Gradient descent (also often called steepest descent) is a first-order

_{1}, ''x''_{2}, and ''x''_{3}. This example shows one iteration of the gradient descent.
Consider the nonlinear system of equations
:$\backslash begin\; 3x\_1-\backslash cos(x\_2x\_3)-\backslash tfrac\; =0\; \backslash \backslash \; 4x\_1^2-625x\_2^2+2x\_2-1\; =\; 0\; \backslash \backslash \; \backslash exp(-x\_1x\_2)+20x\_3+\backslash tfrac\; =0\; \backslash end$
Let us introduce the associated function
:$G(\backslash mathbf)\; =\; \backslash begin\; 3x\_1-\backslash cos(x\_2x\_3)-\backslash tfrac\; \backslash \backslash \; 4x\_1^2-625x\_2^2+2x\_2-1\; \backslash \backslash \; \backslash exp(-x\_1x\_2)+20x\_3+\backslash tfrac\; \backslash \backslash \; \backslash end,$
where
:$\backslash mathbf\; =\backslash begin\; x\_1\; \backslash \backslash \; x\_2\; \backslash \backslash \; x\_3\; \backslash \backslash \; \backslash end.$
One might now define the objective function
:$\backslash beginF(\backslash mathbf)\; \&=\; \backslash frac\; G^\backslash mathrm(\backslash mathbf)\; G(\backslash mathbf)\; \backslash \backslash \&=\backslash frac\; \backslash left[\; \backslash left\; (3x\_1-\backslash cos(x\_2x\_3)-\backslash frac\; \backslash right)^2\; +\; \backslash left(4x\_1^2-625x\_2^2+2x\_2-1\; \backslash right)^2\; +\backslash right.\backslash \backslash \; \&\backslash qquad\backslash left.\; \backslash left(\backslash exp(-x\_1x\_2)\; +\; 20x\_3\; +\; \backslash frac\; \backslash right)^2\; \backslash right],\backslash end$
which we will attempt to minimize. As an initial guess, let us use
:$\backslash mathbf^=\; \backslash mathbf\; =\; \backslash begin\; 0\; \backslash \backslash \; 0\; \backslash \backslash \; 0\; \backslash \backslash \; \backslash end.$
We know that
:$\backslash mathbf^=\backslash mathbf-\backslash gamma\_0\; \backslash nabla\; F(\backslash mathbf)\; =\; \backslash mathbf-\backslash gamma\_0\; J\_G(\backslash mathbf)^\backslash mathrm\; G(\backslash mathbf),$
where the Jacobian matrix $J\_G$ is given by
:$J\_G(\backslash mathbf)\; =\; \backslash begin\; 3\; \&\; \backslash sin(x\_2x\_3)x\_3\; \&\; \backslash sin(x\_2x\_3)x\_2\; \backslash \backslash \; 8x\_1\; \&\; -1250x\_2+2\; \&\; 0\; \backslash \backslash \; -x\_2\backslash exp\; \&\; -x\_1\backslash exp(-x\_1x\_2)\; \&\; 20\backslash \backslash \; \backslash end.$
We calculate:
:$J\_G(\backslash mathbf)\; =\; \backslash begin\; 3\; \&\; 0\; \&\; 0\backslash \backslash \; 0\; \&\; 2\; \&\; 0\backslash \backslash \; 0\; \&\; 0\; \&\; 20\; \backslash end,\; \backslash qquad\; G(\backslash mathbf)\; =\; \backslash begin\; -2.5\backslash \backslash \; -1\backslash \backslash \; 10.472\; \backslash end.$
Thus
:$\backslash mathbf^=\; \backslash mathbf-\backslash gamma\_0\; \backslash begin\; -7.5\backslash \backslash \; -2\backslash \backslash \; 209.44\; \backslash end,$
and
:$F(\backslash mathbf)\; =\; 0.5\; \backslash left(\; (-2.5)^2\; +\; (-1)^2\; +\; (10.472)^2\; \backslash right)\; =\; 58.456.$
Now, a suitable $\backslash gamma\_0$ must be found such that
:$F\backslash left\; (\backslash mathbf^\backslash right\; )\; \backslash le\; F\backslash left\; (\backslash mathbf^\backslash right\; )\; =\; F(\backslash mathbf).$
This can be done with any of a variety of line search
In optimization (mathematics), optimization, the line search strategy is one of two basic iteration, iterative approaches to find a maxima and minima, local minimum \mathbf^* of an objective function f:\mathbb R^n\to\mathbb R. The other approach i ...

