Assume that p and k are as in (A1). Then, the two following assertions hold. A.3 The case α < 1 for the Power Descent algorithm Let α = 1, η (0,1], κbe such that (α 1)κ 0and let the initial probability measure µ1 M1(T) be such that Ψα(µ1) < . A common way to approximate intractable integrals of the form (16) is to resort to Importance Sampling methods and in that case we are also interested in ensuring that the support of the variational approximation q Q (with q = µk in our case) is included in the support of p. Seeking to solve the Variational Inference optimation problem inf Dα(µK||P) for α < 1 enables this to happen, as opposed to the case α 1 for which the α-divergenve exhibits the so-called mode-seeking property [2, 3, 4]. As a whole, well-chosen samplers and variance reduction methods appear to be a necessity even in the case α = 1 so that the obtained Monte Carlo estimator of θ 7 bµ,α(θ)do not suffer from a too large variance.

This paper focuses on α-divergence minimisation methods for Variational Inference. We consider the case where the posterior density is approximated by a mixture model and we investigate algorithms optimising the mixture weights of this mixture model by α-divergence minimisation, without any information on the underlying distribution of its mixture components parameters. The Power Descent, defined for all α = 1, is one such algorithm and we establish in our work the full proof of its convergence towards the optimal mixture weights when α < 1. Since the α-divergence recovers the widely-used exclusive Kullback-Leibler when α 1, we then extend the Power Descent to the case α = 1 and show that we obtain an Entropic Mirror Descent. This leads us to investigate the link between Power Descent and Entropic Mirror Descent: first-order approximations allow us to introduce the Rényi Descent, a novel algorithm for which we prove an O(1/N) convergence rate. Lastly, we compare numerically the behavior of the unbiased Power Descent and of the biased Rényi Descent and we discuss the potential advantages of one algorithm over the other.

We provide practical, efficient, and nonparametric methods for auditing the fairness of deployed classification and regression models. Whereas previous work relies on a fixed-sample size, our methods are sequential and allow for the continuous monitoring of incoming data, making them highly amenable to tracking the fairness of real-world systems. We also allow the data to be collected by a probabilistic policy as opposed to sampled uniformly from the population. This enables auditing to be conducted on data gathered for another purpose. Moreover, this policy may change over time and different policies may be used on different subpopulations. Finally, our methods can handle distribution shift resulting from either changes to the model or changes in the underlying population. Our approach is based on recent progress in anytime-valid inference and game-theoretic statistics--the "testing by betting" framework in particular. These connections ensure that our methods are interpretable, fast, and easy to implement. We demonstrate the efficacy of our approach on three benchmark fairness datasets.

Graph Neural Networks (GNNs) have achieved great success on a node classification task. Despite the broad interest in developing and evaluating GNNs, they have been assessed with limited benchmark datasets. As a result, the existing evaluation of GNNs lacks fine-grained analysis from various characteristics of graphs. Motivated by this, we conduct extensive experiments with a synthetic graph generator that can generate graphs having controlled characteristics for fine-grained analysis. Our empirical studies clarify the strengths and weaknesses of GNNs from four major characteristics of real-world graphs with class labels of nodes, i.e., 1) class size distributions (balanced vs. imbalanced), 2) edge connection proportions between classes (homophilic vs. heterophilic), 3) attribute values (biased vs. random), and 4) graph sizes (small vs. large). In addition, to foster future research on GNNs, we publicly release our codebase that allows users to evaluate various GNNs with various graphs. We hope this work offers interesting insights for future research.

Probabilistic context-free grammars (PCFGs) and dynamic Bayesian networks (DBNs) are widely used sequence models with complementary strengths and limitations. While PCFGs allow for nested hierarchical dependencies (tree structures), their latent variables (non-terminal symbols) have to be discrete. In contrast, DBNs allow for continuous latent variables, but the dependencies are strictly sequential (chain structure). Therefore, neither can be applied if the latent variables are assumed to be continuous and also to have a nested hierarchical dependency structure. In this paper, we present Recursive Bayesian Networks (RBNs), which generalise and unify PCFGs and DBNs, combining their strengths and containing both as special cases. RBNs define a joint distribution over tree-structured Bayesian networks with discrete or continuous latent variables. The main challenge lies in performing joint inference over the exponential number of possible structures and the continuous variables. We provide two solutions: 1) For arbitrary RBNs, we generalise inside and outside probabilities from PCFGs to the mixed discrete-continuous case, which allows for maximum posterior estimates of the continuous latent variables via gradient descent, while marginalising over network structures.

Multi-agent reinforcement learning (MARL) has emerged as a useful approach to solving decentralised decision-making problems at scale. Research in the field has been growing steadily with many breakthrough algorithms proposed in recent years. In this work, we take a closer look at this rapid development with a focus on evaluation methodologies employed across a large body of research in cooperative MARL. By conducting a detailed meta-analysis of prior work, spanning 75 papers accepted for publication from 2016 to 2022, we bring to light worrying trends that put into question the true rate of progress. We further consider these trends in a wider context and take inspiration from single-agent RL literature on similar issues with recommendations that remain applicable to MARL. Combining these recommendations, with novel insights from our analysis, we propose a standardised performance evaluation protocol for cooperative MARL. We argue that such a standard protocol, if widely adopted, would greatly improve the validity and credibility of future research, make replication and reproducibility easier, as well as improve the ability of the field to accurately gauge the rate of progress over time by being able to make sound comparisons across different works. Finally, we release our meta-analysis data publicly on our project website for future research on evaluation 3 accompanied by our open-source evaluation tools repository4.