Gradient descent

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Gradient descent is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent.

Contents

Gradient descent is generally attributed to Cauchy, who first suggested it in 1847. [1] Hadamard independently proposed a similar method in 1907. [2] [3] Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, [4] with the method becoming increasingly well-studied and used in the following decades, [5] [6] also often called steepest descent.

Description

Illustration of gradient descent on a series of level sets Gradient descent.svg
Illustration of gradient descent on a series of level sets

Gradient descent is based on the observation that if the multi-variable function is defined and differentiable in a neighborhood of a point , then decreases fastest if one goes from in the direction of the negative gradient of at . It follows that, if

for a small enough, then . In other words, the term is subtracted from because we want to move against the gradient, toward the local minimum. With this observation in mind, one starts with a guess for a local minimum of , and considers the sequence such that

We have a monotonic sequence

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. With certain assumptions on the function (for example, convex and Lipschitz) and particular choices of (e.g., chosen either via a line search that satisfies the Wolfe conditions, or the Barzilai–Borwein method [7] [8] shown as following),

convergence to a local minimum can be guaranteed. When the function is convex, all local minima are also global minima, so in this case gradient descent can converge to the global solution.

This process is illustrated in the adjacent picture. Here 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 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.

An analogy for understanding gradient descent

Fog in the mountains Okanogan-Wenatchee National Forest, morning fog shrouds trees (37171636495).jpg
Fog in the mountains

The basic intuition behind gradient descent can be illustrated by a hypothetical scenario. A person is stuck in the mountains and is trying to get down (i.e., trying to find the global minimum). There is heavy fog such that visibility is extremely low. Therefore, the path down the mountain is not visible, so they must use local information to find the minimum. They can use the method of gradient descent, which involves looking at the steepness of the hill at their current position, then proceeding in the direction with the steepest descent (i.e., downhill). If they were trying to find the top of the mountain (i.e., the maximum), then they would proceed in the direction of steepest ascent (i.e., uphill). Using this method, they would eventually find their way down the mountain or possibly get stuck in some hole (i.e., local minimum or saddle point), like a mountain lake. However, assume also that the steepness of the hill is not immediately obvious with simple observation, but rather it requires a sophisticated instrument to measure, which the person happens to have at the moment. It takes quite some time to measure the steepness of the hill with the instrument, thus they should minimize their use of the instrument if they wanted to get down the mountain before sunset. The difficulty then is choosing the frequency at which they should measure the steepness of the hill so not to go off track.

In this analogy, the person represents the algorithm, and the path taken down the mountain represents the sequence of parameter settings that the algorithm will explore. The steepness of the hill represents the slope of the error surface at that point. The instrument used to measure steepness is differentiation (the slope of the error surface can be calculated by taking the derivative of the squared error function at that point). The direction they choose to travel in aligns with the gradient of the error surface at that point. The amount of time they travel before taking another measurement is the step size.

Examples

Gradient descent has problems with pathological functions such as the Rosenbrock function 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 zigzagging slowly with small step sizes towards the minimum.

Banana-SteepDesc.gif

The zigzagging nature of the method is also evident below, where the gradient descent method is applied to

Gradient ascent (contour).png Gradient ascent (surface).png

Choosing the step size and descent direction

Since using a step size that is too small would slow convergence, and a too large would lead to divergence, finding a good setting of is an important practical problem. Philip Wolfe also advocated for using "clever choices of the [descent] direction" in practice. [9] Whilst using a direction that deviates from the steepest descent direction may seem counter-intuitive, the idea is that the smaller slope may be compensated for by being sustained over a much longer distance.

To reason about this mathematically, let's use a direction and step size and consider the more general update:

.

Finding good settings of and requires a little thought. First of all, we would like the update direction to point downhill. Mathematically, letting denote the angle between and , this requires that To say more, we need more information about the objective function that we are optimising. Under the fairly weak assumption that is continuously differentiable, we may prove that: [10]

 

 

 

 

(1)

This inequality implies that the amount by which we can be sure the function is decreased depends on a trade off between the two terms in square brackets. The first term in square brackets measures the angle between the descent direction and the negative gradient. The second term measures how quickly the gradient changes along the descent direction.

In principle inequality ( 1 ) could be optimized over and to choose an optimal step size and direction. The problem is that evaluating the second term in square brackets requires evaluating , and extra gradient evaluations are generally expensive and undesirable. Some ways around this problem are:

Usually by following one of the recipes above, convergence to a local minimum can be guaranteed. When the function is convex, all local minima are also global minima, so in this case gradient descent can converge to the global solution.