algorithms. One might also simply guess $\backslash gamma\_0=0.001,$ which gives
:$\backslash mathbf^=\backslash begin\; 0.0075\; \backslash \backslash \; 0.002\; \backslash \backslash \; -0.20944\; \backslash \backslash \; \backslash end.$
Evaluating the objective function at this value, yields
:$F\; \backslash left\; (\backslash mathbf^\backslash right\; )\; =\; 0.5\; \backslash left\; ((-2.48)^2\; +\; (-1.00)^2\; +\; (6.28)^2\; \backslash right\; )\; =\; 23.306.$
The decrease from $F(\backslash mathbf)=58.456$ to the next step's value of
:$F\backslash left\; (\backslash mathbf^\backslash right\; )\; =23.306$
is a sizable decrease in the objective function. Further steps would reduce its value further, until an approximate solution to the system was found.

line search
In optimization (mathematics), optimization, the line search strategy is one of two basic iteration, iterative approaches to find a maxima and minima, local minimum \mathbf^* of an objective function f:\mathbb R^n\to\mathbb R. The other approach i ...

, finding the locally optimal step size $\backslash gamma$ on every iteration. Performing the line search can be time-consuming. Conversely, using a fixed small $\backslash gamma$ can yield poor convergence.
Methods based on Newton's method in optimization, Newton's method and inversion of the Hessian matrix, 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 Broyden–Fletcher–Goldfarb–Shanno algorithm, 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
In optimization (mathematics), optimization, the line search strategy is one of two basic iteration, iterative approaches to find a maxima and minima, local minimum \mathbf^* of an objective function f:\mathbb R^n\to\mathbb R. The other approach i ...

algorithm, to find the "best" value of $\backslash gamma.$ For extremely large problems, where the computer-memory issues dominate, a limited-memory method such as Limited-memory BFGS, 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 $x\text{'}(t)=-\backslash nabla\; f(x(t))$ to a gradient flow. In turn, this equation may be derived as an optimal controller for the control system $x\text{'}(t)\; =\; u(t)$ with $u(t)$ given in feedback form $u(t)\; =\; -\backslash nabla\; f(x(t))$. However, don't let this analogy to mean that the convergence of the gradient descent in the discrete setting follows from results known for the gradient flow: while the gradient flow has very good convergence guarantees, the situation in the discrete setting is more complicated and one has to be careful in choosing a good learning rate. More generally, it is misleading to think that the good properties of some ODE will automatically guarantee good properties of a corresponding discrete version.

Using gradient descent in C++, Boost, Ublas for linear regression

Series of Khan Academy videos discusses gradient ascent

* Archived at [https://ghostarchive.org/varchive/youtube/20211211/IHZwWFHWa-w Ghostarchive] and th

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{{DEFAULTSORT:Gradient Descent Mathematical optimization First order methods Optimization algorithms and methods Gradient methods

iterative
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...

optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimization problems of sorts arise i ...

algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

for finding a local minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...

of a differentiable function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. The idea is to take repeated steps in the opposite direction of the gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

(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
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (math ...

of that function; the procedure is then known as gradient ascent.
Gradient descent is generally attributed to Cauchy
Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...

, who first suggested it in 1847. Hadamard
Jacques Salomon Hadamard ForMemRS (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis
of the function
.
Hue represents the argument, brightness the magnitud ...

independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry
Haskell Brooks Curry (; September 12, 1900 – September 1, 1982) was an American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
The United S ...

in 1944, with the method becoming increasingly well-studied and used in the following decades.
Description

Gradient descent is based on the observation that if the multi-variable function $F(\backslash mathbf)$ isdefined
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories, intensional definitions (which try to give the sense of a term) and extensional definitions ...

and differentiable
In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

in a neighborhood of a point $\backslash mathbf$, then $F(\backslash mathbf)$ decreases ''fastest'' if one goes from $\backslash mathbf$ in the direction of the negative gradient of $F$ at $\backslash mathbf,\; -\backslash nabla\; F(\backslash mathbf)$. It follows that, if
:$\backslash mathbf\_\; =\; \backslash mathbf\_n-\backslash gamma\backslash nabla\; F(\backslash mathbf\_n)$
for a small enough step size or learning rate
In machine learning and statistics, the learning rate is a Hyperparameter (machine learning), tuning parameter in an Mathematical optimization, optimization algorithm that determines the step size at each iteration while moving toward a minimum of ...

