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This process is illustrated in the adjacent picture. Here, <math>f</math> is assumed to be defined on the plane, and that its graph has a [[Bowl (vessel)|bowl]] shape. The blue curves are the [[contour line]]s, that is, the regions on which the value of <math>f</math> 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 <math>f</math> is minimal.
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[[File:Okanogan-Wenatchee National Forest, morning fog shrouds trees (37171636495).jpg|thumb|Fog in the mountains]]
The basic intuition behind gradient descent can be illustrated by a hypothetical scenario. People are stuck in the mountains and are 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 persons happen 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.
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In this analogy, the persons represent 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 function at that point. The instrument used to measure steepness is [[Differentiation (mathematics)|differentiation]]. The direction they choose to travel in aligns with the [[gradient]] of the function at that point. The amount of time they travel before taking another measurement is the step size.
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Since using a step size <math>\eta</math> that is too small would slow convergence, and a <math>\eta</math> too large would lead to overshoot and divergence, finding a good setting of <math>\eta</math> is an important practical problem. [[Philip Wolfe (mathematician)|Philip Wolfe]] also advocated using "clever choices of the [descent] direction" in practice.<ref>{{cite journal |last1=Wolfe |first1=Philip |title=Convergence Conditions for Ascent Methods |journal=SIAM Review |date=April 1969 |volume=11 |issue=2 |pages=226–235 |doi=10.1137/1011036 }}</ref> While 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.
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==Solution of a linear system==
[[File:Steepest descent.png|thumb|380px|The steepest descent algorithm applied to the [[Wiener filter]]<ref>Haykin, Simon S. Adaptive filter theory. Pearson Education India, 2008. - p. 108-142, 217-242</ref>]]
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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]] <math>\kappa(\mathbf{A})</math> of the system matrix <math>\mathbf{A}</math> (the ratio of the maximum to minimum [[eigenvalues]] of {{nowrap|<math>\mathbf{A}^\top \mathbf{A}</math>)}}, 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.<ref name=":0" />
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In steepest descent applied to solving <math> \mathbf{A x} = \mathbf{b} </math>, where <math> \mathbf{A} </math> is symmetric positive-definite, the residual vectors <math> \mathbf{r}_k = \mathbf{b} - \mathbf{A}\mathbf{x}_k </math> are orthogonal across iterations:
:<math>
\langle \mathbf{r}_{k+1}
</math>
▲characteristic zig-zag path. The poor conditioning of <math> \mathbf{A} </math> is the primary cause of the slow convergence, and orthogonality of successive residuals reinforces this alternation.
[[File:Steepest descent convergence path for A = 2 2, 2 3.png|thumb|Convergence path of steepest descent method for A = [[2, 2], [2, 3]]]]
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==Solution of a non-linear system==
Gradient descent can also be used to solve a system of [[nonlinear equation]]s. Below is an example that shows how to use the gradient descent to solve for three unknown variables, ''x''<sub>1</sub>, ''x''<sub>2</sub>, and ''x''<sub>3</sub>. This example shows one iteration of the gradient descent.
