Combining gradient compression with adaptive optimizers is a highly desirable goal in distributed learning, with potential benefits in both fewer communication rounds and less per-round communication. In spite of preliminary empirical promise, certain major challenges in the convergence analysis of such methods have stayed open: handling compression based approximation of both first and second moments (pre-conditioner) which appear as a ratio; avoiding dependence on the number of parameters, which is extremely large in modern deep models; and providing high-probability guarantees instead of in-expectation, which can hide high variance behavior. In this work, we introduce a family of Sketched Adaptive Distributed Learning (SADL) algorithms which can use suitable unbiased gradient sketching for compression with suitable adaptive optimization algorithms. As our main contribution, we provide theoretical convergence guarantees of SADL algorithms which addresses all of the existing challenges. In particular, our guarantees hold with high probability, picks up only a logarithmic dependence on the number of parameters, and the first and second moment approximation is handled precisely yielding a dependence on the intrinsic dimension of the loss Hessian, which is significantly smaller than the full dimensionality of deep learning models. Empirically, the SADL algorithms are shown to be competitive with and often outperform baselines on both vision and language tasks, in both supervised fine-tuning and training-from-scratch regimes. Further, the SADL algorithms are also competitive with the state-of-the-art communication-efficient distributed learning algorithms based on error feedback.

In this paper, we show that applying adaptive methods directly to distributed minimax problems can result in non-convergence due to inconsistency in locally computed adaptive stepsizes. To address this challenge, we propose D-AdaST, a Distributed Adaptive minimax method with Stepsize Tracking. The key strategy is to employ an adaptive stepsize tracking protocol involving the transmission of two extra (scalar) variables. This protocol ensures the consistency among stepsizes of nodes, eliminating the steady-state error due to the lack of coordination of stepsizes among nodes that commonly exists in vanilla distributed adaptive methods, and thus guarantees exact convergence. For nonconvex-strongly-concave distributed minimax problems, we characterize the specific transient times that ensure time-scale separation of stepsizes and quasi-independence of networks, leading to a near-optimal convergence rate of $\tilde{\mathcal{O}} \left( \epsilon ^{-\left( 4+\delta \right)} \right)$ for any small $\delta > 0$, matching that of the centralized counterpart. To our best knowledge, D-AdaST is the *first* distributed adaptive method achieving near-optimal convergence without knowing any problem-dependent parameters for nonconvex minimax problems. Extensive experiments are conducted to validate our theoretical results.

Adaptive optimization methods, which perform local optimization with a metric constructed from the history of iterates, are becoming increasingly popular for training deep neural networks. Examples include AdaGrad, RMSProp, and Adam. We show that for simple overparameterized problems, adaptive methods often find drastically different solutions than gradient descent (GD) or stochastic gradient descent (SGD). We construct an illustrative binary classification problem where the data is linearly separable, GD and SGD achieve zero test error, and AdaGrad, Adam, and RMSProp attain test errors arbitrarily close to half. We additionally study the empirical generalization capability of adaptive methods on several state-of-the-art deep learning models. We observe that the solutions found by adaptive methods generalize worse (often significantly worse) than SGD, even when these solutions have better training performance. These results suggest that practitioners should reconsider the use of adaptive methods to train neural networks.

The classical analysis of Stochastic Gradient Descent (SGD) with polynomially decaying stepsize $\eta_t = \eta/\sqrt{t}$ relies on well-tuned $\eta$ depending on problem parameters such as Lipschitz smoothness constant, which is often unknown in practice. In this work, we prove that SGD with arbitrary $\eta > 0$, referred to as untuned SGD, still attains an order-optimal convergence rate $\widetilde{\mathcal{O}}(T^{-1/4})$ in terms of gradient norm for minimizing smooth objectives. Unfortunately, it comes at the expense of a catastrophic exponential dependence on the smoothness constant, which we show is unavoidable for this scheme even in the noiseless setting. We then examine three families of adaptive methods -- Normalized SGD (NSGD), AMSGrad, and AdaGrad -- unveiling their power in preventing such exponential dependency in the absence of information about the smoothness parameter and boundedness of stochastic gradients. Our results provide theoretical justification for the advantage of adaptive methods over untuned SGD in alleviating the issue with large gradients.