$\backslash gamma\; \backslash in\; \backslash R\_$, then $F(\backslash mathbf)\backslash geq\; F(\backslash mathbf)$. In other words, the term $\backslash gamma\backslash nabla\; F(\backslash mathbf)$ is subtracted from $\backslash mathbf$ because we want to move against the gradient, toward the local minimum. With this observation in mind, one starts with a guess $\backslash mathbf\_0$ for a local minimum of $F$, and considers the sequence $\backslash mathbf\_0,\; \backslash mathbf\_1,\; \backslash mathbf\_2,\; \backslash ldots$ such that
:$\backslash mathbf\_=\backslash mathbf\_n-\backslash gamma\_n\; \backslash nabla\; F(\backslash mathbf\_n),\backslash \; n\; \backslash ge\; 0.$
We have a monotonic
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

sequence
:$F(\backslash mathbf\_0)\backslash ge\; F(\backslash mathbf\_1)\backslash ge\; F(\backslash mathbf\_2)\backslash ge\; \backslash cdots,$
so, hopefully, the sequence $(\backslash mathbf\_n)$ converges to the desired local minimum. Note that the value of the ''step size'' $\backslash gamma$ is allowed to change at every iteration. With certain assumptions on the function $F$ (for example, $F$ convex
Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to:
Science and technology
* Convex lens
A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...

and $\backslash nabla\; F$ LipschitzLipschitz, Lipshitz, or Lipchitz is an Ashkenazi Jewish surname. The surname has many variants, including: Lifshitz (Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lipschütz), Lipshutz, Lüpschütz; ...

) and particular choices of $\backslash gamma$ (e.g., chosen either via a line search
In optimization (mathematics), optimization, the line search strategy is one of two basic iteration, iterative approaches to find a maxima and minima, local minimum \mathbf^* of an objective function f:\mathbb R^n\to\mathbb R. The other approach i ...

that satisfies the Wolfe conditions In the unconstrained optimization (mathematics), 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 (mathematician), Philip Wol ...

, or the Barzilai–Borwein method shown as following),
: $\backslash gamma\_\; =\; \backslash frac$
Convergent series, convergence to a local minimum can be guaranteed. When the function $F$ is convex
Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to:
Science and technology
* Convex lens
A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...

, 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, $F$ is assumed to be defined on the plane, and that its graph has a Bowl (vessel), 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.
An analogy for understanding gradient descent

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 (mathematics), 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 thegradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

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 (mathematics), pathological functions such as the Rosenbrock function shown here. :$f(x\_,\; x\_)\; =\; (\; 1\; -\; x\_\; )^2\; +\; 100\; (\; x\_\; -\; x\_^2\; )^2.$ The Rosenbrock function has a narrow-curved valley that 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. The Whiplash gradient descent, which is a closed-loop algorithm, successfully minimizes this function. The zigzagging nature of the method is also evident below, where the gradient ascent method is applied to :$F(x,y)=\backslash sin\backslash left(\backslash frac\; x^2\; -\; \backslash frac\; y^2\; +\; 3\; \backslash right)\backslash cos\backslash left(2\; x+1-e^y\backslash right).$Choosing the step size and descent direction

Since using a step size $\backslash gamma$ that is too small would slow convergence, and a $\backslash gamma$ too large would lead to divergence, finding a good setting of $\backslash gamma$ is an important practical problem. Philip Wolfe (mathematician), Philip Wolfe also advocated for using "clever choices of the [descent] direction" in practice. 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 $\backslash mathbf\_n$ and step size $\backslash gamma\_n$ and consider the more general update: :$\backslash mathbf\_\; =\; \backslash mathbf\_n-\backslash gamma\_n\backslash ,\backslash mathbf\_n$. Finding good settings of $\backslash mathbf\_n$ and $\backslash gamma\_n$ requires a little thought. First of all, we would like the update direction to point downhill. Mathematically, letting $\backslash theta\_n$ denote the angle between $\backslash nabla\; F(\backslash mathbf)$ and $\backslash mathbf\_n$, this requires that $\backslash cos\; \backslash theta\_n\; >\; 0.$ To say more, we need more information about the objective function that we are optimising. Under the fairly weak assumption that $F$ is continuously differentiable, we may prove that: This inequality implies that the amount by which we can be sure the function $F$ 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 () could be optimized over $\backslash mathbf\_n$ and $\backslash gamma\_n$ to choose an optimal step size and direction. The problem is that evaluating the second term in square brackets requires evaluating $\backslash nabla\; F(\backslash mathbf\_n\; -\; t\; \backslash gamma\_n\; \backslash mathbf\_n)$, and extra gradient evaluations are generally expensive and undesirable. Some ways around this problem are: * Forgo the benefits of a clever descent direction by setting $\backslash mathbf\_n\; =\; \backslash nabla\; F(\backslash mathbf)$, and useline search
In optimization (mathematics), optimization, the line search strategy is one of two basic iteration, iterative approaches to find a maxima and minima, local minimum \mathbf^* of an objective function f:\mathbb R^n\to\mathbb R. The other approach i ...

to find a suitable step-size $\backslash gamma\_n$, such as one that satisfies the Wolfe conditions In the unconstrained optimization (mathematics), 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 (mathematician), Philip Wol ...

