Communication cost and privacy are two major considerations in federated learning (FL). For communication cost, gradient compression by sketching the clients' transmitted model updates is often used for reducing per round communication. For privacy, the Gaussian mechanism (GM), which consists of clipping updates and adding Gaussian noise, is commonly used to guarantee client level differential privacy. Existing literature on private FL analyzes privacy of sketching and GM in an isolated manner, illustrating that sketching provides privacy determined by the sketching dimension and that GM has to supply any additional desired privacy. In this paper, we introduce the Sketched Gaussian Mechanism (SGM), which directly combines sketching and the Gaussian mechanism for privacy.

Score-based generative models (SGMs) have emerged as one of the most popular classes of generative models. A substantial body of work now exists on the analysis of SGMs, focusing either on discretization aspects or on their statistical performance. In the latter case, bounds have been derived, under various metrics, between the true data distribution and the distribution induced by the SGM, often demonstrating polynomial convergence rates with respect to the number of training samples. However, these approaches adopt a largely approximation theory viewpoint, which tends to be overly pessimistic and relatively coarse. In particular, they fail to fully explain the empirical success of SGMs or capture the role of the optimization algorithm used in practice to train the score network. To support this observation, we first present simple experiments illustrating the concrete impact of optimization hyperparameters on the generalization ability of the generated distribution. Then, this paper aims to bridge this theoretical gap by providing the first algorithmic-and data-dependent generalization analysis for SGMs. In particular, we establish bounds that explicitly account for the optimization dynamics of the learning algorithm, offering new insights into the generalization behavior of SGMs. Our theoretical findings are supported by empirical results on several datasets.

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In contrast, classical graphical methods like A* search are able to solve long-horizon tasks, but assume that the state space is abstracted away from raw sensory input. Recent works have attempted to combine the strengths of deep learning and classical planning; however, dominant methods in this domain are stillquite brittle andscale poorly withthesizeoftheenvironment.

Score-based generative models (SGMs) need to approximate the scores $\nabla \log p_t$ of the intermediate distributions as well as the final distribution $p_T$ of the forward process. The theoretical underpinnings of the effects of these approximations are still lacking. We find precise conditions under which SGMs are able to produce samples from an underlying (low-dimensional) data manifold $\mathcal{M}$. This assures us that SGMs are able to generate the right kind of samples. For example, taking $\mathcal{M}$ to be the subset of images of faces, we provide conditions under which the SGM robustly produces an image of a face, even though the relative frequencies of these images might not accurately represent the true data generating distribution. Moreover, this analysis is a first step towards understanding the generalization properties of SGMs: Taking $\mathcal{M}$ to be the set of all training samples, our results provide a precise description of when the SGM memorizes its training data.