Solution of a linear system

The steepest descent algorithm applied to the Wiener filter Steepest descent.png
The steepest descent algorithm applied to the Wiener filter

Gradient descent can be used to solve a system of linear equations

reformulated as a quadratic minimization problem. If the system matrix is real symmetric and positive-definite, the quadratic function to minimize commonly is

so that

For a general real matrix , linear least squares define

In traditional linear least squares for real and the Euclidean norm is used, in which case

The line search minimization, finding the locally optimal step size on every iteration, can be performed analytically for quadratic functions, and explicit formulas for the locally optimal are known. [5] [12]

For example, for real symmetric and positive-definite matrix , a simple algorithm can be as follows, [5]

To avoid multiplying by twice per iteration, we note that implies , which gives the traditional algorithm, [13]

The method is rarely used for solving linear equations, with the conjugate gradient method being one of the most popular alternatives. The number of gradient descent iterations is commonly proportional to the spectral condition number of the system matrix (the ratio of the maximum to minimum eigenvalues of ), while the convergence of conjugate gradient method is typically determined by a square root of the condition number, i.e., is much faster. Both methods can benefit from preconditioning, where gradient descent may require less assumptions on the preconditioner. [13]

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 the nonlinear system of equations

Let us introduce the associated function

where

One might now define the objective function

which we will attempt to minimize. As an initial guess, let us use

We know that

where the Jacobian matrix is given by

We calculate:

Thus

and

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

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 the objective function at this value, yields

The decrease from to the next step's value of

is a sizable decrease in the objective function. Further steps would reduce its value further, until an approximate 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 Fréchet derivative of the functional to be minimized to determine the descent direction. [6]

That gradient descent works in any number of dimensions (finite number at least) can be seen as a consequence of the Cauchy-Schwarz inequality. That article proves that the magnitude of the inner (dot) product of two vectors of any dimension is maximized when they are colinear. In the case of gradient descent, that would be when the vector of independent variable adjustments is proportional to the gradient vector of partial derivatives.

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 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 and inversion of the Hessian using conjugate gradient techniques can be better alternatives. [14] [15] 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 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 applying Euler's method for solving ordinary differential equations to a gradient flow. In turn, this equation may be derived as an optimal controller [16] for the control system with given in feedback form .

Modifications

Gradient descent can converge to a local minimum and slow down in a neighborhood of a saddle point. Even for unconstrained quadratic minimization, gradient descent develops a zig-zag pattern of subsequent iterates as iterations progress, resulting in slow convergence. Multiple modifications of gradient descent have been proposed to address these deficiencies.

Fast gradient methods

Yurii Nesterov has proposed [17] a simple modification that enables faster convergence for convex problems and has been since further generalized. For unconstrained smooth problems the method is called the fast gradient method (FGM) or the accelerated gradient method (AGM). Specifically, if the differentiable function is convex and is Lipschitz, and it is not assumed that is strongly convex, then the error in the objective value generated at each step by the gradient descent method will be bounded by . Using the Nesterov acceleration technique, the error decreases at . [18] It is known that the rate for the decrease of the cost function is optimal for first-order optimization methods. Nevertheless, there is the opportunity to improve the algorithm by reducing the constant factor. The optimized gradient method (OGM) [19] reduces that constant by a factor of two and is an optimal first-order method for large-scale problems. [20]

For constrained or non-smooth problems, Nesterov's FGM is called the fast proximal gradient method (FPGM), an acceleration of the proximal gradient method.

Momentum or heavy ball method

Trying to break the zig-zag pattern of gradient descent, the momentum or heavy ball method uses a momentum term in analogy to a heavy ball sliding on the surface of values of the function being minimized, [5] or to mass movement in Newtonian dynamics through a viscous medium in a conservative force field. [21] Gradient descent with momentum remembers the solution update at each iteration, and determines the next update as a linear combination of the gradient and the previous update. For unconstrained quadratic minimization, a theoretical convergence rate bound of the heavy ball method is asymptotically the same as that for the optimal conjugate gradient method. [5]

This technique is used in Stochastic gradient descent#Momentum and as an extension to the backpropagation algorithms used to train artificial neural networks. [22] [23]

Extensions

Gradient descent can be extended to handle constraints by including a projection onto the set of constraints. This method is only feasible when the projection is efficiently computable on a computer. Under suitable assumptions, this method converges. This method is a specific case of the forward-backward algorithm for monotone inclusions (which includes convex programming and variational inequalities). [24]

See also

Related Research Articles

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. 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 after him. Hesse originally used the term "functional determinants".

Gauss–Newton algorithm

The Gauss–Newton algorithm is used to solve non-linear least squares problems. It is a modification of Newton's method for finding a minimum of a function. Unlike Newton's method, the Gauss–Newton algorithm can only be used to minimize a sum of squared function values, but it has the advantage that second derivatives, which can be challenging to compute, are not required.

Stochastic gradient descent is an iterative method for optimizing an objective function with suitable smoothness properties. It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient by an estimate thereof. Especially in high-dimensional optimization problems this reduces the computational burden, achieving faster iterations in trade for a lower convergence rate.