. A more economic way Backtracking line search of choosing learning rates, has both good theoretical guarantees and experimental results. Note that one does not need to choose $\backslash mathbf\_n$ to be the gradient, but any direction which has positive intersection product with the gradient is OK.
* Assuming that $F$ is twice-differentiable, use its Hessian $\backslash nabla^2\; F$ to estimate $\backslash ,\; \backslash nabla\; F(\backslash mathbf\_n\; -\; t\; \backslash gamma\_n\; \backslash mathbf\_n)\; -\; \backslash nabla\; F(\backslash mathbf\_n)\backslash ,\; \_2\; \backslash approx\; \backslash ,\; t\; \backslash gamma\_n\; \backslash nabla^2\; F(\backslash mathbf\_n)\; \backslash mathbf\_n\backslash ,\; .$Then choose $\backslash mathbf\_n$ and $\backslash gamma\_n$ by optimising inequality ().
* Assuming that $\backslash nabla\; F$ is LipschitzLipschitz, Lipshitz, or Lipchitz is an Ashkenazi Jewish surname. The surname has many variants, including: Lifshitz (Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lipschütz), Lipshutz, Lüpschütz; ...

, use its Lipschitz constant $L$ to bound $\backslash ,\; \backslash nabla\; F(\backslash mathbf\_n\; -\; t\; \backslash gamma\_n\; \backslash mathbf\_n)\; -\; \backslash nabla\; F(\backslash mathbf\_n)\backslash ,\; \_2\; \backslash leq\; L\; t\; \backslash gamma\_n\; \backslash ,\; \backslash mathbf\_n\backslash ,\; .$ Then choose $\backslash mathbf\_n$ and $\backslash gamma\_n$ by optimising inequality ().
* Build a custom model of $\backslash max\_\; \backslash frac$ for $F$. Then choose $\backslash mathbf\_n$ and $\backslash gamma\_n$ by optimising inequality ().
* Under stronger assumptions on the function $F$ such as Convex function, convexity, more #Fast gradient methods, advanced techniques may be possible.
Usually by following one of the recipes above, convergent series, convergence to a local minimum can be guaranteed. When the function $F$ is convex
Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to:
Science and technology
* Convex lens
A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...

, all local minima are also global minima, so in this case gradient descent can converge to the global solution.
Solution of a linear system

Gradient descent can be used to solve a system of linear equations :$A\backslash mathbf-\backslash mathbf=0$ reformulated as a quadratic minimization problem. If the system matrix $A$ is real Symmetric matrix, symmetric and positive-definite matrix, positive-definite, an objective function is defined as the quadratic function, with minimization of :$F(\backslash mathbf)=\backslash mathbf^T\; A\backslash mathbf-2\backslash mathbf^T\backslash mathbf,$ so that :$\backslash nabla\; F(\backslash mathbf)=2(A\backslash mathbf-\backslash mathbf).$ For a general real matrix $A$, linear least squares define :$F(\backslash mathbf)=\backslash left\backslash ,\; A\backslash mathbf-\backslash mathbf\backslash right\backslash ,\; ^2.$ In traditional linear least squares for real $A$ and $\backslash mathbf$ the Euclidean norm is used, in which case :$\backslash nabla\; F(\backslash mathbf)=2A^T(A\backslash mathbf-\backslash mathbf).$ Theline search
In optimization (mathematics), optimization, the line search strategy is one of two basic iteration, iterative approaches to find a maxima and minima, local minimum \mathbf^* of an objective function f:\mathbb R^n\to\mathbb R. The other approach i ...