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==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.<ref name="AK82">{{cite book |first1=G. P. |last1=Akilov |first2=L. V. |last2=Kantorovich |author-link2=Leonid Kantorovich |title=Functional Analysis |publisher=Pergamon Press |edition=2nd |isbn=0-08-023036-9 |year=1982 }}</ref>
That gradient descent works in any number of dimensions (finite number at least) can be seen as a consequence of the [[
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 modified via momentums<ref>{{Cite journal |last1=Abdulkadirov |first1=Ruslan |last2=Lyakhov |first2=Pavel |last3=Nagornov |first3=Nikolay |date=January 2023 |title=Survey of Optimization Algorithms in Modern Neural Networks |journal=Mathematics |language=en |volume=11 |issue=11 |pages=2466 |doi=10.3390/math11112466 |doi-access=free |issn=2227-7390}}</ref> ([[Nesterov]], Polyak,<ref>{{Cite journal |last1=Diakonikolas |first1=Jelena |last2=Jordan |first2=Michael I. |date=January 2021 |title=Generalized Momentum-Based Methods: A Hamiltonian Perspective |url=https://epubs.siam.org/doi/10.1137/20M1322716 |journal=SIAM Journal on Optimization |language=en |volume=31 |issue=1 |pages=915–944 |doi=10.1137/20M1322716 |arxiv=1906.00436 |issn=1052-6234}}</ref> and
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.<ref>{{cite book |first1=W. H. |last1=Press |author-link1 = William H. Press |first2=S. A. |last2=Teukolsky |author-link2 = Saul Teukolsky |first3=W. T. |last3=Vetterling |first4=B. P. |last4=Flannery |author-link4 = Brian P. Flannery |title=Numerical Recipes in C: The Art of Scientific Computing |url=https://archive.org/details/numericalrecipes00pres_0 |url-access=registration |edition=2nd |publisher=[[Cambridge University Press]] |___location=New York |year=1992 |isbn=0-521-43108-5 }}</ref><ref>{{cite book |first=T. |last=Strutz |title=Data Fitting and Uncertainty: A Practical Introduction to Weighted Least Squares and Beyond |edition=2nd |publisher=Springer Vieweg |year=2016 |isbn=978-3-658-11455-8 }}</ref> 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]] algorithm, to find the "best" value of <math>\eta.</math> 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.
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==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
===Fast gradient methods===
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==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 [[
Gradient descent is a special case of [[mirror descent]] using the squared Euclidean distance as the given [[Bregman divergence]].<ref>{{cite web | url=https://tlienart.github.io/posts/2018/10/27-mirror-descent-algorithm/ | title=Mirror descent algorithm }}</ref>
==
The properties of gradient descent depend on the properties of the objective function and the variant of gradient descent used (for example, if a [[line search]] step is used). The assumptions made affect the convergence rate, and other properties, that can be proven for gradient descent.<ref name=":1">{{cite arXiv|last=Bubeck |first=Sébastien |title=Convex Optimization: Algorithms and Complexity |date=2015 |class=math.OC |eprint=1405.4980 }}</ref> For example, if the objective is assumed to be [[Strongly convex function|strongly convex]] and [[Lipschitz continuity|lipschitz smooth]], then gradient descent converges linearly with a fixed step size.<ref name="auto"/> Looser assumptions lead to either weaker convergence guarantees or require a more sophisticated step size selection.<ref name=":1" />
== Examples ==
* [[Yang–Mills flow]]
* [[Yang–Mills–Higgs flow]]
* [[Seiberg–Witten flow]]
==See also==
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{{Reflist|30em}}
==
*{{cite book |first1=Stephen |last1=Boyd |author-link=Stephen P. Boyd |first2=Lieven |last2=Vandenberghe |chapter=Unconstrained Minimization |title=Convex Optimization |___location=New York |publisher=Cambridge University Press |year=2004 |isbn=0-521-83378-7 |chapter-url=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=471 |pages=457–520 }}
*{{cite book |first1=Edwin K. P. |last1=Chong |first2=Stanislaw H. |last2=Żak |chapter=Gradient Methods |title=An Introduction to Optimization |edition=Fourth |___location=Hoboken |publisher=Wiley |year=2013 |isbn=978-1-118-27901-4 |pages=131–160 |chapter-url=https://books.google.com/books?id=iD5s0iKXHP8C&pg=PA131 }}
*{{cite book |first=David M. |last=Himmelblau |title=Applied Nonlinear Programming |___location=New York |publisher=McGraw-Hill |year=1972 |isbn=0-07-028921-2 |chapter=Unconstrained Minimization Procedures Using Derivatives |pages=63–132 }}
==
{{Commons category|Gradient descent}}
* [
* [https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/gradient-and-directional-derivatives/v/gradient Series of Khan Academy videos discusses gradient ascent]
* [http://neuralnetworksanddeeplearning.com/chap1.html#learning_with_gradient_descent Online book teaching gradient descent in deep neural network context]
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