathcal {A}. \tag {2}
+$$
+
+This constraint implies the first constraint of (1), as  $P_F$  sums to 1 over  $s' \in \mathrm{Ch}(s)$ .
+
+The function  $F$  is typically parametrized as a neural network  $F_{\theta}$  with parameters  $\theta$ , and (if using the DB objective) the policy  $P_F$  as a network producing logits of  $P_F(s' \mid s; \theta)$  given  $s$  as input. The parameters  $\theta$  are optimized to minimize some discrepancy between the left and right sides of (1) or (2). A typical choice is the squared log-ratio; for example, the FM objective at a state  $s$  is
+
+$$
+\mathcal {L} _ {\mathrm {F M}} (s) = \left(\log F _ {\theta} (s) - \log \sum_ {s ^ {\prime} \in \operatorname {C h} (s)} F _ {\theta} \left(s ^ {\prime}\right)\right) ^ {2}. \tag {3}
+$$
+
+The choice of states  $s$  at which this objective is evaluated and optimized is made by a training policy  $\pi$ . For example,  $\pi$  could select the states  $s$  seen in trajectories sampled from  $P_F$  (on-policy training), but could also use off-policy exploration techniques, such as tempering, replay buffers, or Thompson sampling (Rector-Brooks et al., 2023). Because the objective can be simultaneously minimized to zero at all  $s$  for a sufficiently expressive  $F_{\theta}$ , the global optimum of the objective is independent of the choice of training policy  $\pi$ , as long as  $\pi$  has full support. This capacity for off-policy training without differentiating through the sampling procedure is a key advantage of GFlowNets over on-policy RL algorithms and over other hierarchical variational methods (Malkin et al., 2023).
+
+Connections with RL. In the case of tree-structured state spaces, GFlowNets are closely connected to entropy-regularized RL methods (soft Q-learning; Haarnoja et al., 2017): identifying the log-flow function with a value function, the FM/DB objectives are analogous to temporal difference learning (Sutton & Barto, 2018) and TB, along with its variant SubTB (Madan et al., 2023), to path consistency learning (Nachum et al., 2017). As a diversity-seeking agent, a GFlowNet can also be understood as way to train a quantal response agent; see §A for more discussion.
+
+How restrictive is the tree structure? Any environment that has a non-tree DAG structure – i.e., where multiple trajectories may lead to the same state – can be converted to a tree-structured environment by augmenting each state with the history (the trajectory followed to reach the state). This implicitly multiplies the reward of each terminal state by the number of trajectories that lead to it (while keeping the optimal policy independent of the history). This alternative way to reward the state may be desired in some applications (e.g., zero-sum games) for which the path to the solution matters as much as the outcome.
+
+# 2.2 PAST APPROACHES TO GFLOWNETS IN STOCHASTIC ENVIRONMENTS
+
+Pan et al. (2023) propose a generalization of GFlowNets to stochastic environments (i.e., where the state and choice of action nondeterministically yield the subsequent state), following an approach described in Bengio et al. (2023). We now review their formulation, which we refer to as 'stochastic GFlowNets', restating it in a suitable language to motivate our method.
+
+In stochastic environments, every state  $s$  is associated with a set of possible actions  $\mathcal{A}_s$ , and the environment provides a stochastic transition function - a distribution  $P_{\mathrm{env}}(s' \mid s, a)$ , understood as the likelihood of arriving in state  $s'$  when taking action  $a$  at state  $s$ . In stochastic GFlowNets, the state space  $\mathcal{S}$  is augmented with a collection of hypothetical states  $(s, a)$  for  $s \in S \setminus \mathcal{X}$  and  $a \in \mathcal{A}_s$ . The augmented DAG  $G$  contains two kinds of edges:
+
+- Edges  $s \rightarrow (s, a)$  for each  $s \in S$  and  $a \in \mathcal{A}_s$ , which we call agent edges;  
+- Edges  $(s, a) \to s'$  for each hypothetical  $s'$  in the support of  $P_{\mathrm{env}}(\cdot \mid s, a)$ ,  $a \in \mathcal{A}_s$ , and  $s' \in \mathrm{Ch}(s)$ , which we call environment edges.
+
+Stochastic GFlowNets directly apply the training algorithms applicable to deterministic GFlowNets (e.g., DB) to the augmented DAG  $G$ , with the only modification being that the forward policy  $P_F$  is free to be learned only on agent edges, while on environment edges it is fixed to the transition function. Formally, for all  $s \in S \setminus X$  and  $a \in \mathcal{A}_s$ , one learns  $P_F((s, a) \mid s)$  (denoted  $P_F(a \mid s)$  for short), while  $P_F(s' \mid (s, a))$  is fixed to  $P_{\mathrm{env}}(s' \mid s, a)$ .
+
+The environment policy  $P_{\mathrm{env}}$ , which appears in the loss, may be assumed to be known, but may also be approximated using a neural network trained by a maximum-likelihood objective on observed environment transitions, jointly with the agent policy.
+
+Violated desiderata in stochastic GFlowNets. By construction, if a stochastic GFlowNet satisfies the DB constraints, then the policy  $P_F$  samples proportionally to the reward  $R$ . In this way, stochastic GFlowNets are a minimal modification of GFlowNets that can function in stochastic environments. However, there exist stochastic environments and reward functions for which no stochastic GFlowNet policy  $P_F(a \mid s)$  can satisfy the constraints (Fig. 1a). Two consequences of this are the impossibility of minimizing the loss to zero for all transitions, even for a perfectly expressive policy model, and the resulting dependence of the global optimum on the choice of training policy  $\pi$ . Thus stochastic GFlowNets satisfy D0, but not D1 (as noted by Bengio et al. (2023)) and D2 below.
+
+The generalization of GFlowNet constraints and objectives to stochastic environments that we propose satisfies the following desiderata:
+
+D0. If the environment's transition function  $P_{\mathrm{env}}$  is deterministic, one should recover deterministic GFlowNet constraints and objectives.  
+D1. Satisfiability: A perfectly expressive model should be able to minimize the generalized FM/DB losses to 0 for all states/actions in the DAG simultaneously. Consequently, the set of global optima of the loss should not depend on the choice of full-support training policy.  
+D2. Uniqueness: If  $G$  is a tree, then the global optimum of the loss should be unique.  
+D3. Equilibrium: In a game where two GFlowNet agents alternate actions, there should be a unique pair of policies for the two players such that each policy is optimal for its respective loss.
+
+As noted in §2.1, deterministic GFlowNets satisfy D1 and D2. D0 is a common-sense property, as deterministic environments are special cases of stochastic environments. D1 (satisfiability) is essential for off-policy training, while D2 (uniqueness) is desirable in game-playing agents. The meaning of D3 will be detailed in §3.2.
+
+# 3 METHOD:EXPECTED AND ADVERSARIAL FLOW NETWORKS
+
+# 3.1 EXPECTED FLOW NETWORKS
+
+In this section, we define expected flow networks (EFlowNets) on tree-structured spaces, which encompasses the problems we study, in particular, two-player games with memory. We then show that EFlowNets satisfy the desiderata D0-D2 above.
+
+Expected flow networks (EFlowNets) assume the following are given:
+
+- A tree  $G = (\mathcal{S}, \mathcal{A})$ , with initial state  $s_0$  and set of terminal states  $\mathcal{X}$ , and a reward function  $R: \mathcal{X} \to \mathbb{R}_{>0}$ .  
+- A partition of the nonterminal states into two disjoint sets,  $S \setminus X = S_{\text{agent}} \sqcup S_{\text{env}}$ , called the agent states and environment states, respectively.  
+- A distribution  $P_{\mathrm{env}}(\cdot \mid s)$  over the children of every environment state  $s \in S_{\mathrm{env}}$ .
+
+Observe that if  $S_{\mathrm{env}} = \emptyset$ , then the input data for an EFlowNet is the same as the input data for a GFlowNet on a tree-structured space. This setting also generalizes that of stochastic GFlowNets
+
+in §2.2, where all transitions link agent states  $s$  to environment states  $(s,a)$  - called 'hypothetical states' by Pan et al. (2023) - or vice versa.
+
+An agent policy is a collection of distributions  $P_{\mathrm{agent}}(\cdot \mid s)$  over the children of every agent state  $s \in S_{\mathrm{agent}}$ . Together,  $P_{\mathrm{agent}}$  and  $P_{\mathrm{env}}$  determine a forward policy on  $G$ . We define the expected detailed balance (EDB) constraints relating  $P_{\mathrm{agent}}$ ,  $P_{\mathrm{env}}$ , and a state flow function  $F: S \to \mathbb{R}_{>0}$ :
+
+$$
+F (s) P _ {\text {a g e n t}} \left(s ^ {\prime} \mid s\right) = F \left(s ^ {\prime}\right) \quad \forall s \in \mathcal {S} _ {\text {a g e n t}}, s ^ {\prime} \in \operatorname {C h} (s), \tag {4}
+$$
+
+$$
+F (s) = \mathbb {E} _ {s ^ {\prime} \sim P _ {\mathrm {e n v}} \left(s ^ {\prime} \mid s\right)} F \left(s ^ {\prime}\right) \quad \forall s \in S _ {\text {e n v}}, \tag {5}
+$$
+
+$$
+F (x) = R (x) \quad \forall x \in X. \tag {6}
+$$
+
+These constraints satisfy the desiderata D0-D2, as summarized in the following proposition.
+
+Proposition 1. There exists a unique pair of state flow function  $F$  and agent policy  $P_{\mathrm{agent}}$  satisfying constraints (4), (5), and (6). If  $S_{\mathrm{env}} = \emptyset$ , then this pair satisfies the detailed balance constraints (2).
+
+EFlowNets marginalize over the uncertainty of the environment's transitions: they aim to sample each action in proportion to the expected total reward available if the action is taken (see Prop. 5). A connection between EFlowNets and Luce quantal response agents is made in  $\S \mathrm{A}$
+
+Training EFlowNets: From constraints to losses. Just as in deterministic environments, when training EFlowNets, we parametrize the state flow and agent policy as neural networks  $F_{\theta}$  and  $P_{\mathrm{agent}}(\cdot \mid \cdot ;\theta)$ . The EDB constraints can be turned into squared log-ratio losses in the same manner that the FM constraint (1) is converted into the loss (3) and optimized by gradient descent.
+
+In problems where the number of environment transitions is large and computing the expectation on the right side of (5) is costly, it may be replaced by an alternative constraint by introducing a distribution  $Q(s' \mid s) = \frac{F(s') F_{\mathrm{env}}(s'|s)}{F(s)}$ . This quantity sums to 1 over the  $s'$  if and only if (5) is satisfied. Thus (5) is equivalent to the following constraint on  $Q$ :
+
+$$
+F (s) Q \left(s ^ {\prime} \mid s\right) = F \left(s ^ {\prime}\right) P _ {\text {e n v}} \left(s ^ {\prime} \mid s\right). \tag {7}
+$$
+
+Enforcing this constraint requires learning an additional distribution  $Q(s' \mid s; \theta)$ , but does not require summation over children. The conversion from (5) to (7) resembles that from FM (1) to DB (2).
+
+Just like deterministic GFlowNets, the globally optimal agent policy in an EFlowNet is unique and does not depend on the distribution of states at which the objectives are optimized, as long as it has full support. However, the choice of training policy can be an important hyperparameter that can affect the rate of convergence and the local minimum reached in a function approximation setting. We describe the choices we make in the experiment sections below.
+
+Just like in stochastic GFlowNets (§2.2), the environment policy  $P_{\mathrm{env}}$  can be either assumed to be known or learned, jointly with the policy, from observations of the environment's transitions.
+
+# 3.2 ADVERSARIAL FLOW NETWORKS
+
+We now consider the application of EFlowNets to multiagent settings. Although our experiments are in the domain of two-player games, we define adversarial flow networks (AFlowNets) in their full generality, with  $n$  agents. AFlowNets with  $n$  agents, or players, depend on the following information:
+
+- A tree  $G = (\mathcal{S}, \mathcal{A})$ , with initial state  $s_0$  and set of terminal states  $\mathcal{X}$ , and a collection of reward functions  $R_1, \ldots, R_n: \mathcal{X} \to \mathbb{R}_{>0}$ .  
+- A partition of the nonterminal states into disjoint sets,  $S \setminus \mathcal{X} = S_1 \sqcup \dots \sqcup S_n$ .
+
+This data defines a fully observed sequential game, where  $s \in S_i$  means that player  $i$  is to play at  $s$ . An agent policy for player  $i$  is a collection of distributions  $P_{i}(\cdot \mid s)$  over  $\mathrm{Ch}(s)$  for every  $s \in S_i$ .
+
+The input data for an AFlowNet also defines a collection of EFlowNets, one for each player  $i$ . The EFlowNet  $\mathcal{E}_i$  for player  $i$  has the same underlying graph  $G$ , with  $S_{\mathrm{agent}} = S_i$  and  $S_{\mathrm{env}} = \bigsqcup_{j \neq i} S_j = S \setminus (\mathcal{X} \cup S_i)$ , and reward function  $R_i$ . That is, each player is viewed as an agent in an EFlowNet whose 'environment' is given by the other players' policies. (We also remark that the case  $n = 1$  recovers a regular (deterministic) GFlowNet.)
+
+The policy  $P_{i}$  of player  $i$  can be optimized using the EFlowNet training objective given fixed values of the other players' policies  $P_{j}$  ( $j \neq i$ ), and by Proposition 1, there is a unique global optimum for  $P_{i}$ . However, remarkably, there exists a unique collection of policies  $P_{1}, \ldots, P_{n}$  such that each  $P_{i}$  that jointly satisfy the EFlowNet constraints for each player.
+
+Proposition 2. There exist unique agent policies  $P_{1},\ldots ,P_{n}$  and state flow functions  $F_{1},\ldots ,F_{n}: S\to \mathbb{R}_{>0}$  such that  $P_{i}$  and  $F_{i}$  satisfy the EDB constraints with respect to the EFlowNet  $\mathcal{E}_i$  for all  $i$ .
+
+We also have a characterization of the joint optimum in the case of two-agent AFlowNets:
+
+Proposition 3. Suppose that in a 2-player AFlowNet, the agent policies  $P_{1}, P_{2}$  and state flow functions  $F_{1}, F_{2}$  are jointly optimal in the sense of Prop. 2. Then the function  $F(s) = F_{1}(s)F_{2}(s)$  is a flow on  $G$ , i.e., satisfies the FM constraint (1), with respect to the reward  $R(x) = R_{1}(x)R_{2}(x)$ .
+
+# 3.3 BRANCH-ADJUSTED AFLOWNETS FOR TWO-PLAYER ZERO-SUM GAMES.
+
+An important application of AFlowNets is two-player zero-sum games. We will now describe a way to turn the game outcomes into rewards that allows a simpler and more efficient training objective.
+
+Specifically, we consider a two-player, complete-information game with tree-shaped state space  $G$ , in which player 1 moves first (i.e.,  $s_0 \in S_1$ ) and the players alternate moves, i.e., every complete trajectory  $s_0 \to s_1 \to \dots \to s_n = x$  has  $s_i \in S_1$  if and only if  $i$  is even. We assume that the game ends in a win for player 1, a win for player 2, or a draw. A naive way to define the rewards for players 1 and 2 is the following, which ensures the log-rewards sum to zero at every terminal state:
+
+$$
+R _ {i} ^ {\circ} (x) = \left\{ \begin{array}{l l} e ^ {\lambda} & \text {i f p l a y e r i w i n s ,} \\ 1 & \text {i f t h e g a m e e n d s i n a d r a w ,} \\ e ^ {- \lambda} & \text {i f p l a y e r i l o s e s .} \end{array} \right. \tag {8}
+$$
+
+However, we find that AFlowNets trained with this reward often exhibit suboptimal behaviour in complex games: the agent may avoid a move that leads directly to a winning terminal state (high reward) in favour of a move with a large downstream subtree. We therefore define an alternative reward function that favours shorter winning trajectories. If  $s_0 \to s_1 \to \dots \to s_n = x$  is the trajectory leading to  $x$ , then the branch-adjusted reward for player  $i$  is defined as
+
+$$
+R _ {i} (x) = \frac {R _ {i} ^ {\circ} (x)}{B _ {i} (x)}, \quad B _ {i} (x) := \prod_ {k: s _ {k} \in S _ {k}} | \operatorname {C h} \left(s _ {k}\right) |. \tag {9}
+$$
+
+That is,  $B_{i}(x)$  is the product of the branching factors (numbers of children) of the states  $s_k$  on the trajectory at which player  $i$  is to move. Besides delivering a higher reward for less selective trajectories and being empirically essential for good game-playing performance, this correction is necessary to derive the simplified objective below, which critically uses  $R_1^\circ(x)R_2^\circ(x) = 1$ .
+
+A 'trajectory balance' for branch-adjusted AFlowNets. A limitation of objectives such as FM, DB, and their EFlowNet and AFlowNet generalizations are their slow credit assignment (propagation of a reward signal) over long trajectories, which results from these losses being local to a state or transition. This limitation motivated the trajectory balance (TB) loss for GFlowNets (Malkin et al., 2022), which delivers a gradient update to all policy probabilities along a complete trajectory.
+
+While the GFlowNet TB objective does not appear to generalize to expected flow networks, we derive an objective of a TB-like flavour for branch-adjusted AFlowNets.
+
+Proposition 4. In a 2-player AFlowNet with alternating moves satisfying  $R_1^\circ(x)R_2^\circ(x) = 1$ :
+
+(a) Suppose that the agent policies  $P_{1}, P_{2}$  and state flow functions  $F_{1}, F_{2}$  are jointly optimal in the sense of Prop. 2. Then there exists a scalar  $Z$ , independent of  $x$ , such that for every complete trajectory  $s_{0} \rightarrow s_{1} \rightarrow \dots \rightarrow s_{n} = x$ ,
+
+$$
+Z \prod_ {i: s _ {i} \in S _ {1}} P _ {1} \left(s _ {i + 1} \mid s _ {i}\right) = R _ {1} (x) B _ {2} (x) \prod_ {i: s _ {i} \in S _ {2}} P _ {2} \left(s _ {i + 1} \mid s _ {i}\right). \tag {10}
+$$
+
+(b) Conversely, if the constraint (10) holds for some constant  $Z$  and policies  $P_{1}$  and  $P_{2}$ , then  $P_{1}$  and  $P_{2}$  are the jointly optimal AFlowNet policies.
+
+The constraint (10) can be converted into a training objective  $\mathcal{L}_{\mathrm{TB}}$  – the squared log-ratio between the left and right sides – and optimized with respect to the policy parameters and the scalar  $Z$  (parametrized through  $\log Z$  for stability) for complete trajectories (game simulations) sampled from a training policy. Prop. 2 and Prop. 4(a) guarantee that the constraints are satisfiable, while Prop. 4(b) guarantees that the policies satisfying the constraints are unique.
+
+Training AFlowNets. AFlowNets are trained by optimizing the EFlowNet objectives of each agent independently. The states at which the objectives are optimized are chosen by a training policy, which may either sample the agents' policies to produce training trajectories (on-policy self-play) or use off-policy exploration. The joint global optimum, where all agents optimize their losses to 0, is unique and independent of the training policy due to Prop. 2. See Alg. 1.
+
+A significant benefit of AFlowNets over methods like Silver et al. (2018) is that they do not require expensive rollout procedures (i.e., MCTS) to generate games. MCTS performs a number of simulations – each of which requires a forward pass – for every state in a game. AFlowNets, on the other hand, only require a single forward pass per state. Consequently, AFlowNets can be trained on more games given a similar computational budget.
+
+# 4 EXPERIMENTS
+
+We conduct experiments to investigate whether EFlowNets can effectively learn in stochastic environments compared to related methods (§4.1) and whether AFlowNets are effective learners of adversarial gameplay, as measured by their performance against contemporary approaches (§4.2).
+
+# 4.1 GENERATIVE MODELING IN STOCHASTIC ENVIRONMENTS
+
+We evaluate EFlowNets in a protein design task from Jain et al. (2022). The GFlowNet policy autoregressively generates an 8-symbol DNA sequence  $x$  and receives a reward of  $R(x) = f(x)^{\beta}$ , where  $f(x)$  is a proxy model estimating binding affinity to a target protein and  $\beta$  is a hyperparameter controlling the reward distribution's entropy. In Pan et al. (2023), the problem was made stochastic by letting the environment replace the symbol appended by the policy to the right of a partial sequence with a uniformly random symbol with probability  $\alpha$ . Thus  $\alpha = 0$  gives a deterministic environment and  $\alpha = 1$  a fully stochastic environment, where the policy's actions have no effect.
+
+We extend the published code of Pan et al. (2023) with an implementation of the EFlowNet objective. Besides the stochastic GFlowNet (Stoch-GFN) formulation from §2.2, we compare with the two strongest
+
+![](images/bf8b4a6086fa8c5114b561c4692cadff783dc4817e8a7a13c5fa5ab7350444ec.jpg)  
+Figure 2: Results on the TFBind task (five seeds per setting). EFlowNets tend to find more diverse high-reward states, especially when the reward is peaky and environment stochasticity is high, making the Stoch-GFN constraints unsatisfiable.
+
+bases considered in that work: a “naive” GFlowNet that ignores the environment’s transitions (GFN), and a discrete soft actor-critic (SAC; Haarnoja et al., 2018). We use the hyperparameters from the existing implementation for all methods (except SAC, which we reimplemented because code was not available) and report the same primary metrics: the mean reward of the top-100 sequences among 2048 sampled from a trained model and the number of diverse modes found, as measured by the sphere exclusion algorithm from Jain et al. (2022). A model of the environment’s transition distribution is learned, consistent with Pan et al. (2023).
+
+The results and error ranges, with different values of the stochasticity  $\alpha$  and reward exponent  $\beta$ , are shown in Fig. 2. When the reward is peaky (larger  $\beta$ ), EFlowNets outperform other algorithms in both diversity and top-100 reward. This is consistent with situations such as those in Fig. 1a, where the Stoch-GFN constraints are unsatisfiable, being more common when the reward is peaky, as the environment's random actions place smoothness constraints on the sampling distribution. Of note, our implementation of SAC performs better than what is reported in Pan et al. (2023) and often better than Stoch-GFN, which was previously considered only with the flat-reward setting of  $\beta = 3$ .
+
+# 4.2 ADVERSARIAL GAMES
+
+The game-playing capabilities of AFlowNets are evaluated in 2-player games. Rewards are modeled as described in §3.3, and the AFlowNets trained with rewards defined by (8) with a given  $\lambda$  are denoted AFlowNet $_{\lambda}$ . We evaluate how the efficacy of an agent changes with various values of  $\lambda$ .
+
+Algorithm 1: Branch-adjusted AFlowNet Training  
+Data:  $\lambda$  , batch size  $n$  number of trajectories  $K$  number of steps  $L$  buffer capacity  $M$  model and training hyperparameters   
+Initialize AFlowNet policies  $P_{1},P_{2}$  log Z, and replay buffer B with capacity M   
+for  $i = 1$  to  $N$  do Generate  $K$  trajectories (sampling from AFlowNet's policies  $P_{1},P_{2})$  Add trajectories to  $B / / B$  is a FIFO queue for  $i = 1$  to  $L$  do  $\{\tau_j\}_{j = 1}^n\gets$  Sample randomly from  $B$ $\mathcal{L}\leftarrow \frac{1}{n}\sum_{j = 1}^{n}\mathcal{L}_{\mathrm{TB}}(\log Z,P_1,P_2,\tau_j,\lambda)$  Gradient update on  $\nabla \mathcal{L}$  with respect to log Z and policy parameters
+
+AFlowNets are trained using the TB loss to play tic-tac-toe and Connect-4. For each, we run a tournament against an open-source AlphaZero implementation (Thakoor et al., 2016) and a uniform random agent. For tic-tac-toe we also include a tree-search agent which uses AlphaZero's value function, alpha-beta pruning, and a search depth of 3. To compare the agents, we compute their Elo over the course of training using BayesElo (Coulom, 2008; Diaz & Buck-Kaeffer, 2023). The training procedure is outlined in Alg. 1 and details are in  $\S D$ .
+
+Figure 3 (left) illustrates the Elo of the agents in tic-tac-toe over training steps and time. It is clear that AFlowNets quickly achieve a competitive Elo with AlphaZero. It is worth noting that  $\mathrm{AFlowNet}_2$  and  $\mathrm{AFlowNet}_{15}$  achieved and Elo of  $334.8\pm 15.5$  and  $231.1\pm 91.3$  , whereas  $\mathrm{AFlowNet}_{10}$  achieved an Elo of  $338.4\pm 14.1$  As such, it appears that  $\lambda$  has a diminishing return on game performance in tic-tac-toe.
+
+The parameter  $\lambda$  has a large effect on the performance of AFlowNets in Connect-4 (cf. Fig. 3 (right)). The AFlowNets with  $\lambda \in \{10,15\}$  achieve the highest Elo of all tested agents. AFlowNets win almost every game against AlphaZero and achieve an Elo score roughly 800 points higher. Additional tournament results and further analysis are available in §E.
+
+As in Prasad et al. (2018), we take advantage of the fact that Connect-4 is solved (Allis, 1988) to obtain perfect values for arbitrary positions. We compare the moves selected by the AFlowNet with the values computed by a perfect Connect-4 solver (Pons (2019)) over the course of training. Fig. 4 shows an evaluation of the AFlowNet's performance using this metric (and baseline values for a random uniform agent). The AFlowNets learn to play optimal moves in  $>80\%$  of board states after 3 hours of training on one RTX8000 GPU.
+
+![](images/ee90006d107897b1dd074174670aa42e299531f2b279ad0dbafb634769d74bf1.jpg)
+
+![](images/635b5cfe13119d49500bd321ee8932e3e94418c24fa7e088f38408de3aeb1edb.jpg)  
+Figure 3: Elo as a function of training steps and training time. As a convention, random uniform baseline agents represent an Elo of 0. AFlowNets achieve similar Elo to AlphaZero in tic-tac-toe and AFlowNets quickly learn to outperform AlphaZero in Connect-4.
+
+![](images/7a489bdec74b68bd95b5377a3e5d2f977391e662b01efea34ccf2120c88571a7.jpg)  
+Figure 4: Move quality for the AFlowNet (for a set of 10240 randomly generated Connect-4 boards) over the course of training. An optimal move leads to the quickest win or slowest loss. An inaccuracy is a non-optimal move that has the same sign as the optimal move (e.g., leading to a win but not as quickly). A blunder leads from a winning state to either a drawing or losing state.
+
+Table 1: Summary of the main differences between AFlowNet and AlphaZero training.  
+
+<table><tr><td></td><td>AFlowNet</td><td>AlphaZero</td></tr><tr><td>Action sampling</td><td>single forward pass</td><td>MCTS</td></tr><tr><td>Objective input</td><td>complete trajectory</td><td>single state</td></tr><tr><td>States per optim. step</td><td>batch size × traj. length</td><td>batch size</td></tr></table>
+
+Differences between AFlowNets and AlphaZero methodologies. Distinctions exist between our approach and AlphaZero that make direct comparisons between the methods challenging, summarized in Table 1. Most importantly, the batch-adjusted AFlowNet objective depends on an entire game simulation, while the AFlowNet value function updates are performed at individual states. In addition, the game simulations in AFlowNets are obtained using a single policy rollout, without Monte Carlo tree search. Thus, generation of training data with AFlowNets is faster than with AlphaZero, assuming the base model architectures are of a similar scale.
+
+**Demo.** We invite the reader to play Connect-4 anonymously against a pretrained AFlowNet agent at the following URL: https://bit.ly/demoafn.
+
+# 5 RELATED WORK
+
+Stochasticity in GFlowNets. GFlowNets have been used as diversity-seeking samplers in various settings with deterministic environment transitions. In particular, they have been interpreted as hierarchical variational inference algorithms (Malkin et al., 2023; Zimmermann et al., 2023) and correspondingly applied to modeling of Bayesian posteriors (Deleu et al., 2022; van Krieken et al., 2022; Hu et al., 2023). GFlowNets can be trained with stochastic rewards (Bengio et al., 2023), and Deleu et al. (2022; 2023); Liu et al. (2022) take advantage of this property to train samplers of Bayesian posteriors using small batches of observed data. Zhang et al. (2023b) proposed to match the uncertainty in a stochastic reward in a manner resembling distributional RL (Bellemare et al., 2017); however, the stochasticity is introduced only at terminal states, while we consider stochasticity in intermediate transitions. The stochastic modelling of Pan et al. (2023), as we have argued, is insufficient to capture desired sampling behaviours in the problems we consider.
+
+RL in stochastic environments and games. Learning in an environment with stochastic transition dynamics and against adversaries has long been a task of RL (Sutton & Barto, 2018). While AlphaZero (Silver et al., 2018) has achieved state-of-the-art performance in chess, Shogi, and Go, it does not explicitly model stochastic transition dynamics. Stochastic MuZero (Antonoglou et al., 2022) is a model-based stochastic planning method that learns a model of the environment and a policy at the same time, allowing it to perform well in a variety of stochastic games. Both AlphaZero and MuZero use Monte Carlo tree search for policy and value estimation (Silver et al., 2018; Antonoglou et al., 2022; Schrittwieser et al., 2020). Our EFlowNets bear similarities to a recently introduced approach that integrates over environment uncertainty in RL (Yang et al., 2023).
+
+# 6 DISCUSSION
+
+This paper extends GFlowNets to stochastic and adversarial environments. Expected flow networks learn in settings with stochastic transitions while maintaining desirable convergence properties, and adversarial flow networks pit EFlowNets against themselves in self-play. We successfully applied these algorithms to a stochastic generative modeling problem and to two real-world zero-sum games.
+
+In future work, we intend to scale these methods to larger game spaces (e.g., chess and Go). Such scaling is likely to require algorithmic improvements to address the limitations of our method. While we derived an efficient 'trajectory balance' for branch-adjusted AFlowNets, trajectory-level objectives suffer from high variance for long trajectories and have a high memory cost. Although these limitations did not surface in our experiments, it would be interesting to consider interpolations between expected DB and TB, akin to subtrajectory balance for GFlowNets (Madan et al., 2023).
+
+Other future work can consider generalizations to incomplete-information games and cooperative multi-agent settings. For games with continuous action spaces, one can analyze the continuous-time and infinite-agent (mean-field) limits. Beyond games, GFlowNets and EFlowNets, in which node flows are computed by aggregation over children – either summation (4) or expectation (5) – may fall into a more general class of probabilistic flow networks that encompass a range of samplers trainable by local consistency objectives, reminiscent of the manner in which the language of circuits unifies probabilistic models with tractable inference (Choi et al., 2020; Vergari et al., 2021).
+
+# ACKNOWLEDGMENTS
+
+The authors thank Moksh Jain for help with baseline code for the experiments in §4.1 and Manfred Diaz for help with the Elo computations in §4.2. We also thank Quentin Bertrand and Juan Duque for their comments on a draft of the paper.
+
+YB acknowledges funding from CIFAR, NSERC, Intel, and Samsung.
+
+GG acknowledges funding from CIFAR.
+
+The research was enabled in part by computational resources provided by the Digital Research Alliance of Canada (https://alliancecan.ca), Mila (https://mila.quebec), and NVIDIA.
+
+# REFERENCES
+
+Victor Allis. A knowledge-based approach of connect-four. J. Int. Comput. Games Assoc., 11:165, 1988.  
+Ioannis Antonoglou, Julian Schrittwieser, Sherjil Ozair, Thomas K Hubert, and David Silver. Planning in stochastic environments with a learned model. International Conference on Learning Representations (ICLR), 2022.  
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+
+# A GFLOWNETS AS QUANTAL RESPONSE AGENTS
+
+A GFlowNet policy as a Luce agent. GFlowNets in tree-structured spaces are closely related to probabilistic models of imperfect agent behaviour in game theory known as quantal response agents (McKelvey & Palfrey, 1996; 1995; Goeree et al., 2020). A particular kind of quantal response agent uses the Luce ratio rule (Luce, 1959) to sample strategies in proportion to their expected payoff. GFlowNet trajectories  $\tau$  can be seen as (pure) strategies of an agent in a one-player game, and the reward  $R(x)$  as the payoff for a trajectory  $\tau$  leading to  $x \in \mathcal{X}$ . A GFlowNet that samples trajectories proportionally to the rewards of their last states, i.e.,  $P_F(\tau = (s_0 \to s_1 \to \dots \to s_n = x)) \propto R(x)$ , is thus a Luce quantal response agent. On the level of individual actions at a given state  $s$ , the policy is that of a Luce agent that treats  $F(s')$  – the total reward accessible from  $s'$  – as the payoff for a transition  $s \to s'$ .
+
+EFlowNets learn a marginalized quantal response policy. Next, we show that EFlowNets are Bayesian (model-averaging) analogues of Luce agents that marginalize out the uncertainty of the environment's transitions.
+
+Define a (pure) environment strategy as an induced subgraph  $G_{\mathrm{env}}$  of  $G$  whose vertex set  $V(G_{\mathrm{env}}) \subseteq S$  has the following properties:
+
+- If  $s \in V(G_{\mathrm{env}})$  and  $s \neq s_0$ , then the parent of  $s$  is in  $V(G_{\mathrm{env}})$ .  
+- If  $s \in V(G_{\mathrm{env}}) \cap S_{\mathrm{env}}$ , then exactly one child of  $s$  is in  $V(G_{\mathrm{env}})$ .
+
+- If  $s \in V(G_{\mathrm{env}}) \cap S_{\mathrm{agent}}$ , then all children of  $s$  are in  $V(G_{\mathrm{env}})$ .
+
+It is clear from the first property that any such  $G_{\mathrm{env}}$  is a tree. An environment strategy thus amounts to a predetermined choice of action at every state that can be reached if the environment takes the actions chosen by the strategy. The environment policy  $P_{\mathrm{env}}$  determines a distribution over environment strategies, where the child of each agent state  $s$  in  $G_{\mathrm{agent}}$  is sampled from the policy independently for each  $s$ , i.e.,
+
+$$
+P _ {\text {e n v}} \left(G _ {\text {e n v}}\right) = \prod_ { \begin{array}{l} s \in V \left(G _ {\text {e n v}}\right) \cap \mathcal {S} _ {\text {e n v}} \\ s ^ {\prime} \in \operatorname {C h} (s) \cap V \left(G _ {\text {e n v}}\right) \end{array} } P _ {\text {e n v}} \left(s ^ {\prime} \mid s\right). \tag {11}
+$$
+
+The environment strategy is a source of uncertainty for the agent. Any given  $G_{\mathrm{env}}$  is a tree that contains  $s_0$  and some subset  $\chi_{G_{\mathrm{env}}}$  of the terminal states. Viewing  $G_{\mathrm{env}}$  as a (deterministic) GFlowNet in the sense of §2.1, there is a unique policy  $P_F^{G_{\mathrm{env}}}$ , and corresponding state flow  $F^{G_{\mathrm{env}}}$  on  $G_{\mathrm{env}}$  that samples proportionally to the reward  $R$  restricted to  $\chi_{G_{\mathrm{env}}}$ .
+
+The following proposition shows that the optimal EFlowNet policy averages the stepwise utilities (i.e., total accessible rewards) of deterministic-environment GFlowNets  $P_F^{G_{\mathrm{env}}}$  weighted by their likelihood under  $P_{\mathrm{env}}$ .
+
+Proposition 5. Suppose that  $P_{\mathrm{agent}}$  satisfies the EDB constraints. Then, for any  $s \in S_{\mathrm{agent}}$  and  $s' \in \mathrm{Ch}(s)$ ,
+
+$$
+P _ {\text {a g e n t}} \left(s ^ {\prime} \mid s\right) \propto \mathbb {E} _ {G _ {\text {e n v}} \sim P _ {\text {e n v}}} \left[ F ^ {G _ {\text {e n v}}} \left(s ^ {\prime}\right) \mid s \in V (G _ {\text {e n v}}) \right],
+$$
+
+where the expectation is taken over the distribution over strategies determined by  $P_{\mathrm{env}}$ , restricted to the strategies that contain the state  $s$ .
+
+AFlowNets and agent quantal response equilibrium. In two-player games with unadjusted rewards  $R_1^\circ$ ,  $R_2^\circ$  and rewards  $R_1$  and  $R_2$  defined using the branching factor adjustment (9), we also have the following characterization of the optimal state flows:
+
+$$
+\begin{array}{l} B _ {i} (s) F _ {i} (s) = \mathbb {E} _ {\substack {s = s _ {1} \rightarrow s _ {2} \rightarrow \dots s _ {n} = x \in \chi\\s _ {k + 1} \sim \mathcal {U} [ \mathrm {Ch} \left(s _ {k}\right) ] \text {if } s _ {k} \in \mathcal {S} _ {i}\\s _ {k + 1} \sim P _ {j} \left(s _ {k + 1} \mid s _ {k}\right) \text {if } s _ {k} \in \mathcal {S} _ {j} (j \neq i)}} \left[ R _ {i} ^ {\circ} (x) \right] \tag{12} \\ = \sum_ {s = s _ {1} \rightarrow s _ {2} \rightarrow \dots s _ {n} = x \in \mathcal {X}} \left[ \prod_ {k: s _ {k} \in \mathcal {S} _ {i}} \frac {1}{| \operatorname {C h} (s _ {k}) |} \prod_ {k: s _ {k} \in \mathcal {S} _ {j}, j \neq i} P _ {j} \left(s _ {k + 1} \mid s _ {k}\right)\right] R _ {i} ^ {\circ} (x), \\ \end{array}
+$$
+
+where the notation  $B_{i}(s)$  is extended to nonterminal states  $s$  using the same definition (9). This is easily derived by recursion from the EDB constraints and (9) in a similar way to the proof of Prop. 5. Because  $P_{i}(s^{\prime} \mid s) \propto F_{i}(s^{\prime})$  for  $s \in S_{i}$ , the expression (12) characterizes the policy  $P_{i}$  via a form of agent quantal response (McKelvey & Palfrey, 1998), in which the action probability of an agent is proportional to its expected reward under future actions of the opponent (sampled from its policy) and the agent itself (here, sampled uniformly).
+
+# B PROOFS
+
+Proposition 1. There exists a unique pair of state flow function  $F$  and agent policy  $P_{\mathrm{agent}}$  satisfying constraints (4), (5), and (6). If  $S_{\mathrm{env}} = \emptyset$ , then this pair satisfies the detailed balance constraints (2).
