Post-hoc Counterfactual Generation with Supervised Autoencoder

Abstract

Nowadays, AI is increasingly being used in many fields to automate decisions that largely affect the daily lives of humans. The inherent complexity of these systems makes them so-called black-box models. Explainable Artificial Intelligence (XAI) aims at solving this issue by providing methods to overcome this lack of transparency. Counterfactual explanation is a common and well-known class of explanations that produces actionable and understandable explanations for end-users. However, generating realistic and useful counterfactuals remains a challenge. In this work, we investigate the problem of generating counterfactuals that are both close to the data distribution, and to the distribution of the target class. Our objective is to obtain counterfactuals with likely values (i.e. realistic). We propose a model agnostic method for generating realistic counterfactuals by using class prototypes. The novelty of this approach is that these class prototypes are obtained using a supervised auto-encoder. Then, we performed an empirical evaluation across several interpretability metrics, that shows competitive results with the state-of-the-art method.

6 Supplementary Material

This section describes used architectures and hyperparameters.

Supervised Autoencoder: This architecture is composed of an auto-encoder part, and two dense layers (classification layers) next to encoder. The final output is the concatenation of the last dense layer and the decoder output. Auto-encoder part is the same as those used by Van Looveren and Klaise [6]. It consists of an encoder with two convolution layers, the first two contains 16 filters of size $3\times 3$ and ReLU activations and are followed by a $2\times 2$ max-pooling layers and finally a convolution layer with a filter of size $3\times 3$ and linear activation. The decoder part takes encoded instances as input and pass them to a convolutional layer with 16 filters of size $3\times 3$ and RelU activation, then to a $2\times 2$ upsampling layer and, a convolutional layer with 16 filters of size $3\times 3$ and RelU activation again and finally a convolution layer with a filter of size $3\times 3$ and linear activation. The classification part is composed of 2 dense layers stacked next to the encoder. It takes flatten encoded instances from the encoder and pass them to a dense layer of size 128 with RelU activation and $L_{1}$ regularization. This dense layer is followed by a softmax layer with 10 units. Loss is defined as summation of reconstruction loss and classification loss as shown in Sect. 3. To fix $\lambda$ of Eq. 3, we train our model for different lambda values on the training set, and choose the best trade-off between accuracy and reconstruction error on the test set. These results are shown in Table 2, and best performances are obtained for $\lambda = 10$.

Table 2. Accuracy and reconstruction error for each $\lambda$ on test set

As labels are one-hot, classification loss will be a categorical cross-entropy, and reconstruction loss will be defined as a mean squared error loss. Training is achieve with a batch size of 128, and for 25 epochs, optimizer is set as Adam optimizer.

Baseline Autoencoder: This architecture is the same as autoencoder part of supervised autoencoder. Loss is defined as mean squared error, and training is achieved with a batch size of 128, and for 18 epochs, optimizer is set as Adam optimizer. We achieve to train our model, on the training set for a reconstruction error of 0.0016 on test set.

Counterfactual Search Hyperparameters: Hyperparameters values are fixed to those used by Van Looveren et al. [6] ($\gamma = 100$, $\kappa = 0$, $c=1$, $\beta =0.1$, $\theta = 100$, $K = 5$).

Autoencoder for Evaluation: We used the same autoencoder architecture as Van Looveren et al. [6]. The training set is the same as the one used for supervised and baseline autoencoder.