Beyond Noise: Privacy-Preserving Decentralized Learning with Virtual Nodes

Decentralized learning (DL) enables collaborative learning without a server and without training data leaving the users' devices. However, the models shared in DL can still be used to infer training data. Conventional privacy defenses such as differential privacy and secure aggregation fall short in effectively safeguarding user privacy in DL. We introduce Shatter, a novel DL approach in which nodes create virtual nodes (VNs) to disseminate chunks of their full model on their behalf. This enhances privacy by (i) preventing attackers from collecting full models from other nodes, and (ii) hiding the identity of the original node that produced a given model chunk. We theoretically prove the convergence of Shatter and provide a formal analysis demonstrating how Shatter reduces the efficacy of attacks compared to when exchanging full models between participating nodes. We evaluate the convergence and attack resilience of Shatter with existing DL algorithms, with heterogeneous datasets, and against three standard privacy attacks, including gradient inversion. Our evaluation shows that Shatter not only renders these privacy attacks infeasible when each node operates 16 VNs but also exhibits a positive impact on model convergence compared to standard DL. This enhanced privacy comes with a manageable increase in communication volume.

Asynchronous SGD on Graphs: a Unified Framework for Asynchronous Decentralized and Federated Optimization

Decentralized and asynchronous communications are two popular techniques to speedup communication complexity of distributed machine learning, by respectively removing the dependency over a central orchestrator and the need for synchronization. Yet, combining these two techniques together still remains a challenge. In this paper, we take a step in this direction and introduce Asynchronous SGD on Graphs (AGRAF SGD) -- a general algorithmic framework that covers asynchronous versions of many popular algorithms including SGD, Decentralized SGD, Local SGD, FedBuff, thanks to its relaxed communication and computation assumptions. We provide rates of convergence under much milder assumptions than previous decentralized asynchronous works, while still recovering or even improving over the best know results for all the algorithms covered.

Generalization Error of First-Order Methods for Statistical Learning with Generic Oracles

In this paper, we provide a novel framework for the analysis of generalization error of first-order optimization algorithms for statistical learning when the gradient can only be accessed through partial observations given by an oracle. Our analysis relies on the regularity of the gradient w.r.t. the data samples, and allows to derive near matching upper and lower bounds for the generalization error of multiple learning problems, including supervised learning, transfer learning, robust learning, distributed learning and communication efficient learning using gradient quantization. These results hold for smooth and strongly-convex optimization problems, as well as smooth non-convex optimization problems verifying a Polyak-Lojasiewicz assumption. In particular, our upper and lower bounds depend on a novel quantity that extends the notion of conditional standard deviation, and is a measure of the extent to which the gradient can be approximated by having access to the oracle. As a consequence, our analysis provides a precise meaning to the intuition that optimization of the statistical learning objective is as hard as the estimation of its gradient. Finally, we show that, in the case of standard supervised learning, mini-batch gradient descent with increasing batch sizes and a warm start can reach a generalization error that is optimal up to a multiplicative factor, thus motivating the use of this optimization scheme in practical applications.

Stochastic Gradient Descent under Markovian Sampling Schemes

We study a variation of vanilla stochastic gradient descent where the optimizer only has access to a Markovian sampling scheme. These schemes encompass applications that range from decentralized optimization with a random walker (token algorithms), to RL and online system identification problems. We focus on obtaining rates of convergence under the least restrictive assumptions possible on the underlying Markov chain and on the functions optimized. We first unveil the theoretical lower bound for methods that sample stochastic gradients along the path of a Markov chain, making appear a dependency in the hitting time of the underlying Markov chain. We then study Markov chain SGD (MC-SGD) under much milder regularity assumptions than prior works. We finally introduce MC-SAG, an alternative to MC-SGD with variance reduction, that only depends on the hitting time of the Markov chain, therefore obtaining a communication-efficient token algorithm.

(S)GD over Diagonal Linear Networks: Implicit Regularisation, Large Stepsizes and Edge of Stability

In this paper, we investigate the impact of stochasticity and large stepsizes on the implicit regularisation of gradient descent (GD) and stochastic gradient descent (SGD) over diagonal linear networks. We prove the convergence of GD and SGD with macroscopic stepsizes in an overparametrised regression setting and characterise their solutions through an implicit regularisation problem. Our crisp characterisation leads to qualitative insights about the impact of stochasticity and stepsizes on the recovered solution. Specifically, we show that large stepsizes consistently benefit SGD for sparse regression problems, while they can hinder the recovery of sparse solutions for GD. These effects are magnified for stepsizes in a tight window just below the divergence threshold, in the ``edge of stability'' regime. Our findings are supported by experimental results.