+
+Proof of Proposition 1. If the EDB constraints are satisfied, then  $F$  satisfies a recurrence:
+
+$$
+F (s) = \left\{ \begin{array}{l l} \sum_ {s ^ {\prime} \in \operatorname {C h} (s)} F \left(s ^ {\prime}\right) & s \in S _ {\text {a g e n t}} \\ \mathbb {E} _ {s ^ {\prime} \sim P _ {\text {e n v}} \left(s ^ {\prime} \mid s\right)} F \left(s ^ {\prime}\right) & s \in S _ {\text {e n v}} \\ R (s) & s \in X \end{array} , \right. \tag {13}
+$$
+
+where the first case  $(s\in S_{\mathrm{agent}})$  follows from summing (4) over  $s^\prime$  . The uniqueness of  $F(s)$  can easily be seen, e.g., by induction on the length of the longest path from  $s$  to a terminal state.
+
+Because  $R(x) > 0$  for all  $x \in \mathcal{X}$ , and the recurrence preserves the positivity (i.e.,  $F(s') > 0$  for all  $s' \in \mathrm{Ch}(s)$  implies  $F(s) > 0$ ), we have  $F(s) > 0$  for all  $s$ . Therefore, one can recover the unique  $P_{\mathrm{agent}}$  that satisfies (4) jointly with  $F$  via  $P_{\mathrm{agent}}(s' \mid s) = \frac{F(s')}{F(s)}$ .
+
+Finally, if  $S_{\mathrm{env}} = \emptyset$ , then constraint (5) is vacuous, and the remaining constraints exactly recover (2).
+
+Proposition 2. There exist unique agent policies  $P_{1}, \ldots, P_{n}$  and state flow functions  $F_{1}, \ldots, F_{n}: S \to \mathbb{R}_{>0}$  such that  $P_{i}$  and  $F_{i}$  satisfy the EDB constraints with respect to the EFlowNet  $\mathcal{E}_i$  for all  $i$ .
+
+Proof of Proposition 2. As in the proof of Proposition 1, we give a recurrence on the flows:
+
+$$
+F _ {i} (s) = \left\{ \begin{array}{l l} \sum_ {s ^ {\prime} \in \operatorname {C h} (s)} F _ {i} \left(s ^ {\prime}\right) & s \in \mathcal {S} _ {i} \\ \frac {\sum_ {s ^ {\prime} \in \operatorname {C h} (s)} F _ {i} \left(s ^ {\prime}\right) F _ {j} \left(s ^ {\prime}\right)}{\sum_ {s ^ {\prime} \in \operatorname {C h} (s)} F _ {j} \left(s ^ {\prime}\right)} & s \in \mathcal {S} _ {j}, j \neq i. \\ R _ {i} (s) & s \in \mathcal {X} \end{array} \right. \tag {14}
+$$
+
+This recurrence uniquely determines the state flows  $F_{i}$  and therefore the policies  $P_{i}$ . It remains to see that if the flows satisfy (14), then each  $F_{i}$  satisfies the recurrence (13). The cases  $s \in S_{i}$  and  $s \in X$  are clear. For the case  $s \in S_{j}, j \neq i$ , observe that
+
+$$
+\frac {\sum_ {s ^ {\prime} \in \mathrm {C h} (s)} F _ {i} \left(s ^ {\prime}\right) F _ {j} \left(s ^ {\prime}\right)}{\sum_ {s ^ {\prime} \in \mathrm {C h} (s)} F _ {j} \left(s ^ {\prime}\right)} = \sum_ {s ^ {\prime} \in \mathrm {C h} (s)} \frac {F _ {i} \left(s ^ {\prime}\right) F _ {j} \left(s ^ {\prime}\right)}{\sum_ {s ^ {\prime \prime} \in \mathrm {C h} (s)} F _ {j} \left(s ^ {\prime \prime}\right)} = \sum_ {s ^ {\prime} \in \mathrm {C h} (s)} F _ {i} \left(s ^ {\prime}\right) P _ {j} \left(s ^ {\prime} \mid s\right) = \mathbb {E} _ {s ^ {\prime} \sim P _ {j} \left(s ^ {\prime} \mid s\right)} F _ {i} \left(s ^ {\prime}\right),
+$$
+
+which coincides with the second case of the recurrence (13).
+
+![](images/9560080ebb7849c2568c4b21a7e04566fc5f1ecf9e7cb27800801a13f6992ade.jpg)
+
+Proposition 3. Suppose that in a 2-player AFlowNet, the agent policies  $P_{1}, P_{2}$  and state flow functions  $F_{1}, F_{2}$  are jointly optimal in the sense of Prop. 2. Then the function  $F(s) = F_{1}(s)F_{2}(s)$  is a flow on  $G$ , i.e., satisfies the FM constraint (1), with respect to the reward  $R(x) = R_{1}(x)R_{2}(x)$ .
+
+Proof of Proposition 3. Because  $F_{1}(x) = R_{1}(x)$  and  $F_{2}(x) = R_{2}(x)$  for all  $x \in \mathcal{X}$ , we have  $F(x) = R(x)$  for all  $x \in \mathcal{X}$ .
+
+For  $s \in S \setminus \mathcal{X}$ , we must show that  $F(s) = \sum_{s' \in \mathrm{Ch}(s)} F(s')$ . Suppose without loss of generality that  $s \in S_1$ . Then
+
+$$
+F _ {1} (s) F _ {2} (s) = F _ {1} (s) \mathbb {E} _ {s ^ {\prime} \sim P _ {1} (s ^ {\prime} | s)} F _ {2} (s ^ {\prime}) = F _ {1} (s) \sum_ {s ^ {\prime} \in \mathrm {C h} (s)} \frac {F _ {1} (s ^ {\prime})}{F _ {1} (s)} F _ {2} (s ^ {\prime}) = \sum_ {s ^ {\prime} \in \mathrm {C h} (s)} F _ {1} (s ^ {\prime}) F _ {2} (s ^ {\prime}),
+$$
+
+which completes the proof.
+
+![](images/5e2e553349d8f5dc3c0ec7ab0b63243070b395baa406860e839a3dfcbb6da796.jpg)
+
+Proposition 4. In a 2-player AFlowNet with alternating moves satisfying  $R_1^\circ(x)R_2^\circ(x) = 1$ :
+
+(a) Suppose that the agent policies  $P_{1}, P_{2}$  and state flow functions  $F_{1}, F_{2}$  are jointly optimal in the sense of Prop. 2. Then there exists a scalar  $Z$ , independent of  $x$ , such that for every complete trajectory  $s_{0} \rightarrow s_{1} \rightarrow \dots \rightarrow s_{n} = x$ ,
+
+$$
+Z \prod_ {i: s _ {i} \in S _ {1}} P _ {1} \left(s _ {i + 1} \mid s _ {i}\right) = R _ {1} (x) B _ {2} (x) \prod_ {i: s _ {i} \in S _ {2}} P _ {2} \left(s _ {i + 1} \mid s _ {i}\right). \tag {10}
+$$
+
+(b) Conversely, if the constraint (10) holds for some constant  $Z$  and policies  $P_{1}$  and  $P_{2}$ , then  $P_{1}$  and  $P_{2}$  are the jointly optimal AFlowNet policies.
+
+Proof of Proposition 4. Part (a). We first extend the definition of  $B_i$  to nonterminal states: if  $s_0 \to s_1 \to \dots \to s_m = s$  is any trajectory, define  $B_i(s) \coloneqq \prod_{0 \leq i < m: s_i \in S_i} |\operatorname{Ch}(s_i)|$ .
+
+We claim that for all states  $s$ ,  $F_{1}(s)F_{2}(s) = \frac{1}{B_{1}(s)B_{2}(s)}$ . This holds at terminal states  $x$ , since  $F_{1}(x)F_{2}(x) = R_{1}(x)R_{2}(x) = \frac{R_{1}^{\circ}(x)R_{2}^{\circ}(x)}{B_{1}(x)B_{2}(x)} = \frac{1}{B_{1}(x)B_{2}(x)}$ . By Prop. 3,  $F_{1}(s)F_{2}(s)$  is a flow, so it suffices to show that  $\frac{1}{B_{1}(s)B_{2}(s)}$  also satisfies (1) for  $s \in \mathcal{X} \setminus \mathcal{S}$ . Without loss of generality, suppose  $s \in S_{1}$  and let  $s_0 \to \dots \to s_i = s$  be the trajectory leading to  $s$ . Then
+
+$$
+\sum_ {s ^ {\prime} \in \operatorname {C h} (s)} \frac {1}{B _ {1} \left(s ^ {\prime}\right) B _ {2} \left(s ^ {\prime}\right)} = \sum_ {s ^ {\prime} \in \operatorname {C h} (s)} \frac {1}{B _ {1} (s) | \operatorname {C h} (s) | \cdot B _ {2} (s)} = \frac {1}{B _ {1} (s) B _ {2} (s)},
+$$
+
+establishing the claim.
+
+Returning to the proposition, rearranging factors and using the definition (9), it is necessary to show that
+
+$$
+\frac {R _ {1} ^ {\circ} (x) B _ {2} (x) \prod_ {i : s _ {i} \in \mathcal {S} _ {2}} P _ {2} \left(s _ {i + 1} \mid s _ {i}\right)}{B _ {1} (x) \prod_ {i : s _ {i} \in \mathcal {S} _ {1}} P _ {1} \left(s _ {i + 1} \mid s _ {i}\right)}
+$$
+
+is independent of  $x$ . We have, using the above claim,
+
+$$
+\begin{array}{l} \frac {R _ {1} (x) B _ {2} (x) \prod_ {i : s _ {i} \in \mathcal {S} _ {2}} P _ {2} (s _ {i + 1} \mid s _ {i})}{\prod_ {i : s _ {i} \in \mathcal {S} _ {1}} P _ {1} (s _ {i + 1} \mid s _ {i})} = \frac {R _ {1} (x) \prod_ {i : s _ {i} \in \mathcal {S} _ {2}} | \operatorname {C h} (s _ {i}) | \frac {F _ {2} (s _ {i + 1})}{F _ {2} (s _ {i})}}{\prod_ {i : s _ {i} \in \mathcal {S} _ {1}} \frac {F _ {1} (s _ {i + 1})}{F _ {1} (s _ {i})}} \\ = \frac {R _ {1} (x) \prod_ {i : s _ {i} \in \mathcal {S} _ {2}} | \operatorname {C h} (s _ {i}) | \frac {F _ {1} (s _ {i}) B _ {1} (s _ {i}) B _ {2} (s _ {i})}{F _ {1} (s _ {i + 1}) B _ {1} (s _ {i + 1}) B _ {2} (s _ {i + 1})}}{\prod_ {i : s _ {i} \in \mathcal {S} _ {1}} \frac {F _ {1} (s _ {i + 1})}{F _ {1} (s _ {i})}} \\ = \frac {R _ {1} (x) \prod_ {i : s _ {i} \in S _ {2}} \frac {F _ {1} \left(s _ {i}\right)}{F _ {1} \left(s _ {i + 1}\right)}}{\prod_ {i : s _ {i} \in S _ {1}} \frac {F _ {1} \left(s _ {i + 1}\right)}{F _ {1} \left(s _ {i}\right)}} \\ = R _ {1} (x) \prod_ {i <   n \text {e v e n}} \frac {F _ {1} \left(s _ {i}\right)}{F _ {1} \left(s _ {i + 1}\right)} \prod_ {i <   n \text {o d d}} \frac {F _ {1} \left(s _ {i}\right)}{F _ {1} \left(s _ {i + 1}\right)} \\ = F _ {1} \left(s _ {n}\right) \prod_ {i = 0} ^ {n - 1} \frac {F _ {1} \left(s _ {i}\right)}{F _ {1} \left(s _ {i + 1}\right)} \quad = F _ {1} \left(s _ {0}\right), \\ \end{array}
+$$
+
+which is independent of  $x$ .
+
+Part (b). Because jointly optimal AFlowNet policies  $P_{1}, P_{2}$  exist by Prop. 2, and they satisfy (10) by part (a) of this proposition, it suffices to show that the constraint (10) uniquely determines  $P_{1}$  and  $P_{2}$  for all pairs of reward functions  $(R_{1}, R_{2})$  for which  $R_{1}(x)R_{2}(x) = \frac{1}{B_{1}(x)B_{2}(x)}$  for all  $x \in \mathcal{X}$ .
+
+We prove this by strong induction on the number of states in  $G$ . The base case  $|S| = 1$  (there is a unique state which is both initial and terminal) is trivial: the products are empty and the constraint reads  $Z = R_{1}(x)$ .
+
+Now suppose that the constraint uniquely determines  $P_{1}$  and  $P_{2}$  for all reward functions satisfying  $R_{1}(x)R_{2}(x) = \frac{1}{B_{1}(x)B_{2}(x)}$  on graphs with fewer than  $N$  states, for some  $N > 1$ , and consider an AFlowNet  $G = (S, \mathcal{A})$  with  $N$  states, for which (10) holds and the reward functions satisfy  $R_{1}(x)R_{2}(x) = \frac{1}{B_{1}(x)B_{2}(x)}$ . It is easy to see that there exists a state  $s \in S \setminus \mathcal{X}$  such that all children of  $s$  are terminal; select one such state  $s$ .
+
+Suppose that  $s \in S_1$ . We construct a new graph  $G'$  and rewards  $R_1', R_2'$  by deleting the children of  $s$ , making  $s$  a terminal state, and modifying the reward function so that  $R_1'(s) = \sum_{x \in \mathrm{Ch}(s)} R_1(s)$ , setting  $R_2'(s)$  so as to preserve  $R_1'(s)R_2'(s) = \frac{1}{B_1(x)B_2(x)}$ , and setting  $R_i'(x) = R_i(x)$  for all other terminal states  $x$ . Thus the graph  $G'$  is a two-player AFlowNet with alternating turns and satisfying the constraint on rewards.
+
+We claim that the constraint (10) for  $G, R_1, R_2$  and a pair of policies  $P_1, P_2$  on  $G$  implies the constraint for  $G', R_1', R_2'$  and the same policies restricted to the states in  $G'$ . For all terminal states in  $G'$  inherited from  $G$ , the constraint is unchanged. For the new terminal state  $s$ , we sum the constraints on  $G$  for the children  $x_1, \ldots, x_K \in \mathrm{Ch}(s)$ . Letting  $s_0 \to s_1 \to \dots \to s_n = s$  be the trajectory leading to  $s$ , we have:
+
+$$
+\begin{array}{l} \sum_ {k = 1} ^ {K} \left[ Z \prod_ {i <   n: s _ {i} \in \mathcal {S} _ {1}} P _ {1} \left(s _ {i + 1} \mid s _ {i}\right) P _ {1} \left(x _ {k} \mid s\right) \right] = \sum_ {k = 1} ^ {K} \left[ R _ {1} \left(x _ {k}\right) B _ {2} \left(x _ {k}\right) \prod_ {i: s _ {i} \in \mathcal {S} _ {2}} P _ {2} \left(s _ {i + 1} \mid s _ {i}\right) \right] \\ \left(Z \prod_ {i <   n: s _ {i} \in S _ {1}} P _ {1} \left(s _ {i + 1} \mid s _ {i}\right)\right) \sum_ {k = 1} ^ {K} P _ {1} \left(x _ {k} \mid s\right) = \left(B _ {2} (s) \prod_ {i: s _ {i} \in S _ {2}} P _ {2} \left(s _ {i + 1} \mid s _ {i}\right)\right) \sum_ {k = 1} ^ {K} R _ {1} \left(x _ {k}\right) \\ Z \prod_ {i <   n: s _ {i} \in \mathcal {S} _ {1}} P _ {1} (s _ {i + 1} \mid s _ {i}) = B _ {2} (s) \prod_ {i: s _ {i} \in \mathcal {S} _ {2}} P _ {2} (s _ {i + 1} \mid s _ {i}) R _ {1} ^ {\prime} (s), \\ \end{array}
+$$
+
+which is precisely the constraint for  $G'$  at the state  $s$ . So the constraint (10) is satisfied on  $G'$ .
+
+Since  $G'$  has fewer than  $N$  states, by the induction hypothesis,  $P_{1}$  and  $P_{2}$  are uniquely determined on  $G'$  and are therefore equal to the jointly optimal AFlowNet policies. It remains to show that  $P_{1}(\cdot \mid s)$  is uniquely determined. Indeed, the only factor on the left side of (10) that varies between
+
+children  $x$  of  $s$  is  $P_{1}(x \mid s)$ , while on the right side, the only such factor is  $R_{1}(x)$ . It follows that if the constraint is satisfied, then  $P_{1}(x \mid s) \propto R_{1}(x)$ , which uniquely determines  $P_{1}(\cdot \mid s)$ .
+
+The case  $s \in S_2$  is analogous.
+
+![](images/812bbbc5a5a251635e3a535ee1176b9b21f68869957e9522b2d7db31c08f81df.jpg)
+
+Proposition 5. Suppose that  $P_{\mathrm{agent}}$  satisfies the EDB constraints. Then, for any  $s \in S_{\mathrm{agent}}$  and  $s' \in \mathrm{Ch}(s)$ ,
+
+$$
+P _ {\text {a g e n t}} \left(s ^ {\prime} \mid s\right) \propto \mathbb {E} _ {G _ {\text {e n v}} \sim P _ {\text {e n v}}} \left[ F ^ {G _ {\text {e n v}}} \left(s ^ {\prime}\right) \mid s \in V \left(G _ {\text {e n v}}\right) \right],
+$$
+
+where the expectation is taken over the distribution over strategies determined by  $P_{\mathrm{env}}$ , restricted to the strategies that contain the state  $s$ .
+
+Proof of Proposition 5. We first note that the expression inside the expectation is well-defined, since if  $s \in V(G_{\mathrm{env}}) \cap S_{\mathrm{agent}}$ , then all children of  $s$  are also in  $V(G_{\mathrm{env}})$ .
+
+Now suppose that  $P_{\mathrm{agent}}$  satisfies EDB jointly with a flow function  $F$ . By (4), we have  $P_{\mathrm{agent}}(s' \mid s) \propto F(s')$ , and  $s' \in V(G_{\mathrm{env}})$  is equivalent to  $s \in V(G_{\mathrm{env}})$  for  $s \in S_{\mathrm{agent}}$  and  $s' \in \mathrm{Ch}(s)$ . Therefore, it would suffice to show that
+
+$$
+F (s) = \mathbb {E} _ {G _ {\text {e n v}} \sim P _ {\text {e n v}}} \left[ F ^ {G _ {\text {e n v}}} (s) \mid s \in V (G _ {\text {e n v}}) \right]
+$$
+
+for all  $s\in S\setminus \mathcal{X}$
+
+To do so, we show that the expression on the right side satisfies the recurrence (13). We consider three cases:
+
+- If  $s \in S_{\text{agent}}$ , then the child set of  $s$  in any  $G_{\text{env}}$  containing  $s$  is the same as its child set in  $G$ . It follows that
+
+$$
+\begin{array}{l} \mathbb {E} _ {G _ {\text {e n v}} \sim P _ {\text {e n v}}} \left[ F ^ {G _ {\text {e n v}}} (s) \mid s \in V (G _ {\text {e n v}}) \right] = \mathbb {E} _ {G _ {\text {e n v}} \sim P _ {\text {e n v}}} \left[ \sum_ {s ^ {\prime} \in \operatorname {C h} (s)} F ^ {G _ {\text {e n v}}} (s ^ {\prime}) \mid s \in V (G _ {\text {e n v}}) \right] \\ = \sum_ {s ^ {\prime} \in \operatorname {C h} (s)} \mathbb {E} _ {G _ {\text {e n v}} \sim P _ {\text {e n v}}} \left[ F ^ {G _ {\text {e n v}}} \left(s ^ {\prime}\right) \mid s \in V \left(G _ {\text {e n v}}\right) \right] \\ = \sum_ {s ^ {\prime} \in \operatorname {C h} (s)} \mathbb {E} _ {G _ {\text {e n v}} \sim P _ {\text {e n v}}} \left[ F ^ {G _ {\text {e n v}}} \left(s ^ {\prime}\right) \mid s ^ {\prime} \in V \left(G _ {\text {e n v}}\right) \right], \\ \end{array}
+$$
+
+showing the first case of the recurrence.
+
+- If  $s \in S_{\mathrm{env}}$ , then in any  $G_{\mathrm{env}}$  containing  $s$ ,  $s$  has a unique child  $s'$  and  $F^{G_{\mathrm{env}}}(\bar{s}) = F^{G_{\mathrm{env}}}(\bar{s}')$ . We decompose the expectation into terms depending on the child of  $s$  that is present in  $G_{\mathrm{env}}$ :
+
+$$
+\begin{array}{l} \mathbb{E}_{G_{\mathrm{env}}\sim P_{\mathrm{env}}}\left[F^{G_{\mathrm{env}}}(s)\mid s\in V(G_{\mathrm{env}})\right] = \mathbb{E}_{\substack{s^{\prime}\in \mathrm{Ch}(s)\\ s^{\prime}\sim P_{\mathrm{env}}(s^{\prime}\in G_{\mathrm{env}}|s\in G_{\mathrm{env}})}}\mathbb{E}_{G_{\mathrm{env}}\sim P_{\mathrm{env}}}\left[F^{G_{\mathrm{env}}}(s)\mid s^{\prime}\in V(G_{\mathrm{env}})\right] \\ = \mathbb {E} _ {s ^ {\prime} \sim P _ {\text {e n v}} (s ^ {\prime} | s)} \mathbb {E} _ {G _ {\text {e n v}} \sim P _ {\text {e n v}}} \left[ F ^ {G _ {\text {e n v}}} (s ^ {\prime}) \mid s ^ {\prime} \in V (G _ {\text {e n v}}) \right], \\ \end{array}
+$$
+
+which shows the second case of the recurrence.
+
+- The case  $s \in \mathcal{X}$  is simple, since  $F(s') = F^{G_{\mathrm{env}}}(s') = R(s')$  for all  $G_{\mathrm{env}}$  containing  $s$ .
+
+![](images/322f6e867977263b56706645a373460ec853fb18578eee66d5db238ad0270ffc.jpg)
+
+# C GAME SPECIFICATION
+
+# C.1 TIC-TAC-TOE
+
+Two players alternate between placing X tiles and O tiles in a  $3 \times 3$  grid. If any player reaches a board with three of their pieces connected by a straight line, they win. Although simplistic, tic-tac-toe has a small enough state space that the ground truth EDB-based flow values can be computed and compared to the learned values. This allows for the unique opportunity to verify that the AFlowNet has converged to the predicted stable point. The stable point can be found by recursively visiting each state in the game and backpropagating the rewards, flows, and probabilities.
+
+# C.2 CONNECT-4
+
+There are two players who alternate between placing yellow and red tokens (for the sake of simplicity, we use X tiles and O tiles) in a  $6 \times 7$  grid (6 rows, 7 columns). Each token is placed at the top of the grid and falls to the lowest unoccupied point in the column. If any player reaches a board with four of their pieces connected in a straight line, they win. Connect-4 has a far larger state space than tic-tac-toe: it is quite difficult for even humans to learn, although the first player has been proven to have a winning strategy (Allis, 1988). As such, it is computationally infeasible to compute, or to store, the optimal policies at all states in the AFlowNet.
+
+# D TRAINING DETAILS
+
+For 1, we collect trajectories as sequences of tuples (state, mask, curr_player, action, done, log Reward):
+
+- state: the current state of the environment  
+- mask: binary mask of legal moves over the action space  
+- curr_player: the player whose turn it is to make the action  
+- action: the sampled action at the given state  
+- done: whether the action resulted in a terminal state  
+- logreward: the log reward if done
+
+As architecture, we use a convolutional neural network composed of residual blocks inspired by the AlphaZero architecture (Silver et al., 2018; Thakoor et al., 2016) with a few modifications. We remove the batch normalization layers as the population statistics varied too much between training and evaluation. We use only the policy head (using one for each side, e.g., playing as "X" or "O") and increase the number of filters it has as recommended by Prasad et al. (2018). We replace ReLU activations by Leaky ReLU activations, reduce the number of residual blocks and reduce the number of filters in each block (128 instead of 256) as tic-tac-toe/Connect-4 are simpler than chess/Go. Finally, we include a single differentiable parameter log Z.
+
+The main training hyperparameters are:
+
+- num_trajecories_epoch: number of trajectories added to the buffer every epoch  
+- batch_size: batch size used for training and trajectory generation  
+- num_steps: number of optimization steps per epoch  
+- replay_buffer_capacity: the maximum capacity of the replay buffer  
+- learning_rate: learning rate for the policy network  
+- learning_rate_Z: learning rate for the log Z (we find that a higher value helps training)  
+- num_residual_blocks: number of residual blocks in the architecture
+
+Specific values are included in Table 2.
+
+# D.1 TRAINING/EVALUATION POLICY
+
+During training, to generate the training trajectories, we sample actions using the softmax of the policy logits of the AFlowNet with a temperature coefficient of 1.5. At test time, (i.e., for the tournaments), when it is the AFlowNet's turn to play, we select the move corresponding to  $\arg \max_{s' \in Ch(s)} P_i(s' \mid s)$ , where  $i$  is the index of the player to make a move at  $s$ .
+
+# D.2 ALPHAZERO TRAINING
+
+AlphaZero is trained as per the specifications of Thakoor et al. (2016). The batch size of AlphaZero is changed to 512 to match the batch size of AFlowNets. Training discrepancies include the number of examples gathered and retained over each iteration, which is significantly different between the AFlowNet implementation and AlphaZero. Naturally, it is quite difficult to compare AlphaZero and AFlowNets exactly because we do not have an MCTS analogue. Additionally, AlphaZero trains on single transitions whereas AFlowNets with TB loss train on entire trajectories. This leads to a discrepancy in what is considered a training example: a transition versus a trajectory. The number of Monte Carlo tree search iterations for AlphaZero is 25 for both tic-tac-toe and Connect-4.
+
+<table><tr><td>Hyperparameter</td><td>Tic-tac-toe</td><td>Connect-4</td></tr><tr><td>num-trajectories_epoch</td><td>10240</td><td>10240</td></tr><tr><td>batch_size</td><td>512</td><td>1024</td></tr><tr><td>num_steps</td><td>500</td><td>250</td></tr><tr><td>replay_buffer_capacity</td><td>10240</td><td>250000</td></tr><tr><td>learning_rate</td><td>1e-3</td><td>1e-3</td></tr><tr><td>learning_rate_Z</td><td>5e-2</td><td>5e-2</td></tr><tr><td>num_residual_blocks</td><td>10</td><td>15</td></tr><tr><td>GPU</td><td>1xRTX3090Ti</td><td>1xRTX8000</td></tr></table>
+
+Table 2: Hyperparameters used for training the AFlowNets on on tic-tac-toe and Connect-4
+
+![](images/9f4afa09b39efc871314b57228c700410f54318dbe64d71acf8253908bc6aac4.jpg)  
+Figure 5: Graphs of learning performance over various training runs. (Left) The average MAE of learned node flows (not in log space) compared to ground truth flows computed algorithmically. (Middle) Average MAE for learned edge flows. (Right) Loss rate of the three training regimes against a random uniform opponent.
+
+![](images/4da0bc8816b33288815f37f2e70a58156d7515beb9e68fd6940cf484ca50f613.jpg)
+
+![](images/d1fc8d3ea90fddc14417059886bd7a05d2babaf1a3c5900c770aeb9091e90c32.jpg)
+
+# E ADDITIONAL RESULTS FROM ADVERSARIAL GAMES
+
+# E.1 CONVERGENCE OF EFLOWNETS/AFLOWNETS TO UNIQUE OPTIMUM
+
+In addition to testing game-playing performance, we aim to investigate whether EFlowNets/AFlowNets are capable of learning the correct flows (i.e., those corresponding to the unique optimum). The game tree of tic-tac-toe is small enough that ground truth flows and policies (for a fixed opponent and the stable-point optimum) can be computed algorithmically by backtracking recursively from terminal states.
+
+We train neural networks, see §D for details, to evaluate the following configurations:
+
+(1) EFlowNet with a fixed stochastic opponent (policy consists of choosing each legal action with equal probability) from one perspective (e.g., always plays X).  
+(2) EFlowNet versus a fixed stochastic opponent, learning both perspectives.  
+(3) AFlowNet learning using the EDB objective with off-policy self-play.
+
+Figure 5 illustrates the learning performance of the three tested configurations. For all configurations, the ground truth flows/policies that satisfy the EDB constraints are learned, as evidenced by the left and middle graphs. Importantly, this is even the case for training through self-play, where the AFlowNet converges to the actual stable-point optimum.
+
+Interestingly, the AFlowNet does not need to learn the exact flows to achieve strong game-playing performance. For example, here the AFlowNet is able to always win or draw against a uniform agent after less than 5000 steps even though its MAE relative to the correct flows still decreases substantially in the next 20000 steps. Additionally, the EFlowNet formulation is not enough to obtain a robust game playing agent, particularly when the agent it is playing against does not play well.
+
+![](images/187328a9f12dfb81a8f752602048267e2f9fa652083e9b1b68cb6f05c23d23ac.jpg)  
+Figure 6: Graphs of learning performance over various training runs for AFN, DQN, SoftDQN and AlphaZero. (Left) The percent of optimal moves (for TicTacToe solved through minimax) over all states. (Right) Loss rate of the algorithms against a random uniform opponent.
+
+# E.2 COMPARISON TO DQN AND SOFTDQN
+
+We include additional results comparing DQN and SoftDQN to AFlowNets on TicTacToe in Fig. 6. Getting standard RL algorithms to work in multi-agent settings (learning through self-play) is non trivial and does not work in many common RL libraries.
+
+To remedy this, and to ensure as fair of a comparison as possible, we implement DQN and its soft equivalent inside our framework (i.e. using the same environment as AFN, the same architecture, etc.). Notably, to achieve an agent that could consistently beat a uniform opponent, it was necessary to use a minimax version of DQN similar to (Fan et al., 2020) for sequential games (i.e. the q-update is based on the negation of the maximum  $q$ -value of the opponent). For the soft version (denoted SoftDQN), we sample trajectories using the softmax of the Q-values and similarly use the softmax for the updates (once again in a minimax fashion).
+
+# E.3 TIC-TAC-TOE TOURNAMENT
+
+To test the performance of AFlowNets against state of the art methods such as AlphaZero, we train a popular open-source AlphaZero implementation Thakoor et al. (2016) to play tic-tac-toe, pitting the agents against each other and baselines in a tournament. In the set of baselines we include a uniform opponent and a tree-search agent<sup>2</sup>. By changing the value of  $\lambda$  for the AFlowNet, we also test how the learned policy changes with varying rewards. We proceed to test the performance of EDB-based AFlowNets and TB-based AFlowNets.
+
+Results with EDB-based AFlowNets Some selected results of the tournament are listed in Table 3. Note that AlphaZero is trained with Monte Carlo tree search (MCTS), but is tested in the tournament with MCTS both on and off. This ensures a fair comparison of inference-time game-playing capabilities as AFlowNets have no such tree-search mechanism to generate a policy.
+
+It is clear from the tournament results in Table 3 that AlphaZero and AFlowNet agents are capable of perfect play in tic-tac-toe, drawing nearly every game that was played. The biggest differences in performance come from playing against the uniform and tree-search agents. It is clear that  $\mathrm{AFlowNet}_{10}$  performed the best, winning or drawing all games against the uniform and tree-search agents. Interestingly, while a higher  $\lambda$  produces a better X-playing agent, the same is not true for agents playing second: against the tree-search and uniform agents, a lower  $\lambda$  corresponds to a better score. As such, there may be a diminishing return with higher values of  $\lambda$ , perhaps encouraging overly-risky behaviour.
+
+Training with MCTS seems to greatly improve the speed of convergence, as AlphaZero converges to a stable Elo after only a few training steps, see Figure 7. In comparison, AFlowNets require more training steps, optimization steps, and training examples to reach a similar level of performance
+
+Table 3: Selected results of AFlowNets (AFlowNet $\lambda$ ) pitted against AlphaZero (A0) and baselines in tic-tac toe. The agents listed in the rows are playing first as X, the agents listed in the columns are playing second as O. AFlowNets are trained five times with different seeds. The results of the tournament represent the mean and standard deviation of the games over the different seeds, where wins, draws, and losses are given two points, one point, and zero points respectively. Each element in the table is the result from the perspective of the X-playing agent in the row. For example, A0 playing X achieved a score of  $44.2 \pm 5.1$  against the uniform agent.  
+
+<table><tr><td>×↓ | ○→</td><td>AFlowNet2</td><td>AFlowNet10</td><td>AFlowNetAFlowNet15</td><td>A0</td><td>A0+MCTS</td><td>Uniform</td><td>Tree Search</td></tr><tr><td>AFlowNet2</td><td>-</td><td>25 ± 0</td><td>35 ± 13.7</td><td>25 ± 0</td><td>25 ± 0</td><td>45.6 ± 1.5</td><td>45.4 ± 6.4</td></tr><tr><td>AFlowNet10</td><td>25 ± 0</td><td>-</td><td>35 ± 13.7</td><td>25 ± 0</td><td>25 ± 0</td><td>49.8 ± 1.1</td><td>47.6 ± 5.4</td></tr><tr><td>AFlowNet15</td><td>15 ± 13.7</td><td>15 ± 13.7</td><td>-</td><td>30 ± 20.9</td><td>23.2 ± 18.3</td><td>43 ± 2.6</td><td>36 ± 22.0</td></tr><tr><td>A0</td><td>25 ± 0</td><td>25 ± 0</td><td>35 ± 13.7</td><td>-</td><td>25 ± 0</td><td>44.2 ± 5.1</td><td>50 ± 0</td></tr><tr><td>A0+MCTS</td><td>25 ± 0</td><td>25 ± 0</td><td>38.4 ± 13.7</td><td>25 ± 0</td><td>-</td><td>47.4 ± 2.1</td><td>49.8 ± 0.4</td></tr><tr><td>Uniform</td><td>7.2 ± 2.3</td><td>6.8 ± 0.5</td><td>13.8 ± 7.6</td><td>11 ± 7.5</td><td>10.6 ± 3.6</td><td>-</td><td>22.8 ± 5.6</td></tr><tr><td>Tree Search</td><td>0 ± 0</td><td>0 ± 0</td><td>15 ± 22.4</td><td>0 ± 0</td><td>0.4 ± 0.9</td><td>37.8 ± 3.6</td><td>-</td></tr></table>
+
+![](images/68fc02a10a91cf31e57b96cbe19ee0a015eb91a942c2b8283b521a91313641b8.jpg)  
+Figure 7: Elo over optimization steps while training a EDB-based AFlowNet agent in tic-tac-toe. The Elo of the agents after the full course of training are  $363.2 \pm 7.7$ ,  $357.2 \pm 12.7$ ,  $356.9 \pm 11.6$ ,  $344.3 \pm 4.6$ , and  $317.6 \pm 14.3$  for AFlowNet<sub>10</sub>, AFlowNet<sub>2</sub>, AFlowNet<sub>15</sub>, AlphaZero+MCTS, and AlphaZero respectively. Other AFlowNet agents and error bars for the baseline models are omitted for clarity.
+
+to AlphaZero in terms of Elo.  $\mathrm{AFlowNet}_{10}$  achieves the highest Elo, reinforcing its tournament performance in Table 3. Again, it appears that a larger  $\lambda$  produces a worse AFlowNet agent, with  $\mathrm{AFlowNet}_{15}$  achieving the lowest Elo of all AFlowNets. The AlphaZero agents also achieve high Elo scores, with AlphaZero+MCTS achieving the better score of the two.
+
+AlphaZero converges to a stable Elo after a small number of optimization steps whereas AFlowNets using the EDB constraint require an order of magnitude more steps to reach a similar Elo (about 50k steps versus 5k to 10k steps). A similar trend holds for training time, with AFlowNets requiring about 15 times longer to reach similar Elo scores (AlphaZero+MCTS took 38 seconds to reach an Elo of 46 whereas the first AFlowNet to reach a similar Elo of 51 took 558 seconds). While AFlowNets can certainly learn to play tic-tac-toe effectively, they clearly require far more training time and computation to achieve similar levels of performance to AlphaZero. We have not tested AlphaZero without MCTS in training, nor AFlowNets with tree search, so the comparison naturally favors AlphaZero given the power of tree search.
+
+Table 4: Selected results of AFlowNets (AFlowNet $_t$ ) pitted against AlphaZero (A0) and a random uniform baseline in Connect-4. The agents listed in the rows are playing first and the agents listed in the columns are playing second. AFlowNets are trained three times with different seeds. The results of the tournament represent the mean and standard deviation of the games over the different seeds, where wins, draws, and losses are given two points, one point, and zero points respectively. Each element in the table is the result from the perspective of the agent that played first in the row. For example, A0 playing first achieved a score of  $49.2 \pm 1.8$  against the uniform agent.
+
+<table><tr><td>×↓ | ○→</td><td>AFlowNet2</td><td>AFlowNet10</td><td>AFlowNet15</td><td>A0</td><td>A0+MCTS</td><td>Uniform</td></tr><tr><td>AFlowNet2</td><td>-</td><td>0 ± 0</td><td>5 ± 11.2</td><td>50 ± 0</td><td>32.8 ± 19.4</td><td>50 ± 0</td></tr><tr><td>AFlowNet10</td><td>50 ± 0</td><td>-</td><td>20 ± 27.4</td><td>50 ± 0</td><td>46.2 ± 8.5</td><td>50 ± 0</td></tr><tr><td>AFlowNet15</td><td>50 ± 0</td><td>35 ± 22.4</td><td>-</td><td>50 ± 0</td><td>50 ± 0</td><td>50 ± 0</td></tr><tr><td>A0</td><td>0 ± 0</td><td>0 ± 0</td><td>0 ± 0</td><td>-</td><td>25 ± 0</td><td>49.2 ± 1.8</td></tr><tr><td>A0+MCTS</td><td>8.8 ± 8.7</td><td>0.8 ± 1.8</td><td>0 ± 0</td><td>25 ± 0</td><td>-</td><td>49 ± 1.0</td></tr><tr><td>Uniform</td><td>0 ± 0</td><td>0 ± 0</td><td>0 ± 0</td><td>8.4 ± 2.6</td><td>1.2 ± 1.6</td><td>-</td></tr></table>
+
+Results with TB-based AFlowNets The results thus far have focused on EDB-trained agents, but it is important to demonstrate the performance of TB-based agents as well. Figure 3 illustrates the Elo over three training runs with different seeds of  $\mathrm{AFlowNet}_{10}$  and the baseline models. Clearly,  $\mathrm{AFlowNet}_{10}$  matches the Elo of AlphaZero and converges quickly. Compared to Figure 7, it appears that TB-based agents converge about as quickly as AlphaZero, about 10 times faster than the EDB-based agents. Interestingly, the Elo of the TB-based AFlowNets is lower than the Elo of the EDB-based AFlowNets.
