Gradient descent - AbsoluteAstronomy.com

Gradient descent is a first-order optimization algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

(or of the approximate gradient) of the function at the current point. If instead one takes steps proportional to the positive of the gradient

Gradient

, one approaches a local maximum of that function; the procedure is then known as gradient ascent.

Gradient descent is also known as steepest descent, or the method of steepest descent. When known as the latter, gradient descent should not be confused with the method of steepest descent for approximating integrals.

Description

Gradient descent is based on the observation that if the multivariable function

is defined and differentiable

Differentiable function

In calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain...

in a neighborhood of a point

, then

decreases fastest if one goes from

in the direction of the negative gradient of

. It follows that, if

for

a small enough number, then

. With this observation in mind, one starts with a guess

for a local minimum of

, and considers the sequence

such that

We have

so hopefully the sequence

converges to the desired local minimum. Note that the value of the step size

is allowed to change at every iteration.

This process is illustrated in the picture to the right. Here

is assumed to be defined on the plane, and that its graph has a bowl

Bowl (vessel)

A bowl is a common open-top container used in many cultures to serve food, and is also used for drinking and storing other items. They are typically small and shallow, although some, such as punch bowls and salad bowls, are larger and often intended to serve many people.Bowls have existed for...

shape. The blue curves are the contour line

Contour line

A contour line of a function of two variables is a curve along which the function has a constant value. In cartography, a contour line joins points of equal elevation above a given level, such as mean sea level...

s, that is, the regions on which the value of

is constant. A red arrow originating at a point shows the direction of the negative gradient at that point. Note that the (negative) gradient at a point is orthogonal to the contour line going through that point. We see that gradient descent leads us to the bottom of the bowl, that is, to the point where the value of the function

is minimal.

Examples

Gradient descent has problems with pathological functions such as the Rosenbrock function

Rosenbrock function

In mathematical optimization, the Rosenbrock function is a non-convex function used as a performance test problem for optimization algorithms introduced by Howard H. Rosenbrock in 1960. It is also known as Rosenbrock's valley or Rosenbrock's banana function.The global minimum is inside a long,...

shown here.

The Rosenbrock function has a narrow curved valley which contains the minimum. The bottom of the valley is very flat. Because of the curved flat valley the optimization is zig-zagging slowly with small stepsizes towards the minimum.

The "Zig-Zagging" nature of the method is also evident below, where the gradient ascent method is applied to

Limitations

For some of the above examples, gradient descent is relatively slow close to the minimum: Technically, its asymptotic rate of convergence is inferior to other methods. For poorly conditioned convex problems, gradient descent increasingly 'zigzags' as the gradients point nearly orthogonally to the shortest direction to a minimum point. For more details, see Comments below:

For non-differentiable functions, gradient methods are ill-defined. For locally Lipschitz

Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...

problems and especially for convex minimization problems, bundle methods of descent

Subgradient method

Subgradient methods are iterative methods for solving convex minimization problems. Originally developed by Naum Z. Shor and others in the 1960s and 1970s, subgradient methods are convergent when applied even to a non-differentiable objective function...

are well-defined. Non-descent methods, like subgradient projection methods, may also be used.

Solution of a linear system

Gradient descent can be used to solve a system of linear equations, reformulated as a quadratic minimization problem, e.g., using linear least squares

Linear least squares

In statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model...

. Solution of

in the sense of linear least squares is defined as minimizing the function

In traditional linear least squares for real

and

the Euclidean norm is used, in which case

In the case that

is real, square, symmetric and positive definite, a different popular choice of the norm is

which produces a different equation with a better condition number

Condition number

In the field of numerical analysis, the condition number of a function with respect to an argument measures the asymptotically worst case of how much the function can change in proportion to small changes in the argument...

In either case, the line search minimization, finding the locally optimal step size

on every iteration, can be performed analytically, and explicit formulas for the locally optimal

are known. https://hpcrd.lbl.gov/~meza/papers/Steepest%20Descent.pdf

Solution of a non-linear system

Gradient descent can also be used to solve a system of nonlinear equations. Below is an example that shows how to use the gradient descent to solve for three unknown variables, x₁, x₂, and x₃. This example shows one iteration of the gradient descent.

Consider a nonlinear system of equations:

suppose we have the function

where

and the objective function

With initial guess

We know that

where

The Jacobian matrix

Then evaluating these terms at

and

So that

and

Now a suitable

must be found such that

. This can be done with any of a variety of line search algorithms. One might also simply guess

which gives

evaluating at this value,

The decrease from

to the next step's value of

is a sizable decrease in the objective function. Further steps would reduce its value until a solution to the system was found.

Comments

Gradient descent works in spaces of any number of dimensions, even in infinite-dimensional ones. In the latter case the search space is typically a function space

Function space

In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

, and one calculates the Gâteaux derivative

Gâteaux derivative

In mathematics, the Gâteaux differential or Gâteaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gâteaux, a French mathematician who died young in World War I, it is defined for functions between locally convex topological vector...

of the functional to be minimized to determine the descent direction.

The gradient descent can take many iterations to compute a local minimum with a required accuracy, if the curvature

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

in different directions is very different for the given function. For such functions, preconditioning, which changes the geometry of the space to shape the function level sets like concentric circles, cures the slow convergence. Constructing and applying preconditioning can be computationally expensive, however.

The gradient descent can be combined with a line search, finding the locally optimal step size

on every iteration. Performing the line search can be time-consuming. Conversely, using a fixed small

can yield poor convergence.

Methods based on Newton's method

Newton's method in optimization

In mathematics, Newton's method is an iterative method for finding roots of equations. More generally, Newton's method is used to find critical points of differentiable functions, which are the zeros of the derivative function.-Method:...

and inversion of the Hessian

Hessian matrix

In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...

using conjugate gradient techniques can be better alternatives. Generally, such methods converge in fewer iterations, but the cost of each iteration is higher. An example is the BFGS method

BFGS method

In numerical optimization, the Broyden–Fletcher–Goldfarb–Shanno method is a method for solving nonlinear optimization problems ....

which consists in calculating on every step a matrix by which the gradient vector is multiplied to go into a "better" direction, combined with a more sophisticated line search algorithm, to find the "best" value of

For extremely large problems, where the computer memory issues dominate, a limited-memory method such as L-BFGS should be used instead of BFGS or the steepest descent.

Gradient descent can be viewed as Euler's method for solving ordinary differential equations

of a gradient flow.

A computational example

The gradient descent algorithm is applied to find a local minimum of the function f(x)=x⁴−3x³+2, with derivative f'(x)=4x³−9x². Here is an implementation in the Python programming language.

From calculation, we expect that the local minimum occurs at x=9/4

x_old = 0
x_new = 6 # The algorithm starts at x=6
eps = 0.01 # step size
precision = 0.00001

def f_prime(x):
return 4 * x**3 - 9 * x**2

while abs(x_new - x_old) > precision:
x_old = x_new
x_new = x_old - eps * f_prime(x_new)
print "Local minimum occurs at ", x_new

The above piece of code has to be modified with regard to step size according to the system at hand and convergence can be made faster by using an adaptive step size. In the above case the step size is not adaptive. It stays at 0.01 in all the directions which can sometimes cause the method to fail by diverging from the minimum.