This chapter describes methods based on gradient information that achieve faster rates than basic algorithms such as those described in Chapter 3. These accelerated gradient methods, most notably the heavy-ball method and Nesterov’s optimal method, use the concept of momentum which means that each step combines information from recent gradient values but also earlier steps. These methods are described and analyzed using an analysis based on Lyapunov functions. The cases of convex and strongly convex functions are analyzed separately. We motivate these methods using continuous-time limits, which link gradient methods to dynamical systems described by differential equations. We mention also the conjugate gradient method, which was developed separately from the other method but which also makes use of momentum. Finally, we discuss the concept of lower bounds on algorithmic complexity, introducing a function on which no method based on gradients can attain convergence faster than a certain given rate.

Here, we describe methods for minimizing a smooth function over a closed convex set, using gradient information. We first state results that characterize optimality of points in a way that can be checked, and describe the vital operation of projection onto the feasible set. We next describe the projected gradient algorithm, which is in a sense the extension of the steepest-descent method to the constrained case, analyze its convergence, and describe several extensions. We next analyze the conditional-gradient method (also known as “Frank-Wolfe”) for the case in which the feasible set is compact and demonstrate sublinear convergence of this approach when the objective function is convex.

Here, we discuss concepts of duality for convex optimization problems, and algorithms that make use of these concepts. We define the Lagrangian function and its augmented Lagrangian counterpart. We use the Lagrangian to derive optimality conditions for constrained optimization problems in which the constraints are expressed as linear algebraic conditions. We introduce the dual problem, and discuss the concepts of weak and strong duality, and show the existence of positive duality gaps in certain settings. Next, we discuss the dual subgradient method, the augmented Lagrangian method, and the alternating direction method of multipliers (ADMM), which are useful for several types of data science problems.

In this introductory chapter, we outline the ways in which various problems in data analysis can be formulated as optimization problems. Specifically, we discuss least squares problems, problems in matrix optimization (particularly those involving low-rank matrices), linear and kernel support vector machines, binary and multiclass logistic regression, and deep learning. We also outline the scope of the remainder of the book.

We describe the stochastic gradient method, the fundamental algorithm for several important problems in data science, including deep learning. We give several example problems for which this method is suitable, then described its operation for the simple problem of computing a mean of a collection of values. We related it to a classical method, the Kaczmarz method for solving a system of linear equalities and inequalities. Next, we describe the key assumptions to be used in convergence analysis, then describe the convergence rates attainable by several variants of stochastic gradient under several scenarios. Finally, we discuss several aspects of practical implementation of stochastic gradient, including minibatching and acceleration.

We outline theoretical foundations for smooth optimization problems. First, we define the different types of minimizers (solutions) of unconstrained optimization problems. Next, we state Taylor’s theorem, the fundamental theorem of smooth optimization, which allows us to approximate general smooth functions by simpler (linear or quadratic) functions based on information at the current point. We show how minima can be characterized by optimality conditions involving the gradient or Hessian, which can be checked in practice. Finally, we define the convexity of sets and functions, an important property that arises often in practice and that can be exploited by the algorithms described in the remainder of the book.

This chapter describes the coordinate descent approach, in which a single variable (or a block of variables) is updated at each iteration, usually based on partial derivative information for those variables, while the remainder are left unchanged. We describe two problems in machine learning for which this approach has potential advantages relative to the approaches described in previous chapters (which make use of the full gradient), and present convergence analyses for the randomized and cyclic versions of this approach. We show that convergence rates of block coordinate descent methods can be analyzed in a similar fashion to the basic single-component methods.

Here, we describe algorithms for minimizing nonsmooth functions and composite nonsmooth functions, which are the sum of a smooth function and a (usually elementary) nonsmooth function. We start with the subgradient descent method, whose search direction is the minimum-norm element of the subgradient. We then discuss the subgradient method, which steps along an arbitrary direction drawn from the subdifferential. Next, we describe proximal-gradient algorithms for nonsmooth composite optimization, which make use of the gradient of the smooth part of the function and the proximal operator associated with the nonsmooth part. Finally, we describe the proximal point method, a framework optimization that is valuable both as a fundamental method in its own right and as a building block for the augmented Lagrangian approach described in the next chapter.

First derivatives (gradients) are needed for most of the algorithms described in the book. Here, we describe how these gradients can be computed efficiently for functions that have the form of arising in deep learning. The reverse mode of automatic differentiation, often called “back-propagation” in the machine learning community, is described for several problems with nested-composite and progressive structure that arises in neural network training. We provide another perspective on these techniques, based on a constrained optimization formulation and optimality conditions for this formulation.

Here, we define subgradients and subdifferentials of nonsmooth functions. These are a generalization of the concept of gradients for smooth functions, that can be used as the basis of algorithms. We relate subgradients to directional derivatives and to the normal cones associated with convex sets. We introduce composite nonsmooth functions that arise in regularized optimization formulations of data analysis problems and describe optimality conditions for minimizers of these functions. Finally, we describe proximal operators and the Moreau envelope, objects associated with nonsmooth functions that are the basis of algorithms for nonsmooth optimization described in the next chapter.

In this section, we discuss fundamental methods, mostly based on gradient information, that yield descent, that is, the function value decreases at each iteration. We start with the most basic method, the steepest-descent method, analyzing its convergence under different convexity/nonconvexity assumptions on the objective function. We then discuss more general descent methods, based on descent directions other than the negative gradient, showing conditions on the search direction and the steplength that allow convergence results to be proved. We also discuss a method that also makes use of Hessian information, showing that it can find a point satisfying approximate second-order optimality conditions and finding an upper bound on the number of iterations required to do so. We then discuss mirror descent, a class of gradient methods based on more general distance metrics that are particularly useful in optimizing over the unit simplex – a problem that arises often in data science. We conclude by discussing the PL condition, a generalization of the strong convexity condition that allows linear convergence rates to be proved.