+
+There appears to be little difference in the performance of a TB-trained AFlowNet with different values of  $\lambda$  in tic-tac-toe. This is in contrast to the EDB-trained AFlowNets which seemed to be affected by the setting of  $\lambda$ . Similarly to the EDB-based agents, almost every game in the tournament was a draw, with small differences between the performance of an agent against the baselines dictating the differences in Elo.
+
+# E.4 CONNECT-4 TOURNAMENT
+
+We also run a tournament in Connect-4 against AlphaZero and a uniform random baseline. Again, we vary  $\lambda$  to test how the reward structure changes agent performance. The tournament results in Table 4 indicate that the effect of  $\lambda$  is similar to the experiments in tic-tac-toe. A higher  $\lambda$  produces a better agent, however the diminishing return in these experiments relates to a reduction in benefit when increasing lambda rather than a decrease in Elo. This supports the idea that the reward structure affects the behaviour of an agent. The results of the Connect-4 tournament corroborate the Elo results of Figure 3. The Elos of agents  $\mathrm{AFlowNet}_2$ ,  $\mathrm{AFlowNet}_{10}$ , and  $\mathrm{AFlowNet}_{15}$  are  $1190.8 \pm 64.2$ ,  $1700.1 \pm 60.0$ , and  $1835.3 \pm 154.9$ . Clearly,  $\mathrm{AFlowNet}_{15}$  is the best agent.

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+# EXPLAINING KERNEL CLUSTERING VIA DECISION TREES
+
+Maximilian Fleissner
+
+Technical University of Munich  
+fleissner@cit.tum.de
+
+Leena C. Vankadara
+
+Amazon Research  
+Tübingen  
+vleena@amazon.com
+
+Debarghya Ghoshdastidar
+
+Technical University of Munich ghoshdas@cit.tum.de
+
+# ABSTRACT
+
+Despite the growing popularity of explainable and interpretable machine learning, there is still surprisingly limited work on inherently interpretable clustering methods. Recently, there has been a surge of interest in explaining the classic k-means algorithm, leading to efficient algorithms that approximate k-means clusters using axis-aligned decision trees. However, interpretable variants of k-means have limited applicability in practice, where more flexible clustering methods are often needed to obtain useful partitions of the data. In this work, we investigate interpretable kernel clustering, and propose algorithms that construct decision trees to approximate the partitions induced by kernel k-means, a nonlinear extension of k-means. We further build on previous work on explainable k-means and demonstrate how a suitable choice of features allows preserving interpretability without sacrificing approximation guarantees on the interpretable model.
+
+# 1 INTRODUCTION
+
+The increasing predictive power of machine learning has made it a popular tool in many scientific fields. Sensitive applications such as healthcare or autonomous driving however require more than just good accuracy—it is also crucial for a model's decisions to be interpretable (Tjoa & Guan, 2020; Varshney & AleMZadeh, 2017). Unfortunately, popular machine learning models are not transparent and are often referred to as "black box" approaches. The demand for explainable machine learning has led to the development of several tools over the last few years, albeit mostly for supervised learning. Methods such as LIME or Shapley values (Ribeiro et al., 2016; Lundberg & Lee, 2017) are designed to explain the prediction of any given machine learning model. However, posthoc-explainability often is unreliable and has been criticized for not providing insight into the underlying model itself (Rudin, 2019). Counterfactual explanations on the other hand explicitly show the change in prediction had the input variables been different (Wachter et al., 2017), but do not lead to easily interpretable models. Hence, Rudin (2019) calls on researchers and practitioners to "Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead".
+
+Decision trees are at the heart of several inherently interpretable machine learning models (Molnar, 2020; Breiman, 2017), and have recently also gained significant interest in the field of clustering (Bertsimas et al., 2018; Fraiman et al., 2013; Ghattas et al., 2017). Through recursive partitioning of the data based on individual features, they provide interpretability on a global scale by identifying the important features, while at the same time also allowing us to retrace the path of individual decisions through the tree. Despite the extensive empirical research on decision trees for clustering, Moshkovitz et al. (2020) were the first to introduce a clustering model based on decision trees that also satisfies worst-case approximation guarantees. They propose the Iterative Mistake Minimization (IMM) algorithm which approximates a given  $k$ -means clustering by a decision tree with  $k$  leaves, where each leaf represents a cluster. The quality of the resulting interpretable clustering is measured by the price of explainability, defined as the ratio between the cost of the decision tree and the optimal  $k$ -means cost. IMM returns a decision tree with price of explainability of order  $O(k^2)$ , implying that the interpretable clusters achieve a cost not much worse than the optimal partition.
+
+Various interpretable clustering solutions for the  $k$ -means problem with theoretical guarantees have emerged (see Appendix A for an overview). However, despite these advancements, a fundamental
+
+![](images/7550b25a2afa7e7695ea905ec9ff2cbcb3515d0a2ffe3a9e72ee1206be59afb1.jpg)  
+Figure 1:  $k$ -means does not perform well on halfmoons data that is not linearly separable, and explainable  $k$ -means naturally inherits its flaws. Kernel  $k$ -means perfectly finds the clusters and hence, its interpretable variant (proposed Kernel IMM) returns an axis-aligned decision tree with good clustering.
+
+![](images/cfd6d0487a8255f235454e68e3271de63581537425c1e99e58d77815cf99c15c.jpg)
+
+![](images/3d82099a4459906c3ad24d3584b1daaf046d66ee240ed08fedb510ef6dd1d5a1.jpg)
+
+![](images/4544382e19dad239283c1577480b3058a3a023b29649b7dab92f9224b45d1829.jpg)
+
+issue persists: the  $k$ -means cost is ill-suited for real-world datasets, as it fails to identify clusters that are not linearly separable. Clustering methods deployed in practice tend to be more complex, and consequently even less interpretable than standard  $k$ -means. The kernel  $k$ -means algorithm (Dhillon et al., 2004) that extends the standard  $k$ -means by implicitly mapping data to a reproducing kernel Hilbert space (RKHS), is particularly notable. It allows to discern clusters beyond Voronoi partitions of the input space, but this increased flexibility further diminishes the model's interpretability.
+
+Works on interpretable kernel methods are scarce. Chau et al. (2022) propose efficient methods for computing Shapley values for kernel regression, which provide post-hoc explanations. Wu et al. (2019) enforce interpretability in kernel dimension reduction by linearly projecting the data into a low-dimensional space before computing kernel matrices—in spite of operating on fewer features, the kernel step of this approach is not interpretable. As such, the interpretability of kernel methods remains an open problem. We resolve this in the context of clustering by providing decision tree based interpretable approximations of the kernel  $k$ -means algorithm with provable guarantees.
+
+Our contributions. While a significant body of work on decision trees for explaining  $k$ -means has emerged, these algorithms cannot directly be translated to the kernel setting. In particular, there are cases where explainable  $k$ -means does not lead to axis-aligned decision trees for any choice of features for the popular Gaussian kernel. We prove this in Section 3 by introducing the notions of interpretable feature maps and interpretable decision trees, which provide a theoretical characterization of the obstacles that interpretability faces in kernel clustering. Building on these insights in Section 4, we demonstrate how suitably chosen surrogate features can resolve this issue, and we derive a kernelized variant of the IMM algorithm (Moshkovitz et al., 2020) that preserves interpretability. These surrogate features exist for several important product kernels, including the Gaussian and Laplace kernel. Crucially, the proposed algorithm (Kernel IMM) also comes with worst-case guarantees on the price of explainability for a class of interpretable Taylor kernels that include the Gaussian kernel. By incorporating information encoded in the nonlinearity of kernel  $k$ -means, the resulting decision trees lead to significantly better results than explainable  $k$ -means, as Figure 1 illustrates. In Section 5, we further build on previous work (Frost et al., 2020) to derive two kernelized algorithms (Kernel ExKMC and Kernel Expand) that can be used to further refine the partitions obtained from Kernel IMM (by adding more leaves to the tree). Both can also be run on an empty tree, but do not admit worst-case bounds in this case. We conclude by demonstrating the empirical performance of the proposed methods in Section 6. The algorithms are included in Table 1.
+
+Table 1: Overview of our algorithms ( $x_{i}$  denotes  $i$ -th coordinate of point  $x \in \mathbb{R}^{d}$ )  
+
+<table><tr><td>Algorithm</td><td>Axis-aligned cuts</td><td>Worst price of explainability</td></tr><tr><td>Kernel for additive kernels</td><td>xi∈[θ1, θ2]</td><td>O(k2) (Remark 1, Appendix H)</td></tr><tr><td>for interpretable Taylor kernels</td><td>xi∈[θ1, θ2]</td><td>O(dk2) (Theorem 4)</td></tr><tr><td>for distance-based product kernels</td><td>xi∈[θ1, θ2]</td><td>O(dk2·Cdata) (Theorem 5)</td></tr><tr><td>Kernel ExKMC on empty tree</td><td>xi≥θ or xi∈[θ1, θ2]</td><td>Provably unbounded (Theorem 6)</td></tr><tr><td>Kernel Expand on empty tree</td><td>xi≥θ or xi∈[θ1, θ2]</td><td>Provably unbounded (Remark 2)</td></tr></table>
+
+# 2 BACKGROUND AND PRELIMINARIES
+
+Consider a dataset of  $n$  points,  $X = \{x^{(1)}, x^{(2)}, \ldots, x^{(n)}\} \subseteq \mathbb{R}^d$ . We use  $x^{(i)}$  to denote the  $i$ -th data point in  $X$ , while the notation  $x_i$  denoting the  $i$ -th coordinate of any  $x \in \mathbb{R}^d$  appears more frequently.
+
+$k$ -means and kernel  $k$ -means. For a partition of  $X$  into  $k$  disjoint clusters  $C_1, \ldots, C_k$  with means  $c_1, \ldots, c_k \in \mathbb{R}^d$ , the  $k$ -means cost of the partition is defined as
+
+$$
+\operatorname {c o s t} \left(C _ {1}, \dots , C _ {k}\right) = \sum_ {l = 1} ^ {k} \sum_ {x \in C _ {l}} \| x - c _ {l} \| ^ {2}. \tag {1}
+$$
+
+The optimal  $k$ -means cost of  $X$  is the minimum achievable cost, over all partitions  $C_1, \ldots, C_k$ . We define  $\mathcal{M} = \{c_1, \ldots, c_k\}$  as the set of cluster centers obtained from some algorithm designed to minimize the  $k$ -means cost (1). Since finding the optimal set of centers is NP-hard (Dasgupta, 2008), in practice  $\mathcal{M}$  may not necessarily consist of the  $k$  optimal cluster centers. Kernel  $k$ -means clustering (Dhillon et al., 2004) replaces the squared Euclidean distance in (1) by a kernel-based distance. A kernel is a symmetric, positive definite function  $K: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$ . Every kernel is implicitly associated with a feature space  $\mathcal{H}$ , called the reproducing kernel Hilbert space (RKHS), and a feature map  $\psi: \mathbb{R}^d \to \mathcal{H}$  satisfying  $K(x, y) = \langle \psi(x), \psi(y) \rangle$  for all  $x, y \in \mathbb{R}^d$ . For algorithms that only require knowledge of inner products between points  $\psi(x)$  and  $\psi(y)$ , there is no need to compute the feature map explicitly, but one may instead use the kernel  $K(x, y)$  to evaluate  $\langle \psi(x), \psi(y) \rangle$ . Thus the (implicit) feature map  $\psi$  transforms the data non-linearly to a high dimensional space that provides additional flexibility in the model, without increasing the computational complexity. This is commonly known as the kernel trick. Given data  $X \subseteq \mathbb{R}^d$  and a kernel  $K$  on  $\mathbb{R}^d$ , kernel  $k$ -means attempts to find a partitioning  $\mathcal{C} = \{C_1, \ldots, C_k\}$  of  $X$  that minimizes
+
+$$
+\operatorname {c o s t} \left(C _ {1}, \dots , C _ {k}\right) = \sum_ {l = 1} ^ {k} \sum_ {x \in C _ {l}} \| \psi (x) - c _ {l} \| ^ {2} \tag {2}
+$$
+
+where the centers  $c_{1},\ldots ,c_{k}$  are the means of each cluster in the feature space  $\mathcal{H}$ . Appendix B shows how the cost (2) is expressed in terms of kernel  $K$  and also includes the kernel  $k$ -means algorithm.
+
+Explainable  $k$ -means. The goal of explainable  $k$ -means clustering is to approximate a given partition  $C_1, \ldots, C_k$  of  $X \subseteq \mathbb{R}^d$ , with centers  $\mathcal{M}$ , by an inherently interpretable model. Axis-aligned decision trees with  $k$  leaves have emerged as a natural choice for this purpose, where every leaf of the tree  $T$  corresponds to one cluster. Intuitively, one could think that the tree can simply be obtained by using a supervised learning algorithm with the cluster assignments as labels. However, Moshkovitz et al. (2020, Section 3) show that such algorithms can have arbitrarily bad worst-case approximation guarantees on the cost of the clustering induced by the decision tree, which we denote by  $cost(T, X)$ . This observation has sparked the development of more sophisticated approaches (Moshkovitz et al., 2020; Esfandiari et al., 2022; Charikar & Hu, 2022; Gamlath et al., 2021; Laber & Murtinho, 2021), all of which construct the tree  $T$  in a top-down manner: At every internal node  $u$ , an axis-aligned cut  $x_i \geq \theta$  partitions the data at node  $u$ , while ensuring that every cluster center  $c_l \in \mathcal{M}$  ends up in exactly one leaf of  $T$ . This last requirement is crucial to obtain approximation guarantees. While several works restrict  $T$  to have exactly  $k$  leaves, few also allow more than  $k$  leaves for better approximation (Makarychev & Shan, 2022; Frost et al., 2020). Appendix A provides a review of works on explainable  $k$ -means, while Appendix C gives a more detailed description of the Iterative Mistake Minimization (IMM) algorithm (Moshkovitz et al., 2020) that we build on in Section 4.
+
+# 3 EXPLAINABLE KERNEL K-MEANS VIA CUTS ON FEATURE MAPS
+
+The objective of this paper is to present an inherently interpretable model that approximates the kernel  $k$ -means algorithm. In this section, we make the first steps towards this goal. Standard supervised approaches for learning decision trees are questionable since they, as discussed earlier, can be arbitrarily bad for approximating even the standard  $k$ -means clustering. On the other hand, the aforementioned iterative decision tree construction admits good worst case guarantees in the context of  $k$ -means. Hence, a natural starting point would be to **kernelize** it. Unfortunately however,
+
+the method heavily relies on explicitly characterizing the cluster centers. In kernel clustering, these cluster centers  $c_{1},\ldots ,c_{k}$  reside in the RKHS  $\mathcal{H}$  instead of  $\mathbb{R}^d$ , and typically do not have a pre-image in the input space  $\mathbb{R}^d$ . Thus, operating on centers necessitates explicit computation of the feature maps, which is in contrast to the typical use of the kernel trick.
+
+A naive kernelization of existing algorithms would hence aim to operate in  $\mathcal{H}$  and find axis-aligned cuts using the coordinates of the projection  $\psi(x)$ . This approach has two fundamental limitations. For one, it requires an explicit characterization of the RKHS  $\mathcal{H}$  or the feature map  $\psi$ —the two famously implicit entities in the kernel trick. Explicit characterizations of  $\mathcal{H}$  or  $\psi$  however are not known for most kernels, and even if they are known, the dimension of  $\mathcal{H}$  could be infinite (Steinwart et al., 2006), making it practically impossible to iterate over all axes. This problem does not pose a practical concern when clustering a finite set of points. Any kernel matrix  $K \in \mathbb{R}^{n \times n}$ , computed on data  $X = \{x^{(1)}, \ldots, x^{(n)}\}$ , can be decomposed as  $K = \Phi \Phi^{\top}$  for some  $\Phi = [\phi_1(x^{(j)}) \dots \phi_D(x^{(j)})]_{j=1}^n \in \mathbb{R}^{n \times D}$  through Cholesky or eigendecomposition, providing a data-dependent feature map that we denote as  $\phi: X \to \mathbb{R}^D$ .
+
+The second concern is how to translate axis-aligned cuts  $\phi_j(x) \geqslant \theta$  back to axis-aligned cuts in  $\mathbb{R}^d$ . Here,  $\phi_j$  denotes the  $j$ -th coordinate of  $\phi: X \to \mathbb{R}^D$ . This issue is critically important in the context of interpretability, and can be decomposed into two principle concerns: Firstly, if  $\phi_j(x)$  is a function of more than one coordinate  $x_i$ , then a cut  $\phi_j(x) \geqslant \theta$  cannot correspond to an axis-aligned partition of  $\mathbb{R}^d$ . Secondly, even if  $\phi_j(x)$  depends on exactly one coordinate  $x_j$ , the cut  $\phi_j(x) \geqslant \theta$  cannot correspond to a cut  $x_j \geqslant \theta'$  unless  $\phi_j$  is monotonic.
+
+These conditions are not trivial and, as we show below, the popular Gaussian kernel does not satisfy either of the two conditions, whereas some additive kernels satisfy both. To formally show this, we introduce the notion of interpretable feature maps.
+
+Definition 1. (Interpretable feature maps) Let  $\phi : X \to \mathbb{R}^D$  be a feature map defined on the dataset  $X \subseteq \mathbb{R}^d$  such that  $K(x, y) = \langle \phi(x), \phi(y) \rangle = \sum_{j=1}^{D} \phi_j(x) \phi_j(y)$  for all  $x, y \in X$ . We say that the feature map  $\phi = (\phi_1, \ldots, \phi_D)$  is interpretable if each  $\phi_j(x)$  depends exactly on one coordinate of  $x$ .
+
+The following result shows that no feature map for the Gaussian kernel can be interpretable, even if we allow the feature maps to be data-dependent.
+
+Theorem 1. (The Gaussian kernel cannot have interpretable feature maps) Consider the Gaussian kernel  $K(x,y) = e^{-\gamma \| x - y\| _2^2}$  in  $d > 1$  dimensions. There exists a dataset  $X$  such that for any feature map  $\phi :X\to \mathbb{R}^{D}$  satisfying  $\langle \phi (x),\phi (y)\rangle = K(x,y)$  for all  $x,y\in X$ , there exists some  $j\in [D]$  such that  $\phi_j$  depends on more than just one input dimension of  $x\in X$ .
+
+The short proof is given in Appendix D.1. In Section 4, we resolve the inherent non-interpretability of feature maps for the Gaussian kernel by resorting to suitably chosen surrogate features. However, this still leaves the concern about monotonicity of  $\phi_j$ . This too cannot hold for the Gaussian kernel.
+
+Theorem 2. (The Gaussian kernel does not admit monotonic features) Consider the one-dimensional Gaussian kernel  $K(x,y) = e^{-\gamma |x - y|^2}$ . Then, there exists a dataset  $X \subseteq \mathbb{R}$  such that for any feature map  $\phi : X \to \mathbb{R}^D$  there exists a component  $\phi_j, j \in [D]$  that is not monotonic.
+
+The proof is included in Appendix D.2. Theorem 2 suggests that the existing notion of interpretability, specified through threshold cuts  $x_{i} \geqslant \theta$ , is too restrictive for deriving interpretable decision trees for kernels. Hence, we propose to expand the definition of interpretability. Although the partitions at each node should still align with the axes of  $\mathbb{R}^d$  (this ensures that the important features can be identified), we allow the partition at the node to be any interval and its complement.
+
+Definition 2. (Interpretable decision trees) Consider a decision tree  $T$  that partitions a dataset  $X \subset \mathbb{R}^d$  into  $k$  leaves. We call  $T$  an interpretable decision tree if at every node  $u \in T$ , there exists some  $i \in [d]$ ,  $\theta_1 < \theta_2 \in \mathbb{R}$  such that the data  $X^u$  that arrives at  $u$  is split into two disjoint subsets
+
+$$
+X _ {L} ^ {u} = \{x \in X ^ {u}: x _ {i} \in [ \theta_ {1}, \theta_ {2} ] \} \quad a n d \quad X _ {R} ^ {u} = \{x \in X ^ {u}: x _ {i} \notin [ \theta_ {1}, \theta_ {2} ] \}. \tag {3}
+$$
+
+To conclude this section, we remark that several important additive kernels do in fact admit feature maps that give rise to interpretable decision trees.
+
+Remark 1. (Certain additive kernels have interpretable and monotonic features) Additive kernels on  $\mathbb{R}^d$  refer to kernels that can be decomposed as  $K(x,y) = \sum_{i=1}^{d} K_i(x,y)$ , where each  $K_i$  is
+
+Algorithm 1 Kernel IMM for interpretable Taylor, or distance-based product kernels
+
+Input: Data  $X$ , integer  $k$ , kernel  $K$  with surrogate feature map  $\phi$ . For interpretable Taylor kernels see Definition 3, for distance-based kernels see Equation (4).
+
+Output: Interpretable decision tree  $T$  that partitions  $X$  into  $k$  leaves
+
+Get reference clustering  $C_1, \ldots, C_k \gets \text{KERNEL } k$ -MEANS(X, K, k)
+
+Compute surrogate features  $\Phi = \left[\phi_1(x)\dots \phi_d(x)\right]$  and centers  $\mathcal{M} = \{c_1,\ldots ,c_k\}$  under  $\phi$
+
+Construct a decision tree  $T \gets \mathrm{IMM}(\Phi, y, \mathcal{M})$ , where  $y$  denotes the cluster assignments.
+
+Translate threshold cuts of  $T$  back to a decision tree on  $X$ .
+
+a kernel on  $\mathbb{R}$ . Additive kernels are commonly used for comparing histograms, e.g. in computer vision (Maji et al., 2012; Vedaldi & Zisserman, 2012). In Appendix H, we consider three popular additive kernels and derive feature maps  $\phi$  for which any threshold cut  $\phi_j(x) \geq \theta$  translates to an axis-aligned cut  $x_j \in [\theta_1, \theta_2]$ . Hence, one may run IMM on the map  $\phi$  and directly translate the tree from the feature space to an interpretable decision tree.
+
+# 4 KERNELITERATIVEMISTAKEMINIMIZATION(KERNEL IMM)
+
+In this section, we present an interpretable kernel  $k$ -means algorithm that is applicable to a wide range of bounded product kernels, which include the Gaussian kernel  $K(x,y) = e^{-\gamma \| x - y\|_2^2}$  as well as other distance-based product kernels such as the Laplace kernel  $K(x,y) = e^{-\gamma \| x - y\|_1}$ . We focus on the aforementioned idea of constructing interpretable decision trees via axis-aligned cuts of suitably chosen feature maps  $\phi$ , thereby leveraging existing approximation guarantees. The base interpretable  $k$ -means method that we "kernelize" is the IMM algorithm, primarily due to its simpler analysis. We believe the same construction can also leverage worst-case approximation guarantees for other explainable  $k$ -means algorithms (with tighter guarantees). The proposed Kernel IMM algorithm is described in Algorithm 1, and illustrated in Figure 2.
+
+Surrogate features. Bounded product kernels on  $\mathbb{R}^d$  can be expressed as  $K(x,y) = \prod_{i=1}^{d} K_i(x,y)$ , where each  $K_i$  is a kernel on  $\mathbb{R}$ , operating on the  $i$ -th coordinate of data. We assume that  $K_i(x,y) \leq K_i(x,x) = 1$  for all  $x,y \in \mathbb{R}^d$ . The RKHS of a product kernel  $K$  is given by the tensorization  $\mathcal{H} = \mathcal{H}_1 \otimes \ldots \otimes \mathcal{H}_d$ , where  $\mathcal{H}_i$  is the RKHS of kernel  $K_i$ . The implicit feature map  $\psi: \mathbb{R}^d \to \mathcal{H}$  associated with the product kernel  $K$  can be related to the feature maps  $\psi_i: \mathbb{R} \to \mathcal{H}_i$  of the  $d$  individual kernels  $K_i$  via  $\langle \psi(x), \psi(y) \rangle = \prod_{i=1}^{d} \langle \psi_i(x_i), \psi_i(y_i) \rangle$ . In order to preserve axis-aligned structures, we propose to use a surrogate feature map  $\phi = (\phi_1, \dots, \phi_d)$ , where each  $\phi_i$  is a finite-dimensional approximation to  $\psi_i$ . Notably,  $\phi$  does not approximate a map to  $\mathcal{H}$  but instead to the Cartesian space  $\mathcal{H}_1 \times \ldots \times \mathcal{H}_d$ , and  $\langle \phi(x), \phi(y) \rangle \approx \sum_{i=1}^{d} \langle \psi_i(x_i), \psi_i(y_i) \rangle \neq \langle \psi(x), \psi(y) \rangle$ . Crucially however, we will show later that the additional cost induced by this step can be bounded. By construction,  $\phi = (\phi_i)_{i=1}^d$  is an interpretable feature map, and we may compute the set of cluster centers  $\mathcal{M}$  in the surrogate feature space associated with  $\phi$ . As described in 1, we let IMM operate on  $\mathcal{M}$ . Finally, interpretability of  $\phi$  ensures that axis-aligned cuts of  $\phi$  do in fact correspond to axis-aligned partitions of the input space  $\mathbb{R}^d$ . For the Gaussian kernel, our surrogate features  $\phi_i$  are Taylor approximations to  $\psi_i$ . We also present an alternative, quite general choice for  $\phi_i$  that works for all distance-based product kernels, but leads to data-dependent approximation guarantees.
+
+Interpretable Taylor kernels. We now define a class of kernels over  $\mathbb{R}$  that we refer to as interpretable Taylor kernels, and construct surrogate features for these kernels.
+
+Definition 3. (Interpretable Taylor kernels and their surrogate features) Let  $K_{i}$  be a bounded kernel on a connected domain  $\mathcal{X}_i \subseteq \mathbb{R}$ , of the form  $K_{i}(z,z^{\prime}) = f(z)f(z^{\prime})g(zz^{\prime})$  for some differentiable function  $f$  and an analytic function  $g$ . Denote by  $g^{(j)}$  the  $j$ -th derivative of  $g$ . Assume that  $g^{(j)}(0) \geq 0$  for all  $j \in \mathbb{N}$  and that  $z^{j-1}(zf'(z) + jf(z)) = 0$  has at most one solution  $z$  on  $\mathcal{X}_i$ . Then, we call  $K_{i}$  an interpretable Taylor kernel. Furthermore, assume  $K$  is a product of  $d$  interpretable Taylor kernels  $K_{1},\ldots,K_{d}$ , defined on a connected domain  $\mathcal{X} \subset \mathbb{R}^{d}$ . Given an integer  $M$ , we define the surrogate feature map for  $K$  as  $\phi(x) = (\phi_i(x_i))_{i=1}^d$ , where each  $\phi_i : \mathbb{R} \to \mathbb{R}^{M+1}$  is given by  $\phi_i(z) = (\phi_{i,j}(z))_{j=0}^M$  where we define  $\phi_{i,j}(z) = z^j f(z)\sqrt{g^{(j)}(0)/j!}$  for all  $j \leq M$ .
+
+![](images/f2adb8c151a2600df3029fc19d7d13e8556d2003dd21a65a48128afdca3fe108.jpg)  
+Figure 2: A schematic of Kernel IMM for Interpretable Taylor kernels.
+
+The Gaussian kernel with bandwidth  $\gamma > 0$  is a product of interpretable Taylor kernels, because with  $g(z) = \exp(2z)$  and  $f(z) = \exp(-\gamma z^2)$ ,  $zf'(z) + jf(z) = 0$  has at most one solution on  $(0, \infty)$ . This is sufficient because we can always assume data  $X$  that is clustered using the Gaussian kernel to be contained in  $(0, \infty)^d$ , as shifting does not affect the clustering cost.
+
+Theorem 3. (Threshold cuts in the surrogate feature space of interpretable Taylor kernels yield interpretable decision trees) Let  $K$  be product of interpretable Taylor kernels, and let  $\phi$  denote its surrogate feature map. Then, a threshold cut of the form  $\phi_{i,j}(x_i) \geqslant \theta$  leads to an interpretable decision tree in  $\mathbb{R}^d$ .
+
+Our choice of surrogate features doesn't only provide interpretable decision trees, but also preserves worst-case bounds on the cost. To formalize this, we generalize the price of explainability to kernels.
+
+Definition 4. (The price of explainability) For a set  $X \subset \mathbb{R}^d$  and an interpretable decision tree  $T$  as defined in 2, the price of explainability of  $T$  on  $X$  is defined as the ratio  $p(T, X) = \text{cost}(T, X) / \text{cost}_{opt}(X)$ , where  $\text{cost}_{opt}(X)$  is the optimal kernel  $k$ -means cost. The price of explainability of the data set  $X$  is given by  $p(X) = \min_T p(T, X)$ , where the minimum is taken over all interpretable decision trees.
+
+The Kernel IMM algorithm admits worst-case guarantees quite similar to the ones obtained in the linear setting, albeit with an additional factor of order  $d$ . This is a result of operating on the surrogate features that decouple the  $d$  input dimensions (see Appendix E).
+
+Theorem 4. (Price of explainability for interpretable Taylor kernels) Let  $C_1, \ldots, C_k$  be the clusters of a dataset  $X \subset \mathbb{R}^d$  derived from kernel  $k$ -means, where the kernel  $K$  is an interpretable Taylor kernel. Denote by  $\Phi = \phi(X)$  the surrogate features for  $K$  on  $X$ . Then, the interpretable decision tree obtained from Kernel IMM on  $\Phi$  satisfies  $p(T, X) = O\left(dk^2\right) + \frac{O(dk^2\delta)}{\text{cost}_{opt}(X)}$ , where  $\delta = \delta(M)$  depends on the order  $M$  of the surrogate feature map and  $\lim_{M \to \infty} \delta(M) = 0$  for any dataset  $X$ . Thus, by adaptively choosing  $M$  such that  $\delta = O(\text{cost}_{opt})$ , we obtain  $O(dk^2)$  bounds.
+
+The proof can be found in Appendix F. For interpretable Taylor kernels, Kernel IMM operates in a feature space of dimension  $O(dM)$ . In practice, often even low orders  $M \leq 5$  of the Taylor-based surrogate features provide enough flexibility to obtain good interpretable clusterings, and hence Kernel IMM computationally is not much more expensive than standard IMM.
+
+Extension to all distance-based product kernels. To conclude this section, we present an extension of Kernel IMM to the entire class of distance-based product kernels  $K(x,y) = \prod_{i=1}^{d} h(|x_i - y_i|)$ , where  $h$  is a decreasing function on the positive real line, with  $h(0) = 1$ . Examples include the Laplace kernel, for which  $h(t) = \exp(-t)$ . For these kernels, we simply set
+
+$$
+\phi_ {i, j} (x) = K _ {i} \left(x, x ^ {(j)}\right) = h \left(\left| x _ {i} - x _ {i} ^ {(j)} \right|\right) \tag {4}
+$$
+
+where  $x^{(j)}$  denotes the  $j$ th point in the dataset  $X$ . Because  $\phi_{i,j}(x) \leq \theta$  if and only if  $|x_i - x_i^{(j)}| \geq h^{-1}(\theta)$ , this leads to an interpretable decision tree. However, the worst-case bounds obtained from IMM do not translate directly.
+
+Theorem 5. (Price of explainability for distance-based product kernels) Let  $C_1, \ldots, C_k$  be the clusters of a dataset  $X \subset \mathbb{R}^d$  derived from kernel  $k$ -means, where the kernel  $K$  is a distance-based product kernel. Denote by  $\Phi = \phi(X)$  the surrogate features for  $K$  on  $X$ , where  $\phi_{i,j}(x) = K_i(x, x^{(j)})$ . Then, the interpretable decision tree obtained from Kernel IMM on  $\Phi$  satisfies  $p(T, X) = O(Cdk^2)$ , where  $C$  depends on the dataset  $X$  and the kernel  $K$ .
+
+We discuss this in Appendix G. Despite the data-dependent approximation guarantee, we observe in practice that the surrogate features defined in (4) perform well, sometimes even outperforming the surrogate Taylor features of the Gaussian kernel.
+
+# 5 GreEDy COST MINIMIZATION
+
+While the previous two sections cover several important kernels, there may nonetheless be cases where Kernel IMM is not applicable (because no suitable surrogate features exist), or not powerful enough to approximate kernel  $k$ -means (due to its restriction to just  $k$  leaves, as we illustrate in Figure 3). To address these issues, we propose kernelized variants of the ExKMC algorithm (Frost et al., 2020), which greedily add new leaves. This does not require computations in the feature space: Any set of reference clusters  $C_1, \ldots, C_k$  obtained from kernel  $k$ -means implicitly comes with a set of reference centers  $\mathcal{M} = \{c_1, \ldots, c_k\} \subset \mathcal{H}$ . The kernel trick ensures that the distance between the feature map of any point  $\psi(x) \in \mathcal{H}$  and a cluster center  $c_l$  can be evaluated without explicitly computing the center as
+
+$$
+\left\| \psi (x) - c _ {l} \right\| ^ {2} = K (x, x) + \frac {1}{\left| C _ {l} \right| ^ {2}} \sum_ {y, z \in C _ {l}} K (y, z) - \frac {2}{\left| C _ {l} \right|} \sum_ {y \in C _ {l}} K (x, y) \tag {5}
+$$
+
+Kernel ExKMC (Algorithm 2). The algorithm proceeds sequentially. For every node  $u$  of the decision tree, let  $X^u$  denote the set of points that reach the node  $u$ . A threshold cut  $(i, \theta)$  is chosen, separating  $X^u$  into two subsets  $X_L^u = X^u \cap \{x_i \leq \theta\}$  and  $X_R^u = X^u \cap \{x_i > \theta\}$ . The Kernel ExKMC algorithm chooses  $(i, \theta)$  as the minimizer of the cost function
+
+$$
+\operatorname {c o s t} _ {e x k m c} (u, i, \theta) = \min  _ {j, l \in [ k ]} \sum_ {x \in X _ {L} ^ {u}} \| \psi (x) - c _ {j} \| ^ {2} + \sum_ {x \in X _ {R} ^ {u}} \| \psi (x) - c _ {l} \| ^ {2} \tag {6}
+$$
+
+where the squared distances can be computed using (5). One reference center is chosen for each of the two child nodes of  $u$ . To decide on which node to split, we choose the one that maximizes the difference between the cost of not splitting, given by  $\min_{j\in [k]}\sum_{x\in X^u}\| \psi (x) - c_j\| ^2$ , and  $cost_{exkmc}(i,\theta)$ .
+
+Since all relevant computations happen in the input space, and no explicit feature map  $\psi$  is needed, the algorithm is applicable to arbitrary kernels and can be used to refine any given partition  $\hat{C}_1,\ldots ,\hat{C}_p$ . Unfortunately, we can prove that Kernel ExKMC on an empty tree does not admit approximation guarantees when limited to  $k$  leaves.
+
+Theorem 6 (ExKMC does not admit no worst-case bounds). For any  $m \in \mathbb{N}$  there exists a data set  $X \subset \mathbb{R}$  such that the price of explainability on  $X$  is  $p(X) = 1$ , IMM attain this minimum, but ExKMC (initialized on an empty tree) constructs a decision tree  $T$  with  $p(T, X) \geq m$ .
+
+![](images/63c9e8875911df07aca4c7aea2772abfac8cb86c6e21eaf438f6084cfaec288c.jpg)  
+Figure 3: Standard  $k$ -means is ill-suited for clustering certain datasets, and this translates to explainable  $k$ -means (not plotted here). Kernel  $k$ -means recovers the ground truth well. However, Kernel IMM is restricted to 3 leaves and not powerful enough to approximate it. To resolve this, we suggest Kernel ExKMC and Kernel Expand, which extend the tree to 6 leaves.
+
+![](images/dccfeaf6b04c496e53de1210c20335506b31649bd03649df009a788207174bda.jpg)
+
+![](images/d7370718aaac9772e4b66b04f195380f3d6c344266e9b946f69c5fb6dd292b57.jpg)
+
+![](images/0032cdf4b6e8a8a019405a637ef5992b19b9a3473320ece1395c8411df503f55.jpg)
+
+![](images/be65f5bfc177c8722d1925fad1409fcfb3df6ec28c93e120b15d9bfbf23d0394.jpg)
+
+# Algorithm 2 Kernel ExKMC
+
+Input: Number of leaves  $m\in \mathbb{N}$  reference clusters  $C_1\dots ,C_k$
+
+decision tree  $T$  with leaves  $\hat{C}_1,\dots ,\hat{C}_p$
+
+Output: Tree with  $m$  leaves  $\mathcal{L}$
+
+Initialize the set of leafs as  $\mathcal{L} = \{\hat{C}_1,\dots ,\hat{C}_p\}$  while  $|\mathcal{L}| <   m$  do
+
+Choose node  $X^u \in \mathcal{L}$  and a cut  $(i, \theta)$  minimizing (6)
+
+Update  $\mathcal{L}$  by replacing  $X^u\in \mathcal{L}$  by its two child nodes  $X_{L}^{u},X_{R}^{u}$
+
+end while
+
+# Algorithm 3 Kernel Expand
+
+Input: Number of leaves  $m \in \mathbb{N}$ ,
+
+reference clusters  $C_1\ldots ,C_k$
+
+decision tree  $T$  with leaves  $\hat{C}_1,\dots ,\hat{C}_p$
+
+Output: Tree with  $m$  leaves  $\mathcal{L}$
+
+Initialize the set of leafs as  $\mathcal{L} = \{\hat{C}_1,\dots ,\hat{C}_p\}$
+
+while  $|\mathcal{L}| <   m$  do
+
+Choose data  $X^u \in \mathcal{L}$  and a cut  $(i, \theta)$  minimizing (7)
+
+Update  $\mathcal{L}$  by replacing  $X^u\in \mathcal{L}$  by its two child nodes  $X_L^u$ $X_{R}^{u}$
+
+end while
+
+The proof can be found in Appendix J. If Kernel ExKMC is run as a refinement of Kernel IMM (which constructs an interpretable decision tree in the sense of Definition 2 with two-sided threshold cuts) then we suggest sticking to this expanded notion of interpretability, and Kernel ExKMC at every node chooses  $(i,\theta_1,\theta_2)\in [d]\times \mathbb{R}^2$  and checks whether  $x_{i}\in [\theta_{1},\theta_{2}]$  or not.
