We introduce a new and increasingly relevant setting for distributed optimization in machine learning, where the data defining the optimization are unevenly distributed over an extremely large number of nodes. The goal is to train a high-quality centralized model. We refer to this setting as Federated Optimization. In this setting, communication efficiency is of the utmost importance and minimizing the number of rounds of communication is the principal goal. A motivating example arises when we keep the training data locally on users' mobile devices instead of logging it to a data center for training. In federated optimziation, the devices are used as compute nodes performing computation on their local data in order to update a global model. We suppose that we have extremely large number of devices in the network --- as many as the number of users of a given service, each of which has only a tiny fraction of the total data available. In particular, we expect the number of data points available locally to be much smaller than the number of devices. Additionally, since different users generate data with different patterns, it is reasonable to assume that no device has a representative sample of the overall distribution. We show that existing algorithms are not suitable for this setting, and propose a new algorithm which shows encouraging experimental results for sparse convex problems. This work also sets a path for future research needed in the context of optimization.
When an explicit strongly convex, but not necessarily smooth, regularizer is added to the average loss , it is possible to write down its (Fenchel) dual and the dual variables live in @math -dimensional space. Applying RCD leads to an algorithm for solving known under the name Stochastic Dual Coordinate Ascent @cite_74 . This method has gained broad popularity with practicioners, likely due to the fact that for a number of loss functions, the method comes without the need to tune any hyper-parameters. The work @cite_74 was first to show that by applying RCD @cite_98 to the dual problem, one also solves the primal problem . For a theoretical and computational comparison of applying RCD to the primal versus the dual problems, see @cite_37 .
@cite_98 @cite_37 @cite_74
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Two types of low cost-per-iteration gradient descent methods have been extensively studied in parallel. One is online or stochastic gradient descent (OGD SGD), and the other is randomzied coordinate descent (RBCD). In this paper, we combine the two types of methods together and propose online randomized block coordinate descent (ORBCD). At each iteration, ORBCD only computes the partial gradient of one block coordinate of one mini-batch samples. ORBCD is well suited for the composite minimization problem where one function is the average of the losses of a large number of samples and the other is a simple regularizer defined on high dimensional variables. We show that the iteration complexity of ORBCD has the same order as OGD or SGD. For strongly convex functions, by reducing the variance of stochastic gradients, we show that ORBCD can converge at a geometric rate in expectation, matching the convergence rate of SGD with variance reduction and RBCD. Randomized coordinate descent (RCD) methods are state-of-the-art algorithms for training linear predictors via minimizing regularized empirical risk. When the number of examples ( @math ) is much larger than the number of features ( @math ), a common strategy is to apply RCD to the dual problem. On the other hand, when the number of features is much larger than the number of examples, it makes sense to apply RCD directly to the primal problem. In this paper we provide the first joint study of these two approaches when applied to L2-regularized ERM. First, we show through a rigorous analysis that for dense data, the above intuition is precisely correct. However, we find that for sparse and structured data, primal RCD can significantly outperform dual RCD even if @math , and vice versa, dual RCD can be much faster than primal RCD even if @math . Moreover, we show that, surprisingly, a single sampling strategy minimizes both the (bound on the) number of iterations and the overall expected complexity of RCD. Note that the latter complexity measure also takes into account the average cost of the iterations, which depends on the structure and sparsity of the data, and on the sampling strategy employed. We confirm our theoretical predictions using extensive experiments with both synthetic and real data sets. We present an improved analysis of mini-batched stochastic dual coordinate ascent for regularized empirical loss minimization (i.e. SVM and SVM-type objectives). Our analysis allows for flexible sampling schemes, including where data is distribute across machines, and combines a dependence on the smoothness of the loss and or the data spread (measured through the spectral norm).