minimization, finding the locally optimal step size $\backslash gamma$ on every iteration, can be performed analytically for quadratic functions, and explicit formulas for the locally optimal $\backslash gamma$ are known.
For example, for real Symmetric matrix, symmetric and positive-definite matrix, positive-definite matrix $A$, a simple algorithm can be as follows,
:$\backslash begin\; \&\; \backslash text\; \backslash \backslash \; \&\; \backslash qquad\; \backslash mathbf\; :=\; \backslash mathbf\; -\; \backslash mathbf\; \backslash \backslash \; \&\; \backslash qquad\; \backslash gamma\; :=\; /\; \backslash \backslash \; \&\; \backslash qquad\; \backslash mathbf\; :=\; \backslash mathbf\; +\; \backslash gamma\; \backslash mathbf\; \backslash \backslash \; \&\; \backslash qquad\; \backslash hbox\; \backslash mathbf^\backslash mathsf\; \backslash mathbf\; \backslash text\; \backslash \backslash \; \&\; \backslash text\; \backslash \backslash \; \&\; \backslash text\; \backslash mathbf\; \backslash text\; \backslash end$
To avoid multiplying by $A$ twice per iteration,
we note that $\backslash mathbf\; :=\; \backslash mathbf\; +\; \backslash gamma\; \backslash mathbf$ implies $\backslash mathbf\; :=\; \backslash mathbf\; -\; \backslash gamma\; \backslash mathbf$, which gives the traditional algorithm,
:$\backslash begin\; \&\; \backslash mathbf\; :=\; \backslash mathbf\; -\; \backslash mathbf\; \backslash \backslash \; \&\; \backslash text\; \backslash \backslash \; \&\; \backslash qquad\; \backslash gamma\; :=\; /\; \backslash \backslash \; \&\; \backslash qquad\; \backslash mathbf\; :=\; \backslash mathbf\; +\; \backslash gamma\; \backslash mathbf\; \backslash \backslash \; \&\; \backslash qquad\; \backslash hbox\; \backslash mathbf^\backslash mathsf\; \backslash mathbf\; \backslash text\; \backslash \backslash \; \&\; \backslash qquad\; \backslash mathbf\; :=\; \backslash mathbf\; -\; \backslash gamma\; \backslash mathbf\; \backslash \backslash \; \&\; \backslash text\; \backslash \backslash \; \&\; \backslash text\; \backslash mathbf\; \backslash text\; \backslash end$
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 $\backslash kappa(A)$ of the system matrix $A$ (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 Preconditioner, preconditioning, where gradient descent may require less assumptions on the preconditioner.
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, ''x''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. 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 aModifications

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 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 $F$ is convex and $\backslash nabla\; F$ isLipschitzLipschitz, Lipshitz, or Lipchitz is an Ashkenazi Jewish surname. The surname has many variants, including: Lifshitz (Lifschitz), Lifshits, Lifshuts, Lefschetz; Lipschitz, Lipshitz, Lipshits, Lopshits, Lipschutz (Lipschütz), Lipshutz, Lüpschütz; ...

, and it is not assumed that $F$ is Convex function#Strongly convex functions, strongly convex, then the error in the objective value generated at each step $k$ by the gradient descent method will be Big O notation, bounded by $\backslash mathcal\backslash left(\backslash tfrac\backslash right)$. Using the Nesterov acceleration technique, the error decreases at $\backslash mathcal\backslash left(\backslash tfrac\backslash right)$. It is known that the rate $\backslash mathcal\backslash left(\backslash right)$ for the decrease of the loss function, 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) reduces that constant by a factor of two and is an optimal first-order method for large-scale problems.
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, or to mass movement in Newtonian dynamics through a viscous medium in a conservative force field. 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. This technique is used in Stochastic gradient descent#Momentum, stochastic gradient descent and as an extension to the backpropagation algorithms used to train artificial neural networks.Extensions

Gradient descent can be extended to handle Constraint (mathematics), constraints by including a Projection (linear algebra), 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, forward-backward algorithm for monotone inclusions (which includes convex programming and Variational inequality, variational inequalities).See also

* Backtracking line search * Conjugate gradient method * Stochastic gradient descent * Rprop * Delta rule *Wolfe conditions In the unconstrained optimization (mathematics), 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 (mathematician), Philip Wol ...

* Preconditioning
* Broyden–Fletcher–Goldfarb–Shanno algorithm
* Davidon–Fletcher–Powell formula
* Nelder–Mead method
* Gauss–Newton algorithm
* Hill climbing
* Quantum annealing
* TFNP#CLS, Continuous Local Search
References

Further reading

* * *External links

Using gradient descent in C++, Boost, Ublas for linear regression

Series of Khan Academy videos discusses gradient ascent

* Archived at [https://ghostarchive.org/varchive/youtube/20211211/IHZwWFHWa-w Ghostarchive] and th

Wayback Machine

{{DEFAULTSORT:Gradient Descent Mathematical optimization First order methods Optimization algorithms and methods Gradient methods