+
+Kernel Expand (Algorithm 3). We suggest another algorithm, Kernel Expand, that also adds new leaves to an existing tree, but maximizes the purity of each leaf. This is achieved by minimizing
+
+$$
+\operatorname {c o s t} _ {\exp} (i, \theta) = \min  _ {j, l \in [ k ]} \left(| \{x \in X _ {L} ^ {u}: c (x) \neq c _ {j} \} | + | \{x \in X _ {R} ^ {u}: c (x) \neq c _ {l} \} |\right) \tag {7}
+$$
+
+where  $c(x)$  denotes the cluster center of  $x$  according to the reference partition  $\mathcal{M}$ . To decide on which node to split, we again choose the one that maximizes the difference between the cost of not splitting, given by  $\min_{j\in [k]}|\{x\in X^u:c(x)\neq c_j\}|$ , and  $cost_{expand}(i,\theta)$ .
+
+Remark 2. (Kernel Expand does not admit worst-case bounds) In the same way as Kernel ExKMC, Kernel Expand does not admit worst-case bounds when initialized without Kernel IMM. This is obvious from the example in Section 3 of (Moshkovitz et al., 2020).
+
+Finally, it is important to note that while adding more leaves improves the clustering, it gradually reduces the interpretability of the tree. Thus, there is a trade-off between explainability and accuracy in Kernel ExKMC and Kernel Expand.
+
+# 6 EXPERIMENTS
+
+We validate our algorithms on a number of benchmark datasets, including three synthetic clustering datasets, Pathbased, Aggregation and Flame (Franti & Sieranoja) and real datasets, Iris (Fisher, 1936) and Wisconsin breast cancer (Street et al., 1993). On all five datasets, we evaluate Kernel IMM as well as the greedy cost minimization algorithms from Section 5, Kernel ExKMC and Kernel Expand, both of which refine Kernel IMM by adding more leaves. All experiments reported here use either the Laplace or the Gaussian kernel. In Appendix H, we derive suitable approximate feature maps and evaluate Kernel IMM for the additive  $\chi^2$  kernel, which is well-suited for clustering histograms or distributions. These results underline the usefulness of our method beyond distance-based product
+
+![](images/c4fb5c5b11328fb9bbedfdb2495a4f2ab6d39d2b453b305d85150c72e18ab23b.jpg)  
+Figure 4: We verify the approximation properties of our algorithms by computing the price of explainability (left plot). We also compare the clusters obtained on  $k$ -means and IMM, as well as kernel  $k$ -means and our three algorithms to the ground truth via the Rand index (right plot).
+
+![](images/458a510e945d3778c83f55ea4e02ea059d180f5f08f8f3e04e860d286d557e63.jpg)
+
+kernels. Finally, we also compare Kernel IMM to CART, which performs similarly well. Details on all experiments can be found in Appendix I. Although Kernel IMM can lead to sub-optimal solutions, its greedy refinements mostly result in a price of explainability close to 1 (see Figure 3 for the Pathbased dataset). Figure 4 reports the price of explainability of the three proposed explainable kernel clustering methods (closer to 1 is better). Since the ground truth is known in all datasets considered here, Figure 4 also reports the adjusted Rand index (Pedregosa et al., 2011) that measures the agreement between the interpretable clusters and the true labels (a higher value is better). This shows that kernel  $k$ -means is naturally superior to  $k$ -means and explainable  $k$ -means, and one can observe that Kernel ExKMC and Kernel Expand mostly achieve Rand index as good as that of kernel  $k$ -means, establishing that our methods provide both interpretable and accurate clusters. In addition, Kernel IMM improves over  $k$ -means and IMM in recovering the ground truth.
+
+# 7 CONCLUSION AND DISCUSSION
+
+In this paper, we contribute to the increasing need for interpretable clustering methods by explaining kernel  $k$ -means using decision trees. By operating on carefully chosen features, interpretability can be preserved, while still allowing for approximation guarantees. We characterize the obstacles that interpretability faces in kernel clustering, and find that interpretable feature maps play a key role. These maps may also be useful to understand the interpretability of other kernel methods, which still rely almost exclusively on post-hoc explainability.
+
+On the price of explainability for kernels. We observe below that the price of explainability for kernel  $k$ -means, as defined in Definition 4, may not be finite for arbitrary kernels.
+
+Theorem 7 (Unbounded price of explainability). Consider either the quadratic kernel  $K(x, y) = \langle x, y \rangle^2$  or the  $\epsilon$ -neighborhood kernel  $K(x, y) = \mathbf{1}_{[0, \epsilon]}(\|x - y\|_2)$ . For both, there exists a dataset  $X$  such that  $p(X) = \infty$ .
+
+The above result, proved in Appendix K, is in stark contrast to standard  $k$ -means, where IMM always ensures a bounded price of explainability. Our experiments further show that the price of explainability is close to 1 even in cases where the interpretable clustering model is quite different from the reference partition. In fact, if the kernel  $k$ -means cost is  $\Omega(n)$  for a dataset of size  $n$  and the kernel is bounded, then the price of explainability is  $O(1)$ . This indicates that typical metrics should be adjusted depending on the kernel and the data. While the Rand index may be used if a ground truth is known, a fundamental question remains unsolved: Under what conditions on the data can a decision tree achieve a good agreement with the nonlinear partitions of kernel  $k$ -means? Another question is whether approximation guarantees can be improved. It is very likely that specific kernels benefit from algorithms tailored to their cost functions. Finally, investigating lower bounds (beyond the ones a linear kernel inherits from existing results on explainable  $k$ -means) also poses an interesting direction for future work.
+
+Ethics statement. While we do not expect negative societal impacts of our method, we remark that clustering algorithms themselves can be biased, and this can translate to its interpretable approximations.  
+Reproducibility. We add further details on the experiments in Appendix I. In addition, all proofs for our theoretical results can be found in the appendix, and have been referenced in the main paper.
+
+# ACKNOWLEDGMENTS
+
+This paper is supported by the DAAD programme Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the Federal Ministry of Education and Research, and the German Research Foundation (Research Grant GH 257/4-1).
+
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+
+# A RELATED WORK ON EXPLAINABLE CLUSTERING
+
+There are several papers that suggest improvements of the original IMM algorithm (Moshkovitz et al., 2020). Most rely on random threshold cuts to obtain better worst-case bounds on the price of explainability for  $k$ -means and  $k$ -medians (Makarychev & Shan, 2022; Esfandiari et al., 2022; Makarychev & Shan, 2023). As of today, the sharpest known upper bound on the price of explainability for  $k$ -means is  $O(k\log \log k)$  (Gupta et al., 2023). It is also known that the price of explainability for  $k$ -means is  $\Omega (k)$  (Gamlath et al., 2021), leaving a small gap. While the above works provide dimension-independent guarantees, it is also known that in low dimensions, even better guarantees can be provided (Laber & Murtinho, 2021; Charikar & Hu, 2022). A different line of research has loosened the restriction to decision trees with  $k$  leaves, leading to both improved practical performance (Frost et al., 2020) and better worst-case bounds on the price of explainability (Makarychev & Shan, 2022). In addition, Laber et al. (2023) investigate shallow decision trees for explainable clustering, while Deng et al. (2023) show that depth reduction is impossible for explainable  $k$ -means and  $k$ -medians. There has also been some research on other cost functions such as  $k$ -centers (Laber & Murtinho, 2021) or general  $\| \cdot \|_p$  norms (Gamlath et al., 2021). However, to the best of our knowledge, there has not yet been an extension to the closely related kernel  $k$ -means problem, a gap that we aim to bridge in this work. Other works on interpretable clustering investigate polyhedral descriptions of the clusters (Lawless & Gunluk, 2023), choose similarity-based prototypes for each cluster (Carrizosa et al., 2022), or fit sparse oblique trees to describe the clusters (Gabadolla & Carreira-Perpiñán, 2022).
+
+# B THE KERNEL K-MEANS ALGORITHM
+
+We first remind the reader of the following result.
+
+Lemma 1. Let  $K$  be a kernel operating on data  $X \subset \mathbb{R}^d$ . Denote  $\phi$  for a (possibly data-dependent) feature map of  $K$ . Let  $C_1, \ldots, C_k$  be a partition of  $X$  into  $k$  clusters with means  $c_1, \ldots, c_k \in \mathcal{H}$ . Then for any  $x \in X$ , we have
+
+$$
+\underset {l \in [ k ]} {\arg \min } \| \phi (x) - c _ {l} \| ^ {2} = \underset {l \in [ k ]} {\arg \min } \left(\frac {1}{| C _ {l} | ^ {2}} \sum_ {y, z \in C _ {l}} K (y, z) - \frac {2}{| C _ {l} |} \sum_ {y \in C _ {l}} K (x, y)\right) \tag {8}
+$$
+
+Proof. First recall the identity  $\| a - b \|^2 = \| a \|^2 + \| b \|^2 - 2\langle a, b \rangle$  and note that for any cluster  $C_l$ , its mean is given by
+
+$$
+c _ {l} = \frac {1}{| C _ {l} |} \sum_ {x \in C _ {l}} \phi (x)
+$$
+
+Expressing everything in terms of the kernel  $K$ , we see that
+
+$$
+\| \phi (x) - c _ {l} \| ^ {2} = K (x, x) + \frac {1}{| C _ {l} | ^ {2}} \sum_ {y, z \in C _ {l}} K (y, z) - \frac {2}{| C _ {l} |} \sum_ {y \in C _ {l}} K (x, y)
+$$
+
+Note that the first term does not depend on  $l \in [k]$ .
+
+![](images/617fecc14ffc4a69804f518c395d4a76f7950373734904ee01f1d597b2423fc1.jpg)
+
+Clearly, equation 8 allows computing distances in the feature space by simply evaluating  $K$ .
+
+![](images/7c8d53a4ddc053da20358be21e9775137b1db4b040ee7346ab718a5a09222faf.jpg)  
+Figure 5: The threshold cut illustrated by the black vertical line defines a decision tree  $T$  with 2 leaves. While some points do not end up in the same leaf as their corresponding center, the tree  $T$  clearly does a good job in approximating the two clusters.
+
+Algorithm 4 Kernel  $k$ -means  
+Input: Kernel  $K$  for data  $X\subset \mathbb{R}^d$  , integer  $k\in \mathbb{N}$    
+Output: Partitioning of  $X$  into  $k$  clusters  $\mathcal{C} = (C_1,\dots ,C_k)$    
+Initialize the clusters  $C_1,\ldots ,C_k$    
+Converged  $\leftarrow$  FALSE   
+while Converged  $=$  FALSE do for  $x\in X$  do Assign  $x$  to  $C_l^\prime$  such that  $l = \mathrm{argmin}_{j\in [k]}\| \phi (x) - c_j\|^2$  using equation (8) end for if  $C_l^{\prime} = C_l$  for all  $l\in [k]$  then Converged  $\leftarrow$  TRUE end if Update  $\mathcal{C} = (C_1',\dots ,C_k')$    
+end while
+
+# C ITERATIVE MISTAKE MINIMIZATION (IMM) ALGORITHM
+
+In this section, we review the IMM algorithm (Moshkovitz et al., 2020). Consider data  $X \subset \mathbb{R}^d$  and centers  $\mathcal{M}$  obtained from the  $k$ -means algorithm. For any  $x \in X$ , we denote  $c(x) \in \mathcal{M}$  for the correct center according to the  $k$ -means algorithm. The algorithm constructs a decision tree  $T$  that sequentially the input space  $\mathbb{R}^d$  into  $k$  axis-aligned cells using threshold cuts of the form  $x_i \geq \theta$ . At every node  $u$  of the tree, IMM chooses the threshold cut that minimizes the number of mistakes. A mistake happens if a point  $x$  that arrives at node  $u$  together with its correct cluster center  $c(x) \in \mathcal{M}$  is now separated from this center as a result of the cut. This happens when  $x_i \leq \theta$  but  $c_i(x) > \theta$  or vice versa. For any point  $x \in X$ , we denote  $\mu(x)$  for the center that ends up in the same leaf of  $T$  as  $x$ . This is not necessarily  $c(x)$ , as we illustrate in Figure 5.
+
+Let us also briefly recap the theoretical analysis of IMM for  $k$ -means, as presented by Moshkovitz et al. (2020). For each node  $u$  of the tree  $T$  built by IMM, denote by  $\mathcal{M}^u$  the remaining set of centers that arrive at the node. The tree  $T$  induces a partitioning  $\hat{C}_1, \ldots, \hat{C}_k$ . Writing  $t(u)$  for the number of mistakes at a node  $u \in T$  and denoting  $D(u) = \max_{a, b \in \mathcal{M}^u} \| a - b \|^2$ , it holds that
+
+$$
+\begin{array}{l} c o s t (\hat {C} _ {1}, \ldots , \hat {C} _ {k}) \leq \sum_ {l = 1} ^ {k} \sum_ {x \in \hat {C} _ {l}} \| x - \mu (x) \| ^ {2} \\ \leq \sum_ {l = 1} ^ {k} \sum_ {x \in \hat {C} _ {l}} \left(2 \| x - c (x) \| ^ {2} + 2 \| c (x) - \mu (x) \| ^ {2}\right) \tag {9} \\ \leq 2 \cdot c o s t (C _ {1}, \dots , C _ {k}) + 2 \sum_ {u \in T} t (u) D (u) \\ \end{array}
+$$
+
+The main argument in IMM is now given by the following result.
+
+Lemma 2. Let  $u \in T$  be any node of the IMM decision tree  $T$ . Denote  $X_{cor}^{u}$  for the subset of points  $x$  that arrive at  $u$  together with their correct center  $c(x)$ .
+
+1. For any  $i \in [d]$ , it holds that
+
+$$
+t (u) \cdot \max  _ {a, b \in \mathcal {M} ^ {u}} \left(a _ {i} - b _ {i}\right) ^ {2} \leq 4 k \cdot \sum_ {x \in X _ {c o r} ^ {u}} \left(x _ {i} - c _ {i} (x)\right) ^ {2} \tag {10}
+$$
+
+2. This directly implies
+
+$$
+t (u) D (u) \leq 4 k \cdot \sum_ {x \in X _ {c o r} ^ {u}} \| x _ {i} - c _ {i} (x) \| ^ {2} \tag {11}
+$$
+
+The proof of the first statement is based on the observation that along every axis  $i \in [d]$ , and for any threshold cut halfway between two centers  $a, b \in \mathcal{M}^u$  (projected to the  $i$ -th axis), at least  $t(u)$  mistakes are made, by definition of  $t(u)$ . The second statement then directly follows from the first by summing over all  $d$  coordinates. Together with Equation 9 and the fact that the maximum depth of the IMM tree is  $k$ , Lemma 2 yields  $O(k^2)$  bounds on the price of explainability. Let us use this opportunity to point out two things.
+
+- Since only threshold cuts halfway between two projected centers are considered in the proof of Lemma 2, the algorithm actually preserves worst-case bounds even when we only check the number of mistakes along these  $O(dm)$  threshold cuts, instead of along all possible  $O(dn)$  cuts. In other words, the IMM algorithm (as well as our kernelized version) can be run in data-independent time.  
+- The IMM algorithm is essentially a supervised learning algorithm, recreating a partition  $C_1, \ldots, C_k$  no matter how optimal it may be. This observation is crucial when we analyze Kernel IMM on our surrogate features (with respect to a surrogate kernel), for which the reference clustering may not even be close to the optimal partition with respect to this surrogate kernel.
+
+# D PROOFS FROM SECTION 3
+
+# D.1 PROOF OF THEOREM 1
+
+Theorem. (The Gaussian kernel cannot have interpretable feature maps) Consider the Gaussian kernel in  $d > 1$  dimensions. There exists a dataset  $X$  such that for any feature map  $\phi : X \to \mathbb{R}^D$  satisfying  $\langle \phi(x), \phi(y) \rangle = K(x, y)$  for all  $x, y \in X$ , there exists some  $j \in [D]$  such that  $\phi_j$  depends on more than just one input dimension of  $x \in X$ .
+
+Proof. We present a simple dataset  $X$  such that no feature map for the Gaussian kernel  $K(x, y) = \exp(-\gamma \|x - y\|^2)$  can depend only on one input coordinate in each of its elements  $\phi_1, \ldots, \phi_D$ , no matter how large  $D$  may be. Consider three vectors in  $\mathbb{R}^2$ , given by
+
+$$
+x ^ {(1)} = \left( \begin{array}{c} 0 \\ 0 \end{array} \right), x ^ {(2)} = \left( \begin{array}{c} 1 \\ 0 \end{array} \right), x ^ {(3)} = \left( \begin{array}{c} 1 \\ 1 \end{array} \right)
+$$
+
+For sufficiently large  $\gamma > 0$ , we know that  $K_{1,2} + K_{2,3} < 1$ . Assume there exists some interpretable feature map  $\phi: X \to \mathbb{R}^D$ . Then, we can write  $\phi(x) = (f(x), g(x))$ , where  $f(x)$  is a function of  $x_1$  and  $g(x)$  is a function of  $x_2$ . By our choice of points, we may write
+
+$$
+\begin{array}{l} \phi (x ^ {(1)}) = (a, b) \\ \phi (x ^ {(2)}) = (c, b) \\ \phi (x ^ {(3)}) = (c, d) \\ \end{array}
+$$
+
+for some vectors  $a, b, c, d$ . This implies
+
+$$
+\begin{array}{l} \boldsymbol {K} _ {1, 3} = a ^ {T} c + b ^ {T} d \\ = a ^ {T} c + \left\| b \right\| ^ {2} + \left(b ^ {T} d - \left\| b \right\| ^ {2}\right) \\ = \boldsymbol {K} _ {1, 2} + \left(b ^ {T} d + \| c \| ^ {2} - 1\right) \\ = \boldsymbol {K} _ {1, 2} + \boldsymbol {K} _ {2, 3} - 1 \\ <   0 \\ \end{array}
+$$
+
+a contradiction to  $K_{1,3} \geq 0$ . For other  $\gamma > 0$ , simply rescale the data accordingly.
+
+![](images/5de111ba13489e9ce35b8987d69f84776a79c39591bb62b9916f2b1b9da56af4.jpg)
+
+# D.2 PROOF OF THEOREM 2
+
+Theorem. Consider the one-dimensional Gaussian kernel  $K(x,y) = \exp (-|x - y|^2)$ . Then, there exists a dataset  $X\subset \mathbb{R}$  such that for any feature map  $\phi = (\phi_j)_{j = 1}^D$  there exists a component  $\phi_j$  that is not monotonic.
+
+Proof. Consider the Gaussian kernel  $K(x,y) = \exp \left(- (x - y)^2\right)$  on  $\mathbb{R}$ . Choose a dataset  $X = \{x^{(1)}, x^{(2)}, x^{(3)}\}$  with  $x^{(1)} < x^{(2)} < x^{(3)}$  and corresponding kernel matrix  $\pmb{K}$  satisfying
+
+$$
+\boldsymbol {K} _ {1, 2} + \boldsymbol {K} _ {2, 3} <   \boldsymbol {K} _ {1, 3} + 1
+$$
+
+This is surely the case when the points are at a sufficiently large distance. Now assume there exists a feature map  $\phi : X \to \mathbb{R}^D$ ,  $\phi(x^{(i)}) = (v_i, w_i)$  where  $(v_i)$  is non-decreasing and  $(w_i)$  is non-increasing for  $i = 1, 2, 3$ . In other words, we assume there exist two vectors  $v_1, w_1$  as well as non-negative vectors  $\epsilon_1, \epsilon_2, \epsilon_3, \delta_1, \delta_2, \delta_3$  such that, writing
+
+$$
+\boldsymbol {\Phi} = \left( \begin{array}{c c c} v _ {1} & v _ {1} + \epsilon_ {1} & v _ {1} + \epsilon_ {1} + \epsilon_ {2} \\ w _ {1} & w _ {1} - \delta_ {1} & w _ {1} - \delta_ {1} - \delta_ {2} \end{array} \right) = \left( \begin{array}{c c c} v _ {1} & v _ {1} + \epsilon_ {1} & v _ {1} + \epsilon_ {3} \\ w _ {1} & w _ {1} - \delta_ {1} & w _ {1} - \delta_ {3} \end{array} \right)
+$$
+
+it holds that  $\Phi^T\Phi = K$ . This condition gives rise to
+
+$$
+v _ {1} ^ {T} v _ {1} + w _ {1} ^ {T} w _ {1} = 1
+$$
+
+$$
+\left(v _ {1} + \epsilon_ {1}\right) ^ {T} \left(v _ {1} + \epsilon_ {1}\right) + \left(w _ {1} - \delta_ {1}\right) ^ {T} \left(w _ {1} - \delta_ {1}\right) = 1
+$$
+
+$$
+v _ {1} ^ {T} \left(v _ {1} + \epsilon_ {1}\right) + w _ {1} ^ {T} \left(w _ {1} - \delta_ {1}\right) = K _ {1, 2}
+$$
+
+which implies
+
+$$
+v _ {1} ^ {T} \epsilon_ {1} - w _ {1} ^ {T} \delta_ {1} = \boldsymbol {K} _ {1, 2} - 1
+$$
+
+and hence
+
+$$
+\left\| \epsilon_ {1} \right\| ^ {2} + \left\| \delta_ {1} \right\| ^ {2} = 2 - 2 K _ {1, 2}
+$$
+
+Using the same arguments for the two other pairs of points from  $X$ , we derive in the same way
+
+$$
+\left\| \epsilon_ {2} \right\| ^ {2} + \left\| \delta_ {2} \right\| ^ {2} = 2 - 2 K _ {2, 3}
+$$
+
+$$
+\left\| \epsilon_ {3} \right\| ^ {2} + \left\| \delta_ {3} \right\| ^ {2} = 2 - 2 K _ {1, 3}
+$$
+
+Since  $\epsilon_3 = \epsilon_1 + \epsilon_2$  and  $\delta_3 = \delta_1 + \delta_2$ , this implies
+
+$$
+\left\| \epsilon_ {1} \right\| ^ {2} + \left\| \epsilon_ {2} \right\| ^ {2} + 2 \epsilon_ {1} ^ {T} \epsilon_ {2} + \left\| \delta_ {1} \right\| ^ {2} + \left\| \delta_ {2} \right\| ^ {2} + 2 \delta_ {1} ^ {T} \delta_ {2} = 2 - 2 K _ {1, 3}
+$$
+
+However, this is equivalent to
+
+$$
+2 - 2 \boldsymbol {K} _ {1, 2} + 2 - 2 \boldsymbol {K} _ {2, 3} + 2 \left(\epsilon_ {1} ^ {T} \epsilon_ {2} + \delta_ {1} ^ {T} \delta_ {2}\right) = 2 - 2 \boldsymbol {K} _ {1, 3} \iff
+$$
+
+$$
+0 <   \epsilon_ {1} ^ {T} \epsilon_ {2} + \delta_ {1} ^ {T} \delta_ {2} = \boldsymbol {K} _ {1, 2} + \boldsymbol {K} _ {2, 3} - 1 - \boldsymbol {K} _ {1, 3} <   0
+$$
+
+a contradiction.
+
+![](images/e5648a928126cccbaec35fa869dfbd6ba043bd7c5585c04ca6abbb7aa3a7158b.jpg)
+
+# E DECOUPLING DIMENSIONS FOR BOUNDED PRODUCT KERNELS
+
+In this section, we prove that decoupling the input dimensions of bounded product kernels introduces errors no larger than  $O(d)$ . Recall that our surrogate feature map is given by
+
+$$
+\phi (x) = \left(\phi_ {i} (x)\right) _ {i = 1} ^ {d}
+$$
+
+where each  $\phi_{i}$  is a valid feature map for the one-dimensional kernel  $K_{i}$ . This surrogate feature map itself is associated with a surrogate additive kernel:
+
+$$
+\langle \phi (x), \phi (y) \rangle = \sum_ {i = 1} ^ {d} \langle \phi_ {i} (x), \phi_ {i} (y) \rangle = \sum_ {i = 1} ^ {d} K _ {i} (x, y) \neq K (x, y)
+$$
+
+Of course, it is not clear that running IMM with respect to this surrogate kernel is actually sensible and not only ensures interpretability, but also preserves worst-case bounds. The justification is given in the following Lemma.
+
+Lemma 3. Let  $X \subset \mathbb{R}^d$  be a dataset, partitioned into  $k$  clusters  $C_1, \ldots, C_k$ . Denote by  $\text{cost}(C_1, \ldots, C_k)$  the kernel  $k$ -means cost function associated with a distance-based product kernel  $K$  on  $\mathbb{R}^d$ , and by  $\text{cost}_i$  the cost with respect to the feature map  $\phi_i$ . Then
+
+$$
+c o s t (C _ {1}, \dots , C _ {k}) \leq \sum_ {i = 1} ^ {d} c o s t _ {i} (C _ {1}, \dots , C _ {k}) \leq d \cdot c o s t (C _ {1}, \dots , C _ {k})
+$$
+
+Proof. First, use the GM-AM inequality and the fact that  $K_{i}(x,y) \leq 1$  to obtain
+
+$$
+K (x, y) = \prod_ {i = 1} ^ {d} K _ {i} (x, y) \leq \left(\prod_ {i = 1} ^ {d} K _ {i} (x, y)\right) ^ {1 / d} \leq \frac {1}{d} \sum_ {i = 1} ^ {d} K _ {i} (x, y)
+$$
+
+Thus, rewriting the kernel  $k$ -means cost function, we see that
+
+$$
+\begin{array}{l} \operatorname {c o s t} \left(C _ {1}, \dots , C _ {k}\right) = | X | - \sum_ {l = 1} ^ {k} \frac {1}{| C _ {l} |} \sum_ {x, y \in C _ {l}} K (x, y) \\ \geq | X | - \sum_ {l = 1} ^ {k} \frac {1}{| C _ {l} |} \sum_ {x, y \in C _ {l}} \frac {1}{d} \sum_ {i = 1} ^ {d} K _ {i} (x, y) \\ = \frac {1}{d} \sum_ {i = 1} ^ {d} \left(| X | - \sum_ {l = 1} ^ {k} \frac {1}{| C _ {l} |} \sum_ {x, y \in C _ {l}} K _ {i} (x, y)\right) \\ = \frac {1}{d} \sum_ {i = 1} ^ {d} c o s t _ {l} (C _ {1}, \dots , C _ {k}) \\ \end{array}
+$$
+
+This implies the upper bound from Lemma 3. For the lower bound, note that for any  $x,y\in \mathbb{R}^d$
+
+$$
+\begin{array}{l} \left\| \phi (x) - \phi (y) \right\| ^ {2} = 2 \left(1 - K (x, y)\right) \\ = 2 \left(1 - \prod_ {i = 1} ^ {d} K _ {i} (x, y)\right) \\ \leq 2 \left(\sum_ {i = 1} ^ {d} (1 - K _ {i} (x, y))\right) \\ = \sum_ {i = 1} ^ {d} \| \phi_ {i} (x) - \phi_ {i} (y) \| ^ {2} \\ \end{array}
+$$
+
+where we use the fact that  $\left|\prod_{i=1}^{d} a_i - \prod_{i=1}^{d} b_i\right| \leq \left|\sum_{i=1}^{d} a_i - b_i\right|$  for any  $|a_i|, |b_i| \leq 1$ . Rewriting the kernel  $k$ -means cost to make use of this observation, we arrive at
+
+$$
+\begin{array}{l} c o s t (C _ {1}, \dots , C _ {k}) = \sum_ {l = 1} ^ {k} \frac {1}{2 | C _ {l} |} \sum_ {x, y \in C _ {l}} \| \phi (x) - \phi (y) \| ^ {2} \\ \leq \sum_ {i = 1} ^ {d} \sum_ {l = 1} ^ {k} \frac {1}{2 | C _ {l} |} \sum_ {x, y \in C _ {l}} \| \phi_ {i} (x) - \phi_ {i} (y) \| ^ {2} \\ = \sum_ {i = 1} ^ {d} c o s t _ {i} \left(C _ {1}, \dots , C _ {k}\right) \\ \end{array}
+$$
+
+![](images/1f84c5f47e9b918b8bd6eb8b6d123409cb2e389e7fc4d155b2bc5adce4d5c895.jpg)
+
+# F INTERPRETABLE TAYLOR KERNELS
+
+# F.1 PROOF OF THEOREM 3
+
+Theorem. (Threshold cuts in the surrogate feature space of interpretable Taylor kernels yield interpretable decision trees) Let  $K$  be product of interpretable Taylor kernels, and let  $\phi$  denote its surrogate feature map. Then, a threshold cut of the form  $\phi_{i,j}(x_i) \geqslant \theta$  leads to an interpretable decision tree in  $\mathbb{R}^d$ .
+
+Proof. Recall our notion of surrogate features for interpretable Taylor kernels: For all  $i \in [d]$  and  $x \in X$ , we define
+
+$$
+\phi_ {i} (x) = \left(f (x _ {i}) x _ {i} ^ {j} \sqrt {\frac {g ^ {(j)} (0)}{j !}}\right) _ {j = 0} ^ {M}
+$$
+
+and refer to the concatenation  $\phi(x) = (\phi_i(x))_{i=1}^d$  as the surrogate feature map of order  $M$ . Note that  $\phi_{i,j}'(x) \propto x_i^{j-1}$  ( $f'(x_i)x_i + jf(x_i)$ ). For interpretable Taylor kernels, this expression is zero for at most one point  $x$  in the domain of the kernel. Thus, when we run IMM on these surrogate features, the threshold cuts  $\phi_{i,j}(x) \gtrsim \theta$  can be translated to interpretable decision trees in the input space.
+
+# F.2 PROOF OF THEOREM 4
+
+Theorem. Let  $C_1, \ldots, C_k$  be the clusters of a dataset  $X \subset \mathbb{R}^d$  derived from kernel  $k$ -means, where the kernel  $K$  is an interpretable Taylor kernel. Denote by  $\Phi = \phi(X)$  the surrogate features for  $K$  on  $X$ . Then, the interpretable decision tree obtained from Kernel IMM on  $\Phi$  satisfies  $p(T, X) = O\left(dk^2\right) + \frac{O(dk^2\delta)}{\text{cost}_{opt}(X)}$ , where  $\delta = \delta(M)$  depends on the order  $M$  of the surrogate feature map and  $\lim_{M \to \infty} \delta(M) = 0$  for any dataset  $X$ . Thus, by adaptively choosing  $M$  such that  $\delta = O(\text{cost}_{opt})$ , we obtain  $O(dk^2)$  bounds.
+
+Proof. We begin by noting that for all  $i \in [d]$  and  $x, y \in X$ , and any integer  $M$ , Taylor's formula ensures that
+
+$$
+\begin{array}{l} |K_{i}(x,y) - \langle \phi_{i}(x),\phi_{i}(y)\rangle | = f(x_{i})f(y_{i})\cdot \left|\sum_{j = M}^{\infty}\frac{g^{(j)}(0)}{j!} (x_{i}y_{i})^{j}\right| \\ \leq \left(\max  _ {x \in X} | f (x _ {i}) | ^ {2}\right) \cdot \frac {\| g ^ {(j)} \| _ {\infty ([ 0 , x _ {i} y _ {i} ])} \left(x _ {i} y _ {i}\right) ^ {M}}{M !} \\ \end{array}
+$$
+
+Since  $g$  is assumed to be analytic,  $\lim_{M\to \infty}\langle \phi_i(x),\phi_i(y)\rangle = K_i(x,y)$  for all  $x,y$ . Thus, the surrogate features  $\phi_{i}$  approximate the one-dimensional kernels  $K_{i}$  within some  $\delta >0$  that vanishes as  $M\rightarrow \infty$ . Now, fix a dataset  $X$  and let  $\hat{C}_1,\dots \hat{C}_k$  be the interpretable clusters chosen by IMM
+
+on the surrogate features  $\phi$ . Denote by  $C_1, \ldots, C_k$  the reference clusters obtained from unrestricted kernel  $k$ -means with respect to the original, interpretable Taylor kernel  $K$ . Denote by cost the kernel  $k$ -means cost with respect to the product kernel  $K$ , by  $\text{cost}_i$  the costs with respect to the univariate kernels  $K_i$ , and by  $\text{cost}_i$  the cost with respect to the approximate kernel implicitly defined by virtue of the surrogate feature maps  $\phi_i$ . Using Lemma 3 from Appendix E and keeping track of the approximation error, we obtain
+
+$$
+\begin{array}{l} c o s t (\hat {C} _ {1}, \dots , \hat {C} _ {k}) \leq \sum_ {i = 1} ^ {d} c o s t _ {i} (\hat {C} _ {1}, \dots , \hat {C} _ {k}) \\ \leq \sum_ {i = 1} ^ {d} \left(\widetilde {\operatorname {c o s t} _ {i}} (\hat {C} _ {1}, \dots , \hat {C} _ {k}) + O (\delta)\right) \\ \leq O (d \delta) + \sum_ {i = 1} ^ {d} \left(O \left(k ^ {2}\right) \cdot \widetilde {\operatorname {c o s t} _ {i}} \left(C _ {1}, \dots , C _ {k}\right)\right) \\ \leq O (d \delta) + \sum_ {i = 1} ^ {d} O (k ^ {2}) \left(\operatorname {c o s t} _ {i} \left(C _ {1}, \dots , C _ {k}\right) + O (\delta)\right) \\ = O (d k ^ {2} \delta) + \sum_ {i = 1} ^ {d} O (k ^ {2}) \cdot c o s t _ {i} (C _ {1}, \dots , C _ {k}) \\ \leq O \left(d k ^ {2}\right) \cdot c o s t \left(C _ {1}, \dots , C _ {k}\right) + O \left(d k ^ {2} \delta\right) \\ \end{array}
+$$
+
+![](images/68d94e5d8821fbfa2774cd49a8779d91195a1f63ecfb5386b292837fd9acbf08.jpg)
+
+# G DISTANCE-BASED PRODUCT KERNELS
+
+Consider a distance-based product kernel  $K(x,y) = \prod_{i=1}^{d} h(|x_i - y_i|)$  on  $\mathbb{R}^d$ . As pointed out in the main paper, we may define surrogate features via  $\phi_{i,j}(x) = K_i(x,x^{(j)})$  where  $x^{(j)}$  denotes the  $j$ th point in the dataset  $X$ . While these features lead to interpretable decision trees, they do not define features that approximate the one-dimensional kernel: For all  $i \in [d]$ , it must be noted that  $\langle \phi_i(x),\phi_i(y)\rangle \neq K_i(x,y)$  under the standard Euclidean inner product. Of course, if we equip  $\mathbb{R}^n$  with a new inner product given by  $\langle \phi_i(x),\phi_i(y)\rangle = \phi_i(x)^T\pmb{K}_i^{-1}\phi_i(y)$ , then our surrogate features remain valid features for the one-dimensional kernels  $K_i$ . Consequently, when setting  $\phi = (\phi_i)_{i=1}^d$ , we actually map to a space  $\mathcal{V}$  in which the inner product is given by  $\langle u,v\rangle = \sum_{i=1}^{d}u_i^T\pmb{K}_i^{-1}v_i$ . Running explainable clustering algorithms such as IMM in this space is possible, it however introduces additional constants that depend on the kernel matrices  $K_i$ .
+
+Theorem. Let  $C_1, \ldots, C_k$  be the clusters of a dataset  $X \subset \mathbb{R}^d$  derived from kernel  $k$ -means, where the kernel  $K$  is a distance-based product kernel. Denote by  $\Phi = \phi(X)$  the surrogate features for  $K$  on  $X$ , where  $\phi_{i,j}(x) = K_i(x, x^{(j)})$ . Then, the interpretable decision tree obtained from Kernel IMM on  $\Phi$  satisfies  $p(T, X) = O(Cdk^2)$ , where  $C$  depends on the dataset  $X$  and the kernel  $K$ .
+
+Proof. To prove Theorem 5 we adopt the general proof strategy from IMM and again decouple dimensions as discussed in Appendix E. Again,  $cost_{i}$  denotes the cost with respect to each one-dimensional kernel  $K_{i}$ .
+
+$$
+\begin{array}{l} c o s t (\hat {C} _ {1}, \dots , \hat {C} _ {k}) \leq \sum_ {i = 1} ^ {d} c o s t _ {i} (\hat {C} _ {1}, \dots , \hat {C} _ {k}) \\ = \sum_ {l = 1} ^ {k} \sum_ {x \in \hat {C} _ {l}} \| \phi (x) - \mu (x) \| _ {\mathcal {V}} ^ {2} \tag {12} \\ \leq 2 \cdot \operatorname {c o s t} _ {\mathcal {V}} \left(C _ {1}, \dots , C _ {k}\right) + 2 \sum_ {u \in T} t (u) D (u) \\ \end{array}
+$$
+
+Now, we would in principle like to use Lemma 2. However, the space  $\mathcal{V}$  in which Kernel IMM runs is equipped with the non-Euclidean inner product. Thus, the second part (11) of Lemma 2 no longer follows from the first (10). Instead, we arrive at
+
+$$
+\begin{array}{l} t (u) D (u) := t (u) \cdot \max  _ {a, b \in \mathcal {M} ^ {u}} \| a - b \| _ {\mathcal {V}} ^ {2} \\ = t (u) \cdot \max  _ {a, b \in \mathcal {M} ^ {u}} \sum_ {i = 1} ^ {d} \left(a _ {i} - b _ {i}\right) ^ {T} \boldsymbol {K} _ {i} ^ {- 1} \left(a _ {i} - b _ {i}\right) \\ \leq \kappa_ {1} \cdot t (u) \cdot \max  _ {a, b \in \mathcal {M} ^ {u}} (a - b) ^ {T} (a - b) \\ \leq 4 \kappa_ {1} \cdot k \sum_ {x \in X _ {c o r} ^ {u}} (\phi (x) - c (x)) ^ {T} (\phi (x) - c (x)) \\ \leq 4 \kappa_ {1} \kappa_ {2} \cdot k \sum_ {x \in X _ {c o r} ^ {u}} \| \phi (x) - c (x) \| _ {\mathcal {V}} ^ {2} \\ \end{array}
+$$
+
+where we use the first part of Lemma 2 in the second inequality. The constants  $\kappa_{1},\kappa_{2}$  are given by
+
+$$
+\kappa_ {1} = \max  _ {a, b \in \mathcal {M} ^ {u}} \frac {\sum_ {i = 1} ^ {d} (a _ {i} - b _ {i}) ^ {T} \boldsymbol {K} _ {i} ^ {- 1} (a _ {i} - b _ {i})}{\sum_ {i = 1} ^ {d} (a _ {i} - b _ {i}) ^ {T} (a _ {i} - b _ {i})}
+$$
+
+$$
+\kappa_ {2} = \max _ {x \in X} \frac {\sum_ {i = 1} ^ {d} (\phi_ {i} (x) - c _ {i} (x)) ^ {T} (\phi_ {i} (x) - c _ {i} (x))}{\sum_ {i = 1} ^ {d} (\phi_ {i} (x) - c _ {i} (x)) ^ {T} \pmb {K} _ {i} ^ {- 1} (\phi_ {i} (x) - c _ {i} (x))}
+$$
+
+and depend on the data.
