Gradient descent

For the analytical method called "steepest descent", see Method of steepest descent.

Gradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or of the approximate gradient) of the function at the current point. If instead one takes steps proportional to the positive of the 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

Illustration of gradient descent

Gradient descent is based on the observation that if the multivariable function $F(\mathbf{x})$ is defined and differentiable in a neighborhood of a point $\mathbf{a}$, then $F(\mathbf{x})$ decreases fastest if one goes from $\mathbf{a}$ in the direction of the negative gradient of F at $\mathbf{a}$, $-\nabla F(\mathbf{a})$. It follows that, if

$\mathbf{b} = \mathbf{a}-\gamma\nabla F(\mathbf{a})$

for γ > 0 a small enough number, then $F(\mathbf{a})\geq F(\mathbf{b})$. With this observation in mind, one starts with a guess $\mathbf{x}_0$ for a local minimum of F, and considers the sequence $\mathbf{x}_0, \mathbf{x}_1, \mathbf{x}_2, \dots$ such that

$\mathbf{x}_{n+1}=\mathbf{x}_n-\gamma_n \nabla F(\mathbf{x}_n),\ n \ge 0.$

We have

$F(\mathbf{x}_0)\ge F(\mathbf{x}_1)\ge F(\mathbf{x}_2)\ge \cdots,$

so hopefully the sequence $(\mathbf{x}_n)$ converges to the desired local minimum. Note that the value of the step size γ is allowed to change at every iteration.

This process is illustrated in the picture to the right. Here F is assumed to be defined on the plane, and that its graph has a bowl shape. The blue curves are the contour lines, that is, the regions on which the value of F 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 F is minimal.

Examples

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

$f(x_1, x_2) = (1-x_1)^2 + 100(x_2-x_1^2)^2 .\quad$

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 $F(x,y)=\sin\left(\frac{1}{2} x^2 - \frac{1}{4} y^2 + 3 \right) \cos(2 x+1-e^y)$.

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

$A\mathbf{x}-\mathbf{b}=0$

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

$F(x)=\|A\mathbf{x}-\mathbf{b}\|^2.$

In traditional linear least squares for real A and $\mathbf{b}$ the Euclidean norm is used, in which case

$\nabla F(\mathbf{x})=2A^T(A\mathbf{x}-\mathbf{b}).$

In the case that A is real, square, symmetric and positive definite, a different popular choice of the norm is

$\|\mathbf{a}\|^2=\mathbf{a}^TA^{-1}\mathbf{a}$

which produces a different equation with a better condition number:

$\nabla F(\mathbf{x})=2(A\mathbf{x}-\mathbf{b}).$

In either case, the line search minimization, finding the locally optimal step size γ on every iteration, can be performed analytically, and explicit formulas for the locally optimal γ are known. [1]

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:

$\begin{cases} 3x_1-\cos(x_2x_3)-\tfrac{3}{2}=0 \\ 4x_1^2-625x_2^2+2x_2-1=0 \\ \exp(-x_1x_2)+20x_3+\tfrac{10\pi-3}{3}=0 \end{cases}$

suppose we have the function

$G(\mathbf{x}) = \begin{bmatrix} 3x_1-\cos(x_2x_3)-\tfrac{3}{2} \\ 4x_1^2-625x_2^2+2x_2-1 \\ \exp(-x_1x_2)+20x_3+\tfrac{10\pi-3}{3} \\ \end{bmatrix}$

where

$\mathbf{x} =\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \end{bmatrix}$

and the objective function

$F(\mathbf{x}) = \tfrac{1}{2} G^\mathrm{T}(\mathbf{x}) G(\mathbf{x})$

$=\tfrac{1}{2}((3x_1-\cos(x_2x_3)-\tfrac{3}{2})^2 + (4x_1^2-625x_2^2+2x_2-1)^2 + (\exp(-x_1x_2)+20x_3+\tfrac{10\pi-3}{3})^2)$

With initial guess

$\mathbf{x}^{(0)}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \end{bmatrix} =\begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix}$

We know that

$\mathbf{x}^{(1)}=\mathbf{x}^{(0)}-\gamma_0 \nabla F(x^{(0)})$

where

$\nabla F(\mathbf{x}^{(0)})= J_G(\mathbf{x}^{(0)})^\mathrm{T}*G(\mathbf{x}^{(0)})$

The Jacobian matrix $J_G(\mathbf{x}^{(0)})$

$J_G = \begin{bmatrix} 3 & \sin(x_2x_3)x_3 & \sin(x_2x_3)x_2 \\ 8x_1 & -1250x_2+2 & 0 \\ -x_2\exp{(-x_1x_2)} & -x_1\exp(-x_1x_2) & 20\\ \end{bmatrix}$

Then evaluating these terms at $\mathbf{x}^{(0)}$

$J_G \left(\mathbf{x}^{(0)}\right) = \begin{bmatrix} 3 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 20 \end{bmatrix}$

and

$G(\mathbf{x}^{(0)}) = \begin{bmatrix} -2.5\\ -1\\ 10.472 \end{bmatrix}$

So that

$\mathbf{x}^{(1)}=0-\gamma_0 \begin{bmatrix} -7.5\\ -2\\ 209.44 \end{bmatrix}.$

and

$F \left(\mathbf{x}^{(0)}\right) = 0.5((-2.5)^2 + (-1)^2 + (10.472)^2) = 58.456$

An animation showing the first 83 iterations of gradient decent applied to this example. Surfaces are isosurfaces of $F(\mathbf{x}^{(n)})$ at current guess $\mathbf{x}^{(n)}$, and arrows show the direction of descent. Due to a small and constant step size, the convergence is slow.

Now a suitable γ₀ must be found such that $F(\mathbf{x}^{(1)}) \le F(\mathbf{x}^{(0)})$. This can be done with any of a variety of line search algorithms. One might also simply guess γ₀ = 0.001 which gives

$\mathbf{x}^{(1)}=\begin{bmatrix} 0.0075 \\ 0.002 \\ -0.20944 \\ \end{bmatrix}$

evaluating at this value,

$F \left(\mathbf{x}^{(1)}\right) = 0.5((-2.48)^2 + (-1.00)^2 + (6.28)^2) = 23.306$

The decrease from $F(\mathbf{x}^{(0)})=58.456$ to the next step's value of $F(\mathbf{x}^{(1)})=23.306$ 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, and one calculates the Gâteaux derivative 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 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 γ on every iteration. Performing the line search can be time-consuming. Conversely, using a fixed small γ can yield poor convergence.

Methods based on Newton's method and inversion of the Hessian 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 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 γ. 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 $x'(t)=-\nabla f(x(t))$ 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 scripting 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.

See also

• Stochastic gradient descent
• Rprop
• Delta rule
• Wolfe conditions
• Preconditioning
• Nelder–Mead method

References

• Mordecai Avriel (2003). Nonlinear Programming: Analysis and Methods. Dover Publishing. ISBN 0-486-43227-0.
• Jan A. Snyman (2005). Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms. Springer Publishing. ISBN 0-387-24348-8
• Cauchy, Augustin (1847). Méthode générale pour la résolution des systèmes d'équations simultanées. pp. 536-538. http://gallica.bnf.fr/ark:/12148/bpt6k2982c.image.f540.pagination.langEN.