We study distributed optimization algorithms for minimizing the average of convex functions. The applications include empirical risk minimization problems in statistical machine learning where the datasets are large and have to be stored on different machines. We design a distributed stochastic variance reduced gradient algorithm that, under certain conditions on the condition number, simultaneously achieves the optimal parallel runtime, amount of communication and rounds of communication among all distributed first-order methods up to constant factors. Our method and its accelerated extension also outperform existing distributed algorithms in terms of the rounds of communication as long as the condition number is not too large compared to the size of data in each machine. We also prove a lower bound for the number of rounds of communication for a broad class of distributed first-order methods including the proposed algorithms in this paper. We show that our accelerated distributed stochastic variance reduced gradient algorithm achieves this lower bound so that it uses the fewest rounds of communication among all distributed first-order algorithms.
Let @math be the computation cost of the gradient of @math . For typical problems such as logistic regression, we have @math . Let @math be the cost of solving a linear system. We have @math if exact matrix inversion is used and have @math if an @math -approximate inverse is found by an accelerated gradient method @cite_7 .
@cite_7
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We study the minimization of a convex function @math over the set of @math positive semi-definite matrices, but when the problem is recast as @math , with @math and @math . We study the performance of gradient descent on @math ---which we refer to as Factored Gradient Descent (FGD)---under standard assumptions on the original function @math . We provide a rule for selecting the step size and, with this choice, show that the local convergence rate of FGD mirrors that of standard gradient descent on the original @math : i.e., after @math steps, the error is @math for smooth @math , and exponentially small in @math when @math is (restricted) strongly convex. In addition, we provide a procedure to initialize FGD for (restricted) strongly convex objectives and when one only has access to @math via a first-order oracle; for several problem instances, such proper initialization leads to global convergence guarantees. FGD and similar procedures are widely used in practice for problems that can be posed as matrix factorization. To the best of our knowledge, this is the first paper to provide precise convergence rate guarantees for general convex functions under standard convex assumptions. Many conjugate gradient-like methods for solving linear systems @math use recursion formulas for updating residual vectors instead of computing the residuals directly. For such methods it is shown that the difference between the actual residuals and the updated approximate residual vectors generated in finite precision arithmetic depends on the machine precision @math and on the maximum norm of an iterate divided by the norm of the true solution. It is often observed numerically, and can sometimes be proved, that the norms of the updated approximate residual vectors converge to zero or, at least, become orders of magnitude smaller than the machine precision. In such cases, the actual residual norm reaches the level @math times the maximum ratio of the norm of an iterate to that of the true solution. Using exact arithmetic theory to bound the size of the iterates, we give a priori estimates of the size of the final residual for a number of algorithms. Classical stochastic gradient methods are well suited for minimizing expected-value objective functions. However, they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected-value functions, i.e., the problem @math minxEvfv(Ew[gw(x)]). In order to solve this stochastic composition problem, we propose a class of stochastic compositional gradient descent (SCGD) algorithms that can be viewed as stochastic versions of quasi-gradient method. SCGD update the solutions based on noisy sample gradients of @math fv,gw and use an auxiliary variable to track the unknown quantity @math Ewgw(x). We prove that the SCGD converge almost surely to an optimal solution for convex optimization problems, as long as such a solution exists. The convergence involves the interplay of two iterations with different time scales. For nonsmooth convex problems, the SCGD achieves a convergence rate of @math O(k-1 4) in the general case and @math O(k-2 3) in the strongly convex case, after taking k samples. For smooth convex problems, the SCGD can be accelerated to converge at a rate of @math O(k-2 7) in the general case and @math O(k-4 5) in the strongly convex case. For nonconvex problems, we prove that any limit point generated by SCGD is a stationary point, for which we also provide the convergence rate analysis. Indeed, the stochastic setting where one wants to optimize compositions of expected-value functions is very common in practice. The proposed SCGD methods find wide applications in learning, estimation, dynamic programming, etc. We study the performance of a family of randomized parallel coordinate descent methods for minimizing the sum of a nonsmooth and separable convex functions. The problem class includes as a special case L1-regularized L1 regression and the minimization of the exponential loss ("AdaBoost problem"). We assume the input data defining the loss function is contained in a sparse @math matrix @math with at most @math nonzeros in each row. Our methods need @math iterations to find an approximate solution with high probability, where @math is the number of processors and @math for the fastest variant. The notation hides dependence on quantities such as the required accuracy and confidence levels and the distance of the starting iterate from an optimal point. Since @math is a decreasing function of @math , the method needs fewer iterations when more processors are used. Certain variants of our algorithms perform on average only @math arithmetic operations during a single iteration per processor and, because @math decreases when @math does, fewer iterations are needed for sparser problems. We consider the problem of learning a one-hidden-layer neural network: we assume the input @math is from Gaussian distribution and the label @math , where @math is a nonnegative vector in @math with @math , @math is a full-rank weight matrix, and @math is a noise vector. We first give an analytic formula for the population risk of the standard squared loss and demonstrate that it implicitly attempts to decompose a sequence of low-rank tensors simultaneously. Inspired by the formula, we design a non-convex objective function @math whose landscape is guaranteed to have the following properties: 1. All local minima of @math are also global minima. 