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Wasserstein gradient flows on probability measures have found a host of applications in various optimization problems. They typically arise as the continuum limit of exchangeable particle systems evolving by some mean-field interaction involving a gradient-type potential. However, in many problems, such as in multi-layer neural networks, the so-called particles are edge weights on large graphs whose nodes are exchangeable. Such large graphs are known to converge to continuum limits called graphons as their size grows to infinity. We show that the Euclidean gradient flow of a suitable function of the edge weights converges to a novel continuum limit given by a curve on the space of graphons that can be appropriately described as a gradient flow or, more technically, a curve of maximal slope. Several natural functions on graphons, such as homomorphism functions and the scalar entropy, are covered by our setup, and the examples have been worked out in detail. 

In this article, we present a detailed review of current practices and state-of-the-art methodologies in the field of differential privacy (DP), with a focus of advancing DP’s deployment in real-world applications. Key points and high-level contents of the article were originated from the discussions from “Differential Privacy (DP): Challenges Towards the Next Frontier,” a workshop held in July 2022 with experts from industry, academia, and the public sector seeking answers to broad questions pertaining to privacy and its implications in the design of industry-grade systems.This article aims to provide a reference point for the algorithmic and design decisions within the realm of privacy, highlighting important challenges and potential research directions. Covering a wide spectrum of topics, this article delves into the infrastructure needs for designing private systems, methods for achieving better privacy/utility trade-offs, performing privacy attacks and auditing, as well as communicating privacy with broader audiences and stakeholders. 

Federated Learning (FL) is a distributed learning framework, in which the local data never leaves clients’ devices to preserve privacy, and the server trains models on the data via accessing only the gradients of those local data. Without further privacy mechanisms such as differential privacy, this leaves the system vulnerable against an attacker who inverts those gradients to reveal clients’ sensitive data. However, a gradient is often insufficient to reconstruct the user data without any prior knowledge. By exploiting a generative model pretrained on the data distribution, we demonstrate that data privacy can be easily breached. Further, when such prior knowledge is unavailable, we investigate the possibility of learning the prior from a sequence of gradients seen in the process of FL training. We experimentally show that the prior in a form of generative model is learnable from iterative interactions in FL. Our findings demonstrate that additional mechanisms are necessary to prevent privacy leakage in FL.