+
+![](images/17e615f8475ed34f3033b8c42864aceb39d1753350e3b784ca785c5e65b7c198.jpg)
+
+# H KERNEL IMM FOR ADDITIVE KERNELS
+
+Additive kernels are of the form  $K(x,y) = \sum_{i=1}^{d} K_i(x,y)$  where every individual  $K_i$  is a kernel on a suitable subset of  $\mathbb{R}$ , often the positive real line. Feature maps of additive kernels  $K$  can be written as the concatenation of the univariate feature maps associated with each kernel  $K_i$ . If these univariate — and hence interpretable — maps consist of functions that have at most one change of slope, threshold cuts with respect to the feature map lead directly to interpretable decision trees. Conveniently, this is the case for several important additive kernels.
+
+- Consider Hellinger's kernel  $K(x, y) = \sum_{i=1}^{d} \sqrt{x_i y_i}$  on a dataset  $X \subset [0, \infty)^d$ . Then  $K$  admits the simple interpretable (and monotonic) feature map  $\phi(x) = (\sqrt{x_1}, \ldots, \sqrt{x_d})$ .  
+- Let  $K(x,y) = \sum_{i=1}^{d} \min \left( x_i^\beta, y_i^\beta \right)$  be the generalized histogram intersection kernel, for some parameter  $\beta > 0$ . For any dataset  $X \subset [0,\infty)^d$ , there again exists a feature map that yields interpretable decision trees: Given a feature  $i \in [d]$ , denote by  $z_1 < \dots < z_{m_i}$  the  $m_i$  unique values in the set  $\{x_i^\beta \mid x \in X\}$ . For every component  $j \in [m_i]$ , let
+
+$$
+\phi_ {i, j} (x) = \left\{ \begin{array}{l l} \sqrt {z _ {j} - z _ {j - 1}} & , \text {i f} j \geq 2 \text {a n d} x _ {i} ^ {\beta} \geq z _ {j} \\ \sqrt {z _ {j}} & , \text {i f} j = 1 \\ 0 & , \text {e l s e} \end{array} \right. \tag {13}
+$$
+
+Then the concatenation  $\phi = (\phi_{i,j})_{i\in [d],j\in [m_i]}$  is composed of monotone functions. Moreover, it is straightforward to verify that for all  $x,y\in X$  it holds that  $K(x,y) = \langle \phi (x),\phi (y)\rangle$ . Thus,  $\phi$  is a valid feature map.
+
+- As a third example, consider the additive  $\chi^2$  kernel  $K(x,y) = \sum_{i=1}^{d} \frac{2x_i y_i}{x_i + y_i}$  on a dataset  $X \subset (0,\infty)^d$ . Given some  $M \in \mathbb{N}$ , the kernel  $K$  can be approximated from
+
+$$
+\begin{array}{l} K (x, y) = \sum_ {i = 1} ^ {d} 2 x _ {i} y _ {i} \cdot \int_ {0} ^ {1} t ^ {x _ {i} + y _ {i} - 1} d t \\ = \sum_ {i = 1} ^ {d} \int_ {0} ^ {1} \sqrt {2 / t} \cdot x _ {i} t ^ {x _ {i}} \cdot \sqrt {2 / t} \cdot y _ {i} t ^ {y _ {i}} d t \tag {14} \\ \approx \sum_ {i = 1} ^ {d} \frac {1}{M} \sum_ {j = 1} ^ {M} \sqrt {2 M / j} \cdot x _ {i} \left(\frac {j}{M}\right) ^ {x _ {i}} \cdot \sqrt {2 M / j} \cdot y _ {i} \left(\frac {j}{M}\right) ^ {y _ {i}} \\ \end{array}
+$$
+
+We conclude that the concatenation of all functions
+
+$$
+\phi_ {i, j} (x) = \sqrt {\frac {2}{j}} \cdot x _ {i} \left(\frac {j}{M}\right) ^ {x _ {i}}
+$$
+
+defines an approximate feature map for  $K$ . Since each  $\phi_{i,j}$  is strictly increasing for  $x_i < 1 / \log(M / j)$ , and strictly decreasing for  $x_i > 1 / \log(M / j)$ , they give rise to a feature map for which threshold cuts lead to interpretable decision trees. For normalized histograms we find that even low values of  $M \leq 5$  induce the same partitions as the additive  $\chi^2$  kernel, and hence  $O(k^2)$  bounds are preserved when running (Kernel) IMM with respect to the kernel induced by the map  $\phi$ .
+
+# I EXPERIMENTS
+
+Let us now give some details on the experiments. Our code is available on GitHub.
+
+Main experiments for Kernel IMM. We first verify the approximation properties of our proposed methods on the synthetic datasets "Pathbased", "Aggregation" and "Flame" (Fränti & Sieranoja) which have  $k = 3$ ,  $k = 7$  and  $k = 2$  clusters respectively. We start with linear  $k$ -means and IMM on all three, and then run kernel  $k$ -means with both the Laplace as well as the Gaussian kernel over a range of hyperparameters  $\gamma$ , choosing the best agreement with the ground truth as our starting point for Kernel IMM. When the Gaussian kernel is chosen, we run Kernel IMM both on the surrogate Taylor features from Definition 3 with  $M = 5$ , as well as on the surrogate features based on the kernel matrix, as defined in Equation (4), and choose the better one. We then refine the partition induced by Kernel IMM using both Kernel ExKMC as well as Kernel Expand, constructing  $m = 6$ ,  $m = 10$  and  $m = 4$  leaves respectively. Note that at every step, Kernel ExKMC and Kernel Expand only need to check the cost of the threshold cuts at the new nodes (obtained from the previous iteration). Thus, adding  $m$  leaves to an existing tree with  $p$  leaves amounts to  $p + 2m$  iterations over all possible threshold cuts.
+
+We follow the same procedure for the two real world datasets. In Iris (Fisher, 1936), there are three classes with 50 observations each. Every class refers to a type of iris plant. As illustrated in the barplot included in Section 6, kernel  $k$ -means slightly improves over  $k$ -means and this translates to Kernel IMM, Kernel ExKMC and Kernel Expand. The Wisconsin breast cancer dataset consists of 569 observations of benign and malignant cells. The 30-dimensional features describe characteristics of the cell nuclei observed in each image. Interestingly, IMM exactly replicates its suboptimal reference  $k$ -means clustering. Kernel  $k$ -means better identifies the ground truth, and Kernel IMM approximates it well (even achieving a slightly higher agreement with the ground truth). The same is true for Kernel ExKMC and Kernel Expand.
+
+Kernel IMM for the  $\chi^2$  kernel. The additive  $\chi^2$  kernel is evaluated on a toy dataset obtained from a mixture model of four discrete distributions, with values in four bins. Figure 6 shows a plot of the different distributions. For all four distribution, we draw 5 instances of 100 random samples, and compute the fraction of observations in each bin for every instance (thus  $n = 20$  and  $d = 4$ ). We repeat this procedure 100 times. We find that the  $\chi^2$  kernel achieves a Rand index that is consistently higher than the one of  $k$ -means (see Figure 6). This is not very surprising: The denominator of the
+
+![](images/0ab6fd2dba3337d3c5d9c349facda3d2ae34ed2a4cbcf530d9f348d71820212c.jpg)  
+Figure 6: We cluster samples drawn from a mixture model of four discrete distributions by checking the fraction of observations in each of four bins. The true underlying probabilities are shown in the left plot. Over 100 draws of samples, the  $\chi^2$  kernel improves over standard  $k$ -means in recovering the ground truth, as the boxplot on the right shows.
+
+![](images/cc92e410c3ef380e40f5098ad950fd4b2550b8ea0bf59e887267be6f47469602.jpg)
+
+$\chi^2$  kernel accounts for the overall number of observations in each bin, penalizing deviations in less probable bins more than in frequently visited bins — a nonlinear characteristic that standard  $k$ -means lacks.
+
+To provide some more intuition on how Kernel IMM constructs interpretable decision trees, let us now give some additional details for the  $\chi^2$  kernel. Having drawn 5 instances (containing 100 samples) for each distribution, we compute the class probabilities for the four bins and cluster using kernel  $k$ -means. We then map to a higher-dimensional feature space using the features derived for the  $\chi^2$  kernel (see Appendix H). We choose  $M = 5$  features for every dimension, and hence operate in a  $D = Md = 20$ -dimensional space. Assuming the first threshold cut is chosen along the first axis in the feature space, Kernel IMM provides us with some  $\theta$  such that we cut at
+
+$$
+\phi_ {1, 1} (x) \leq \theta \iff \sqrt {2} x _ {1} \left(\frac {1}{5}\right) ^ {x _ {1}} \leq \theta \iff \frac {x _ {1}}{5 ^ {x _ {1}}} \leq \frac {\theta}{\sqrt {2}}
+$$
+
+All we are now left with is identifying which values of  $x_{1}$  satisfy the above inequality, and which do not. Equivalently, we can represent this as an interval (or its complement) in the sample space.
+
+Comparison with CART. Standard decision tree algorithms such as CART also perform well in our experiments. CART achieves a price of explainability very close to the one that Kernel IMM attains (despite it being known that no approximation results exist for CART in the standard  $k$ -means setting). We show a comparison of the results in Table 2. For the Wisconsin breast cancer dataset, CART even improves over unconstrained kernel  $k$ -means (which has, in this case, not found the optimal partition).
+
+Table 2: Comparison of price of explainability (PoE) between CART with one-sided cuts and  $k$  leaves, and Kernel IMM.  
+
+<table><tr><td>Dataset</td><td>Kernel</td><td>PoE (Kernel IMM)</td><td>PoE (CART)</td></tr><tr><td>Pathbased</td><td>Gaussian</td><td>1.06645</td><td>1.07004</td></tr><tr><td>Aggregation</td><td>Laplace</td><td>1.00125</td><td>1.00125</td></tr><tr><td>Flame</td><td>Gaussian</td><td>1.02256</td><td>1.02732</td></tr><tr><td>Iris</td><td>Laplace</td><td>1.00502</td><td>1.00502</td></tr><tr><td>Cancer</td><td>Gaussian</td><td>1.00179</td><td>0.99330</td></tr></table>
+
+# J EXKMC ON AN EMPTY TREE ADMITS NO WORST-CASE GUARANTEES (THEOREM 6)
+
+Theorem. (ExKMC admits no worst-case bounds) For any  $m \in \mathbb{N}$  there exists a data set  $X \subset \mathbb{R}$  such that the price of explainability on  $X$  is  $p(X) = 1$ , IMM attain this minimum, but ExKMC (initialized on an empty tree) constructs a decision tree  $T$  with  $p(T, X) \geq m$ .
+
+Proof. Consider a dataset  $X$  of size  $n$  in  $\mathbb{R}$ , consisting of three clusters with centers given by  $c_{1} = -1$ ,  $c_{2} = 0$ ,  $c_{3} = 1$ . Assume all points of each cluster  $C_i$  lie at a distance of exactly  $\epsilon$  from the center  $c_{i}$ , half of them at  $c_{i} - \epsilon$  and half of them at  $c_{i} + \epsilon$ . The optimal  $k$ -means cost is then given by
+
+$$
+c o s t _ {o p t} = n \epsilon^ {2}
+$$
+
+At the first iteration of ExKMC, suppose it chooses some cut that does not separate points belonging to the same cluster. We may then assume that w.l.o.g.  $(i,\theta) = (1,0.5)$ , which implies  $c_{R} = c_{3}$  and w.l.o.g.  $c_{L} = c_{1}$ . Thus, the total cost of this cut is
+
+$$
+\begin{array}{l} f (i, \theta) = \frac {2 n}{3} \epsilon^ {2} + \sum_ {x \in C _ {2}} \| x - c _ {1} \| ^ {2} \\ = \frac {2 n}{3} \epsilon^ {2} + \frac {n}{6} (1 - \epsilon) ^ {2} + \frac {n}{6} (1 + \epsilon) ^ {2} \\ \end{array}
+$$
+
+This cost can be improved upon when ExKMC cuts along  $(i,\theta) = (1,0)$ , choosing  $c_{1} = c_{L}$ ,  $c_{3} = c_{R}$ , leading to a cost of
+
+$$
+f (i, \theta) = \frac {2 n}{3} \epsilon^ {2} + \frac {n}{3} (1 - \epsilon) ^ {2}
+$$
+
+Since ExKMC can now only add one more leaf, it inevitably separates at least  $\frac{n}{6}$  points from their correct cluster center. Thus, there exists a leaf for which  $\frac{2n}{6} \cdot \frac{n}{3} = \frac{n^2}{9}$  pairs of points are at distance of at least  $1 - 2\epsilon$ , and the final cost of the associated decision tree  $T$  is hence
+
+$$
+c o s t (T) \geq \frac {n \epsilon^ {2}}{2} + \frac {1}{n} \cdot \frac {2 n ^ {2} (1 - 2 \epsilon) ^ {2}}{9} = \frac {n \epsilon^ {2}}{2} + \frac {2 n (1 - 2 \epsilon) ^ {2}}{9}
+$$
+
+The price of explainability of  $T$  is given by
+
+$$
+p (T, X) \geq \frac {n \epsilon^ {2}}{2 n \epsilon^ {2}} + \frac {2 (1 - 2 \epsilon) ^ {2}}{9 \epsilon^ {2}}
+$$
+
+Thus, as  $\epsilon \to 0$ , we see that ExKMC gives rise to decision trees with an arbitrarily large price of explainability  $p(T,X) \geq m$ . Note that IMM recreates the optimal partition perfectly, since it is possible to split clusters without making any mistakes. Thus  $p(X) = 1$ . Clearly, the fact that ExKMC restricts our choice of centers to the set of reference centers limits its performance.
+
+# K THE PRICE OF EXPLAINABILITY CAN BE INFINITE FOR KERNEL CLUSTERING (THEOREM 7)
+
+The price of explainability, as defined in Definition 4, may not be finite for all kernels.
+
+Proposition 1. (Unbounded price for the quadratic kernel) Let  $K(x,y) = \langle x,y\rangle^2$ . Then, there exists a dataset  $X\subset \mathbb{R}^2$  such that  $p(X) = \infty$ .
+
+Proof. Let  $x = (0,1), y = (0, -1), z = (1,0), w = (-1,0) \in \mathbb{R}^2$ . Consider the quadratic kernel  $K(s,t) = (\langle s,t\rangle)^2$  operating on  $X = \{x,y,z,w\}$ . Then quadratic kernel 2-means achieves a cost of zero by partitioning  $X$  into the two clusters  $C_1 = \{x,y\}$ ,  $C_2 = \{z,w\}$ , since  $K(s,t) = 1$  for all  $s,t \in X$  from the same cluster. However, no interpretable decision tree can reproduce this partitioning of  $X$ . Every decision tree  $T$  with two leaves assigns one pair of points  $(s,t)$  from different clusters to the same leaf. Since  $K(s,t) = 0$  for this pair,  $cost(T) > 0$  and thus  $p(X) = \infty$ .
+
+This may not come as a surprise considering the special geometry of the quadratic kernel. However, the following result demonstrates that even distance-based kernels can lead to similar problems. For this, we consider the  $\epsilon$ -neighborhood kernel  $K(x,y) = \mathbf{1}(\| x - y\| < \epsilon)$ . While  $K$  is not a positive definite kernel (its kernel matrix may have eigenvalues), it is commonly used in spectral and kernel clustering nonetheless.
+
+Proposition 2. (Unbounded price for the  $\epsilon$ -neighborhood kernel) Let  $\epsilon > 0$  and let  $K(x, y) = \mathbf{1}(\|x - y\| \leq \epsilon)$ . Then, there exists a dataset  $X$  such that  $p(X) = \infty$ .
+
+Proof. For four points  $x^{(1)}, \ldots, x^{(4)}$ , there are exactly  $d' = 6$  ways of assigning to two of them the value 0, and to the two others the value 1. Similarly, for four other points  $x^{(5)}, \ldots, x^{(8)}$ , there are exactly  $d' = 6$  ways of assigning to two of them the value 0, and to the two others the value -1. There are  $d = (d')^2 = 36$  possible combinations of all these assignments. We define a  $d$ -dimensional cluster  $C_1$  by concatenating all of the combinations for each of the points  $x^{(1)}, \ldots, x^{(4)}$ . Thus, every  $x \in C_1$  is a vector in  $\{0, 1\}^d$ . Similarly, define  $C_2$  by concatenating all of the combinations for each of the points  $x^{(5)}, \ldots, x^{(8)}$ . Thus, the vectors  $y \in C_2$  are  $\{0, -1\}^d$ . Note that for any  $x, x' \in C_1$ , they agree in exactly  $d/3$  dimensions but differ by 1 everywhere else. Thus  $\|x - x'\|^2 = 24$ . The same holds for any pair  $y, y' \in C_2$ . However, for any pair  $x \in C_1, y \in C_2$  from different clusters, they agree on  $d/4 = 9$  dimensions, but are separated by 1 along  $d/2 = 18$  dimensions and are separated by 2 elsewhere. Thus, their distance is  $\|x - y\|^2 = 18 + 36 = 54$ . By choosing the  $\epsilon$ -neighborhood kernel with  $\epsilon = \sqrt{24}$ , we see that  $K(x, x') = K(y, y') = 1$  for all  $x, x' \in C_1, y, y' \in C_2$ . This implies an optimal kernel  $k$ -means cost of zero. However, any threshold cut along any one of the  $d$  dimensions will assign at least one pair  $(x, y)$  from distinct clusters to the same leaf, implying  $K(x, y) = 0$ . Thus, the kernel  $k$ -means cost for this explainable partition is strictly positive. This proves  $p(X) = \infty$ . To produce the same result for other values of  $\epsilon > 0$ , simply rescale the data accordingly.
+
+![](images/799d93d50e6a89ccae9df4b9c0d41b594de85696cfee27cea941fb44270b8176.jpg)

2024/Explaining Kernel Clustering via Decision Trees/images.zip

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2024/Explaining Time Series via Contrastive and Locally Sparse Perturbations/full.md

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+# EXPLAINING TIME SERIES VIA CONTRASTIVE AND LOCALLY SPARSE PERTURBATIONS
+
+Zichuan Liu $^{1,2*}$ , Yingying Zhang $^{2*}$ , Tianchun Wang $^{3*}$ , Zefan Wang $^{2,4}$ , Dongsheng Luo $^{5}$ , Mengnan Du $^{6}$ , Min Wu $^{7}$ , Yi Wang $^{8}$ , Chunlin Chen $^{1\dagger}$ , Lunting Fan $^{2}$ , and Qingsong Wen $^{2\dagger}$
+
+$^{1}$ Nanjing University,  $^{2}$ Ailibaba Group,  $^{3}$ Pennsylvania State University,
+
+$^{4}$ Tsinghua University,  $^{5}$ Florida International University,
+
+$^{6}$ New Jersey Institute of Technology,  $^{7}$ A*STAR,  $^{8}$ The University of Hong Kong
+
+# ABSTRACT
+
+Explaining multivariate time series is a compound challenge, as it requires identifying important locations in the time series and matching complex temporal patterns. Although previous saliency-based methods addressed the challenges, their perturbation may not alleviate the distribution shift issue, which is inevitable especially in heterogeneous samples. We present ContraLSP, a locally sparse model that introduces counterfactual samples to build uninformative perturbations but keeps distribution using contrastive learning. Furthermore, we incorporate sample-specific sparse gates to generate more binary-skewed and smooth masks, which easily integrate temporal trends and select the salient features parsimoniously. Empirical studies on both synthetic and real-world datasets show that ContraLSP outperforms state-of-the-art models, demonstrating a substantial improvement in explanation quality for time series data. The source code is available at https://github.com/zichuan-liu/ContraLSP.
+
+# 1 INTRODUCTION
+
+Providing reliable explanations for predictions made by machine learning models is of paramount importance, particularly in fields like finance (Mokhtari et al., 2019), games (Liu et al., 2023), and healthcare (Amann et al., 2020), where transparency and interpretability are often ethical and legal prerequisites. These domains frequently deal with complex multivariate time series data, yet the investigation into methods for explaining time series models remains an underexplored frontier (Rojat et al., 2021). Besides, adapting explainers originally designed for different data types presents challenges, as their inductive biases may struggle to accommodate the inherently complex and less interpretable nature of time series data (Ismail et al., 2020). Achieving this requires the identification of crucial temporal positions and aligning them with explainable patterns.
+
+In response, the predominant explanations involve the use of saliency methods (Baehrens et al., 2010; Tjoa & Guan, 2020), where the explanatory distinctions depend on how they interact with an arbitrary model. Some works establish saliency maps, e.g., incorporating gradient (Sundararajan et al., 2017; Lundberg et al., 2018) or constructing attention (Garnot et al., 2020; Lin et al., 2020), to better handle time series characteristics. Other surrogate methods, including Shapley (Castro et al., 2009; Lundberg & Lee, 2017) or LIME (Ribeiro et al., 2016), provide insight into the predictions of a model by locally approximating them through weighted linear regression. These methods mainly provide instance-level saliency maps, but the feature inter-correlation often leads to notable generalization errors (Yang et al., 2022).
+
+The most popular class of explanation methods is to use samples for perturbation (Fong et al., 2019; Leung et al., 2023; Lee et al., 2022), usually through different styles to make non-salient features uninformative. Two representative perturbation methods in time series are Dynamask (Crabbé & Van Der Schaar, 2021) and Extrmask (Enguehard, 2023).
+
+Dynamask utilizes meaningful perturbations to incorporate temporal smoothing, while Extrmask generates perturbations of less sense close to zero through neural network learning. However, due to shifts in shape (Zhao et al., 2022), perturbed time series may be out of distribution for the explained model, leading to a loss of faithfulness in the generated explanations. For example, a time series classified as 1 and its different forms of perturbation are shown in Figure 1. We see that the distribution of all classes moves away from 0 at intermediate time, while the 0 and mean perturbations shift in shape. In addition, the blur and learned perturbations are close to the original feature and therefore contain information for classification 1. It may result in a label leaking problem (Jethani et al., 2023), as informative perturbations are introduced. This causes us to think about counterfactuals, i.e., a contrasting perturbation does not affect model inference in non-salient areas.
+
+![](images/b830ca8d46c18cb695bd4f0350dadb313a3a649890dc94c38b72fc271cd844bc.jpg)  
+Figure 1: Illustrating different styles of perturbation. The red line is a sample belonging to class 1 within the two categories, while the dark background indicates the salient features, otherwise non-salient. Other perturbations could be either not uninformative or not in-domain, while ours is counterfactual that is toward the distribution of negative samples.
+
+To address these challenges, we propose a Contrastive and Locally Sparse Perturbations (ContraLSP) framework based on contrastive learning and sparse gate techniques. Specifically, our ContraLSP learns a perturbation function to generate counterfactual and in-domain perturbations through a contrastive learning loss. These perturbations tend to align with negative samples that are far from the current features (Figure 1), rendering them uninformative. To optimize the mask, we employ  $\ell_0$ -regularised gates with injected random noises in each sample for regularization, which encourages the mask to approach a binary-skewed form while preserving the localized sparse explanation. Additionally, we introduce a smooth constraint with a trend function to allow the mask to capture temporal patterns. We summarize our contributions as below:
+
+- We propose ContraLSP as a stronger time series model explanatory tool, which incorporates counterfactual samples to build uninformative in-domain perturbation through contrastive learning.  
+- ContraLSP integrates sample-specific sparse gates as a mask indicator, generating binary-skewed masks on time series. Additionally, we enforce a smooth constraint by considering temporal trends, ensuring consistent alignment of the latent time series patterns.  
+- We evaluate our method through experiments on both synthetic and real-world time series datasets. These datasets comprise either classification or regression tasks and the synthetic one includes ground-truth explanatory labels, allowing for quantitative benchmarking against other state-of-the-art time series explainers.
+
+# 2 RELATED WORK
+
+Time series explainability. Recent literature has delved into the realm of eXplainable Artificial Intelligence (XAI) for multivariate time series (Bento et al., 2021; Ismail et al., 2020; Zhu et al., 2023). Among them, gradient-based methods (Shrikumar et al., 2017; Sundararajan et al., 2017; Lundberg et al., 2018) translate the impact of localized input alterations to feature saliency. Attention-based methods (Lin et al., 2020; Choi et al., 2016) leverage attention layers to produce importance scores that are intrinsically based on attention coefficients. Perturbation-based methods, as the most common form in time series, usually modify the data through baseline (Suresh et al., 2017), generative models (Tonekaboni et al., 2020; Leung et al., 2023), or making the data more uninformative (Crabbé & Van Der Schaar, 2021; Enguehard, 2023). However, these methods provide only an instance-level saliency map, while the inter-sample saliency maps have been studied little in the existing literature (Gautam et al., 2022). Our investigation performs counterfactual perturbations through inter-sample variation, which goes beyond the instance-level saliency methods by focusing on understanding both the overall and specific model's behavior across groups.
+
+Model sparsification. For a better understanding of which part of the features are most influential to the model's behavior, the existing literature enforcing sparsity (Fong et al., 2019) to constrain the model's focus on specific regions. A typical approach is LASSO (Tibshirani, 1996), which selects a subset of the most relevant features by adding an  $\ell_1$  constraint to the loss function. Based on this, several works (Feng & Simon, 2017; Scardapane et al., 2017; Louizos et al., 2018; Yamada et al., 2020) are proposed to employ distinct forms of regularization to encourage the input features to be sparse. All these methods select global informative features that may neglect the underlying correlation between them. To cope with it, local stochastic gates (Yang et al., 2022) consider an instance-wise selector to heterogeneous samples, accommodating cases where salient features vary among samples. Lee et al. (2022) takes a self-supervised way to enhance stochastic gates that encourage the model's sparse explainability meanwhile. However, most of these sparse methods are utilized in tabular feature selection. Different from them, our approach reveals crucial features within the temporal patterns of multivariate time series data, offering local sample explanations.
+
+Counterfactual explanations. Perturbation-based methods are known to have distribution shift problems, leading to abnormal model behaviors and unreliable explanation (Hase et al., 2021; Hsieh et al., 2021). Previous works (Goyal et al., 2019; Teney et al., 2020) have tackled generating reasonable counterfactuals for perturbation-based explanations, which searches pairwise inter-class perturbations in the sample domain to explain the classification models. In the field of time series, Delaney et al. (2021) builds counterfactuals by adapting label-changing neighbors. To alleviate the need for labels in model interpretation, Chuang et al. (2023) uses triplet contrastive representation learning with disturbed samples to train an explanatory encoder. However, none of these methods explored label-free perturbation generation aligned with the sample domain. On the contrary, our method yields counterfactuals with contrastive sample selection to sustain faithful explanations.
+
+# 3 PROBLEM FORMULATION
+
+Let  $\{(\pmb{x}_i, y_i)\}_{i=1}^N$  be a set of multi-variate time series, where  $\pmb{x}_i \in \mathbb{R}^{T \times D}$  is a sample with  $T$  time steps and  $D$  observations,  $y_i \in \mathcal{V}$  is the ground truth.  $\pmb{x}_i[t,d]$  denotes a feature of  $\pmb{x}_i$  in time step  $t$  and observation dimension  $d$ , where  $t \in [1:T]$  and  $d \in [1:D]$ . We let  $\pmb{x} \in \mathbb{R}^{N \times T \times D}$ ,  $y \in \mathcal{V}^N$  be the set of all the samples and that of the ground truth, respectively. We are interested in explaining the prediction  $\hat{y} = f(\pmb{x})$  of a pre-trained black-box model  $f$ . More specifically, our objective is to pinpoint a subset  $\mathcal{S} \subseteq [N \times T \times D]$ , in which the model uses the relevant selected features  $\pmb{x}[\mathcal{S}]$  to optimize its proximity to the target outcome. It can be rewritten as addressing an optimization problem:  $\arg \min_{\mathcal{S}} \mathcal{L}(\hat{y}, f(\pmb{x}[\mathcal{S}]))$ , where  $\mathcal{L}$  represents the cross-entropy loss for  $C$ -classification tasks (i.e.,  $\mathcal{Y} = \{1, \dots, C\}$ ) or the mean squared error for regression tasks (i.e.,  $\mathcal{Y} = \mathbb{R}$ ).
+
+To achieve this goal, we consider finding masks  $\mathbf{m} = \mathbf{1}_{\mathcal{S}} \in \{0, 1\}^{N \times T \times D}$  by learning the samples of perturbed features through  $\Phi(\mathbf{x}, \mathbf{m}) = \mathbf{m} \odot \mathbf{x} + (\mathbf{1} - \mathbf{m}) \odot \mathbf{x}^r$ , where  $\mathbf{x}^r = \varphi_{\theta_1}(\mathbf{x})$  is the counterfactual explanation obtained from a perturbation function  $\varphi_{\theta_1}: \mathbb{R}^{N \times T \times D} \to \mathbb{R}^{N \times T \times D}$ , and  $\theta_1$  is a parameter of the function  $\varphi(\cdot)$  (e.g., neural networks). Thus, existing literature (Fong & Vedaldi, 2017; Fong et al., 2019; Crabbé & Van Der Schaar, 2021) propose to rewrite the above optimization problem by learning an optimal mask as
+
+$$
+\underset {\boldsymbol {m}, \theta_ {1}} {\arg \min } \mathcal {L} (f (\boldsymbol {x}), f \circ \Phi (\boldsymbol {x}, \boldsymbol {m})) + \mathcal {R} (\boldsymbol {m}) + \mathcal {A} (\boldsymbol {m}), \tag {1}
+$$
+
+which promotes proximity between the predictions on the perturbed samples and the original ones in the first term, and restricts the number of explanatory features in the second term (e.g.,  $\mathcal{R}(\boldsymbol{m}) = \|\boldsymbol{m}\|_1$ ). The third term enforces the mask's value to be smooth by penalizing irregular shapes.
+
+Challenges. In the real world, particularly within the healthcare field, two primary challenges are encountered: (i) Current strategies (Fong & Vedaldi, 2017; Louizos et al., 2018; Lee et al., 2022; Enguehard, 2023) of learning the perturbation  $\varphi(\cdot)$  could be either not counterfactual or out of distributions due to unknown data distribution (Jethani et al., 2023). (ii) Under-considering the inter-correlation of samples would result in significant generalization errors (Yang et al., 2022). During training, cross-sample interference among masks  $\{\pmb{m}_i\}_{i=1}^N$  may cause ambiguous sample-specific predictions, while local sparse weights can remove the ambiguity (Yamada et al., 2017). These challenges motivate us to learn counterfactual perturbations that are adapted to each sample individually with localized sparse masks.
+
+![](images/2f6f1ee41782fc09922d5d241ec0e49b38051e483f8b2231c734b093a6fc21a8.jpg)  
+Figure 2: The architecture of ContraLSP. A sample of features  $\boldsymbol{x}_i \in \mathbb{R}^{T \times D}$  is fed simultaneously to a perturbation function  $\varphi(\cdot)$  and to a trend function  $\tau(\cdot)$ . The perturbation function  $\varphi(\cdot)$  uses  $\boldsymbol{x}_i$  to generate counterfactuals  $\boldsymbol{x}_i^r$  that are closer to other negative samples (but within the sample domain) through contrastive learning. In addition,  $\tau(\cdot)$  learns to predict temporal trends, which together with a set of parameters  $\boldsymbol{\mu}_i$  depicts the smooth vectors  $\boldsymbol{\mu}_i^r$ . It acts on the locally sparse gates by injecting noises  $\epsilon_i$  to get the mask  $m_i$ . Finally, the counterfactuals are replaced with perturbed features and the predictions are compared to the original results to determine which features are salient enough.
+
+# 4 OUR METHOD
+
+We now present the Contrastive and Locally Sparse Perturbations (ContraLSP), whose overall architecture is illustrated in Figure 2. Specifically, our ContraLSP learns counterfactuals by means of contrastive learning to augment the uninformative perturbations but maintain sample distribution. This allows perturbed features toward a negative distribution in heterogeneous samples, thus increasing the impact of the perturbation. Meanwhile, a mask selects sample-specific features in sparse gates, which is learned to be constrained with  $\ell_0$ -regularization and temporal trend smoothing. Finally, comparing the perturbed prediction to the original prediction, we subsequently backpropagate the error to learn the perturbation function and adapt the saliency scores contained in the mask.
+
+# 4.1 COUNTERFACTUALS FROM CONTRASTIVE LEARNING
+
+To obtain counterfactual perturbations, we train the perturbation function  $\varphi_{\theta_1}(\cdot)$  through a triplet-based contrastive learning objective. The main idea is to make counterfactual perturbations more uninformative by inversely optimizing a triplet loss (Schroff et al., 2015), which adapts the samples by replacing the masked unimportant regions. Specifically, we take each counterfactual perturbation  $\pmb{x}_i^r = \varphi_{\theta_1}(\pmb{x}_i)$  as an anchor, and partition all samples  $\pmb{x}^r$  into two clusters: a positive cluster  $\Omega^{+}$  and negative one  $\Omega^{-}$ , based on the pairwise Manhattan similarities between these perturbations. Following this partitioning, we select the  $K^{+}$  nearest positive samples from the positive cluster  $\Omega^{+}$ , denoted as  $\{\pmb{x}_{i,k}^{r + }\}_{k = 1}^{K^{+}}$ , which ex
+
+hibits similarity with the anchor features. In parallel, we randomly select  $K^{-}$  subsamples from the negative cluster  $\Omega^{-}$ , denoted as  $\{x_{i,k}^{r^{-}}\}_{k=1}^{K^{-}}$ , where  $K^{+}$  and  $K^{-}$  represent the numbers of positive and negative samples selected, respectively. The strategy of triple sampling is similar to Li et al. (2021), and we introduce the details in Appendix B.
+
+To this end, we obtain the set of triplets  $\mathcal{T}$  with each element being a tuple  $\mathcal{T}_i = \left(\pmb{x}_i^r,\{\pmb{x}_{i,k}^{r+}\}_{k=1}^{K^+},\{\pmb{x}_{i,k}^{r-}\}_{k=1}^{K-}\right)$ . Let the Manhattan distance between the anchor with negative samples be  $\mathcal{D}_{an} = \frac{1}{K^{-}}\sum_{k=1}^{K^{-}}|\pmb{x}_i^r - \pmb{x}_{i,k}^{r-}|$ , and that with positive samples be  $\mathcal{D}_{ap} = \frac{1}{K^+}\sum_{k=1}^{K^+}|\pmb{x}_i^r - \pmb{x}_{i,k}^{r+}|$ .
+
+![](images/b30c2bec39389e4560824142547342db4a50f20fa4928cdec1fde12615dffe29.jpg)  
+Figure 3: Illustration of the impact of triplet loss to generate counterfactual perturbations. The anchor is closer to negatives but farther from positives.
+
+As shown in Figure 3, we aim to ensure that  $\mathcal{D}_{an}$  is smaller than  $\mathcal{D}_{ap}$  with a margin  $b$ , thus making the perturbations counterfactual. Therefore, the objective of optimizing the perturbation function  $\varphi_{\theta_1}(\cdot)$  with triplet-based contrastive learning is given by
+
+$$
+\mathcal {L} _ {\text {c n t r}} \left(\boldsymbol {x} _ {i}\right) = \max  \left(0, \mathcal {D} _ {\text {a n}} - \mathcal {D} _ {\text {a p}} - b\right) + \left\| \boldsymbol {x} _ {i} ^ {r} \right\| _ {1}, \tag {2}
+$$
+
+which encourages the original sample  $\pmb{x}_i$  and the perturbation  $\pmb{x}_i^r$  to be dissimilar. The second regularization limits the extent of counterfactuals. In practice, the margin  $b$  is set to 1 following (Balntas et al., 2016), and we discuss the effects of different distances in Appendix E.1.
+
+# 4.2 SPARSE GATES WITH SMOOTH CONSTRAINT
+
+Logical masks preserve the sparsity of feature selection but introduce a large degree of variance in the approximated Bernoulli masks due to their heavy-tailedness (Yamada et al., 2020). To address this limitation, we apply a sparse stochastic gate to each feature in each sample  $i$ , thus approximating the Bernoulli distribution for the local sample. Specifically, for each feature  $x_{i}[t,d]$ , a sample-specific mask is obtained based on the hard thresholding function by
+
+$$
+\boldsymbol {m} _ {i} [ t, d ] = \min  \left(1, \max  \left(0, \boldsymbol {\mu} _ {i} ^ {\prime} [ t, d ] + \boldsymbol {\epsilon} _ {i} [ t, d ]\right)\right), \tag {3}
+$$
+
+where  $\epsilon_{i}[t,d]\sim \mathcal{N}(0,\delta^{2})$  is a random noise injected into each feature. We fix the Gaussian variance  $\delta^2$  during training. Typically,  $\pmb{\mu}_{i}^{\prime}[t,d]$  is taken as an intrinsic parameter of the sparse gate. However, as a binary-skewed parameter,  $\pmb{\mu}_{i}^{\prime}[t,d]$  does not take into account the smoothness, which may lose the underlying trend in temporal patterns. Inspired by Elfwing et al. (2018) and Biswas et al. (2022), we adopt a sigmoid-weighted unit with the temporal trend to smooth  $\pmb{\mu}_{i}^{\prime}$ . Specifically, we construct the smooth vectors  $\pmb{\mu}_{i}^{\prime}$  as
+
+$$
+\boldsymbol {\mu} _ {i} ^ {\prime} = \boldsymbol {\mu} _ {i} \odot \sigma \left(\tau_ {\theta_ {2}} \left(\boldsymbol {x} _ {i}\right) \boldsymbol {\mu} _ {i}\right) = \frac {\boldsymbol {\mu} _ {i}}{1 + e ^ {- \tau_ {\theta_ {2}} \left(\boldsymbol {x} _ {i}\right) \boldsymbol {\mu} _ {i}}}, \tag {4}
+$$
+
+![](images/ed2bae09b22783bc30fb22c3de86427e50258ae855d55c5ba39ea19cfaafb8ba.jpg)  
+Figure 4: Different temperatures for the sigmoid-weighted unit. The learned trend function  $\tau (\cdot)$  can be better adapted to smooth vectors (red) to hard masks (black).
