Pre-training Transformer models is resource-intensive, and recent studies have shown that sign momentum is an efficient technique for training large-scale deep learning models, particularly Transformers. However, its application in distributed training or federated learning remains underexplored. This paper investigates a novel communication-efficient distributed sign momentum method with local updates. Our proposed method allows for a broad class of base optimizers for local updates, and uses sign momentum in global updates, where momentum is generated from differences accumulated during local steps. We evaluate our method on the pre-training of various GPT-2 models, and the empirical results show significant improvement compared to other distributed methods with local updates. Furthermore, by approximating the sign operator with a randomized version that acts as a continuous analog in expectation, we present an $O(1/\sqrt{T})$ convergence for one instance of the proposed method for nonconvex smooth functions.

We study large deviations and mean-squared error (MSE) guarantees of a general framework of nonlinear stochastic gradient methods in the online setting, in the presence of heavy-tailed noise. Unlike existing works that rely on the closed form of a nonlinearity (typically clipping), our framework treats the nonlinearity in a black-box manner, allowing us to provide unified guarantees for a broad class of bounded nonlinearities, including many popular ones, like sign, quantization, normalization, as well as component-wise and joint clipping. We provide several strong results for a broad range of step-sizes in the presence of heavy-tailed noise with symmetric probability density function, positive in a neighbourhood of zero and potentially unbounded moments. In particular, for non-convex costs we provide a large deviation upper bound for the minimum norm-squared of gradients, showing an asymptotic tail decay on an exponential scale, at a rate $\sqrt{t} / \log(t)$. We establish the accompanying rate function, showing an explicit dependence on the choice of step-size, nonlinearity, noise and problem parameters. Next, for non-convex costs and the minimum norm-squared of gradients, we derive the optimal MSE rate $\widetilde{\mathcal{O}}(t^{-1/2})$. Moreover, for strongly convex costs and the last iterate, we provide an MSE rate that can be made arbitrarily close to the optimal rate $\mathcal{O}(t^{-1})$, improving on the state-of-the-art results in the presence of heavy-tailed noise. Finally, we establish almost sure convergence of the minimum norm-squared of gradients, providing an explicit rate, which can be made arbitrarily close to $o(t^{-1/4})$.

We study high-probability convergence in online learning, in the presence of heavy-tailed noise. To combat the heavy tails, a general framework of nonlinear SGD methods is considered, subsuming several popular nonlinearities like sign, quantization, component-wise and joint clipping. In our work the nonlinearity is treated in a black-box manner, allowing us to establish unified guarantees for a broad range of nonlinear methods. For symmetric noise and non-convex costs we establish convergence of gradient norm-squared, at a rate $\widetilde{\mathcal{O}}(t^{-1/4})$, while for the last iterate of strongly convex costs we establish convergence to the population optima, at a rate $\mathcal{O}(t^{-\zeta})$, where $\zeta \in (0,1)$ depends on noise and problem parameters. Further, if the noise is a (biased) mixture of symmetric and non-symmetric components, we show convergence to a neighbourhood of stationarity, whose size depends on the mixture coefficient, nonlinearity and noise. Compared to state-of-the-art, who only consider clipping and require unbiased noise with bounded $p$-th moments, $p \in (1,2]$, we provide guarantees for a broad class of nonlinearities, without any assumptions on noise moments. While the rate exponents in state-of-the-art depend on noise moments and vanish as $p \rightarrow 1$, our exponents are constant and strictly better whenever $p < 6/5$ for non-convex and $p < 8/7$ for strongly convex costs. Experiments validate our theory, demonstrating noise symmetry in real-life settings and showing that clipping is not always the optimal nonlinearity, further underlining the value of a general framework.

In this paper, we study secure distributed optimization against arbitrary gradient attack in multi-agent networks. In distributed optimization, there is no central server to coordinate local updates, and each agent can only communicate with its neighbors on a predefined network. We consider the scenario where out of $n$ networked agents, a fixed but unknown fraction $\rho$ of the agents are under arbitrary gradient attack in that their stochastic gradient oracles return arbitrary information to derail the optimization process, and the goal is to minimize the sum of local objective functions on unattacked agents. We propose a distributed stochastic gradient method that combines local variance reduction and clipping (CLIP-VRG). We show that, in a connected network, when unattacked local objective functions are convex and smooth, share a common minimizer, and their sum is strongly convex, CLIP-VRG leads to almost sure convergence of the iterates to the exact sum cost minimizer at all agents. We quantify a tight upper bound of the fraction $\rho$ of attacked agents in terms of problem parameters such as the condition number of the associated sum cost that guarantee exact convergence of CLIP-VRG, and characterize its asymptotic convergence rate. Finally, we empirically demonstrate the effectiveness of the proposed method under gradient attacks in both synthetic dataset and image classification datasets.