Newtons method in optimization Method for finding stationary points of a function

In calculus, Newton's method is an iterative method for finding the roots of a differentiable function F, which are solutions to the equation F (x) = 0. As such, Newton's method can be applied to the derivative f of a twice-differentiable function f to find the roots of the derivative, also known as the Critical point (mathematics)s of f. These solutions may be minima, maxima, or saddle points, see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article. This is relevant in optimization, which aims to find (global) minima of the function f.

Conjugate gradient method

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems.

In optimization, the line search strategy is one of two basic iterative approaches to find a local minimum of an objective function . The other approach is trust region.

In (unconstrained) minimization, a backtracking line search, a search scheme based on the Armijo–Goldstein condition, is a line search method to determine the amount to move along a given search direction. It involves starting with a relatively large estimate of the step size for movement along the search direction, and iteratively shrinking the step size until a decrease of the objective function is observed that adequately corresponds to the decrease that is expected, based on the local gradient of the objective function.

In the unconstrained minimization problem, the Wolfe conditions are a set of inequalities for performing inexact line search, especially in quasi-Newton methods, first published by Philip Wolfe in 1969.

In numerical optimization, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is an iterative method for solving unconstrained nonlinear optimization problems. Like the related Davidon–Fletcher–Powell method, BFGS determines the descent direction by preconditioning the gradient with curvature information. It does so by gradually improving an approximation to the Hessian matrix of the loss function, obtained only from gradient evaluations via a generalized secant method.

Nelder–Mead method

The Nelder–Mead method is a commonly applied numerical method used to find the minimum or maximum of an objective function in a multidimensional space. It is a direct search method and is often applied to nonlinear optimization problems for which derivatives may not be known. However, the Nelder–Mead technique is a heuristic search method that can converge to non-stationary points on problems that can be solved by alternative methods.

The Frank–Wolfe algorithm is an iterative first-order optimization algorithm for constrained convex optimization. Also known as the conditional gradient method, reduced gradient algorithm and the convex combination algorithm, the method was originally proposed by Marguerite Frank and Philip Wolfe in 1956. In each iteration, the Frank–Wolfe algorithm considers a linear approximation of the objective function, and moves towards a minimizer of this linear function.

In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducing a condition number of the problem. The preconditioned problem is then usually solved by an iterative method.

Limited-memory BFGS is an optimization algorithm in the family of quasi-Newton methods that approximates the Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS) using a limited amount of computer memory. It is a popular algorithm for parameter estimation in machine learning. The algorithm's target problem is to minimize over unconstrained values of the real-vector where is a differentiable scalar function.

In numerical optimization, the nonlinear conjugate gradient method generalizes the conjugate gradient method to nonlinear optimization. For a quadratic function

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. When the objective function is differentiable, sub-gradient methods for unconstrained problems use the same search direction as the method of steepest descent.

Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. There are many similarities to linear least squares, but also some significant differences. In economic theory, the non-linear least squares method is applied in (i) the probit regression, (ii) threshold regression, (iii) smooth regression, (iv) logistic link regression, (v) Box-Cox transformed regressors.

Online machine learning Method of machine learning

In computer science, online machine learning is a method of machine learning in which data becomes available in a sequential order and is used to update the best predictor for future data at each step, as opposed to batch learning techniques which generate the best predictor by learning on the entire training data set at once. Online learning is a common technique used in areas of machine learning where it is computationally infeasible to train over the entire dataset, requiring the need of out-of-core algorithms. It is also used in situations where it is necessary for the algorithm to dynamically adapt to new patterns in the data, or when the data itself is generated as a function of time, e.g., stock price prediction. Online learning algorithms may be prone to catastrophic interference, a problem that can be addressed by incremental learning approaches.

Gradient boosting Machine learning technique

Gradient boosting is a machine learning technique for regression and classification problems, which produces a prediction model in the form of an ensemble of weak prediction models, typically decision trees. When a decision tree is the weak learner, the resulting algorithm is called gradient boosted trees, which usually outperforms random forest. It builds the model in a stage-wise fashion like other boosting methods do, and it generalizes them by allowing optimization of an arbitrary differentiable loss function.

Coordinate descent is an optimization algorithm that successively minimizes along coordinate directions to find the minimum of a function. At each iteration, the algorithm determines a coordinate or coordinate block via a coordinate selection rule, then exactly or inexactly minimizes over the corresponding coordinate hyperplane while fixing all other coordinates or coordinate blocks. A line search along the coordinate direction can be performed at the current iterate to determine the appropriate step size. Coordinate descent is applicable in both differentiable and derivative-free contexts.

In mathematical optimization, oracle complexity is a standard theoretical framework to study the computational requirements for solving classes of optimization problems. It is suitable for analyzing iterative algorithms which proceed by computing local information about the objective function at various points. The framework has been used to provide tight worst-case guarantees on the number of required iterations, for several important classes of optimization problems.

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Further reading