+
+where  $\tau_{\theta_2}(\cdot):\mathbb{R}^{N\times T\times D}\to \mathbb{R}^{N\times T\times D}$  is a trend function parameterized by  $\theta_{2}$  that plays a role in the sig
+
+moid function as temperature scaling, and  $\mu_{i}$  is a set of parameters initialized randomly. In practice, we use a neural network (e.g., MLP) to implement the trend function  $\tau_{\theta_2}(\cdot)$ , whose details are shown in Appendix D.4. Note that employing a constant temperature may render the mask continuous. However, for a valid mask interpretation, adherence to a discrete property is appropriate (Queen et al., 2023). We illustrate in Figure 4 that a learned temperature (red solid) makes the hard mask smoother and keeps its skewed binary, in contrast to other constant temperatures.
+
+To make the mask more informative in Eq. (1), we follow Yang et al. (2022) by replacing the  $\ell_1$ -regularization into an  $\ell_0$ -like constraint. Consequently, the regularization term  $\mathcal{R}(\cdot)$  can be rewritten using the Gaussian error function (erf) as
+
+$$
+\mathcal {R} \left(\boldsymbol {x} _ {i}, \boldsymbol {m} _ {i}\right) = \| \boldsymbol {m} _ {i} \| _ {0} = \sum_ {t = 1} ^ {T} \sum_ {d = 1} ^ {D} \left(\frac {1}{2} + \frac {1}{2} \operatorname {e r f} \left(\frac {\boldsymbol {\mu} _ {i} ^ {\prime} [ t , d ]}{\sqrt {2} \delta}\right)\right), \tag {5}
+$$
+
+where  $\pmb{\mu}_i^{\prime}$  is obtained from Eq.(4). The full derivations are given in Appendix A. We calculate the empirical expectation over  $m_{i}$  for all samples. Thus, masks  $\pmb{m}$  are learned by the objective
+
+$$
+\underset {\boldsymbol {\mu}, \theta_ {2}} {\arg \min } \mathcal {L} (f (\boldsymbol {x}), f \circ \Phi (\boldsymbol {x}, \boldsymbol {m})) + \frac {\alpha}{N} \sum_ {i = 1} ^ {N} \mathcal {R} \left(\boldsymbol {x} _ {i}, \boldsymbol {m} _ {i}\right), \tag {6}
+$$
+
+where  $\alpha$  is the regular strength. Note that the smooth vectors  $\pmb{\mu}_i^{\prime}$  restrict the penalty term  $\mathcal{A}(\cdot)$  in Eq. (1) for jump saliency over time.
+
+Table 1: Performance on Rare-Time and Rare-Observation experiments w/o different groups.  
+
+<table><tr><td rowspan="2">METHOD</td><td colspan="4">RARE-TIME</td><td colspan="4">RARE-TIME (DIFFGROUPS)</td></tr><tr><td>AUP↑</td><td>AUR↑</td><td>Im/10^4↑</td><td>Sm/10^2↓</td><td>AUP↑</td><td>AUR↑</td><td>Im/10^4↑</td><td>Sm/10^2↓</td></tr><tr><td>FO</td><td>1.00±0.00</td><td>0.13±0.00</td><td>0.46±0.01</td><td>47.20±0.61</td><td>1.00±0.00</td><td>0.16±0.00</td><td>0.53±0.01</td><td>54.89±0.70</td></tr><tr><td>AFO</td><td>1.00±0.00</td><td>0.15±0.01</td><td>0.51±0.01</td><td>55.60±0.85</td><td>1.00±0.00</td><td>0.16±0.00</td><td>0.54±0.01</td><td>57.76±0.72</td></tr><tr><td>IG</td><td>1.00±0.00</td><td>0.13±0.00</td><td>0.46±0.01</td><td>47.61±0.62</td><td>1.00±0.00</td><td>0.15±0.00</td><td>0.53±0.01</td><td>54.62±0.85</td></tr><tr><td>SVS</td><td>1.00±0.00</td><td>0.13±0.00</td><td>0.47±0.01</td><td>47.20±0.61</td><td>1.00±0.00</td><td>0.15±0.00</td><td>0.52±0.02</td><td>54.28±0.84</td></tr><tr><td>DYNAMASK</td><td>0.99±0.01</td><td>0.67±0.02</td><td>8.68±0.11</td><td>37.24±0.48</td><td>0.99±0.01</td><td>0.51±0.00</td><td>5.75±0.13</td><td>47.33±1.02</td></tr><tr><td>EXTRMASK</td><td>1.00±0.00</td><td>0.88±0.00</td><td>16.40±0.13</td><td>13.10±0.78</td><td>1.00±0.00</td><td>0.83±0.03</td><td>13.37±0.78</td><td>27.44±3.68</td></tr><tr><td>CONTRALSP</td><td>1.00±0.00</td><td>0.97±0.01</td><td>19.51±0.30</td><td>4.65±0.71</td><td>1.00±0.00</td><td>0.94±0.01</td><td>18.92±0.37</td><td>4.40±0.60</td></tr><tr><td rowspan="2">METHOD</td><td colspan="4">RARE-OBSERVATION</td><td colspan="4">RARE-OBSERVATION (DIFFGROUPS)</td></tr><tr><td>AUP↑</td><td>AUR↑</td><td>Im/10^4↑</td><td>Sm/10^2↓</td><td>AUP↑</td><td>AUR↑</td><td>Im/10^4↑</td><td>Sm/10^2↓</td></tr><tr><td>FO</td><td>1.00±0.00</td><td>0.13±0.00</td><td>0.46±0.00</td><td>47.39±0.16</td><td>1.00±0.00</td><td>0.14±0.00</td><td>0.50±0.01</td><td>52.13±0.96</td></tr><tr><td>AFO</td><td>1.00±0.00</td><td>0.16±0.00</td><td>0.55±0.01</td><td>56.81±0.39</td><td>1.00±0.00</td><td>0.16±0.01</td><td>0.54±0.02</td><td>56.92±1.24</td></tr><tr><td>IG</td><td>1.00±0.00</td><td>0.13±0.00</td><td>0.46±0.00</td><td>47.82±0.15</td><td>1.00±0.00</td><td>0.13±0.00</td><td>0.47±0.00</td><td>49.90±0.88</td></tr><tr><td>SVS</td><td>1.00±0.00</td><td>0.13±0.00</td><td>0.46±0.00</td><td>47.39±0.16</td><td>1.00±0.00</td><td>0.13±0.00</td><td>0.47±0.01</td><td>49.53±0.84</td></tr><tr><td>DYNAMASK</td><td>0.97±0.00</td><td>0.65±0.00</td><td>8.32±0.06</td><td>22.87±0.58</td><td>0.98±0.00</td><td>0.52±0.01</td><td>6.12±0.10</td><td>30.88±0.70</td></tr><tr><td>EXTRMASK</td><td>1.00±0.00</td><td>0.76±0.00</td><td>13.25±0.07</td><td>9.55±0.39</td><td>1.00±0.00</td><td>0.70±0.04</td><td>10.40±0.54</td><td>32.81±0.88</td></tr><tr><td>CONTRALSP</td><td>1.00±0.00</td><td>1.00±0.00</td><td>20.68±0.03</td><td>0.32±0.16</td><td>1.00±0.00</td><td>0.99±0.00</td><td>20.51±0.07</td><td>0.57±0.20</td></tr></table>
+
+# 4.3 LEARNING OBJECTIVE
+
+In our method, we utilize the preservation game (Fong & Vedaldi, 2017), where the aim is to maximize data masking while minimizing the deviation of predictions from the original ones. Thus, the overall learning objective is to train the whole framework by minimizing the total loss
+
+$$
+\underset {\boldsymbol {\mu}, \theta_ {1}, \theta_ {2}} {\arg \min } \mathcal {L} (f (\boldsymbol {x}), f \circ \Phi (\boldsymbol {x}, \boldsymbol {m})) + \frac {\alpha}{N} \sum_ {i = 1} ^ {N} \mathcal {R} \left(\boldsymbol {x} _ {i}, \boldsymbol {m} _ {i}\right) + \frac {\beta}{N} \sum_ {i = 1} ^ {N} \mathcal {L} _ {\text {c n t r}} \left(\boldsymbol {x} _ {i}\right), \tag {7}
+$$
+
+where  $\{\pmb {\mu},\theta_1,\theta_2\}$  are learnable parameters of the whole framework and  $\{\alpha ,\beta \}$  are hyperparameters adjusting the weight of losses to learn the sparse masks. Note that during the inference phase, we remove the random noises  $\epsilon_{i}$  from the sparse gates and set  $m_{i} = \min (1,\max (0,\pmb{\mu}_{i}\odot \sigma (\tau (\pmb{x}_{i})\pmb{\mu}_{i}))$  for deterministic masks. We summarize the pseudo-code of the proposed ContraLSP in Appendix C.
+
+# 5 EXPERIMENTS
+
+In this section, we evaluate the explainability of the proposed method on synthetic datasets (where truth feature importance is accessible) for both regression (white-box) and classification (black-box), as well as on more intricate real-world clinical tasks. For black-box and real-world experiments, we use 1-layer GRU with 200 hidden units as the target model  $f_{\theta}$  to explain. All performance results for our method, benchmarks, and ablations are reported using mean ± std of 5 repetitions. For each metric in the results, we use ↑ to indicate a preference for higher values and ↓ to indicate a preference for lower values, and we mark **bold** as the best and **underline** as the second best. More details of each dataset and experiment are provided in Appendix D.
+
+# 5.1 WHITE-BOX REGRESSION SIMULATION
+
+Datasets and Benchmarks. Following Crabbé & Van Der Schaar (2021), we apply sparse white-box regressors whose predictions depend only on the known sub-features  $\mathcal{S} = \mathcal{S}^T \times \mathcal{S}^D \subset [ : 1 : T] \times [ :, 1 : D ]$  as salient indices. Besides, we extend our investigation by incorporating heterogeneous samples to explore the influence of inter-samples on masking. Specifically, we consider the subset of samples from two unequal nonlinear groups  $\{S_1, S_2\} \subset S$ , denoted as Diff-Groups. Here,  $S_1$  and  $S_2$  collectively constitute the entire set  $S$ , with each subset having a size of  $|S_1| = |S_2| = |S| / 2 = 50$ . The salient features are represented mathematically as
+
+$$
+[ f (\boldsymbol {x}) ] _ {t} = \left\{ \begin{array}{c c} \sum_ {[: t, d ] \in \mathcal {S}} \left(\boldsymbol {x} [ t, d ]\right) ^ {2} & \text {i f i n} \mathcal {S} \\ 0 & \text {e l s e ,} \end{array} \right. \text {a n d} [ f (\boldsymbol {x}) ] _ {t} = \left\{ \begin{array}{c c} \sum_ {[ i, t, d ] \in \mathcal {S} _ {1}} \left(\boldsymbol {x} _ {i} [ t, d ]\right) ^ {2} & \text {i f i n} \mathcal {S} _ {1} \\ \left(\sum_ {[ j, t, d ] \in \mathcal {S} _ {2}} \boldsymbol {x} _ {j} [ t, d ]\right) ^ {2} & \text {e l i f i n} \mathcal {S} _ {2} \\ 0 & \text {e l s e .} \end{array} \right.
+$$
+
+![](images/a5bdc5faa7f51225eeed7d238bd7dc469f2a3aa8fbeab75a82590d5e2901e36e.jpg)  
+Figure 5: Differences between ContraLSP and Extrmask perturbations on the Rare-Observation (Diffgroups) experiment. We randomly select a sample in each of the two groups and sum all observations. The background color represents the mask value, with darker colors indicating higher values. ContraLSP provides counterfactual information, yet Extrmask's perturbation is close to 0.
+
+In our experiments, we separately examine two scenarios with and without DiffGroups: where setting  $|\mathcal{S}^T| \ll N \times T$  is called Rare-Time and setting  $|\mathcal{S}^D| \ll N \times D$  is called Rare-Observation. These scenarios are recognized in saliency methods due to their inherent complexity (Ismail et al., 2019). In fact, some methods are not applicable to evaluate white-box regression models, e.g., DeepLIFT (Shrikumar et al., 2017) and FIT (Tonekaboni et al., 2020). To ensure a fair comparison, we compare ContraLSP with several baseline methods, including Feature Occlusion (FO) (Suresh et al., 2017), Augmented Feature Occlusion (AFO) (Tonekaboni et al., 2020), Integrated Gradient (IG) (Sundararajan et al., 2017), Shapley Value Sampling (SVS) (Castro et al., 2009), Dynamask (Crabbé & Van Der Schaar, 2021), and Extrmask (Enguehard, 2023). The implementation details of all algorithms are available in Appendix D.5.
+
+Metrics. Since we know the exact cause, we utilize it as the ground truth important for evaluating explanations. Observations causing prediction label changes receive an explanation of 1, otherwise it is 0. To this end, we evaluate feature importance with area under precision (AUP) and area under recall (AUR). To gauge the information of the masks and the sharpness of region explanations, we also use two metrics introduced by Crabbé & Van Der Schaar (2021): the information  $I_{\mathbf{m}}(\mathbf{a}) = -\sum_{[i,t,d] \in \mathbf{a}} \ln(1 - \mathbf{m}_i[t,d])$  and mask entropy  $S_{\mathbf{m}}(\mathbf{a}) = -\sum_{[i,t,d] \in \mathbf{a}} \mathbf{m}_i[t,d] \ln(\mathbf{m}_i[t,d]) + (1 - \mathbf{m}_i[t,d]) \ln(1 - \mathbf{m}_i[t,d])$ , where  $\mathbf{a}$  represents true salient features.
+
+Results. Table 1 summarizes the performance results of the above regressors with rare salient features. AUP does not work as a performance discriminator in sparse scenarios. We find that for all metrics except AUP, our method significantly outperforms all other benchmarks. Moreover, ContraLSP identifies a notably larger proportion of genuinely important features in all experiments, even close to precise attribution, as indicated by the higher AUR. Note that when different groups are present within the samples, the performance of mask-based methods at the baseline significantly deteriorates, while ContraLSP remains relatively unaffected. We present a comparison between the perturbations generated by ContraLSP and Extrmask, as shown in Figure 5. This suggests that employing counterfactuals for learning contrastive inter-samples leads to less information in non-salient areas and highlights the mask more compared to other methods. We display the saliency maps for rare experiments, which are shown in the Appendix G. Our method accurately captures the important features with some smoothing in this setting, indicating that the sparse gates are working. We also explore in Appendix F whether different perturbations keep the original data distribution.
+
+# 5.2 BLACK-BOX CLASSIFICATION SIMULATION
+
+Datasets and Benchmarks. We reproduce the Switch-Feature and State experiments from Tonekaboni et al. (2020). The Switch-Feature data introduces complexity by altering features using a Gaussian Process (GP) mixture model. For the State dataset, we introduce intricate temporal dynamics using a non-stationary Hidden Markov Model (HMM) to generate multivariate altering observations with time-dependent state transitions. These alterations influence the predictive distribution, highlighting the importance of identifying key features during state transitions. Therefore, an accurate generator for capturing temporal dynamics is essential in this context. For a further description of the datasets, see Appendix D.2. For the benchmarks, in addition to the previous ones, we also use FIT, DeepLIFT, GradSHAP (Lundberg & Lee, 2017), LIME (Ribeiro et al., 2016), and RETAIN (Choi et al., 2016).
+
+Table 2: Performance on Switch Feature and State data.  
+
+<table><tr><td rowspan="2">METHOD</td><td colspan="4">SWITCH-Feature</td><td colspan="4">STATE</td></tr><tr><td>AUP ↑</td><td>AUR ↑</td><td>Im/104↑</td><td>Sm/103↓</td><td>AUP ↑</td><td>AUR ↑</td><td>Im/104↑</td><td>Sm/103↓</td></tr><tr><td>FO</td><td>0.89±0.03</td><td>0.37±0.02</td><td>1.86±0.14</td><td>15.60±0.28</td><td>0.90±0.05</td><td>0.30±0.01</td><td>2.73±0.15</td><td>28.07±0.54</td></tr><tr><td>AFO</td><td>0.82±0.06</td><td>0.41±0.02</td><td>2.00±0.14</td><td>17.32±0.29</td><td>0.84±0.08</td><td>0.36±0.03</td><td>3.16±0.27</td><td>34.03±1.10</td></tr><tr><td>IG</td><td>0.91±0.02</td><td>0.44±0.03</td><td>2.21±0.17</td><td>16.87±0.52</td><td>0.93±0.02</td><td>0.34±0.03</td><td>3.17±0.28</td><td>30.19±1.22</td></tr><tr><td>GRADSHAP</td><td>0.88±0.02</td><td>0.38±0.02</td><td>1.92±0.13</td><td>15.85±0.40</td><td>0.88±0.06</td><td>0.30±0.02</td><td>2.76±0.20</td><td>28.18±0.96</td></tr><tr><td>DEEPLIFT</td><td>0.91±0.02</td><td>0.44±0.02</td><td>2.23±0.16</td><td>16.86±0.52</td><td>0.93±0.02</td><td>0.35±0.03</td><td>3.20±0.27</td><td>30.21±1.19</td></tr><tr><td>LIME</td><td>0.94±0.02</td><td>0.40±0.02</td><td>2.01±0.13</td><td>16.09±0.58</td><td>0.95±0.02</td><td>0.32±0.03</td><td>2.94±0.26</td><td>28.55±1.53</td></tr><tr><td>FIT</td><td>0.48±0.03</td><td>0.43±0.02</td><td>1.99±0.11</td><td>17.16±0.50</td><td>0.45±0.02</td><td>0.59±0.02</td><td>7.92±0.40</td><td>33.59±0.17</td></tr><tr><td>RETAIN</td><td>0.93±0.01</td><td>0.33±0.04</td><td>1.54±0.20</td><td>15.08±1.13</td><td>0.52±0.16</td><td>0.21±0.02</td><td>1.56±0.24</td><td>25.01±0.57</td></tr><tr><td>DYNAMASK</td><td>0.35±0.00</td><td>0.77±0.02</td><td>5.22±0.26</td><td>12.85±0.53</td><td>0.36±0.01</td><td>0.79±0.01</td><td>10.59±0.20</td><td>25.11±0.40</td></tr><tr><td>EXTRMASK</td><td>0.97±0.01</td><td>0.65±0.05</td><td>8.45±0.51</td><td>6.90±1.44</td><td>0.87±0.01</td><td>0.77±0.01</td><td>29.71±1.39</td><td>7.54±0.46</td></tr><tr><td>CONTRALSP</td><td>0.98±0.00</td><td>0.80±0.03</td><td>24.23±1.27</td><td>0.91±0.26</td><td>0.90±0.03</td><td>0.81±0.01</td><td>50.09±0.78</td><td>0.50±0.05</td></tr></table>
+
+Table 3: Effects of contrastive perturbations (using the triplet loss) and smoothing constraint (using the trend function) on the Switch-Feature and State datasets.  
+
+<table><tr><td rowspan="2">METHOD</td><td colspan="4">SWITCH-Feature</td><td colspan="4">STATE</td></tr><tr><td>AUP↑</td><td>AUR↑</td><td>Im/104↑</td><td>Sm/103↓</td><td>AUP↑</td><td>AUR↑</td><td>Im/104↑</td><td>Sm/103↓</td></tr><tr><td>CONTRALSP W/O BOTH</td><td>0.92±0.01</td><td>0.79±0.02</td><td>22.08±1.43</td><td>0.78±0.16</td><td>0.76±0.02</td><td>0.74±0.01</td><td>42.26±0.45</td><td>0.14±0.02</td></tr><tr><td>CONTRALSP W/O TRIPLETS LOSS</td><td>0.97±0.01</td><td>0.79±0.02</td><td>22.99±0.84</td><td>1.00±0.21</td><td>0.88±0.03</td><td>0.80±0.01</td><td>49.04±0.75</td><td>0.76±0.07</td></tr><tr><td>CONTRALSP W/O TREND FUNCTION</td><td>0.92±0.01</td><td>0.80±0.01</td><td>24.16±0.69</td><td>0.65±0.10</td><td>0.77±0.02</td><td>0.80±0.01</td><td>42.22±0.50</td><td>0.15±0.02</td></tr><tr><td>CONTRALSP</td><td>0.98±0.00</td><td>0.80±0.03</td><td>24.23±1.27</td><td>0.91±0.26</td><td>0.90±0.03</td><td>0.81±0.01</td><td>50.09±0.78</td><td>0.50±0.05</td></tr></table>
+
+Metrics. We maintain consistency with the ones previously employed.
+
+Results. The performance results on simulated data are presented in Table 2. Across Switch-Feature and State settings, ContraLSP is the best explainer on 7/8 (4 metrics in two datasets) over the strongest baselines. Specifically, when AUP is at the same level, our method achieves high AUR results from its emphasis on producing smooth masks over time, favoring complete subsequence patterns over sparse portions, aligning with human interpretation needs. The reason why Dynamask has a high AUR is that the failure pro
+
+![](images/7afd2e7781166816cd4bbf853d35f7c3a3142d9f8a8393188ae5aad18a4630ad.jpg)  
+Figure 6: Saliency maps produced by various methods for Switch-Feature data.
+
+duces a smaller region of masks, as shown in Figure 6. ContraLSP also has an average  $94.75\%$  improvement in the information content  $I_{m}$  and an average  $90.24\%$  reduction in the entropy  $S_{m}$  over the strongest baselines. This indicates that the contrastive perturbation is superior to perturbation by other means when explaining forecasts based on multivariate time series data.
+
+Ablation study. We further explore these two datasets with the ablation study of two crucial components of the model: (i) let  $\mathcal{L}_{cntr}$  to cancel contrastive learning with the triplet loss and (ii) without the trend function  $\tau_{\theta_2}(\cdot)$  so that  $\pmb{\mu}' = \pmb{\mu}$ . As shown in Table 3, the ContraLSP with both components performs best. Whereas without the use of triplet loss, the performance degrades as the method fails to learn the mask with counterfactuals. Such perturbations without contrastive optimization are not sufficiently uninformative, leading to a lack of distinction among samples. Moreover, equipped with the trend function, ContraLSP improves the AUP by 0.06 and 0.13 on the two datasets, respectively. It indicates that temporal trends introduce context as a smoothing factor, which improves the explanatory ability of our method. To determine the values of  $\alpha$  and  $\beta$  in Eq. (7), we also show different values for parameter combination, which are given in more detail in the Appendix E.2.
+
+# 5.3 MIMIC-III MORTALITY DATA
+
+Dataset and Benchmarks. We use the MIMIC-III dataset (Johnson et al., 2016), which is a comprehensive clinical time series dataset encompassing various vital and laboratory measurements. It is extensively utilized in healthcare and medical artificial intelligence-related research. For more details, please refer to Appendix D.3. We use the same benchmarks as before the classification.
+
+Table 4: Performance report on MIMIC-III mortality by masking  $20\%$  data.  
+
+<table><tr><td rowspan="2">METHOD</td><td colspan="4">AVERAGE SUBSTITUTION</td><td colspan="4">ZERO SUBSTITUTION</td></tr><tr><td>ACC ↓</td><td>CE ↑</td><td>SUFF*10^2 ↓</td><td>COMP*10^2 ↑</td><td>ACC ↓</td><td>CE ↑</td><td>SUFF*10^2 ↓</td><td>COMP*10^2 ↑</td></tr><tr><td>FO</td><td>0.988±0.001</td><td>0.094±0.005</td><td>0.455±0.076</td><td>-0.229±0.059</td><td>0.971±0.003</td><td>0.121±0.008</td><td>-0.539±0.169</td><td>-0.523±0.274</td></tr><tr><td>AFO</td><td>0.989±0.002</td><td>0.097±0.005</td><td>0.185±0.122</td><td>0.008±0.077</td><td>0.972±0.004</td><td>0.120±0.008</td><td>-0.546±0.322</td><td>-0.169±0.240</td></tr><tr><td>IG</td><td>0.988±0.002</td><td>0.096±0.005</td><td>0.273±0.098</td><td>-0.080±0.150</td><td>0.971±0.004</td><td>0.122±0.006</td><td>-0.474±0.228</td><td>-0.385±0.268</td></tr><tr><td>GRADSHAP</td><td>0.987±0.003</td><td>0.095±0.005</td><td>0.400±0.103</td><td>-0.219±0.058</td><td>0.968±0.005</td><td>0.128±0.015</td><td>0.066±0.460</td><td>-0.628±0.377</td></tr><tr><td>DEEPLIFT</td><td>0.987±0.002</td><td>0.095±0.004</td><td>0.303±0.104</td><td>-0.115±0.140</td><td>0.972±0.004</td><td>0.119±0.005</td><td>-0.427±0.193</td><td>-0.482±0.246</td></tr><tr><td>LIME</td><td>0.997±0.001</td><td>0.094±0.005</td><td>0.116±0.122</td><td>-0.028±0.050</td><td>0.988±0.003</td><td>0.099±0.004</td><td>1.688±0.472</td><td>0.254±0.241</td></tr><tr><td>FIT</td><td>0.996±0.01</td><td>0.098±0.004</td><td>-0.139±0.139</td><td>0.375±0.067</td><td>0.987±0.004</td><td>0.108±0.07</td><td>-0.745±0.450</td><td>1.053±0.224</td></tr><tr><td>RETAIN</td><td>0.988±0.001</td><td>0.092±0.005</td><td>0.788±0.046</td><td>-0.425±0.096</td><td>0.971±0.004</td><td>0.119±0.008</td><td>0.072±0.394</td><td>-0.984±0.266</td></tr><tr><td>DYNAMASK</td><td>0.990±0.001</td><td>0.099±0.005</td><td>-0.083±0.089</td><td>0.354±0.064</td><td>0.976±0.004</td><td>0.114±0.007</td><td>-0.422±0.501</td><td>0.609±0.170</td></tr><tr><td>EXTRMASK</td><td>0.982±0.003</td><td>0.118±0.007</td><td>-1.157±0.362</td><td>1.538±0.395</td><td>0.943±0.007</td><td>0.318±0.051</td><td>-6.942±0.531</td><td>10.847±2.055</td></tr><tr><td>CONTRLSP</td><td>0.980±0.002</td><td>0.127±0.007</td><td>-1.792±0.095</td><td>2.386±0.175</td><td>0.928±0.020</td><td>0.357±0.044</td><td>-6.636±0.315</td><td>17.442±2.544</td></tr></table>
+
+![](images/85ddd4cd7d09872462e3c314936d43232c4ae8b5e2517d5e7b79b979e012c375.jpg)  
+Figure 7: Quantitative results on the MIMIC-III mortality experiment, focusing on Accuracy  $\downarrow$ , Cross Entropy  $\uparrow$ , Sufficiency  $\downarrow$ , and Comprehensiveness  $\uparrow$ . We mask a varying percentage of the data (ranging from  $10\%$  to  $60\%$ ) for each patient and replace the masked data with the overall average over time for each feature:  $\overline{\boldsymbol{x}}_i[t,d] = \sum_{t=1}^{T} \boldsymbol{x}_i[t,d]$ . Since some curves are similar, we show representative baselines for clarity.
+
+Metrics. Due to the absence of real attribution features in MIMIC-III, we mask certain portions of the features to assess their importance. We report that performance is evaluated using top mask substitution, as is done in Enguehard (2023). It replaces masked features either with an average over time of this feature  $(\overline{\boldsymbol{x}}_i[t,d] = \frac{1}{T}\sum_{t = 1}^{T}\boldsymbol{x}_i[t,d])$  or with zeros  $(\overline{\boldsymbol{x}}_i[t,d] = 0)$ . The metrics we select are Accuracy (Acc, lower is better), Cross-Entropy (CE, higher is better), Sufficiency (Suff, lower is better), and Comprehensiveness (Comp, higher is better), where the details are in Appendix D.3.
+
+Results. The performance results on MIMIC-III mortality by masking  $20\%$  data are presented in Table 4. We can see that our method outperforms the leading baseline Extrmask on 7/8 metrics (across 4 metrics in two substitutions). Compared to other methods on feature-removal (FO, AFO, FIT) and gradient (IG, DeepLift, GradShap), the gains are greater. The reason could be that the local mask produced by ContraLSP is sparser than others and is replaced by more uninformative perturbations. We show the details of hyperparameter determination for the MIMIC-III dataset, which is deferred to Appendix E.2. Considering that replacement masks different proportions of the data, we also show the average substitution using the above metrics in Figure 7, where  $10\%$  to  $60\%$  of the data is masked for each patient. Our results show that our method outperforms others in most cases. This indicates that perturbations using contrastive learning are superior to those using other perturbations in interpreting forecasts for multivariate time series data.
+
+# 6 CONCLUSION
+
+We introduce ContraLSP, a perturbation-base model designed for the interpretation of time series models. By incorporating counterfactual samples and sample-specific sparse gates, ContraLSP not only offers contractive perturbations but also maintains sparse salient areas. The smooth constraint applied through temporal trends further enhances the model's ability to align with latent patterns in time series data. The performance of ContraLSP across various datasets and its ability to reveal essential patterns make it a valuable tool for enhancing the transparency and interpretability of time series models in diverse fields. However, generating perturbations by the contrasting objective may not bring counterfactuals strong enough, since it is label-free generation. Besides, an inherent limitation of our method is the selection of sparse parameters, especially when dealing with different datasets. Addressing this challenge may involve the implementation of more parameter-efficient tuning strategies, so it would be interesting to explore one of these adaptations to salient areas.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by Alibaba Group through Alibaba Research Intern Program.
+
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+Harini Suresh, Nathan Hunt, Alistair Johnson, Leo Anthony Celi, Peter Szolovits, and Marzyeh Ghassemi. Clinical intervention prediction and understanding with deep neural networks. In MLHC, pp. 322-337, 2017.  
+Damien Teney, Ehsan Abbasnedjad, and Anton van den Hengel. Learning what makes a difference from counterfactual examples and gradient supervision. In ECCV, pp. 580-599, 2020.  
+Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology, 58(1):267-288, 1996.  
+Erico Tjoa and Cuntai Guan. A survey on explainable artificial intelligence (xai): Toward medical xai. IEEE Transactions on Neural Networks and Learning Systems, 32(11):4793-4813, 2020.  
+Sana Tonekaboni, Shalmali Joshi, Kieran Campbell, David K Duvenaud, and Anna Goldenberg. What went wrong and when? Instance-wise feature importance for time-series black-box models. In NeurIPS, pp. 799-809, 2020.  
+Makoto Yamada, Takeuchi Koh, Tomoharu Iwata, John Shawe-Taylor, and Samuel Kaski. Localized lasso for high-dimensional regression. In AISTATS, pp. 325-333, 2017.  
+Yutaro Yamada, Ofir Lindenbaum, Sahand Negahban, and Yuval Kluger. Feature selection using stochastic gates. In ICML, pp. 10648-10659, 2020.  
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+Zhaoyang Zhu, Weiqi Chen, Rui Xia, Tian Zhou, Peisong Niu, Bingqing Peng, Wenwei Wang, Hengbo Liu, Ziqing Ma, Xinyue Gu, et al. Energy forecasting with robust, flexible, and explainable machine learning algorithms. AI Magazine, 44(4):377-393, 2023.
+
+# A REGULARIZATION TERM
+
+Let  $\operatorname{erf}$  be the Gaussian error function defined as  $\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt$ , and let the mask  $\pmb{m}_i$  be obtained with the sigmoid gate output  $\pmb{\mu}_i'$  and an injected noise  $\pmb{\epsilon}_i$  from  $\mathcal{N}(0, \delta^2)$ . Thus, the regularization term for each sample  $\mathcal{R}^{(i)}$  can be expressed by
+
+$$
+\begin{array}{l} \mathcal {R} ^ {(i)} \left(\boldsymbol {x} _ {i}, \boldsymbol {m} _ {i}\right) = \mathbb {E} \left[ \alpha \left\| \boldsymbol {m} _ {i} \right\| _ {0} \right] \\ = \alpha \sum_ {t = 1} ^ {T} \sum_ {d = 1} ^ {D} \mathbb {P} \left(\boldsymbol {\mu} _ {i} ^ {\prime} [ t, d ] + \boldsymbol {\epsilon} _ {i} [ t, d ] > 0\right) \\ = \alpha \sum_ {t = 1} ^ {T} \sum_ {d = 1} ^ {D} [ 1 - \mathbb {P} (\boldsymbol {\mu} _ {i} ^ {\prime} [ t, d ] + \boldsymbol {\epsilon} _ {i} [ t, d ] \leq 0) ] \\ = \alpha \sum_ {t = 1} ^ {T} \sum_ {d = 1} ^ {D} \left[ 1 - \Psi \left(\frac {- \boldsymbol {\mu} _ {i} ^ {\prime} [ t , d ]}{\delta}\right) \right] \tag {8} \\ = \alpha \sum_ {t = 1} ^ {T} \sum_ {d = 1} ^ {D} \Psi \left(\frac {\boldsymbol {\mu} _ {i} ^ {\prime} [ t , d ]}{\delta}\right) \\ = \alpha \sum_ {t = 1} ^ {T} \sum_ {d = 1} ^ {D} \left(\frac {1}{2} - \frac {1}{2} \operatorname {e r f} \left(- \frac {\mu_ {i} ^ {\prime} [ t , d ]}{\sqrt {2} \delta}\right)\right) \\ = \alpha \sum_ {t = 1} ^ {T} \sum_ {d = 1} ^ {D} \left(\frac {1}{2} + \frac {1}{2} \operatorname {e r f} (\frac {\mu_ {i} ^ {\prime} [ t , d ]}{\sqrt {2} \delta})\right), \\ \end{array}
+$$
+
+where  $\Psi (\cdot)$  is the cumulative distribution function, and  $\pmb{\mu}_i^{\prime}[t,d]$  is computed by Eq. (4).
+
+# B TRIPLE SAMPLES SELECTED
+
+In this section, we describe how to generate positive and negative samples for contrastive learning. For each sample  $\boldsymbol{x}_i$ , our goal is to generate the counterfactuals  $\boldsymbol{x}_i^r$  via the perturbation function  $\varphi(\cdot)$ , optimized to be counterfactual for an uninformative perturbed sample. The pseudo-code of the triplet sample selection is shown in Algorithm 1 and elaborated as follows. (i) We start by clustering samples in each batch into the positives  $\Omega^+$  and the negatives  $\Omega^-$  with 2-kmeans, (ii) we select the current sample from each cluster as an anchor, along with  $K^+$  nearest samples from the same cluster as the positive samples, (iii) and we select  $K^-$  random samples from the other cluster yielding negative samples. Note that we use  $S_p$  and  $S_n$  as auxiliary variables representing two sets to select positive and negative samples, respectively.
+
+Algorithm 1 Selection of a triplet sample  
+Input: The set of perturbation time series  $\Omega = \{\pmb{x}_i^r\}_{i=1}^N$  and the current perturbation  $\pmb{x}_i^r$ .  
