Diffusion models are recent state-of-the-art methods for image generation and likelihood estimation. In this work, we generalize continuous-time diffusion models to arbitrary Riemannian manifolds and derive a variational framework for likelihood estimation. Computationally, we propose new methods for computing the Riemannian divergence which is needed for likelihood estimation. Moreover, in generalizing the Euclidean case, we prove that maximizing this variational lowerbound is equivalent to Riemannian score matching. Empirically, we demonstrate the expressive power of Riemannian diffusion models on a wide spectrum of smooth manifolds, such as spheres, tori, hyperboloids, and orthogonal groups. Our proposed method achieves new state-of-the-art likelihoods on all benchmarks.

A.1 Connection to online learning In Section 2 we motivated the update (2) as a way to adjust the size of our prediction sets in response to the realized historical miscoverage frequency. Alternatively, one could also derive (2) as an online gradient descent algorithm with respect to the pinball loss. To be more precise let t:= sup{: Yt 2 Cˆt()}, where we remark that Cˆt( t) can be thought of as the smallest prediction set containing Yt. Because the pinball loss is convex, this gradient descent update falls within a well understood class of algorithms that have been extensively studied in the online learning literature (see e.g. Unfortunately, this notion of regret fails to capture our intuition that t is adaptively tracking the moving target .

What are Russia's gains from the Iran war? 'We are not losers; we are winners' But the calm is deceptive. Two soldiers scour the skies, hands firmly gripping anti-aircraft guns mounted on pick-up trucks parked on a small, dilapidated bridge on a tributary of the Pripyat River. Danger is all around, both in the surrounding land, which still carries the legacy of the 1986 Chornobyl nuclear disaster, with pockets of intense radioactive contamination, and above, where Russian drones and missiles launched from just across the border in Belarus, a short distance to the north, regularly pass overhead. The area is known as the Chornobyl Exclusion Zone (CEZ), a restricted area of approximately 30km (19 miles) in diameter, comparable in size to Luxembourg, established to contain the spread of contamination. Since Russia launched its full-scale invasion of Ukraine on February 24, 2022, briefly occupying the CEZ and the surrounding area, large swaths of it have become militarised, adding another layer of restriction to an already tightly controlled and hazardous environment. Yet despite the CEZ's many dangers, four decades on from the Chornobyl disaster, small communities of scientists, elderly returnees and soldiers have carved out lives among its abandoned buildings, while wildlife thrives in the surrounding forests.

We propose a novel reduction-to-binary (R2B) approach that enforces demographic parity for multiclass classification with non-binary sensitive attributes via a reduction to a sequence of binary debiasing tasks. We prove that R2B satisfies optimality and bias guarantees and demonstrate empirically that it can lead to an improvement over two baselines: (1) treating multiclass problems as multi-label by debiasing labels independently and (2) transforming the features instead of the labels. Surprisingly, we also demonstrate that independent label debiasing yields competitive results in most (but not all) settings.

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a "noising" stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a "denoising" process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a "noising" stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a "denoising" process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.

Federated learning (FL) has emerged as an enabling framework for communicationefficient decentralized training. We study three broadly applicable problem classes in FL: (i) Nondifferentiable nonconvex federated optimization; (ii) Federated bilevel optimization; (iii) Federated minimax problems. Notably, in an implicit sense, both (ii) and (iii) are instances of (i). However, the hierarchical problems in (ii) and (iii) are often complicated by the absence of a closed-form expression for the implicit objective function. Unfortunately, research on these problems has been limited and afflicted by reliance on strong assumptions, including the need for differentiability and L-smoothness of the implicit function. We address this shortcoming by making the following contributions. In (i), by leveraging convolution-based smoothing and Clarke's subdifferential calculus, we devise a randomized smoothing-enabled zeroth-order FL method and derive communication and iteration complexity guarantees for computing an approximate Clarke stationary point. To contend with (ii) and (iii), we devise a unified randomized implicit zeroth-order FL framework, equipped with explicit communication and iteration complexities. Importantly, our method utilizes delays during local steps to skip making calls to the inexact lower-level FL oracle.