gradient descent with a batch size of 500 and with a total loss function, $$\label{eq:combined_loss_AE}
    \mathcal{L}_{\mathrm{tot}} = \mathcal{L}_{\mathrm{auto}} - \alpha \mathcal{L}_{\mathrm{adv}}~.$$

4.  The adversary is trained for one epoch using mini-batch gradient descent with a batch size of 500.

5.  Steps 3 and 4 are repeated until they have been performed a total of 1500 times, with the learning rate decaying every 100 epochs to a factor of 0.75 of its previous value, starting from an initial value of 0.01.

We find this procedure to provide stable and numerically reliable results. Again, the relative weighting between the autoencoder and the adversary is set to $\alpha=100$. The performance of the adversarially-trained autoencoder is shown in Fig. \[fig:AEadv\]. The background distributions shown in Fig. \[fig:AEadvresponse\] have been shaped such that they are independent of the direction of smearing, which results in the ROC curves in Fig. \[fig:AEadvROC\] becoming almost identical. This shows that the method has become independent of uncertainties inherent to the reconstruction of the final-state objects of LHC events.

[0.49]{} ![Autoencoder loss (a) and ROC curves (b) for an adversarial autoencoder trained only on background events. The three background distributions result from the three different directions of smearing.[]{data-label="fig:AEadv"}](figures/AEadv3response.pdf "fig:"){width="\textwidth"}

[0.49]{} 