Non-Asymptotic Convergence Analysis of Inexact Gradient Methods for Machine Learning Without Strong Convexity

Many recent applications in machine learning and data fitting call for the algorithmic solution of structured smooth convex optimization problems. Although the gradient descent method is a natural choice for this task, it requires exact gradient computations and hence can be inefficient when the problem size is large or the gradient is difficult to evaluate. Therefore, there has been much interest in inexact gradient methods (IGMs), in which an efficiently computable approximate gradient is used to perform the update in each iteration. Currently, non-asymptotic linear convergence results for IGMs are typically established under the assumption that the objective function is strongly convex, which is not satisfied in many applications of interest; while linear convergence results that do not require the strong convexity assumption are usually asymptotic in nature. In this paper, we combine the best of these two types of results and establish---under the standard assumption that the gradient approximation errors decrease linearly to zero---the non-asymptotic linear convergence of IGMs when applied to a class of structured convex optimization problems. Such a class covers settings where the objective function is not necessarily strongly convex and includes the least squares and logistic regression problems. We believe that our techniques will find further applications in the non-asymptotic convergence analysis of other first-order methods

Recent Advances in Randomized Methods for Big Data Optimization

In this thesis, we discuss and develop randomized algorithms for big data problems. In particular, we study the finite-sum optimization with newly emerged variance- reduction optimization methods (Chapter 2), explore the efficiency of second-order information applied to both convex and non-convex finite-sum objectives (Chapter 3) and employ the fast first-order method in power system problems (Chapter 4).In Chapter 2, we propose two variance-reduced gradient algorithms – mS2GD and SARAH. mS2GD incorporates a mini-batching scheme for improving the theoretical complexity and practical performance of SVRG/S2GD, aiming to minimize a strongly convex function represented as the sum of an average of a large number of smooth con- vex functions and a simple non-smooth convex regularizer. While SARAH, short for StochAstic Recursive grAdient algoritHm and using a stochastic recursive gradient, targets at minimizing the average of a large number of smooth functions for both con- vex and non-convex cases. Both methods fall into the category of variance-reduction optimization, and obtain a total complexity of O((n+κ)log(1/ε)) to achieve an ε-accuracy solution for strongly convex objectives, while SARAH also maintains a sub-linear convergence for non-convex problems. Meanwhile, SARAH has a practical variant SARAH+ due to its linear convergence of the expected stochastic gradients in inner loops.In Chapter 3, we declare that randomized batches can be applied with second- order information, as to improve upon convergence in both theory and practice, with a framework of L-BFGS as a novel approach to finite-sum optimization problems. We provide theoretical analyses for both convex and non-convex objectives. Meanwhile, we propose LBFGS-F as a variant where Fisher information matrix is used instead of Hessian information, and prove it applicable to a distributed environment within the popular applications of least-square and cross-entropy losses.In Chapter 4, we develop fast randomized algorithms for solving polynomial optimization problems on the applications of alternating-current optimal power flows (ACOPF) in power system field. The traditional research on power system problem focuses on solvers using second-order method, while no randomized algorithms have been developed. First, we propose a coordinate-descent algorithm as an online solver, applied for solving time-varying optimization problems in power systems. We bound the difference between the current approximate optimal cost generated by our algorithm and the optimal cost for a relaxation using the most recent data from above by a function of the properties of the instance and the rate of change to the instance over time. Second, we focus on a steady-state problem in power systems, and study means of switching from solving a convex relaxation to Newton method working on a non-convex (augmented) Lagrangian of the problem

Conjugate gradient acceleration of iteratively re-weighted least squares methods

Iteratively Re-weighted Least Squares (IRLS) is a method for solving minimization problems involving non-quadratic cost functions, perhaps non-convex and non-smooth, which however can be described as the infimum over a family of quadratic functions. This transformation suggests an algorithmic scheme that solves a sequence of quadratic problems to be tackled efficiently by tools of numerical linear algebra. Its general scope and its usually simple implementation, transforming the initial non-convex and non-smooth minimization problem into a more familiar and easily solvable quadratic optimization problem, make it a versatile algorithm. However, despite its simplicity, versatility, and elegant analysis, the complexity of IRLS strongly depends on the way the solution of the successive quadratic optimizations is addressed. For the important special case of $\textit{compressed sensing}$ and sparse recovery problems in signal processing, we investigate theoretically and numerically how accurately one needs to solve the quadratic problems by means of the $\textit{conjugate gradient}$ (CG) method in each iteration in order to guarantee convergence. The use of the CG method may significantly speed-up the numerical solution of the quadratic subproblems, in particular, when fast matrix-vector multiplication (exploiting for instance the FFT) is available for the matrix involved. In addition, we study convergence rates. Our modified IRLS method outperforms state of the art first order methods such as Iterative Hard Thresholding (IHT) or Fast Iterative Soft-Thresholding Algorithm (FISTA) in many situations, especially in large dimensions. Moreover, IRLS is often able to recover sparse vectors from fewer measurements than required for IHT and FISTA.Comment: 40 page

Simba: A Scalable Bilevel Preconditioned Gradient Method for Fast Evasion of Flat Areas and Saddle Points

The convergence behaviour of first-order methods can be severely slowed down when applied to high-dimensional non-convex functions due to the presence of saddle points. If, additionally, the saddles are surrounded by large plateaus, it is highly likely that the first-order methods will converge to sub-optimal solutions. In machine learning applications, sub-optimal solutions mean poor generalization performance. They are also related to the issue of hyper-parameter tuning, since, in the pursuit of solutions that yield lower errors, a tremendous amount of time is required on selecting the hyper-parameters appropriately. A natural way to tackle the limitations of first-order methods is to employ the Hessian information. However, methods that incorporate the Hessian do not scale or, if they do, they are very slow for modern applications. Here, we propose Simba, a scalable preconditioned gradient method, to address the main limitations of the first-order methods. The method is very simple to implement. It maintains a single precondition matrix that it is constructed as the outer product of the moving average of the gradients. To significantly reduce the computational cost of forming and inverting the preconditioner, we draw links with the multilevel optimization methods. These links enables us to construct preconditioners in a randomized manner. Our numerical experiments verify the scalability of Simba as well as its efficacy near saddles and flat areas. Further, we demonstrate that Simba offers a satisfactory generalization performance on standard benchmark residual networks. We also analyze Simba and show its linear convergence rate for strongly convex functions