+Output: Triple sample  $\mathcal{T}_i = (\pmb{x}_i^r, \{\pmb{x}_{i,k}^{r+}\}_{k=1}^{K^+}, \{\pmb{x}_{i,k}^{-}\}_{k=1}^{K^-})$   
+Initialize a positive set  $S_p = \{\}$  and a negative set  $S_n = \{\}$   
+Clustering positive and negative samples  $\{\Omega^+, \Omega^-\} \gets 2$ -kmeans( $\Omega$ )  
+for  $\Omega^*$  in  $\{\Omega^+, \Omega^-\}$  do  
+    Select Anchor  $\pmb{x}_i^r \in \Omega^*$ $\Omega^* \gets \Omega^* \setminus \{\pmb{x}_i^r\}$   
+    for  $k \gets 1$  to  $K^+$  do  
+         $\pmb{x}_{i,k}^{r+} = \Omega^*. \mathrm{Top}(\pmb{x}_i^r)$ $\Omega^* \gets \Omega^* \setminus \{\pmb{x}_{i,k}^{r+}\}$ ,  $S_p \gets S_p \cup \pmb{x}_{i,k}^{r+}$   
+end for  
+for  $k \gets 1$  to  $K^-$  do  
+     $\pmb{x}_{i,k}^{r-} = \mathrm{random}(\Omega \setminus \Omega^*)$ $\Omega^* \gets \Omega^* \setminus \{\pmb{x}_{i,k}^{r-}\}$ ,  $S_n \gets S_n \cup \pmb{x}_{i,k}^{r-}$   
+end for  
+end for  
+Output: Triple sample  $(\pmb{x}_i^r, \{\pmb{x}_{i,k}^{r+}\}_{k=1}^{K^+}, \{\pmb{x}_{i,k}^{-}\}_{k=1}^{K^-})$
+
+# C PSEUDO CODE
+
+Algorithm 2 The pseudo-code of our ContraLSP  
+Input: Multi-variate time series  $\{\pmb {x}_i\}_{i = 1}^N$  , black-box model  $f$  , sparsity hyper-parameters  $\{\alpha ,\beta \}$  Gaussian noise  $\delta$  , total training epochs  $E$  , learning rate  $\gamma$    
+Output: Masks m to explain   
+Training:   
+Initialize the indicator vectors  $\mu = \{\pmb {\mu}_i\}_{i = 1}^N$  of sparse perturbation   
+Initialize a perturbation function  $\varphi_{\theta_1}(\cdot)$  and a trend function  $\tau_{\theta_2}(\cdot)$    
+for  $e\gets 1$  to  $E$  do   
+for  $i\gets 1$  to  $N$  do Get time treads  $\{\tau_{\theta_2}(\pmb {x}_i[(:,d)])\}_{d = 1}^D$  in each observations  $\pmb {x}_i[:,d]$  Compute  $\pmb {\mu}_i^{\prime}\gets \pmb {\mu}_i\odot \sigma (\tau (\pmb {x}_i)\pmb {\mu}_i)$  Sample  $\epsilon_{i}$  from the Gaussian distribution  $\mathcal{N}(0,\delta)$  Compute instance-wise masks  $m_i\gets \min (1,\max (0,\pmb {\mu}_i' + \epsilon_i))$  Get counterfactual features  $\pmb{x}_i^r\gets \varphi_{\theta_1}(\pmb {x}_i)$  Compute the triplet loss  $\mathcal{L}_{cntr}$  via Alg. 1 and Eq. (2) Compute the regularization term  $\mathcal{R}(\pmb {x}_i,\pmb {m}_i)$  via Eq. (5)   
+end for   
+Get perturbations  $\Phi (\pmb {x},\pmb {m})\gets \pmb {m}\odot \pmb {x} + (\pmb {1} - \pmb {m})\odot \pmb{x}^{r}$    
+Construct the total loss function:  $\tilde{\mathcal{L}} = \mathcal{L}\left(f(\pmb {x}),f\circ \Phi (\pmb {x},\pmb {m})\right) + \frac{\alpha}{N}\sum_{i = 1}^{N}\mathcal{R}(\pmb {x}_i,\pmb {m}_i) + \frac{\beta}{N}\sum_{i = 1}^{N}\mathcal{L}_{cntr}(\pmb {x}_i)$  Update  $\mu \gets \mu -\gamma \nabla_{\mu}\widetilde{\mathcal{L}},\theta_1\gets \theta_1 - \gamma \nabla_{\theta_1}\widetilde{\mathcal{L}},\theta_2\gets \theta_2 - \gamma \nabla_{\theta_2}\widetilde{\mathcal{L}}$    
+end for   
+Store  $\mu ,\varphi_{\theta_1}(\cdot),\tau_{\theta_2}(\cdot)$    
+Inference: Compute final masks  $m\gets \min (1,\max (0,\pmb {\mu}\odot \sigma (\tau (\pmb {x})\pmb {\mu}))$    
+Return: Masks m
+
+# D EXPERIMENTAL SETTINGS AND DETAILS
+
+# D.1 WHITE-BOX REGRESSION DATA
+
+As this experiment relies on a white-box approach, our sole responsibility is to create the input sequences. As detailed by Crabbé & Van Der Schaar (2021), each feature sequence is generated using an ARMA process:
+
+$$
+\boldsymbol {x} _ {i} [ t, d ] = 0. 2 5 \boldsymbol {x} _ {i} [ t - 1, d ] + 0. 1 \boldsymbol {x} _ {i} [ t - 2, d ] + 0. 0 5 \boldsymbol {x} _ {i} [ t - 3, d ] + \epsilon_ {i} ^ {\prime}, \tag {9}
+$$
+
+where  $\epsilon_i^{\prime} \sim \mathcal{N}(0,1)$ . We generate 100 sequence samples for each observation  $d$  within the range of  $d \in [1:50]$  and time  $t$  within the range of  $t \in [1:50]$ , and set the sample size  $|S_1| = |S_1| = |S| / 2 = 50$  in different group experiments.
+
+In the experiment involving Rare-Time, we identify 5 time steps as salient in each sample, where consecutive time steps are randomly selected and differently for different groups. The salient observation instances are defined as  $S^D = [(:,13:38]$  without different groups and as  $S_1^D = [(:,1:25],S_2^D = [(:,13:38]$  with different groups.
+
+In the experiment involving Rare-Observation, we identify 5 salient observations in each sample without replacement from  $[1:50]$ , whereas in different groups  $S_1^D$  and  $S_2^D$  are 5 different observations randomly selected respectively. The salient time instances are defined as  $S^T = [:,13:38]$  without different groups, and as  $S_1^T = [:,1:25], S_2^T = [:,13:38]$  with different groups.
+
+# D.2 BLACK-BOX CLASSIFICATION DATA
+
+Data generation on the Switch-Feature experiment. We generate this dataset closely following Tonekaboni et al. (2020), where the time series states are generated via a two-state HMM with equal
+
+initial state probabilities of  $\left[\frac{1}{3},\frac{1}{3},\frac{1}{3}\right]$  and the following transition probabilities
+
+$$
+\left[ \begin{array}{c c c} 0. 9 5 & 0. 0 2 & 0. 0 3 \\ 0. 0 2 & 0. 9 5 & 0. 0 3 \\ 0. 0 3 & 0. 0 2 & 0. 9 5 \end{array} \right].
+$$
+
+The emission probability is a GP mixture, which is governed by an RBF kernel with 0.2 and uses means  $\mu_{1} = [0.8, -0.5, -0.2]$ ,  $\mu_{2} = [0.0, -1.0, 0.0]$ ,  $\mu_{3} = [-0.2, -0.2, 0.8]$  in each state. The output  $y_{i}$  at every step is designed as
+
+$$
+\boldsymbol {p} _ {i} [ t ] = \left\{ \begin{array}{l l} \frac {1}{1 + e ^ {- \boldsymbol {x} _ {i} [ t , 1 ]}}, & \text {i f} \boldsymbol {s} _ {i} [ t ] = 0 \\ \frac {1}{1 + e ^ {- \boldsymbol {x} _ {i} [ t , 2 ]}}, & \text {e l i f} \boldsymbol {s} _ {i} [ t ] = 1, \quad \text {a n d} \quad y _ {i} [ t ] \sim \operatorname {B e r n o u l l i} (\boldsymbol {p} _ {i} [ t ]), \\ \frac {1}{1 + e ^ {- \boldsymbol {x} _ {i} [ t , 3 ]}}, & \text {e l i f} \boldsymbol {s} _ {i} [ t ] = 2 \end{array} \right.
+$$
+
+where  $s_i[t]$  is a single state at each time that controls the contribution of a single feature to the output, and we set 100 states:  $t \in [1:100]$ . We generate 1000 time series samples using this approach. Then we employ a single-layer GRU trained using the Adam optimizer with a learning rate of  $10^{-4}$  for 50 epochs to predict  $y_i$  based on  $x_i$ .
+
+Data generation on the State experiment. We generate this dataset following Tonekaboni et al. (2020) and Enguehard (2023). The random states of the time series are generated using a two-state HMM with  $\pi = [0.5, 0.5]$  and the following transition probabilities
+
+$$
+\left[ \begin{array}{c c} 0. 1 & 0. 9 \\ 0. 1 & 0. 9 \end{array} \right].
+$$
+
+The emission probability is a multivariate Gaussian, where means are  $\mu_{1} = [0.1, 1.6, 0.5]$  and  $\mu_{2} = [-0.1, -0.4, -1.5]$ . The label  $y_{i}[t]$  is generated only using the last two observations, while the first one is irrelevant. Thus, the output  $y_{i}$  at every step is defined as
+
+$$
+\boldsymbol {p} _ {i} [ t ] = \left\{ \begin{array}{l l} \frac {1}{1 + e ^ {- \boldsymbol {x} _ {i} [ t , 1 ]}} & \text {i f} \boldsymbol {s} _ {i} [ t ] = 0 \\ \frac {1}{1 + e ^ {- \boldsymbol {x} _ {i} [ t , 2 ]}} & \text {e l i f} \boldsymbol {s} _ {i} [ t ] = 1 \end{array} \right., \quad \text {a n d} \quad y _ {i} [ t ] \sim \operatorname {B e r n o u l l i} (\boldsymbol {p} _ {i} [ t ]),
+$$
+
+where  $s_i[t]$  is either 0 or 1 at each time, and we generate 200 states:  $t \in [1:200]$ . We also generate 1000 time series samples using this approach and employ a single-layer GRU with 200 units trained by the Adam optimizer with a learning rate of  $10^{-4}$  for 50 epochs to predict  $y_i$  based on  $x_i$ .
+
+# D.3 MIMIC-III DATA
+
+For this experiment, we opt for adult ICU admission data sourced from the MIMIC-III dataset (Johnson et al., 2016). The objective is to predict in-hospital mortality of each patient based on 48 hours of data  $(T = 48)$ , and we need to explain the prediction model (the true salient features are unknown). For each patient, we used features and data processing consistent with Tonekaboni et al. (2020). We summarize all the observations in Table 5, with a total of  $D = 31$ . Patients with complete 48-hour blocks missing for specific features are excluded, resulting in 22,988 ICU admissions. The predicted model we train is a single-layer RNN consisting of 200 GRU cells. It undergoes training for 80 epochs using the Adam optimizer with a learning rate of 0.001.
+
+Table 5: List of clinical observations at each time for the risk predictor model.  
+
+<table><tr><td>DATA CLASS</td><td>NAME</td></tr><tr><td>STATIC OBSERVATIONS</td><td>AGE, GENDER, ETHNICITY, FIRST ADMISSION TO THE ICU</td></tr><tr><td>LAB OBSERVATIONS</td><td>LACTATE, MAGNESIUM, PHOSPHATE, PLATELET, POTASSIUM, PTT, INR, PT, SODIUM, BUN, WBC</td></tr><tr><td>VITAL OBSERVATIONS</td><td>HEARTRATE, DIASBP, SYSBP, MEANBP, RESPRATE, SPO2, GLUCOSE, TEMP</td></tr></table>
+
+In this task, we introduce the same metrics as Enguehard (2023), which are detailed as follows: (i) Accuracy (Acc) means the prediction accuracy while salient features selected by the model are removed, so a lower value is preferable. (ii) Cross-Entropy (CE) represents the entropy between
+
+Table 6: Experimental settings for ContraLSP across all datasets.  
+
+<table><tr><td>PARAMETER</td><td>RATE-TIME</td><td>RATE-OBSERVATION</td><td>SWITCH-Feature</td><td>STATE</td><td>MIMIC-III</td></tr><tr><td>LEARNING RATE γ</td><td>0.1</td><td>0.1</td><td>0.01</td><td>0.01</td><td>0.1</td></tr><tr><td>OPTIMIZER</td><td>ADAM</td><td>ADAM</td><td>ADAM</td><td>ADAM</td><td>ADAM</td></tr><tr><td>MAX EPOCHS E</td><td>200</td><td>200</td><td>500</td><td>500</td><td>200</td></tr><tr><td>α</td><td>0.1</td><td>0.1</td><td>1.0</td><td>2.0</td><td>0.005</td></tr><tr><td>β</td><td>0.1</td><td>0.1</td><td>2.0</td><td>1.0</td><td>0.01</td></tr><tr><td>δ</td><td>0.5</td><td>0.5</td><td>0.8</td><td>0.5</td><td>0.5</td></tr><tr><td>K+</td><td>|Ω+|/5</td><td>|Ω+|/5</td><td>|Ω+|/5</td><td>|Ω+|/5</td><td>50</td></tr><tr><td>K-</td><td>|Ω-|/5</td><td>|Ω-|/5</td><td>|Ω-|/5</td><td>|Ω-|/5</td><td>50</td></tr></table>
+
+Table 7: The specific structure of the trend function.  
+
+<table><tr><td>No.</td><td>STRUCTURE</td></tr><tr><td>1ST OBS.</td><td>MLP[LINEAR(T, 32), RELU, LINEAR(32, T)]</td></tr><tr><td>2ND OBS.</td><td>MLP[LINEAR(T, 32), RELU, LINEAR(32, T)]</td></tr><tr><td>...</td><td>...</td></tr><tr><td>DTH OBS.</td><td>MLP[LINEAR(T, 32), RELU, LINEAR(32, T)]</td></tr></table>
+
+the predictions of perturbed features with the original features. It quantifies the information loss when crucial features are omitted, with a higher value being preferable. (iii) Sufficiency (Suff) is the average change in predicted class probabilities relative to the original values, with lower values being preferable. (iv) Comprehensiveness (Comp) is the average difference of target class prediction probability when most salient features are removed. It reflects how much the removal of features hinders the prediction, so a higher value is better.
+
+# D.4 DETAILS OF OUR METHOD
+
+We list hyperparameters for each experiment performed in Table 6, and for the triplet loss, the marginal parameter  $b$  is consistently set to 1. The size of  $K^{+}$  and  $K^{-}$  are chosen to depend on the number of positive and negative samples ( $|\Omega^{+}|$  and  $|\Omega^{-}|$ ). In the perturbation function  $\varphi_{\theta_1}(\cdot)$ , we use a single-layer bidirectional GRU, which corresponds to a generalization of the fixed perturbation. In the trend function  $\tau_{\theta_2}(\cdot)$ , we employ an independent MLP for each observation  $d$  to find its trend, whose details are shown in Table 7. Please refer to our codebase<sup>1</sup> for additional details on these hyperparameters and implementations.
+
+# D.5 DETAILS OF BENCHMARKS
+
+We compare our method against ten popular benchmarks, including FO (Suresh et al., 2017), AFO (Tonekaboni et al., 2020), IG (Sundararajan et al., 2017), GradSHAP (Lundberg & Lee, 2017) (SVS (Castro et al., 2009) in regression), FIT (Tonekaboni et al., 2020), DeepLIFT (Shrikumar et al., 2017), LIME (Shrikumar et al., 2017), RETAIN (Choi et al., 2016), Dynamask (Crabbé & Van Der Schaar, 2021), and Extrmask (Enguehard, 2023), whereas the implementation of benchmarks is based on open source codes time interpret² and DynaMask³. All hyperparameters follow the code provided by the authors.
+
+# E ADDITIONAL ABLATION STUDY
+
+# E.1 EFFECT OF DISTANCE TYPE IN CONTRASTIVE LEARNING.
+
+For the instance-wise similarity, we can consider various losses to maximize the distance between the anchor with positive or negative samples in Eq. (2). We evaluate three typical distance metrics in Rare-Time and Rare-Observation datasets: Manhattan distance, Euclidean distance, and cosine
+
+Table 8: Performance of ContraLSP with different contrastive loss types on rare experiments.  
+
+<table><tr><td rowspan="2">DISTANCE TYPE IN Lcntr</td><td colspan="4">RARE-TIME</td><td colspan="4">RARE-OBSERVATION</td></tr><tr><td>AUP ↑</td><td>AUR ↑</td><td>Im/104↑</td><td>Sm/102↓</td><td>AUP ↑</td><td>AUR ↑</td><td>Im/104↑</td><td>Sm/102↓</td></tr><tr><td>MANHATTAN DISTANCE</td><td>1.00±0.00</td><td>0.97±0.01</td><td>19.51±0.30</td><td>4.65±0.71</td><td>1.00±0.00</td><td>1.00±0.00</td><td>20.68±0.03</td><td>0.32±0.16</td></tr><tr><td>EUCLIDEAN DISTANCE</td><td>1.00±0.00</td><td>0.97±0.02</td><td>19.67±0.52</td><td>4.97±0.55</td><td>1.00±0.00</td><td>1.00±0.01</td><td>20.72±0.06</td><td>0.69±0.17</td></tr><tr><td>COSINE SIMILARITY</td><td>1.00±0.00</td><td>0.96±0.02</td><td>18.41±0.64</td><td>5.87±0.74</td><td>1.00±0.00</td><td>0.98±0.01</td><td>19.22±0.06</td><td>0.98±0.23</td></tr></table>
+
+similarity. The results presented in Table 8 indicate that the Manhattan distance is slightly better than the other evaluated losses.
+
+# E.2 EFFECT OF REGULARIZATION FACTOR.
+
+We conduct ablations on the black-box classification data using our method to determine which values of  $\alpha$  and  $\beta$  should be used in Eq. (7). For each parameter combination, we employed five distinct seeds, and the experimental results for Switch-Feature and State are presented in Table 9 and Table 10, respectively. Higher values of AUP and AUR are preferred, and the underlined values represent the best parameter pair associated with these metrics. Those Tables indicate that the  $\ell$ -regularized mask  $m$  is most appropriate when  $\alpha$  is set to 1.0 and 2.0 for both Switch-Feature and State data, allowing for the retention of a small but highly valuable subset of features. Moreover, to force  $\varphi_{\theta_1}(\cdot)$  to learn counterfactual perturbations from other distinguishable samples,  $\beta$  is best set to 2.0 and 1.0, respectively. Otherwise, the perturbation may contain crucial features of the current sample, thereby impacting the classification.
+
+We also perform ablation on the MIMIC-III dataset for parameters  $\alpha$  and  $\beta$  using our method. We employ Accuracy and Cross-Entropy as metrics and show the average substitution in Table 11. This Table shows that  $\beta$  is best set to 0.01 to learn counterfactual perturbations. Note that the results are better when lower values of  $\alpha$  are used, but over-regularizing  $m$  close to 0 may not be beneficial. Notably, lower values of  $\alpha$  yield superior results, but excessively regularizing  $m$  toward 0 may prove disadvantageous (Enguehard, 2023). Therefore, we select  $\alpha = 0.005$  and  $\beta = 0.01$  as deterministic parameters on the MIMIC-III dataset.
+
+Table 9: Effects of  $\alpha$  and  $\beta$  on the Switch-Feature data. Underlining is the best.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">α=0.1</td><td colspan="2">α=0.5</td><td colspan="2">α=1.0</td><td colspan="2">α=2.0</td><td colspan="2">α=5.0</td></tr><tr><td>AUP</td><td>AUR</td><td>AUP</td><td>AUR</td><td>AUP</td><td>AUR</td><td>AUP</td><td>AUR</td><td>AUP</td><td>AUR</td></tr><tr><td>β=0.1</td><td>0.53±0.05</td><td>0.28±0.18</td><td>0.26±0.07</td><td>0.01±0.00</td><td>0.18±0.07</td><td>0.01±0.00</td><td>0.12±0.05</td><td>0.01±0.00</td><td>0.14±0.06</td><td>0.01±0.00</td></tr><tr><td>β=0.5</td><td>0.56±0.03</td><td>0.97±0.01</td><td>0.91±0.06</td><td>0.44±0.28</td><td>0.52±0.20</td><td>0.02±0.01</td><td>0.19±0.05</td><td>0.02±0.00</td><td>0.16±0.09</td><td>0.01±0.00</td></tr><tr><td>β=1.0</td><td>0.55±0.02</td><td>0.97±0.01</td><td>0.89±0.02</td><td>0.87±0.02</td><td>0.98±0.01</td><td>0.56±0.10</td><td>0.71±0.27</td><td>0.09±0.09</td><td>0.28±0.12</td><td>0.02±0.00</td></tr><tr><td>β=2.0</td><td>0.54±0.02</td><td>0.97±0.01</td><td>0.86±0.02</td><td>0.89±0.02</td><td>0.98±0.01</td><td>0.80±0.03</td><td>0.99±0.00</td><td>0.68±0.06</td><td>0.50±0.32</td><td>0.05±0.07</td></tr><tr><td>β=5.0</td><td>0.54±0.02</td><td>0.97±0.01</td><td>0.87±0.02</td><td>0.89±0.02</td><td>0.97±0.01</td><td>0.80±0.03</td><td>0.99±0.00</td><td>0.69±0.05</td><td>0.99±0.00</td><td>0.37±0.09</td></tr></table>
+
+Table 10: Effects of  $\alpha$  and  $\beta$  on the State data. Underlining is the best.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">α=0.1</td><td colspan="2">α=0.5</td><td colspan="2">α=1.0</td><td colspan="2">α=2.0</td><td colspan="2">α=5.0</td></tr><tr><td>AUP</td><td>AUR</td><td>AUP</td><td>AUR</td><td>AUP</td><td>AUR</td><td>AUP</td><td>AUR</td><td>AUP</td><td>AUR</td></tr><tr><td>β=0.1</td><td>0.54±0.01</td><td>0.99±0.00</td><td>0.67±0.03</td><td>0.79±0.05</td><td>0.69±0.05</td><td>0.01±0.00</td><td>0.32±0.14</td><td>0.01±0.00</td><td>0.53±0.08</td><td>0.01±0.00</td></tr><tr><td>β=0.5</td><td>0.52±0.01</td><td>0.96±0.00</td><td>0.66±0.01</td><td>0.90±0.01</td><td>0.77±0.02</td><td>0.85±0.01</td><td>0.88±0.03</td><td>0.79±0.03</td><td>0.77±0.11</td><td>0.08±0.14</td></tr><tr><td>β=1.0</td><td>0.52±0.02</td><td>0.96±0.00</td><td>0.66±0.01</td><td>0.91±0.00</td><td>0.77±0.03</td><td>0.87±0.01</td><td>0.90±0.02</td><td>0.82±0.01</td><td>0.88±0.09</td><td>0.23±0.29</td></tr><tr><td>β=2.0</td><td>0.52±0.01</td><td>0.96±0.00</td><td>0.65±0.02</td><td>0.92±0.00</td><td>0.77±0.02</td><td>0.88±0.01</td><td>0.89±0.02</td><td>0.82±0.01</td><td>0.97±0.01</td><td>0.70±0.01</td></tr><tr><td>β=5.0</td><td>0.52±0.01</td><td>0.96±0.00</td><td>0.65±0.01</td><td>0.91±0.00</td><td>0.76±0.02</td><td>0.88±0.01</td><td>0.89±0.03</td><td>0.82±0.01</td><td>0.97±0.01</td><td>0.70±0.02</td></tr></table>
+
+# F DISTRIBUTION ANALYSIS OF PERTURBATIONS
+
+To investigate whether the perturbed samples are within the original dataset's distribution, we first compute the distribution of the original samples by kernel density estimation $^{4}$  (KDE). Subsequently,
+
+Table 11: Effects of  $\alpha$  and  $\beta$  on MIMIC-III mortality. We mask  $20\%$  data and replace the masked data with the overall average over time for each feature. Underlining is the best.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">α = 0.001</td><td colspan="2">α = 0.005</td><td colspan="2">α = 0.01</td><td colspan="2">α = 0.1</td><td colspan="2">α = 1.0</td></tr><tr><td>Acc</td><td>CE</td><td>Acc</td><td>CE</td><td>Acc</td><td>CE</td><td>Acc</td><td>CE</td><td>Acc</td><td>CE</td></tr><tr><td>β = 0.001</td><td>0.982±0.003</td><td>0.124±0.007</td><td>0.983±0.003</td><td>0.122±0.007</td><td>0.984±0.002</td><td>0.120±0.006</td><td>0.993±0.001</td><td>0.094±0.004</td><td>0.997±0.001</td><td>0.087±0.004</td></tr><tr><td>β = 0.005</td><td>0.981±0.002</td><td>0.123±0.007</td><td>0.984±0.002</td><td>0.123±0.006</td><td>0.984±0.003</td><td>0.121±0.007</td><td>0.993±0.002</td><td>0.095±0.006</td><td>0.996±0.001</td><td>0.087±0.005</td></tr><tr><td>β = 0.01</td><td>0.980±0.003</td><td>0.124±0.007</td><td>0.980±0.002</td><td>0.127±0.007</td><td>0.984±0.002</td><td>0.121±0.007</td><td>0.994±0.002</td><td>0.094±0.004</td><td>0.996±0.001</td><td>0.087±0.004</td></tr><tr><td>β = 0.1</td><td>0.980±0.002</td><td>0.127±0.007</td><td>0.980±0.003</td><td>0.127±0.007</td><td>0.983±0.003</td><td>0.123±0.007</td><td>0.992±0.002</td><td>0.098±0.006</td><td>0.997±0.001</td><td>0.087±0.005</td></tr><tr><td>β = 1.0</td><td>0.981±0.002</td><td>0.127±0.006</td><td>0.981±0.003</td><td>0.128±0.008</td><td>0.983±0.002</td><td>0.123±0.007</td><td>0.989±0.002</td><td>0.106±0.007</td><td>0.996±0.001</td><td>0.088±0.005</td></tr></table>
+
+Table 12: Difference between the distribution of different perturbations and the original distribution.  
+
+<table><tr><td rowspan="2">PERTURBATION TYPE</td><td colspan="2">RARE-TIME</td><td colspan="2">RARE-OBSERVATION</td></tr><tr><td>KDE-SCORE ↑</td><td>KL-DIVERGENCE ↓</td><td>KDE-SCORE ↑</td><td>KL-DIVERGENCE ↓</td></tr><tr><td>ZERO PERTURBATION</td><td>-25.242</td><td>0.0523</td><td>-23.377</td><td>0.0421</td></tr><tr><td>MEAN PERTURBATION</td><td>-30.805</td><td>0.0731</td><td>-26.421</td><td>0.0589</td></tr><tr><td>EXTRMASK PERTURBATION</td><td>-22.532</td><td>0.0219</td><td>-19.102</td><td>0.0104</td></tr><tr><td>CONTRALSP PERTURBATION</td><td>-23.290</td><td>0.0393</td><td>-22.732</td><td>0.0386</td></tr></table>
+
+![](images/b9be7a7e066c2f18c25c803fd22ac2717a386851320c03115009d985aa75a1d3.jpg)
+
+![](images/f8967a992656f3e5f23875c5641a9c47efd34b97486f3de3e6b5553c488cc224.jpg)
+
+![](images/c88ea324be19d8965441ef6a2723c37e0da7797f328ad60c2c1ad23b7b5074ad.jpg)
+
+![](images/5607f074d4ac2e0c0a8c2623ff3f52b2dda16a7408c6431dfa7a2bcf2585b43d.jpg)
+
+![](images/b928487cbea9eac283b6f6bd3b086d3956571c7d57627fd125bfb994878b4597.jpg)  
+Figure 8: Saliency maps produced by various methods for Rare-Time experiment.
+
+![](images/38fd43f9bfb9ff211374bdc06206fea463541726e34700db4d41beb688d374c8.jpg)
+
+![](images/784afa90f8b2756e646033b804bc56be08e65452d47470c4f58ab2a03f81a658.jpg)
+
+![](images/2b295c02d2e8a497a9b044897f04cfb3b92f55c213dbfa58687ac0d72a8acbbb.jpg)
+
+we assess the log-likelihood of each perturbed sample under the original distribution, called as KDE-score, where closer to 0 indicates a higher likelihood of perturbed samples originating from the original distribution. Additionally, we quantify the KL-divergence between the distribution of perturbed samples and original samples, where a smaller KL means that the two distributions are closer. We conduct experiments on the Rare-Time and Rare-Observation datasets and the results are shown in Table 12. It demonstrates that our ContraLSP's perturbation is more akin to the original distribution compared to the zero and mean perturbation. Furthermore, Extrmask performs best because it generates perturbations only from current samples, and therefore the generated perturbations are not guaranteed to be uninformative. This conclusion aligns with the visualization depicted in Figure 1.
+
+# G ILLUSTRATIONS OF SALIENCY MAPS
+
+Saliency maps represent a valuable technique for visualizing the significance of features, and previous works (Alqaraawi et al., 2020; Tonekaboni et al., 2020; Leung et al., 2023), particularly in multivariate time series analysis, have demonstrated their utility in enhancing the interpretative aspects of the results. We also demonstrate the saliency maps of the benchmarks and our method for each dataset: (i) the saliency maps for the rare experiments are shown in Figure 8, 9, 11, and 10, (ii)
+
+![](images/ab1c602fdb6517f2bdd5ff51be277518d6462ddfed3ec32637754bd307139921.jpg)
+
+![](images/b321232adfbfd2e3b5fc327cf5b69ae7d8a3234ebbee0784593519bc3df11316.jpg)
+
+![](images/15a69388e9f89c5d949abb6227a07a39fa0b89a8feeb1d5f62a99a8dd48594b6.jpg)
+
+![](images/a95cb6e7a7efd8b205026836759032a557578e40d546224ad5213654c6fafde5.jpg)
+
+![](images/2f5e79bc20112ec61fb0cc1b2f278562edc9f38cc461d2bdf7dbbb91ded30eed.jpg)  
+Figure 9: Saliency maps produced by various methods for Rare-Observation experiment.
+
+![](images/50e2bb2011b09c7ed0ee40d088830849099d286d918e1eb363d482e7613ce579.jpg)
+
+![](images/71d6a36f7e34ea73861fbc36360d1708cb5a16fb280fb9ffd733d3107277496f.jpg)
+
+![](images/1a34797fd5db15b3a7dd9a3a9a68850c22787ec0468a0d975e6ef392a65dfc95.jpg)
+
+![](images/2c12796ff25615cca13aa76b9534a2890d74e3919c53366c5caedc813521ff0e.jpg)  
+Figure 10: Saliency maps produced by various methods for Rare-Time (Diffgroups) experiment.
+
+the Switch-Feature and State saliency maps are shown in Figure 12 and Figure 13, respectively, (iii) and the saliency maps for the MIMIC-III mortality are in Figure 14.
+
+![](images/3645e00fc8c806534d82a53217949f53a3b2c1d326e82f65336c9f0ff87030aa.jpg)  
+Figure 11: Saliency maps produced by various methods for Rare-Observation (Diffgroups) experiment.
+
+![](images/6428a2994c365befec9530ba9df7815ad84ed0f2908bf323dfab280bc4d96754.jpg)  
+Figure 12: Saliency maps produced by various methods for Switch-Feature data.
+
+![](images/e8d8b96f4e9076d4b9a417a3541987858b9f251b3f9ed1aa58f1712e75c07f00.jpg)  
+Figure 13: Saliency maps produced by various methods for State data.
+
+![](images/e0ca93ff2404fa330336ab6c56eb2c814af800c96eb7716ce79ff9eb221baa75.jpg)  
+Figure 14: Saliency maps produced by various methods for MIMIC-III Mortality data.

2024/Explaining Time Series via Contrastive and Locally Sparse Perturbations/images.zip

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2024/Exploring Diffusion Time-steps for Unsupervised Representation Learning/eb88b99c-2ad3-4064-a4bc-2d7994130e04_content_list.json

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+    {
+        "type": "text",
+        "text": "EXPLORING DIFFUSION TIME-STEPS FOR UNSUPERVISED REPRESENTATION LEARNING",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Zhongqi Yue<sup>1</sup>, Jiankun Wang<sup>1</sup>, Qianru Sun<sup>2</sup>, Lei Ji<sup>3</sup>, Eric I-Chao Chang<sup>3</sup>, Hanwang Zhang<sup>1,4</sup>  \n<sup>1</sup>Nanyang Technological University, <sup>2</sup>Singapore Management University, <sup>3</sup>Microsoft Research Asia, <sup>4</sup>Skywork AI zhongqi.yue@ntu.edu.sg, jiankun001@e.ntu.edu.sg, qianrusun@smu.edu.sg, leiji@microsoft.com, eric.i.chang@outlook.com, hanwangzhang@ntu.edu.sg",
+        "bbox": [
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+            167,
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+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
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+            450,
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+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Representation learning is all about discovering the hidden modular attributes that generate the data faithfully. We explore the potential of Denoising Diffusion Probabilistic Model (DM) in unsupervised learning of the modular attributes. We build a theoretical framework that connects the diffusion time-steps and the hidden attributes, which serves as an effective inductive bias for unsupervised learning. Specifically, the forward diffusion process incrementally adds Gaussian noise to samples at each time-step, which essentially collapses different samples into similar ones by losing attributes, e.g., fine-grained attributes such as texture are lost with less noise added (i.e., early time-steps), while coarse-grained ones such as shape are lost by adding more noise (i.e., late time-steps). To disentangle the modular attributes, at each time-step  $t$ , we learn a  $t$ -specific feature to compensate for the newly lost attribute, and the set of all  $\\{1, \\ldots, t\\}$ -specific features, corresponding to the cumulative set of lost attributes, are trained to make up for the reconstruction error of a pre-trained DM at time-step  $t$ . On CelebA, FFHQ, and Bedroom datasets, the learned feature significantly improves attribute classification and enables faithful counterfactual generation, e.g., interpolating only one specified attribute between two images, validating the disentanglement quality. Codes are in https://github.com/yue-zhongqi/diti.",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
+            171,
+            571,
+            339,
+            588
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A good feature representation should faithfully capture the underlying generative attributes in a compact and modular vector space Bengio et al. (2013), enabling not only sample inference (e.g., image classification) but also counterfactual generation Besserve et al. (2020) (e.g., image synthesis of unseen attribute combinations). Over the past decade, discriminative training has been the feature learning mainstream with exceptional performance in inference tasks He et al. (2016; 2020). However, it hardly achieves faithful generation due to the deliberate discard of certain attributes, e.g., class-irrelevant ones in supervised learning or augmentation-related ones in contrastive learning.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, generative Denoising Diffusion Probabilistic Model (DM) Sohl-Dickstein et al. (2015); Song et al. (2021) can retain all the underlying attributes for faithful generation Dhariwal & Nichol (2021), or even extrapolate novel attribute combinations by textual control Rombach et al. (2022) (e.g., \"Teddy bear skating in Time Square\"), outperforming other generative models like VAE Kingma & Welling (2014) and GAN Creswell et al. (2018). This implies that DM effectively captures the modularity of hidden attributes Yue et al. (2021). However, as DM's formulation has no explicit encoders that transform samples into feature vectors, the conventional encoder-decoder feature learning paradigm via reconstruction Higgins et al. (2017) is no longer applicable.",
+        "bbox": [
+            169,
+            708,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Can we extract DM's knowledge about the modular attributes as a compact feature vector? To answer the question, we start by revealing a natural connection between the diffusion time-steps and the hidden attributes. In the forward diffusion process, given an initial sample  $\\mathbf{x}_0$ , a small amount of Gaussian noise  $\\mathcal{N}(\\mathbf{0},\\mathbf{I})$  is incrementally added to the sample across  $T$  time-steps, resulting in a sequence of noisy samples  $\\mathbf{x}_1,\\ldots ,\\mathbf{x}_T$ . In particular, each  $\\mathbf{x}_t$  adheres to its noisy sample distribution  $q(\\mathbf{x}_t|\\mathbf{x}_0)$ , whose mean and covariance matrix  $\\sigma^2\\mathbf{I}$  is denoted as a dot and a circle with radius  $\\sigma$  in Figure 1a, respectively. As  $t$  increases,  $q(\\mathbf{x}_t|\\mathbf{x}_0)$  progressively collapses towards a pure noise by",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+            47
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+        "page_idx": 0
+    },
+    {
+        "type": "page_number",
+        "text": "1",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/583b9f206b52ce870550d7f775b20d3b406b3249e21771a7ab0c3029fb9b9200.jpg",
+        "image_caption": [
+            "Figure 1: (a) Illustration of attribute loss as time-step  $t$  increases in the forward diffusion process. The two axes depict a two-dimensional sample space. (b) DM reconstructed  $\\mathbf{x}_0$ , denoted as  $\\hat{\\mathbf{x}}_0$ , from randomly sampled  $\\mathbf{x}_t$  at various  $t$ . DM is pre-trained on CelebA, from where  $\\mathbf{x}_0$  is drawn."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "reducing mean (i.e., dots moving closer to the origin), and increasing  $\\sigma$  (i.e., larger radii), which enlarges the overlapping area (OVL) of the probability density function of noisy sample distributions. In particular, we theoretically show in Section 4.1 that OVL is correlated with DM's ambiguous reconstruction of different samples, e.g., due to the large overlap between the red circles at  $t_1$ , noisy samples drawn from the two distributions are reconstructed ambiguously as either A or B, losing the attribute \"expression\" that distinguishes them. As  $t$  increases, more noisy sample distributions exhibit significant overlaps, inducing additional attribute loss, e.g., further losing \"gender\" at  $t_2$  that confuses the identities of A/B and C/D. Eventually, with a large enough  $T$ ,  $\\mathbf{x}_T$  becomes pure noise and all attributes are lost—the reconstruction can be any sample.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This intrinsic connection between diffusion time-steps and modular attributes can act as an effective inductive bias for unsupervised learning. Specifically, DM training can be viewed as learning to reconstruct  $\\mathbf{x}_0$  given  $\\mathbf{x}_t$  at each  $t$  (Section 3.2). However, due to the aforementioned attribute loss, perfect DM reconstruction is impossible, e.g., in Figure 1b, reconstructions exhibit variations in the cumulatively lost attributes. Hence, by contraposition, if a feature enables perfect reconstruction, it must capture the cumulatively lost attributes. This motivates us to learn an expanding set of features to supplement the expanding lost attributes as  $t$  increases. Specifically, we first map  $\\mathbf{x}_0$  to a feature  $\\mathbf{z} = f(\\mathbf{x}_0)$  using an encoder  $f$ . Then we partition  $\\mathbf{z}$  into  $T$  disjoint subsets  $\\{\\mathbf{z}_i\\}_{i=1}^T$ . At each  $t$ , we optimize  $f$  so that  $\\{\\mathbf{z}_i\\}_{i=1}^t$  compensates for a pre-trained DM's reconstruction error. Intuitively by induction, starting from  $t = 0$  with no lost attribute; if  $\\{\\mathbf{z}_i\\}_{i=1}^t$  captures the cumulatively lost attributes till  $t$ ,  $\\mathbf{z}_{t+1}$  must capture the newly lost attribute to enable perfect reconstruction at  $t + 1$ ; until  $t = T - 1$ ,  $\\{\\mathbf{z}_i\\}_{i=1}^{T-1} \\cup \\mathbf{z}_T$  learns all hidden attributes in a compact and modular vector space.",
+        "bbox": [
+            169,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We term our approach as DiTi to highlight our leverage of Diffusion Time-step as an inductive bias for feature learning. We summarize the paper structure and our contributions below:",
+        "bbox": [
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+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- In Section 3.1, we formalize the notion of good feature with a definition of disentangled representation Higgins et al. (2018) and provide a preliminary introduction to DM in Section 3.2.",
+            "- In Section 4.1, we build a theoretical framework that connects diffusion time-steps and the hidden modular attributes, which motivates a simple and practical approach for unsupervised disentangled representation learning, discussed in Section 4.2.",
+            "- In Section 5, by extensive experiments on CelebA Liu et al. (2015), FFHQ Karras et al. (2019) and Bedroom Yu et al. (2015), our DiTi feature brings significant improvements in attribute inference accuracy and enables counterfactual generation, verifying its disentanglement quality."
