Mean-field underdamped Langevin dynamics and its spacetime discretization

We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training mean-field neural networks, maximum mean discrepancy minimization and kernel Stein discrepancy minimization. Our algorithm is based on a novel spacetime discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.

Mean-field Underdamped Langevin Dynamics and its Space-Time Discretization

We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training neural networks in the mean-field regime, density estimation, and kernel Stein discrepancy minimization. Our algorithm is based on a novel space-time discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.

Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin Algorithm

We propose a new method called the Metropolis-adjusted Mirror Langevin algorithm for approximate sampling from distributions whose support is a compact and convex set. This algorithm adds an accept-reject filter to the Markov chain induced by a single step of the mirror Langevin algorithm (Zhang et al., 2020), which is a basic discretisation of the mirror Langevin dynamics. Due to the inclusion of this filter, our method is unbiased relative to the target, while known discretisations of the mirror Langevin dynamics including the mirror Langevin algorithm have an asymptotic bias. We give upper bounds for the mixing time of the proposed algorithm when the potential is relatively smooth, convex, and Lipschitz with respect to a self-concordant mirror function. As a consequence of the reversibility of the Markov chain induced by the algorithm, we obtain an exponentially better dependence on the error tolerance for approximate sampling. We also present numerical experiments that corroborate our theoretical findings.

What is a Fair Diffusion Model? Designing Generative Text-To-Image Models to Incorporate Various Worldviews

Generative text-to-image (GTI) models produce high-quality images from short textual descriptions and are widely used in academic and creative domains. However, GTI models frequently amplify biases from their training data, often producing prejudiced or stereotypical images. Yet, current bias mitigation strategies are limited and primarily focus on enforcing gender parity across occupations. To enhance GTI bias mitigation, we introduce DiffusionWorldViewer, a tool to analyze and manipulate GTI models' attitudes, values, stories, and expectations of the world that impact its generated images. Through an interactive interface deployed as a web-based GUI and Jupyter Notebook plugin, DiffusionWorldViewer categorizes existing demographics of GTI-generated images and provides interactive methods to align image demographics with user worldviews. In a study with 13 GTI users, we find that DiffusionWorldViewer allows users to represent their varied viewpoints about what GTI outputs are fair and, in doing so, challenges current notions of fairness that assume a universal worldview.

Approximate Cross-validation: Guarantees for Model Assessment and Selection

Cross-validation (CV) is a popular approach for assessing and selecting predictive models. However, when the number of folds is large, CV suffers from a need to repeatedly refit a learning procedure on a large number of training datasets. Recent work in empirical risk minimization (ERM) approximates the expensive refitting with a single Newton step warm-started from the full training set optimizer. While this can greatly reduce runtime, several open questions remain including whether these approximations lead to faithful model selection and whether they are suitable for non-smooth objectives. We address these questions with three main contributions: (i) we provide uniform non-asymptotic, deterministic model assessment guarantees for approximate CV; (ii) we show that (roughly) the same conditions also guarantee model selection performance comparable to CV; (iii) we provide a proximal Newton extension of the approximate CV framework for non-smooth prediction problems and develop improved assessment guarantees for problems such as l1-regularized ERM.