2. All global minima of @math correspond to the ground truth parameters. 3. The value and gradient of @math can be estimated using samples. With these properties, stochastic gradient descent on @math provably converges to the global minimum and learn the ground-truth parameters. We also prove finite sample complexity result and validate the results by simulations. We propose a new randomized coordinate descent method for minimizing the sum of convex functions each of which depends on a small number of coordinates only. Our method (APPROX) is simultaneously Accelerated, Parallel, and PROXimal; this is the first time such a method is proposed. In the special case when the number of processors is equal to the number of coordinates, the method converges at the rate @math , where @math is the iteration counter, @math is a data-weighted average degree of separability of the loss function, @math is the average of Lipschitz constants associated with the coordinates and individual functions in the sum, and @math is the distance of the initial point from the minimizer. We show that the method can be implemented without the need to perform full-dimensional vector operations, which is the major bottleneck of accelerated coordinate descent. The fact that the method depends on the average degree of separability, and not on the maximum degree... In this paper we study the problem of minimizing the average of a large number ( @math ) of smooth convex loss functions. We propose a new method, S2GD (Semi-Stochastic Gradient Descent), which runs for one or several epochs in each of which a single full gradient and a random number of stochastic gradients is computed, following a geometric law. The total work needed for the method to output an @math -accurate solution in expectation, measured in the number of passes over data, or equivalently, in units equivalent to the computation of a single gradient of the loss, is @math , where @math is the condition number. This is achieved by running the method for @math epochs, with a single gradient evaluation and @math stochastic gradient evaluations in each. The SVRG method of Johnson and Zhang arises as a special case. If our method is limited to a single epoch only, it needs to evaluate at most @math stochastic gradients. In contrast, SVRG requires @math stochastic gradients. To illustrate our theoretical results, S2GD only needs the workload equivalent to about 2.1 full gradient evaluations to find an @math -accurate solution for a problem with @math and @math . A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the @math -dimensional setting, however, it requires the inversion of an @math covariance matrix, @math , as well as the evaluation of its determinant, @math . In many cases, such as regression using Gaussian processes, the covariance matrix is of the form @math , where @math is computed using a specified covariance kernel which depends on the data and additional parameters (hyperparameters). The matrix @math is typically dense, causing standard direct methods for inversion and determinant evaluation to require @math work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix @math can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an @math algorithm for inversion. More importantly, we show that this factorization enables the evaluation of the determinant @math , permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining @math . Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels. The logistic loss is strictly convex and does not attain its infimum; consequently the solutions of logistic regression are in general off at infinity. This work provides a convergence analysis of gradient descent applied to logistic regression under no assumptions on the problem instance. Firstly, the risk is shown to converge at a rate @math . Secondly, the parameter convergence is characterized along a unique pair of complementary subspaces defined by the problem instance: one subspace along which strong convexity induces parameters to converge at rate @math , and its orthogonal complement along which separability induces parameters to converge in direction at rate @math . We study a general online convex optimization problem. We have a convex set S and an unknown sequence of cost functions c 1 , c 2 ,..., and in each period, we choose a feasible point x t in S, and learn the cost c t (x t ). If the function c t is also revealed after each period then, as Zinkevich shows in [25], gradient descent can be used on these functions to get regret bounds of O(√n). That is, after n rounds, the total cost incurred will be O(√n) more than the cost of the best single feasible decision chosen with the benefit of hindsight, min x Σ ct(x).We extend this to the "bandit" setting, where, in each period, only the cost c t (x t ) is revealed, and bound the expected regret as O(n3 4).Our approach uses a simple approximation of the gradient that is computed from evaluating c t at a single (random) point. We show that this biased estimate is sufficient to approximate gradient descent on the sequence of functions. In other words, it is possible to use gradient descent without seeing anything more than the value of the functions at a single point. The guarantees hold even in the most general case: online against an adaptive adversary.For the online linear optimization problem [15], algorithms with low regrets in the bandit setting have recently been given against oblivious [1] and adaptive adversaries [19]. In contrast to these algorithms, which distinguish between explicit explore and exploit periods, our algorithm can be interpreted as doing a small amount of exploration in each period. Fitting a function by using linear combinations of a large number @math of simple' components is one of the most fruitful ideas in statistical learning. This idea lies at the core of a variety of methods, from two-layer neural networks to kernel regression, to boosting. In general, the resulting risk minimization problem is non-convex and is solved by gradient descent or its variants. Unfortunately, little is known about global convergence properties of these approaches. Here we consider the problem of learning a concave function @math on a compact convex domain @math , using linear combinations of bump-like' components (neurons). The parameters to be fitted are the centers of @math bumps, and the resulting empirical risk minimization problem is highly non-convex. We prove that, in the limit in which the number of neurons diverges, the evolution of gradient descent converges to a Wasserstein gradient flow in the space of probability distributions over @math . Further, when the bump width @math tends to @math , this gradient flow has a limit which is a viscous porous medium equation. Remarkably, the cost function optimized by this gradient flow exhibits a special property known as displacement convexity, which