Increasing concerns about disparate effects of AI have motivated a great deal of work on fair machine learning.

Supplementary material is at the end of this document.

Despite the importance of graph OD and many algorithms being developed for it in recent years, there is no comprehensive benchmark on graph outlier detection, which we believe has hindered the development and understanding of graph OD algorithms.

LLP, prove generalization error bounds associated to two bag generation models, and establish universal consistency with respect to the balanced error rate (BER).

Lastly, we demonstrate the superiority of stochastic proximal methods compared to subgradient-based approaches via extensive experiments.

Models like LASSO and ridge regression are extensively used in practice due to their interpretability, ease of use, and strong theoretical guarantees.

This section provides the detailed proof for Theorem 1 and the lower bounds of the precision and recall. We derive such theorems with the concentration inequalities in probabilistic theory. In this section, we frequently use the spectral norm. U = I when U is an orthogonal matrix). A.2 Proof of Theorem 1 We deal with some require lemmas which are used for the proof of Theorem 1. Lemma 1. Lemma 3. (David-Kahan sin Theorem) F or given symmetric matrices A, B R Assume that A and A + B have non-negative eigenvalues.

Finally, let us consider (b), the general case. A.2 Proposition 2 Proof We will derive the gradients of the unnormalized posterior since In practice, we recommend the log-sum-exp trick for applying Proposition 2. Let us define Again, let us first consider case (a).