Recently, there has been growing interest in developing optimization methods for solving large-scale machine learning problems. Most of these problems boil down to the problem of minimizing an average of a finite set of smooth and strongly convex functions where the number of functions @math is large. Gradient descent method (GD) is successful in minimizing convex problems at a fast linear rate; however, it is not applicable to the considered large-scale optimization setting because of the high computational complexity. Incremental methods resolve this drawback of gradient methods by replacing the required gradient for the descent direction with an incremental gradient approximation. They operate by evaluating one gradient per iteration and executing the average of the @math available gradients as a gradient approximate. Although, incremental methods reduce the computational cost of GD, their convergence rates do not justify their advantage relative to GD in terms of the total number of gradient evaluations until convergence. In this paper, we introduce a Double Incremental Aggregated Gradient method (DIAG) that computes the gradient of only one function at each iteration, which is chosen based on a cyclic scheme, and uses the aggregated average gradient of all the functions to approximate the full gradient. The iterates of the proposed DIAG method uses averages of both iterates and gradients in oppose to classic incremental methods that utilize gradient averages but do not utilize iterate averages. We prove that not only the proposed DIAG method converges linearly to the optimal solution, but also its linear convergence factor justifies the advantage of incremental methods on GD. In particular, we prove that the worst case performance of DIAG is better than the worst case performance of GD.
The other alternative for solving the optimization problem in is the Incremental Aggregated Gradient (IAG) method which is a middle ground between GD and IGD. The IAG method requires one gradient evaluation per iteration, as in IG, while it approximates the gradient of the global objective function @math by the average of the most recent gradient of all instantaneous functions @cite_26 , and it has a linear convergence rate, as in GD. In the IAG method, the functions are chosen in a cyclic order and it takes @math iterations to have a pass over all the available functions. To introduce the update of IAG, recall the definition of @math as the copy of the decision variable @math for the last time that the function @math 's gradient is updated before step @math which can be updated as in . Then, the update of IAG is given by which is identical to the update of SAG in , and the only difference is in the scheme that the index @math is chosen.
@cite_26
2769394111 1494085563 2963487351 1987083649
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. 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. 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. While these methods are widely used in modern nonconvex applications, including training of deep neural networks, whether they are provably superior to GD in the nonconvex setting remains open. This paper studies a simple variant of Nesterov's AGD, and shows that it escapes saddle points and finds a second-order stationary point in ( O (1 ^ 7 4 ) ) iterations, matching the best known convergence rate, which is faster than the ( O (1 ^ 2 ) ) iterations required by GD. To the best of our knowledge, this is the first direct acceleration (single-loop) algorithm that is provably faster than GD in general nonconvex setting---all previous nonconvex accelerated algorithms rely on more complex mechanisms such as nested loops and proximal terms. 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 on each 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. In this paper, we generalize the well-known Nesterov's accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first-order information, similarly to the gradient descent method. We then consider an important class of composite optimization problems and show that the AG method can solve them uniformly, i.e., by using the same aggressive stepsize policy as in the convex case, even if the problem turns out to be nonconvex. We demonstrate that the AG method exhibits an optimal rate of convergence if the composite problem is convex, and improves the best known rate of convergence if the problem is nonconvex. Based on the AG method, we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization. To the best of our knowledge, this is the first time that the convergence of the AG method has been established for solving nonconvex nonlinear programming in the literature.