In this paper, we study the problem of sampling from a distribution under the constraint of differential privacy (DP). Prior works measure the utility of DP sampling with density ratio-based measures such as KL divergence. However, such formulations suffer from two key limitations: 1) they fail to capture the geometric structure of the support, and 2) they are not applicable when the supports of the distributions differ. To deal with these issues, we develop a novel framework for DP sampling with Wasserstein distance as the utility measure. In this formulation, we propose Wasserstein Projection Mechanism (WPM), a minimax optimal mechanism based on Wasserstein projection. Furthermore, we develop efficient algorithms for computing the proposed mechanisms approximately and provide convergence guarantees.

Probabilistic modelling lies at the heart of machine learning.

Learning from pairwise measurements naturally arises from many applications, suchasrankaggregation,ordinalembedding,andcrowdsourcing.

We provide numerical experiments to corroborate our theory.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The paper describes a number of new models for representing a joint distribution over integer-count variables. The authors argue that the default model that arises from Yang et al. is not satisfactory because it can only model negative correlations in order for the distribution to be normalized. They then consider a series of fixes for this including a new truncation method, using a quadratic base measure statistic (which they prove is necessary with everything else fixed), and finally a sub-linear sufficient statistic. This is a well written paper describing some nice solutions for representing count data.

Probabilistic models for binary spike patterns provide a powerful tool for understanding the statistical dependencies in large-scale neural recordings. Maximum entropy (or maxent'') models, which seek to explain dependencies in terms of low-order interactions between neurons, have enjoyed remarkable success in modeling such patterns, particularly for small groups of neurons. However, these models are computationally intractable for large populations, and low-order maxent models have been shown to be inadequate for some datasets. To overcome these limitations, we propose a family of "universal'' models for binary spike patterns, where universality refers to the ability to model arbitrary distributions over all 2 m binary patterns. We construct universal models using a Dirichlet process centered on a well-behaved parametric base measure, which naturally combines the flexibility of a histogram and the parsimony of a parametric model. We derive computationally efficient inference methods using Bernoulli and cascade-logistic base measures, which scale tractably to large populations. We also establish a condition for equivalence between the cascade-logistic and the 2nd-order maxent or "Ising'' model, making cascade-logistic a reasonable choice for base measure in a universal model.

We introduce Dirichlet Process Posterior Sampling (DPPS), a Bayesian non-parametric algorithm for multi-arm bandits based on Dirichlet Process (DP) priors. Like Thompson-sampling, DPPS is a probability-matching algorithm, i.e., it plays an arm based on its posterior-probability of being optimal. Instead of assuming a parametric class for the reward generating distribution of each arm, and then putting a prior on the parameters, in DPPS the reward generating distribution is directly modeled using DP priors. DPPS provides a principled approach to incorporate prior belief about the bandit environment, and in the noninformative limit of the DP posteriors (i.e. Bayesian Bootstrap), we recover Non Parametric Thompson Sampling (NPTS), a popular non-parametric bandit algorithm, as a special case of DPPS. We employ stick-breaking representation of the DP priors, and show excellent empirical performance of DPPS in challenging synthetic and real world bandit environments. Finally, using an information-theoretic analysis, we show non-asymptotic optimality of DPPS in the Bayesian regret setup.

The evidence lower bound (ELBO) is one of the most central objectives for probabilistic unsupervised learning. For the ELBOs of several generative models and model classes, we here prove convergence to entropy sums. As one result, we provide a list of generative models for which entropy convergence has been shown, so far, along with the corresponding expressions for entropy sums. Our considerations include very prominent generative models such as probabilistic PCA, sigmoid belief nets or Gaussian mixture models. However, we treat more models and entire model classes such as general mixtures of exponential family distributions. Our main contributions are the proofs for the individual models. For each given model we show that the conditions stated in Theorem 1 or Theorem 2 of [arXiv:2209.03077] are fulfilled such that by virtue of the theorems the given model's ELBO is equal to an entropy sum at all stationary points. The equality of the ELBO at stationary points applies under realistic conditions: for finite numbers of data points, for model/data mismatches, at any stationary point including saddle points etc, and it applies for any well behaved family of variational distributions.

Creating fair AI systems is a complex problem that involves the assessment of context-dependent bias concerns. Existing research and programming libraries express specific concerns as measures of bias that they aim to constrain or mitigate. In practice, one should explore a wide variety of (sometimes incompatible) measures before deciding which ones warrant corrective action, but their narrow scope means that most new situations can only be examined after devising new measures. In this work, we present a mathematical framework that distils literature measures of bias into building blocks, hereby facilitating new combinations to cover a wide range of fairness concerns, such as classification or recommendation differences across multiple multi-value sensitive attributes (e.g., many genders and races, and their intersections). We show how this framework generalizes existing concepts and present frequently used blocks. We provide an open-source implementation of our framework as a Python library, called FairBench, that facilitates systematic and extensible exploration of potential bias concerns.