implies exponential convergence rates for @math , @math . Surprisingly, this asymptotic theory appears to capture well the behavior for moderate values of @math . Explaining this phenomenon, and understanding the dependence on @math in a quantitative manner remains an outstanding challenge. We examine gradient descent on unregularized logistic regression problems, with homogeneous linear predictors on linearly separable datasets. We show the predictor converges to the direction of the max-margin (hard margin SVM) solution. The result also generalizes to other monotone decreasing loss functions with an infimum at infinity, to multi-class problems, and to training a weight layer in a deep network in a certain restricted setting. Furthermore, we show this convergence is very slow, and only logarithmic in the convergence of the loss itself. This can help explain the benefit of continuing to optimize the logistic or cross-entropy loss even after the training error is zero and the training loss is extremely small, and, as we show, even if the validation loss increases. Our methodology can also aid in understanding implicit regularization n more complex models and with other optimization methods. Accelerated coordinate descent is widely used in optimization due to its cheap per-iteration cost and scalability to large-scale problems. Up to a primal-dual transformation, it is also the same as accelerated stochastic gradient descent that is one of the central methods used in machine learning. In this paper, we improve the best known running time of accelerated coordinate descent by a factor up to @math . Our improvement is based on a clean, novel non-uniform sampling that selects each coordinate with a probability proportional to the square root of its smoothness parameter. Our proof technique also deviates from the classical estimation sequence technique used in prior work. Our speed-up applies to important problems such as empirical risk minimization and solving linear systems, both in theory and in practice. Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in @math iterations, faster than the @math iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second-order stationary point faster than GD, and also the first single-loop algorithm with a faster rate than GD even in the setting of finding a first-order stationary point. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases per step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. We consider the minimization of non-convex quadratic forms regularized by a cubic term, which exhibit multiple saddle points and poor local minima. Nonetheless, we prove that, under mild assumptions, gradient descent approximates the @math to within @math accuracy in @math steps for large @math and @math steps for small @math (compared to a condition number we define), with at most logarithmic dependence on the problem dimension. When we use gradient descent to approximate the Nesterov-Polyak cubic-regularized Newton step, our result implies a rate of convergence to second-order stationary points of general smooth non-convex functions. This paper analyzes the iteration-complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs. More specifically, the objective function is of the form @math where @math is a differentiable function whose gradient is Lipschitz continuous and @math is a closed convex function with bounded domain. The method, basically, consists of applying an accelerated inexact proximal point method for solving approximately a sequence of quadratic penalized subproblems associated to the linearly constrained problem. Each subproblem of the proximal point method is in turn approximately solved by an accelerated composite gradient method. It is shown that the proposed scheme generates a @math -approximate stationary point in at most @math . Finally, numerical results showing the efficiency of the proposed method are also given. We provide tight upper and lower bounds on the complexity of minimizing the average of @math convex functions using gradient and prox oracles of the component functions. We show a significant gap between the complexity of deterministic vs randomized optimization. For smooth functions, we show that accelerated gradient descent (AGD) and an accelerated variant of SVRG are optimal in the deterministic and randomized settings respectively, and that a gradient oracle is sufficient for the optimal rate. For non-smooth functions, having access to prox oracles reduces the complexity and we present optimal methods based on smoothing that improve over methods using just gradient accesses. We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix @math with gradient descent on a factorization of X. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a full dimensional factorization converges to the minimum nuclear norm solution. We present an accelerated gradient method for nonconvex optimization problems with Lipschitz continuous first and second derivatives. In a time @math , the method finds an @math -stationary point, meaning a point @math such that @math . The method improves upon the @math complexity of gradient descent and provides the additional second-order guarantee that @math for the computed @math . Furthermore, our method is Hessian free, i.e., it only requires gradient computations, and is therefore suitable for large-scale applications. In this paper we show how to accelerate randomized coordinate descent methods and achieve faster convergence rates without paying per-iteration costs in asymptotic running time. In particular, we show how to generalize and efficiently implement a method proposed by Nesterov, giving faster asymptotic running times for various algorithms that use standard coordinate descent as a black box. In addition to providing a proof of convergence for this new general method, we show that it is numerically stable, efficiently implementable, and in certain regimes, asymptotically optimal. To highlight the power of this algorithm, we show how it can used to create faster linear system solvers in several regimes: - We show how this method achieves a faster asymptotic runtime than conjugate gradient for solving a broad class of symmetric positive definite systems of equations. - We improve the convergence guarantees for Kaczmarz methods, a popular technique for image reconstruction and solving over determined systems of equations, by accelerating an algorithm of Strohmer and Vershynin. - We achieve the best known running time for solving Symmetric Diagonally Dominant (SDD) system of equations in the unit-cost RAM model, obtaining a running time of O(m log3 2n (log log n)1 2 log((log n) eps)) by accelerating a recent solver by Beyond the independent interest of these solvers, we believe they highlight the versatility of the approach of this paper and we hope that they will open the door for further algorithmic improvements in the future.