For distributed computing environments, we consider the canonical machine learning problem of empirical risk minimization (ERM) with quadratic regularization, and we propose a distributed and communication-efficient Newton-type optimization method. At every iteration, each worker locally finds an Approximate NewTon (ANT) direction, and then it sends this direction to the main driver. The driver, then, averages all the ANT directions received from workers to form a Globally Improved ANT (GIANT) direction. GIANT naturally exploits the trade-offs between local computations and global communications in that more local computations result in fewer overall rounds of communications. GIANT is highly communication efficient in that, for @math -dimensional data uniformly distributed across @math workers, it has @math or @math rounds of communication and @math communication complexity per iteration. Theoretically, we show that GIANT's convergence rate is faster than first-order methods and existing distributed Newton-type methods. From a practical point-of-view, a highly beneficial feature of GIANT is that it has only one tuning parameter---the iterations of the local solver for computing an ANT direction. This is indeed in sharp contrast with many existing distributed Newton-type methods, as well as popular first-order methods, which have several tuning parameters, and whose performance can be greatly affected by the specific choices of such parameters. In this light, we empirically demonstrate the superior performance of GIANT compared with other competing methods.
Among the existing distributed second-order optimization methods, the most notably are DANE @cite_4 , AIDE @cite_18 , and DiSCO @cite_31 . Another similar method is CoCoA @cite_15 @cite_40 @cite_26 , which is analogous to second-order methods in that it involves sub-problems which are local quadratic approximations to the dual objective function. However, despite the fact that CoCoA makes use of the smoothness condition, it does not exploit any explicit second-order information.
@cite_18 @cite_26 @cite_4 @cite_15 @cite_40 @cite_31
2952986481 2963861706 2556660792 2895571900
Distributed optimization methods for large-scale machine learning suffer from a communication bottleneck. It is difficult to reduce this bottleneck while still efficiently and accurately aggregating partial work from different machines. In this paper, we present a novel generalization of the recent communication-efficient primal-dual framework (CoCoA) for distributed optimization. Our framework, CoCoA+, allows for additive combination of local updates to the global parameters at each iteration, whereas previous schemes with convergence guarantees only allow conservative averaging. We give stronger (primal-dual) convergence rate guarantees for both CoCoA as well as our new variants, and generalize the theory for both methods to cover non-smooth convex loss functions. We provide an extensive experimental comparison that shows the markedly improved performance of CoCoA+ on several real-world distributed datasets, especially when scaling up the number of machines. Communication remains the most significant bottleneck in the performance of distributed optimization algorithms for large-scale machine learning. In this paper, we propose a communication-efficient framework, COCOA, that uses local computation in a primal-dual setting to dramatically reduce the amount of necessary communication. We provide a strong convergence rate analysis for this class of algorithms, as well as experiments on real-world distributed datasets with implementations in Spark. In our experiments, we find that as compared to state-of-the-art mini-batch versions of SGD and SDCA algorithms, COCOA converges to the same .001-accurate solution quality on average 25 × as quickly. The scale of modern datasets necessitates the development of efficient distributed optimization methods for machine learning. We present a general-purpose framework for distributed computing environments, CoCoA, that has an efficient communication scheme and is applicable to a wide variety of problems in machine learning and signal processing. We extend the framework to cover general non-strongly-convex regularizers, including L1-regularized problems like lasso, sparse logistic regression, and elastic net regularization, and show how earlier work can be derived as a special case. We provide convergence guarantees for the class of convex regularized loss minimization objectives, leveraging a novel approach in handling non-strongly-convex regularizers and non-smooth loss functions. The resulting framework has markedly improved performance over state-of-the-art methods, as we illustrate with an extensive set of experiments on real distributed datasets. We consider the problem of finding an approximate second-order stationary point of a constrained non-convex optimization problem. We first show that, unlike the unconstrained scenario, the vanilla projected gradient descent algorithm may converge to a strict saddle point even when there is only a single linear constraint. We then provide a hardness result by showing that checking ( , )-second order stationarity is NP-hard even in the presence of linear constraints. Despite our hardness result, we identify instances of the problem for which checking second order stationarity can be done efficiently. For such instances, we propose a dynamic second order Frank--Wolfe algorithm which converges to ( , )-second order stationary points in O ( ^ -2 , ^ -3 ) iterations. The proposed algorithm can be used in general constrained non-convex optimization as long as the constrained quadratic sub-problem can be solved efficiently.