Asynchronous SGD Beats Minibatch SGD Under Arbitrary Delays

The existing analysis of asynchronous stochastic gradient descent (SGD) degrades dramatically when any delay is large, giving the impression that performance depends primarily on the delay. On the contrary, we prove much better guarantees for the same asynchronous SGD algorithm regardless of the delays in the gradients, depending instead just on the number of parallel devices used to implement the algorithm. Our guarantees are strictly better than the existing analyses, and we also argue that asynchronous SGD outperforms synchronous minibatch SGD in the settings we consider. For our analysis, we introduce a novel recursion based on "virtual iterates" and delay-adaptive stepsizes, which allow us to derive state-of-the-art guarantees for both convex and non-convex objectives.

Muffliato: Peer-to-Peer Privacy Amplification for Decentralized Optimization and Averaging

Decentralized optimization is increasingly popular in machine learning for its scalability and efficiency. Intuitively, it should also provide better privacy guarantees, as nodes only observe the messages sent by their neighbors in the network graph. But formalizing and quantifying this gain is challenging: existing results are typically limited to Local Differential Privacy (LDP) guarantees that overlook the advantages of decentralization. In this work, we introduce pairwise network differential privacy, a relaxation of LDP that captures the fact that the privacy leakage from a node $u$ to a node $v$ may depend on their relative position in the graph. We then analyze the combination of local noise injection with (simple or randomized) gossip averaging protocols on fixed and random communication graphs. We also derive a differentially private decentralized optimization algorithm that alternates between local gradient descent steps and gossip averaging. Our results show that our algorithms amplify privacy guarantees as a function of the distance between nodes in the graph, matching the privacy-utility trade-off of the trusted curator, up to factors that explicitly depend on the graph topology. Finally, we illustrate our privacy gains with experiments on synthetic and real-world datasets.

A Continuized View on Nesterov Acceleration for Stochastic Gradient Descent and Randomized Gossip

We introduce the continuized Nesterov acceleration, a close variant of Nesterov acceleration whose variables are indexed by a continuous time parameter. The two variables continuously mix following a linear ordinary differential equation and take gradient steps at random times. This continuized variant benefits from the best of the continuous and the discrete frameworks: as a continuous process, one can use differential calculus to analyze convergence and obtain analytical expressions for the parameters; and a discretization of the continuized process can be computed exactly with convergence rates similar to those of Nesterov original acceleration. We show that the discretization has the same structure as Nesterov acceleration, but with random parameters. We provide continuized Nesterov acceleration under deterministic as well as stochastic gradients, with either additive or multiplicative noise. Finally, using our continuized framework and expressing the gossip averaging problem as the stochastic minimization of a certain energy function, we provide the first rigorous acceleration of asynchronous gossip algorithms.

Decentralized Optimization with Heterogeneous Delays: a Continuous-Time Approach

In decentralized optimization, nodes of a communication network privately possess a local objective function, and communicate using gossip-based methods in order to minimize the average of these per-node objectives. While synchronous algorithms can be heavily slowed down by a few nodes and edges in the graph (the straggler problem), their asynchronous counterparts lack from a sharp analysis taking into account heterogeneous delays in the communication network. In this paper, we propose a novel continuous-time framework to analyze asynchronous algorithms, which does not require to define a global ordering of the events, and allows to finely characterize the time complexity in the presence of (heterogeneous) delays. Using this framework, we describe a fully asynchronous decentralized algorithm to minimize the sum of smooth and strongly convex functions. Our algorithm (DCDM, Delayed Coordinate Dual Method), based on delayed randomized gossip communications and local computational updates, achieves an asynchronous speed-up: the rate of convergence is tightly characterized in terms of the eigengap of the graph weighted by local delays only, instead of the global worst-case delays as in previous analyses.

Concentration of Non-Isotropic Random Tensors with Applications to Learning and Empirical Risk Minimization

Dimension is an inherent bottleneck to some modern learning tasks, where optimization methods suffer from the size of the data. In this paper, we study non-isotropic distributions of data and develop tools that aim at reducing these dimensional costs by a dependency on an effective dimension rather than the ambient one. Based on non-asymptotic estimates of the metric entropy of ellipsoids -- that prove to generalize to infinite dimensions -- and on a chaining argument, our uniform concentration bounds involve an effective dimension instead of the global dimension, improving over existing results. We show the importance of taking advantage of non-isotropic properties in learning problems with the following applications: i) we improve state-of-the-art results in statistical preconditioning for communication-efficient distributed optimization, ii) we introduce a non-isotropic randomized smoothing for non-smooth optimization. Both applications cover a class of functions that encompasses empirical risk minization (ERM) for linear models.