Local SGD Converges Fast and Communicates Little

Mini-batch stochastic gradient descent (SGD) is state of the art in large scale distributed training. The scheme can reach a linear speedup with respect to the number of workers, but this is rarely seen in practice as the scheme often suffers from large network delays and bandwidth limits. To overcome this communication bottleneck recent works propose to reduce the communication frequency. An algorithm of this type is local SGD that runs SGD independently in parallel on different workers and averages the sequences only once in a while. This scheme shows promising results in practice, but eluded thorough theoretical analysis. We prove concise convergence rates for local SGD on convex problems and show that it converges at the same rate as mini-batch SGD in terms of number of evaluated gradients, that is, the scheme achieves linear speedup in the number of workers and mini-batch size. The number of communication rounds can be reduced up to a factor of T^{1/2}---where T denotes the number of total steps---compared to mini-batch SGD. This also holds for asynchronous implementations. Local SGD can also be used for large scale training of deep learning models. The results shown here aim serving as a guideline to further explore the theoretical and practical aspects of local SGD in these applications.Comment: to appear at ICLR 2019, 19 page

Variable Metric Random Pursuit

We consider unconstrained randomized optimization of smooth convex objective functions in the gradient-free setting. We analyze Random Pursuit (RP) algorithms with fixed (F-RP) and variable metric (V-RP). The algorithms only use zeroth-order information about the objective function and compute an approximate solution by repeated optimization over randomly chosen one-dimensional subspaces. The distribution of search directions is dictated by the chosen metric. Variable Metric RP uses novel variants of a randomized zeroth-order Hessian approximation scheme recently introduced by Leventhal and Lewis (D. Leventhal and A. S. Lewis., Optimization 60(3), 329--245, 2011). We here present (i) a refined analysis of the expected single step progress of RP algorithms and their global convergence on (strictly) convex functions and (ii) novel convergence bounds for V-RP on strongly convex functions. We also quantify how well the employed metric needs to match the local geometry of the function in order for the RP algorithms to converge with the best possible rate. Our theoretical results are accompanied by numerical experiments, comparing V-RP with the derivative-free schemes CMA-ES, Implicit Filtering, Nelder-Mead, NEWUOA, Pattern-Search and Nesterov's gradient-free algorithms.Comment: 42 pages, 6 figures, 15 tables, submitted to journal, Version 3: majorly revised second part, i.e. Section 5 and Appendi

Adaptive SGD with Polyak stepsize and Line-search: Robust Convergence and Variance Reduction

The recently proposed stochastic Polyak stepsize (SPS) and stochastic line-search (SLS) for SGD have shown remarkable effectiveness when training over-parameterized models. However, in non-interpolation settings, both algorithms only guarantee convergence to a neighborhood of a solution which may result in a worse output than the initial guess. While artificially decreasing the adaptive stepsize has been proposed to address this issue (Orvieto et al. [2022]), this approach results in slower convergence rates for convex and over-parameterized models. In this work, we make two contributions: Firstly, we propose two new variants of SPS and SLS, called AdaSPS and AdaSLS, which guarantee convergence in non-interpolation settings and maintain sub-linear and linear convergence rates for convex and strongly convex functions when training over-parameterized models. AdaSLS requires no knowledge of problem-dependent parameters, and AdaSPS requires only a lower bound of the optimal function value as input. Secondly, we equip AdaSPS and AdaSLS with a novel variance reduction technique and obtain algorithms that require $\smash{\widetilde{\mathcal{O}}}(n+1/\epsilon)$ gradient evaluations to achieve an $\mathcal{O}(\epsilon)$-suboptimality for convex functions, which improves upon the slower $\mathcal{O}(1/\epsilon^2)$ rates of AdaSPS and AdaSLS without variance reduction in the non-interpolation regimes. Moreover, our result matches the fast rates of AdaSVRG but removes the inner-outer-loop structure, which is easier to implement and analyze. Finally, numerical experiments on synthetic and real datasets validate our theory and demonstrate the effectiveness and robustness of our algorithms

Spectral Preconditioning for Gradient Methods on Graded Non-convex Functions

The performance of optimization methods is often tied to the spectrum of the objective Hessian. Yet, conventional assumptions, such as smoothness, do often not enable us to make finely-grained convergence statements -- particularly not for non-convex problems. Striving for a more intricate characterization of complexity, we introduce a unique concept termed graded non-convexity. This allows to partition the class of non-convex problems into a nested chain of subclasses. Interestingly, many traditional non-convex objectives, including partially convex problems, matrix factorizations, and neural networks, fall within these subclasses. As a second contribution, we propose gradient methods with spectral preconditioning, which employ inexact top eigenvectors of the Hessian to address the ill-conditioning of the problem, contingent on the grade. Our analysis reveals that these new methods provide provably superior convergence rates compared to basic gradient descent on applicable problem classes, particularly when large gaps exist between the top eigenvalues of the Hessian. Our theory is validated by numerical experiments executed on multiple practical machine learning problems