+        ],
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+    {
+        "type": "text",
+        "text": "2 RELATED WORKS",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "DM for Representation Learning. There are three main approaches: 1) DDIM Song et al. (2020) inversion aims to find a latent  $\\mathbf{x}_T$  as feature, by DDIM sampling with the learned DM, can reconstruct the image  $\\mathbf{x}_0$ . However, this process is time-consuming, and the resulting latent is difficult to interpret Hertz et al. (2022). 2) Kwon et al. (2022) uses the bottleneck feature of the U-Net at all time-steps. However, the feature is not compact and difficult for downstream leverage. 3)",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "page_footnote",
+        "text": "$^{1}$ Attribute name is illustrative. Our method uses no attribute supervision, nor explicitly names them.",
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+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/1485e2de09abaf9c359b389c5daa634155f511a2e723bcae8872c77ac4edd0eb.jpg",
+        "image_caption": [
+            "(a) Examples of counterfactual generation"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/105750c54aae52a17ef134d4c9825fe59750b411c9011ddf642c3c8cb462d460.jpg",
+        "image_caption": [
+            "(b) Histogram of classifier weight value"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/55a1ddda8fafc9c60ca2d3d00c6446a6a05ad4e96cac6b4865869407c00f3f89.jpg",
+        "image_caption": [
+            "Figure 2: (a) Counterfactual generations on CelebA by manipulating 16 out of 512 feature dimensions (i.e., simulating the edit of a single  $\\mathbf{z}_i$ ). A disentangled representation enables editing a single attribute (e.g., gender) without affecting others (e.g., lighting) and promotes faithful extrapolation (e.g., no artifacts). (b) Histogram of the classifier weight value. More dimensions of DiTi weights are closed to 1 and 0 (explanations in the text)."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The closest to our works are Preechakul et al. (2022); Zhang et al. (2022). However, they learn a time-step-independent  $\\mathbf{z}$  to supplement DM reconstruction error. Without explicit design to enforce modularity, the learned feature entangles all the attributes. At time-step  $t$ , it leaks information about attributes lost at a later time-step (i.e.,  $t + 1, \\dots, T$ ). Hence, instead of learning the lost attribute  $\\mathbf{z}_t$ , the reconstruction may digress to use the spurious correlation between leaked information and lost attribute. In Section 5, we validate this deficiency by observing failed counterfactual generation and lower attribute prediction accuracy.",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Disentangled Representation separates the modular hidden attributes that generate data, which can be formally defined from a group-theoretical view Higgins et al. (2018). Our work focuses on the unsupervised learning setting. Existing methods typically learn a VAE Burgess et al. (2017) or GAN Jeon et al. (2021) with information bottleneck. Yet they lack explicit inductive bias that is necessary for unsupervised disentanglement Locatello et al. (2019). Our work reveals that the connection between hidden attributes and diffusion time-steps is an effective inductive bias towards disentanglement (Section 4.1).",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PRELIMINARIES",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 DISENTANGLED REPRESENTATION",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A good feature can be formally defined by the classic notion of disentangled representation. We aim to provide a preliminary introduction below, and refer interesting readers to Higgins et al. (2018); Wang et al. (2021) for more formal details. Each sample  $\\mathbf{x} \\in \\mathcal{X}$  in the image space  $\\mathcal{X}$  is generated from a hidden attribute vector  $\\mathbf{z} \\in \\mathcal{Z}$  in the vector space  $\\mathcal{Z}$  through an injective mapping  $\\Phi: \\mathcal{Z} \\to \\mathcal{X}$  (i.e., different samples are generated by different attributes). In particular,  $\\mathcal{Z}$  can be decomposed into the Cartesian product of  $N$  subspaces  $\\mathcal{Z} = \\mathcal{Z}_1 \\times \\ldots \\times \\mathcal{Z}_N$ , where each  $\\mathcal{Z}_i$  is called a modular attribute, as its value can be locally intervened without affecting other modular attributes (e.g., changing \"expression\" without affecting \"gender\" or \"age\"). A disentangled representation is a mapping  $f: \\mathcal{X} \\to \\mathcal{Z}$  that inverses  $\\Phi$ , i.e., given each image  $\\mathbf{x}$ ,  $f(\\mathbf{x})$  recovers the modular attributes as a feature vector  $\\mathbf{z} = [\\mathbf{z}_1, \\dots, \\mathbf{z}_N]$ , which possesses desired properties of a good feature in the following common views:",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Counterfactual generation. A counterfactual image of  $\\mathbf{x}$  by intervening the  $i$ -th modular attribute with  $g_{i}$  (e.g., aging) is denoted as  $g_{i} \\cdot \\mathbf{x} \\in \\mathcal{X}$  (e.g., an older  $\\mathbf{x}$ ). Specifically,  $g_{i} \\cdot \\mathbf{x}$  is generated by first mapping  $\\mathbf{x}$  to  $\\mathbf{z}$  by  $f$ , editing  $\\mathbf{z}_{i}$  according to  $g_{i}$  without changing other attributes, and finally feeding the modified  $\\mathbf{z}$  to  $\\Phi$ . In fact,  $g_{i} \\cdot \\mathbf{x}$  is formally known as group action Judson (1994). Figure 2 shows counterfactual images generated by editing three attributes.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Sample inference. A modular feature facilitates robust inference in downstream tasks about any subset of  $\\{\\mathbf{z}_i\\}_{i=1}^N$ , e.g., a linear classifier can eliminate task-irrelevant environmental attributes by assigning larger values to the dimensions corresponding to  $\\mathbf{z}_i$  and 0 to other dimensions. In Figure 2, by explicitly disentangling the modular attributes (Section 4), our DiTi achieves higher accuracy in",
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+        "page_idx": 2
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            173,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "page_number",
+        "text": "3",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "attribute prediction than PDAE Zhang et al. (2022) (without disentangling), while more dimensions of our classifier weight are closed to 1 (corresponding to  $\\mathbf{z}_i$ ) and 0 (environmental attributes).",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 DENOISING DIFFUSION PROBABILISTIC MODEL",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Denoising Diffusion Probabilistic Model Ho et al. (2020) (DM) is a latent variable model with a fixed forward process, and a learnable reverse process. Details are given below.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Forward Process. It is fixed to a Markov chain that incrementally adds Gaussian noise to each sample  $\\mathbf{x}_0$  in  $T$  time-steps, producing a sequence of noisy samples  $\\mathbf{x}_1,\\ldots ,\\mathbf{x}_T$  as latents of the same dimensionality as the original sample  $\\mathbf{x}_0$ . Given  $\\mathbf{x}_0$  and a variance schedule  $\\beta_{1},\\dots ,\\beta_{T}$  (i.e., how much noise is added at each time-step), each  $\\mathbf{x}_t$  adheres to the following noisy sample distribution:",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nq \\left(\\mathbf {x} _ {t} \\mid \\mathbf {x} _ {0}\\right) = \\mathcal {N} \\left(\\mathbf {x} _ {t}; \\sqrt {\\bar {\\alpha} _ {t}} \\mathbf {x} _ {0}, (1 - \\bar {\\alpha} _ {t}) \\mathbf {I}\\right), \\quad \\text {w h e r e} \\alpha_ {t} := 1 - \\beta_ {t}, \\bar {\\alpha} _ {t} := \\prod_ {s = 1} ^ {t} \\alpha_ {s}. \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Reverse Process. It corresponds to a learned Gaussian transition  $p_{\\theta}(\\mathbf{x}_{t - 1}|\\mathbf{x}_t)$  parameterized by  $\\theta$ , starting at  $p(\\mathbf{x}_T)\\coloneqq \\mathcal{N}(\\mathbf{x}_T;\\mathbf{0},\\mathbf{I})$ . Each  $p_{\\theta}(\\mathbf{x}_{t - 1}|\\mathbf{x}_t)$  is computed in two steps: 1) Reconstruct  $\\mathbf{x}_0$  from  $\\mathbf{x}_t$  with  $u_{\\theta}(\\mathbf{x}_t,t)$ , where  $u_{\\theta}$  is a learnable U-Net Ronneberger et al. (2015). 2) Compute  $q(\\mathbf{x}_{t - 1}|\\mathbf{x}_t,u_{\\theta}(\\mathbf{x}_t,t))$ , which has a closed-form solution given in Appendix. Training is performed by minimizing the reconstruction error made by  $u_{\\theta}(\\mathbf{x}_t,t)$  at each time-step  $t$ :",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {D M} = \\underset {t, \\mathbf {x} _ {0}, \\epsilon} {\\mathbb {E}} \\left[ \\frac {\\bar {\\alpha} _ {t}}{1 - \\bar {\\alpha} _ {t}} \\| \\mathbf {x} _ {0} - u _ {\\theta} (\\sqrt {\\bar {\\alpha} _ {t}} \\mathbf {x} _ {0} + \\sqrt {1 - \\bar {\\alpha} _ {t}} \\epsilon , t) \\| ^ {2} \\right], \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\epsilon \\sim \\mathcal{N}(\\mathbf{0},\\mathbf{I})$  is a random noise. Note that we formulate the U-Net to predict  $\\mathbf{x}_0$  (equivalent to the  $\\epsilon$ -formulation in Ho et al. (2020)). As DM has no explicit feature encoder, we learn one to capture the modular attributes by leveraging the inductive bias in the diffusion time-steps.",
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+    },
+    {
+        "type": "text",
+        "text": "4 METHOD",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 THEORY",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We reveal that in the forward diffusion process, fine-grained attributes are lost at an earlier time-step compared to coarse-grained ones, where the granularity is defined by the pixel-level changes when altering an attribute. We first formalize the notion of attribute loss, i.e., when DM fails to distinguish the noisy samples drawn from two overlapping noisy sample distributions  $q(\\mathbf{x}_t|\\mathbf{x}_0)$  and  $q(\\mathbf{y}_t|\\mathbf{y}_0)$ .",
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+    {
+        "type": "text",
+        "text": "Definition. (Attribute Loss) Given a  $\\theta$ -parameterized DM optimized by Eq. 2, we say that attribute  $\\mathcal{Z}_i$  is lost with degree  $\\tau$  at  $t$  when  $\\mathbb{E}_{\\mathbf{x}_0 \\in \\mathcal{X}}[\\mathrm{Err}(\\mathbf{x}_0, \\mathbf{y}_0 = g_i \\cdot \\mathbf{x}_0, t)] \\geq \\tau$ , where  $\\mathrm{Err}(\\mathbf{x}_0, \\mathbf{y}_0, t) :=$",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\frac {1}{2} \\left[ \\underset {q \\left(\\mathbf {x} _ {t} \\mid \\mathbf {x} _ {0}\\right)} {\\mathbb {E}} \\left[ \\mathbb {1} \\left(\\| \\hat {\\mathbf {x}} _ {0} - \\mathbf {x} _ {0} \\| > \\| \\hat {\\mathbf {x}} _ {0} - \\mathbf {y} _ {0} \\|\\right) \\right] + \\underset {q \\left(\\mathbf {y} _ {t} \\mid \\mathbf {y} _ {0}\\right)} {\\mathbb {E}} \\left[ \\mathbb {1} \\left(\\| \\hat {\\mathbf {y}} _ {0} - \\mathbf {x} _ {0} \\| <   \\| \\hat {\\mathbf {y}} _ {0} - \\mathbf {y} _ {0} \\|\\right) \\right] \\right], \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with  $\\mathbb{1}(\\cdot)$  denoting the indicator function and  $\\hat{\\mathbf{x}}_0 = u_\\theta (\\mathbf{x}_t),\\hat{\\mathbf{y}}_0 = u_\\theta (\\mathbf{y}_t)$  denoting the DM reconstructed samples from  $\\mathbf{x}_t,\\mathbf{y}_t$  , respectively.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Intuitively,  $\\mathrm{Err}(\\mathbf{x}_0,\\mathbf{y}_0,t)$  measures attribute loss degree by how likely a DM falsely reconstructs  $\\mathbf{x}_t$  drawn from  $q(\\mathbf{x}_t|\\mathbf{x}_0)$  closer to  $\\mathbf{y}_0$  instead of  $\\mathbf{x}_0$  (vice versa). Hence, when  $\\mathbf{x}_0$  and  $\\mathbf{y}_0$  differ in attribute  $\\mathbf{z}_i$  modified by  $g_i$ , a larger  $\\mathrm{Err}(\\mathbf{x}_0,\\mathbf{y}_0,t)$  means that the attribute is more likely lost.",
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+    {
+        "type": "text",
+        "text": "Theorem. (Attribute Loss and Time-step) 1) For each  $\\mathcal{Z}_i$ , there exists a smallest time-step  $t(\\mathcal{Z}_i)$  such that  $\\mathcal{Z}_i$  is lost with degree  $\\tau$  at each  $t \\in \\{t(\\mathcal{Z}_i), \\dots, T\\}$ . 2)  $\\exists \\{\\beta_i\\}_{i=1}^T$  such that  $t(\\mathcal{Z}_i) > t(\\mathcal{Z}_j)$  whenever  $\\| \\mathbf{x}_0 - g_i \\cdot \\mathbf{x}_0 \\|$  is first-order stochastic dominant over  $\\| \\mathbf{x}_0 - g_j \\cdot \\mathbf{x}_0 \\|$  with  $\\mathbf{x}_0 \\sim \\mathcal{X}$  uniformly.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, the first part of the theorem states that a lost attribute will not be regained as time-step  $t$  increases, and there is a time-step  $t(\\mathcal{Z}_i)$  when  $\\mathcal{Z}_i$  becomes lost (with degree  $tau$ ) for the first time. The second part states that when  $\\| \\mathbf{x}_0 - g_i \\cdot \\mathbf{x}_0 \\|$  is more likely to take on a larger value than  $\\| \\mathbf{x}_0 - g_j \\cdot \\mathbf{x}_0 \\|$ , the attribute  $\\mathcal{Z}_i$  (i.e., a coarse-grained attribute) is lost at a larger time-step compared to  $\\mathcal{Z}_j$  (i.e., a fine-grained attribute). We have the following proof sketch:",
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+        "page_idx": 3
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_number",
+        "text": "4",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/757e552673ca1bff80bc1fe1d16aa5466b7396a9472e4eef339384387fccaa10.jpg",
+        "image_caption": [
+            "Figure 3: Illustration of our DiTi. We break down Eq. 5 at each time-step. On the right, we show the detailed network design, where  $\\hat{\\mathbf{x}}_0$  denotes the reconstructed  $\\mathbf{x}_0$  by the pre-trained DM."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1) Increasing  $t$  induces further attribute loss. We show in Appendix that  $\\mathrm{Err}(\\mathbf{x}_0, \\mathbf{y}_0, t) = \\frac{1}{2}\\mathrm{OVL}(q(\\mathbf{x}_t|\\mathbf{x}_0), q(\\mathbf{y}_t|\\mathbf{y}_0))$  for an optimal DM, where OVL is the overlapping coefficient Inman & Bradley Jr (1989), i.e., overlapping area of probability density functions (PDFs). In particular,",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {E r r} \\left(\\mathbf {x} _ {0}, \\mathbf {y} _ {0}, t\\right) = \\frac {1}{2} \\operatorname {O V L} \\left(q \\left(\\mathbf {x} _ {t} \\mid \\mathbf {x} _ {0}\\right), q \\left(\\mathbf {y} _ {t} \\mid \\mathbf {y} _ {0}\\right)\\right) = \\frac {1}{2} \\left[ 1 - \\operatorname {e r f} \\left(\\frac {\\left\\| \\sqrt {\\bar {\\alpha} _ {t}} \\left(\\mathbf {y} _ {0} - \\mathbf {x} _ {0}\\right) \\right\\|}{2 \\sqrt {2 \\left(1 - \\bar {\\alpha} _ {t}\\right)}}\\right) \\right]. \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As  $\\bar{\\alpha}_t$  decreases with an increasing  $t$  from Eq. 1, and the error function  $\\operatorname{erf}(\\cdot)$  is strictly increasing,  $\\mathrm{Err}(\\mathbf{x}_0,\\mathbf{y}_0,t)$  is strictly increasing in  $t$  given any  $\\mathbf{x}_0,\\mathbf{y}_0$ . Hence  $\\mathbb{E}_{\\mathbf{x}_0\\in \\mathcal{X}}[\\mathrm{Err}(\\mathbf{x}_0,\\mathbf{y}_0 = g_i\\cdot \\mathbf{x}_0,t)]\\geq \\mathbb{E}_{\\mathbf{x}_0\\in \\mathcal{X}}[\\mathrm{Err}(\\mathbf{x}_0,\\mathbf{y}_0 = g_i\\cdot \\mathbf{x}_0,t(\\mathcal{Z}_i))]$  for every  $t\\geq t(\\mathcal{Z}_i)$ , which completes the proof.",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2) Coarse-grained attributes are lost at a larger  $t$ . Given that  $\\mathrm{erf}(\\cdot)$  is strictly increasing and  $\\| \\mathbf{x}_0 - g_i \\cdot \\mathbf{x}_0 \\|$  is first-order stochastic dominant over  $\\| \\mathbf{x}_0 - g_j \\cdot \\mathbf{x}_0 \\|$ , we have  $\\mathbb{E}_{\\mathbf{x}_0 \\in \\mathcal{X}} [\\mathrm{Err}(\\mathbf{x}_0, g_i \\cdot \\mathbf{x}_0, t)] > \\mathbb{E}_{\\mathbf{x}_0 \\in \\mathcal{X}} [\\mathrm{Err}(\\mathbf{x}_0, g_j \\cdot \\mathbf{x}_0, t)]$  at every time-step  $t$  using Eq. 4. Hence  $t(\\mathcal{Z}_i) > t(\\mathcal{Z}_j)$  under any variance schedule  $\\{\\beta_i\\}_{i=1}^T$  such that  $\\mathcal{Z}_i$  is not lost at  $t(\\mathcal{Z}_j)$ , completing the proof. Note that in practice, DM leverages a large  $T$  with small  $\\{\\beta_i\\}_{i=1}^T$ . This ensures that  $\\mathrm{Err}(\\mathbf{x}_0, \\mathbf{y}_0, t)$  slows increases with  $t$  according to Eq. 4. Hence empirically, this theorem tends to hold.",
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+    {
+        "type": "text",
+        "text": "4.2 PROPOSED APPROACH",
+        "text_level": 1,
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+        "text": "The previous theoretical analysis leads to two interesting groundings: 1) Attribute loss is linked with the error of reconstructing  $\\mathbf{x}_0$  from the noisy  $\\mathbf{x}_t$ . 2) Increasing time-step  $t$  in forward diffusion process causes an expanding cumulative loss of attributes, with fine-grained attributes lost at a smaller  $t$  and coarse-grained ones lost at a larger  $t$ . They serve as an effective inductive bias for unsupervised learning: 1) making up for the reconstruction error can retrieve lost attributes, 2) an expanding set of features captures the expanding cumulatively lost attributes. This motivates our approach below.",
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+        "text": "As illustrated in Figure 3, our model includes a frozen DM U-Net  $u_{\\theta}$  pre-trained on  $\\mathcal{D}$  using Eq. 2 until convergence, a trainable encoder  $f$  (trained to become a disentangled representation) that maps each image (without noise) to a  $d$ -dimensional feature  $\\mathbf{z} \\in \\mathbb{R}^d$ , and a trainable decoder  $g$  that maps  $\\mathbf{z}$  back to image space given  $t$ . In training, given  $\\mathbf{z} = f(\\mathbf{x}_0)$ , we partition it into  $T$  disjoint subsets  $\\{\\mathbf{z}_i\\}_{i=1}^T$ . At each time-step  $t$ , we construct a new feature  $\\bar{\\mathbf{z}}_t = [\\mathbf{z}_1, \\dots, \\mathbf{z}_t, \\mathbf{0}, \\dots, \\mathbf{0}]$ , which is only about  $\\{\\mathbf{z}_i\\}_{i=1}^t$  by masking  $\\mathbf{z}_{t+1}, \\dots, \\mathbf{z}_T$  as  $\\mathbf{0}$ . Our training loss requires  $\\bar{\\mathbf{z}}_t$  to compensate for the reconstruction error made by  $u_{\\theta}$  at each time-step  $t$ , given by:",
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+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} = \\underset {t, \\mathbf {x} _ {0}, \\epsilon} {\\mathbb {E}} \\underbrace {\\left[ \\lambda_ {t} \\| \\mathbf {x} _ {0} - \\left(u _ {\\theta} \\left(\\sqrt {\\bar {\\alpha} _ {t}} \\mathbf {x} _ {0} + \\sqrt {1 - \\bar {\\alpha} _ {t}} \\epsilon , t\\right) + w _ {t} g \\left(\\bar {\\mathbf {z}} _ {t} , t\\right)\\right) \\| ^ {2} \\right]} _ {\\mathcal {L} _ {t}}, \\tag {5}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where  $\\lambda_{t},w_{t}$  are time-step weight and compensate strength. We follow PDAE Zhang et al. (2022) to use fixed values computed from the variance schedule  $\\{\\beta_t\\}_{t = 1}^T$  (details in Appendix).",
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+        "type": "text",
+        "text": "To see the effect of Eq. 5, consider  $t = 1$ , only  $\\mathbf{z}_1$  is used to compensate for reconstruction error, hence  $\\mathbf{z}_1$  effectively captures a (fine-grained) modular attribute  $\\mathcal{Z}_i$  with  $t(\\mathcal{Z}_i) = 1$  (lost at  $t = 1$ ) (such  $\\mathcal{Z}_i$  always exists for some threshold  $\\tau$  in our definition). At  $t = 2$ , the cumulatively lost attributes correspond to  $\\mathcal{Z}_i$  and  $\\mathcal{Z}_j$ , where  $t(\\mathcal{Z}_j) = 2$ . As  $\\mathbf{z}_1$  already captures the attribute  $\\mathcal{Z}_i$ ,",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
+    {
+        "type": "page_number",
+        "text": "5",
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+    {
+        "type": "text",
+        "text": "$\\mathbf{z}_2$  is encouraged to further learn the attribute  $\\mathcal{Z}_j$  to minimize the reconstruction error at  $t = 2$ . Eventually with a large enough  $T$ ,  $\\mathbf{x}_T$  becomes pure noise and all attributes are lost. By the above induction,  $\\{\\mathbf{z}_i\\}_{i=1}^T$  captures all the hidden attributes as a modular feature. In contrast, the most related work PDAE differs from Eq. 5 by replacing  $g(\\bar{\\mathbf{z}}_t)$  with  $g(\\bar{\\mathbf{z}})$ . This means that PDAE uses the full-dimensional feature  $\\mathbf{z}$  to compensate the reconstruction error at all time-steps, which fails to leverage the inductive bias discussed in Section 4.1. We show in Section 5 that this simple change leads to drastic differences in both inference and generation tasks.",
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+    {
+        "type": "text",
+        "text": "Implementation Details. We highlight some design considerations with ablations in Section 5.2.",
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+        "list_items": [
+            "- Number of Subsets  $k$ . In practice, the number of subsets for partitioning  $\\mathbf{z}$  can be smaller than  $T$ , e.g., we use feature dimension  $d = 512$  that is smaller than  $T = 1,000$ . This means that adjacent time-steps may share a modular feature  $\\mathbf{z}_i$ . We compare the choice of  $k$  later.",
+            "- Partition Strategy. The simplest way is to use the same number of feature dimensions for all modular features  $\\mathbf{z}_i$  (i.e., balanced). However, we empirically find that features corresponding to time-step 100-300 require more dimensions to improve convergence of Eq. 5. We design an imbalanced partition strategy and compare it with the balanced one.",
+            "- Optimization Strategy. In training, we experiment detaching the gradient of  $\\mathbf{z}_1, \\ldots, \\mathbf{z}_{t-1}$  to form  $\\bar{\\mathbf{z}}_t$ , i.e., training only  $\\mathbf{z}_t$  at time-step  $t$ . We find that it leads to improved disentanglement quality at the cost of slower convergence."
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 SETTINGS",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Datasets. We choose real-world datasets to validate if DiTi learns a disentangled representation of the generative attributes: 1) Celebrity Faces Attributes (CelebA) Liu et al. (2015) is a large-scale face attributes dataset. Each face image has 40 binary attribute labels, delineating facial features and characteristics, such as expressions, accessories, and lighting conditions. 2) Flickr-Faces-HQ (FFHQ) Karras et al. (2019) contains 70,000 high-quality face images obtained from Flickr. 3) We additionally used the Labeled Faces in the Wild (LFW) dataset Huang et al. (2007) that provides continuous attribute labels. 4) Bedroom is part of the Large-scale Scene Understanding (LSUN) dataset Yu et al. (2015) that contains around 3 million images. We apply off-the-shelf image classifiers following Yang et al. (2021) to determine scene attribute labels.",
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+        "type": "text",
+        "text": "Evaluation Protocols. Our evaluation is based on the properties of a disentangled representation (Section 3.1). For inference, we perform unsupervised learning on CelebA train split or FFHQ and test the linear-probe attribute prediction accuracy on CelebA test split, measured by Average Precision (AP). We also evaluate the challenging attribute regression task using LFW, where the metrics are Pearson correlation coefficient (Pearson's r) and Mean Squared Error (MSE). For counterfactual generation, we extend the conventional interpolation and manipulation technique Preechakul et al. (2022) to tailor for our needs: 1) Instead of interpolating the whole features of an image pair, we interpolate only a feature subset while keeping its complement fixed. With a disentangled representation, only the attribute captured by the subset will change in the generated counterfactuals, e.g., changing only expression between two faces. 2) In manipulation, an image is edited by altering its feature along the direction of an attribute classifier weight, where the classifier is trained on whole features. We constrain the classifier by ProbMask Zhou et al. (2021) to use a small subset of the most discriminative feature dimensions (32 out of 512 dimensions). For a disentangled representation, this should have minimal impact on the generation quality, as each single attribute is encoded in a compact feature subset. Detailed algorithms are in Appendix.",
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+    {
+        "type": "text",
+        "text": "Baselines include 3 representative lineups: 1) Conventional unsupervised disentanglement methods  $\\beta$ -TCVAE Chen et al. (2018) and IB-GAN Jeon et al. (2021); 2) Self-supervised learning methods SimCLR Chen et al. (2020); 3) The most related works Diff-AE Preechakul et al. (2022) and PDAE Zhang et al. (2022) that also learn a representation by making up DM's reconstruction errors.",
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+    {
+        "type": "text",
+        "text": "Implementation Details. We followed the network design of encoder  $f$  and decoder  $g$  in PDAE and adopted its hyper-parameter settings (e.g.,  $\\lambda_t, w_t$  in Eq. 4, details in Appendix). This ensures that any emerged property of disentangled representation is solely from our leverage of the inductive bias in Section 4.1. We also used the same training iterations as PDAE, i.e., 290k iterations on CelebA,",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+    },
+    {
+        "type": "page_number",
+        "text": "6",
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+    {
+        "type": "table",
+        "img_path": "images/9da3549af3a4f89d6a2b72bd74c87a75207921c90c8d29673e7b162bfd5c8f2a.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">CelebA</td><td colspan=\"3\">FFHQ</td></tr><tr><td>AP↑</td><td>Pearson&#x27;s r ↑</td><td>MSE ↓</td><td>AP↑</td><td>Pearson&#x27;s r ↑</td><td>MSE ↓</td></tr><tr><td>β-TCVAE Chen et al. (2018)</td><td>45.0</td><td>0.378</td><td>0.573</td><td>43.2</td><td>0.335</td><td>0.608</td></tr><tr><td>IB-GAN Jeon et al. (2021)</td><td>44.2</td><td>0.307</td><td>0.597</td><td>42.8</td><td>0.260</td><td>0.644</td></tr><tr><td>Diff-AE Preechakul et al. (2022)</td><td>60.3</td><td>0.598</td><td>0.421</td><td>60.5</td><td>0.606</td><td>0.410</td></tr><tr><td>PDAE Zhang et al. (2022)</td><td>60.2</td><td>0.596</td><td>0.410</td><td>59.7</td><td>0.603</td><td>0.416</td></tr><tr><td>SimCLR Chen et al. (2020)</td><td>59.7</td><td>0.474</td><td>0.603</td><td>60.8</td><td>0.481</td><td>0.638</td></tr><tr><td>DiTi (Ours)</td><td>62.3</td><td>0.617</td><td>0.392</td><td>61.4</td><td>0.622</td><td>0.384</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/67dafc349a46461b996af81c60cfc17ca82e5f33c747b2af2eb774230dc05ede.jpg",
+        "image_caption": [
+            "Figure 4: Improvements in attribute classification precision of our DiTi over PDAE (bottom) and over SimCLR (top). Improvements more than  $2\\%$  are highlighted with red bars. Negative values are marked with blue bars."
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "500k iterations on FFHQ, and 540k iterations on Bedroom. Hence our DiTi training is as efficient as PDAE. Our experiments were performed on 4 NVIDIA A100 GPUs.",
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+    {
+        "type": "text",
+        "text": "5.2 INFERENCE",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Comparison with Diff-AE and PDAE. In Table 1, our proposed DiTi significantly outperforms the two related works. We break down the improved accuracy over PDAE on each attribute in Figure 4 bottom, where DiTi is consistently better on all attributes. The result, together with the previous analysis of Figure 2b, validates that DiTi exhibits the sample inference property of a disentangled representation. It also supports our analysis on the baselines' deficiency in Section 2, i.e., they may digress to use spurious correlation to supplement reconstruction error. For example, the most improved attribute is \"Brown Hair\", which strongly correlates with the female gender.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Comparison with SimCLR. SimCLR does not perform well in attribute inference, surpassed even by PDAE. This is because SimCLR learns a representation invariant to augmentations (e.g., color jittering), which disregards its related attributes, such as hair or skin color. This is verified by the large improvements of DiTi over SimCLR on \"Pale Skin\" accuracy in Figure 4 top. The deficiency cannot be addressed by simply removing augmentations, which leads to severe performance degradation (see Appendix). Note that SimCLR slightly outperforms DiTi on some unnoticeable attributes (e.g., earrings). This suggests that some fine-grained attributes are not well disentangled. To tackle this, we will improve model expressiveness (e.g., using Stable Diffusion Rombach et al. (2022) as the frozen DM) and explore more advanced training objectives Song et al. (2023) as future work.",
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+    {
+        "type": "text",
+        "text": "Comparison with  $\\beta$ -TCVAE and IBGAN. It is perhaps not surprising that existing unsupervised disentanglement methods perform poorly. Their previous successes are limited to synthetic datasets, and their models are far less expressive compared to DM. In fact, their pursuit of a disentangled representation by information bottleneck proves to be ever-elusive Locatello et al. (2019).",
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+    {
+        "type": "table",
+        "img_path": "images/efa22c5c87c84527dbdd0b175f0b5ef403e705ffa79a73c578cc2c971f4ced57.jpg",
+        "table_caption": [
+            "Table 1: AP (%) on CelebA attribute classification and Pearson's r, MSE on LFW attribute regression. The first column shows the training dataset."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Imbalance</td><td>Detach</td><td>k</td><td>AP ↑</td><td>Pearson&#x27;s r ↑</td><td>MSE ↓</td></tr><tr><td>✓</td><td></td><td>16</td><td>62.1</td><td>0.619</td><td>0.389</td></tr><tr><td>✓</td><td></td><td>32</td><td>62.0</td><td>0.615</td><td>0.392</td></tr><tr><td>✓</td><td></td><td>64</td><td>62.3</td><td>0.617</td><td>0.392</td></tr><tr><td>✓</td><td></td><td>128</td><td>61.9</td><td>0.616</td><td>0.391</td></tr><tr><td></td><td></td><td>64</td><td>61.9</td><td>0.604</td><td>0.410</td></tr><tr><td></td><td>✓</td><td>64</td><td>62.5</td><td>0.590</td><td>0.422</td></tr><tr><td>✓</td><td>✓</td><td>64</td><td>62.4</td><td>0.604</td><td>0.405</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 2: Ablations on DiTi designs on CelebA. Imbalance and Detach: using our partition and optimization strategy.",
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+    {
+        "type": "text",
+        "text": "Ablation on #Subsets  $k$ . From Table 2 first 4 lines, we observe that inference results are not sensitive to the subset number  $k$  (details in Section 4.2). We choose  $k = 64$  with the highest AP.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    {
+        "type": "page_number",
+        "text": "7",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/f35c124ee4fe0f604612e5f915077ddd98424e8c0aa545fe2ffe31bd3aa2255a.jpg",
+        "image_caption": [
+            "Figure 5: Counterfactual generations on FFHQ by interpolating a feature subset of 3 image pairs  $(\\mathbf{x},\\mathbf{y})$ . We partition  $\\{\\mathbf{z}_i\\}_{i = 1}^T$  according to early  $t\\in (0,T / 3]$ , middle  $t\\in (T / 3,2T / 3]$  and late  $t\\in (2T / 3,T]$ . For each subset, we show results in 2 directions  $(\\mathbf{x}\\rightarrow \\mathbf{y},\\mathbf{y}\\rightarrow \\mathbf{x})$  under 3 interpolation scales (color gradient↑, scale↑)."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "This means that our model can learn up to 64 hidden attributes, whose combinations are certainly diverse enough for our chosen datasets.",
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+    {
+        "type": "text",
+        "text": "Ablation on Partition & Optimization Strategy. We devise an imbalanced partition strategy to allocate more feature dimensions for  $\\mathbf{z}_i$  corresponding to  $t \\in \\{100, \\dots, 300\\}$ , as this time-step range contributes the most to the overall loss (Figure A9). This leads to faster convergence and improved performance in Table 2 (line 3 vs. line 5). For optimization strategy, we tried detaching the gradient of  $\\mathbf{z}_1, \\dots, \\mathbf{z}_{t-1}$  at time-step  $t$  to prevent them from capturing the lost attribute at  $t$ . Yet it leads to slower convergence, which especially hurts the challenging regression task. We leave it as future work to explore more principled ways of improving feature modularity.",
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+    {
+        "type": "text",
+        "text": "5.3 COUNTERFACTUAL GENERATION",
+        "text_level": 1,
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+        "type": "text",
+        "text": "FFHQ Results. Figure 5 shows DiTi generations by interpolating a feature subset while fixing its complement. Different from full-feature interpolation (Figure 7), there are two directions:  $\\mathbf{x} \\rightarrow \\mathbf{y}$  fixes the complement to the corresponding feature values of  $\\mathbf{x}$ , and  $\\mathbf{y} \\rightarrow \\mathbf{x}$  fixes it to those of  $\\mathbf{y}$ . By identifying how images are changing during interpolation (left to right for top, right to left for bottom) and comparing the changes across the 3 columns, we have several observations:",
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+        "list_items": [
+            "1. Interpolating different subsets modifies different attributes. For example, subset of early  $t$  mainly controls micro-expressions, such as mouth corners, cheeks, or eye details. The interpolation hardly changes a person's identity, but certainly moves one's temperament towards the other person. In contrast, subset of middle  $t$  is more responsible for identity-related attributes (e.g., proportion of facial features) and accessories (e.g., eyeglasses). Finally, late  $t$  controls attributes such as hairstyle, pose, and even position of the face in the image (e.g., the bottom right face moves towards center).",
+            "2. From early to late  $t$ , the corresponding feature subset controls a more coarse-grained feature, where the granularity is defined based on the pixel-level changes. For example, changing hairstyle or removing hat (subset of late  $t$ ) modifies more pixels than changing facial feature proportions (subset of middle  $t$ ). This is in line with our theoretical analysis in Section 4.1. Notably, both our feature subset partition and the correspondence between subsets and attributes granularity are known a priori, without relying on manual discovery techniques such as latent traversal.",
+            "3. Our generated counterfactuals have minimal distortion or artifact, and demonstrate modular properties (e.g., subset of late  $t$  controls pose with minimal impact on facial features and accessories). These are strong proofs that DiTi learns a disentangled representation Besserve et al. (2020). We show in Appendix that all generative baselines do not exhibit these traits."
+        ],
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+        "text": "8",
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+    {
+        "type": "image",
+        "img_path": "images/ce0850b4c08cd6dbc412fb4ba20c8a727bafe8638390c43eea0598724b8b8488.jpg",
+        "image_caption": [
+            "Figure 6: Counterfactual generations on Bedroom by manipulating 32 out of 512 feature dimensions.",
+            "Figure 7: Results of interpolating the whole feature on FFHQ. Baselines have distortions during interpolation."
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Bedroom Manipulation. Bedroom is a more challenging dataset, where room scenes are much more complex than facial images (e.g., various layouts, decorations, objects). As shown in Figure 6, DiTi is the only method that generates faithful counterfactuals. 1st row: only DiTi can alter lighting without making significant changes to room furniture. 2nd row: baselines fail to modify the rustic feeling without artifacts or changing room layouts. 3rd row: DiTi makes the room more cluttered by introducing additional objects. Last row: only DiTi adds an additional carpet.",
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+    {
+        "type": "text",
+        "text": "Whole-feature Interpolation. Under this conventional setting, as shown in Figure 7, only DiTi smoothly interpolates between the attributes of each image pair, with minimal distortion and artifact along the trajectories. This means that our DiTi can generate faithful counterfactuals (i.e., valid images in the support of data distribution), which validates our disentanglement quality Besserve et al. (2020). For example, if a feature only captures the attribute \"smile\" (i.e., disentangled), interpolating this feature between a smiling person and a non-smiling one leads to the gradual transition of expression. However, if a feature entangles multiple attributes, such interpolation will alter all of them or even cause distortion when the interpolated feature is out of the data distribution.",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We presented a novel unsupervised method to learn a disentangled representation, which leverages the inductive bias of diffusion time-steps. In particular, we reveal an inherent connection between time-step and hidden modular attributes that generate data faithfully, enabling a simple and effective approach to disentangle the attributes by learning a time-step-specific feature. The learned feature improves downstream inference and enables counterfactual generation, validating its disentanglement quality. As future work, we will seek additional inductive bias to improve disentanglement, e.g., using text as a disentangled template by exploring text-to-image diffusion models, and devise practical optimization techniques to enable faster convergence.",
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+    {
+        "type": "text",
+        "text": "7 ACKNOWLEDGEMENT",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This research is supported by Microsoft Research Asia, the National Research Foundation, Singapore under its AI Singapore Programme (AISG Award No: AISG2-RP-2021-022), MOE AcRF Tier 2 (MOE2019-T2-2-062), Wallenberg-NTU Presidential Postdoctoral Fellowship, the Lee Kong Chian (LKC) Fellowship fund awarded by Singapore Management University, and the DSO Research Grant (Fund Code MG22C03).",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+    },
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+        "type": "page_number",
+        "text": "9",
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+    {
+        "type": "text",
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