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{{Machine learning}}

[[File:Gradient Descent in 2D.webm|thumb|right|Gradient Descent in 2D]]

In mathematics, '''gradientGradient descent''' (alsois oftena calledmethod '''steepestfor descent''')unconstrained [[mathematical optimization]]. It is a [[:Category:First order methods|first-order]] [[Iterative algorithm|iterative]] [[Mathematical optimization|optimization]] [[algorithm]] for finding a [[local minimum]] ofminimizing a [[differentiable function|differentiable]]. The idea is to take repeated steps in the opposite direction of the [[gradient]] (or approximate gradient) of themultivariate 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.

It is particularly useful in machine learning for minimizing the cost or loss function.<ref>{{cite web | url=https://keepnotes.com/document/stanford-university/319-gradient-descent-intuition | title=Gradient descent intuition &#124; Stanford University - KeepNotes }}</ref> Gradient descent should not be confused with [[Local search (optimization)|local search]] algorithms, although all are [[Iterative method|iterative methods]] for [[Global optimization|optimization]].

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 trajectory that maximizes that function; the procedure is then known as ''gradient ascent''.

Gradient descent is generally attributed to [[Augustin-Louis Cauchy]], who first suggested it in 1847.<ref>{{cite journal |first=C. |last=Lemaréchal |author-link=Claude Lemaréchal |title=Cauchy and the Gradient Method |journal=Doc Math Extra |pages=251–254 |year=2012 |url=https://www.math.uni-bielefeld.de/documenta/vol-ismp/40_lemarechal-claude.pdf }}</ref> [[Jacques Hadamard]] independently proposed a similar method in 1907.<ref>{{Cite journal|last=Hadamard|first=Jacques|date=1908|title=Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées|journal=Mémoires présentés par divers savants éstrangers à l'Académie des Sciences de l'Institut de France|volume=33}}</ref><ref>{{cite journal |last1=Courant |first1=R. |title=Variational methods for the solution of problems of equilibrium and vibrations |journal=Bulletin of the American Mathematical Society |date=1943 |volume=49 |issue=1 |pages=1–23 |doi=10.1090/S0002-9904-1943-07818-4 |doi-access=free }}</ref> Its convergence properties for non-linear optimization problems were first studied by [[Haskell Curry]] in 1944,<ref>{{cite journal |first=Haskell B. |last=Curry |title=The Method of Steepest Descent for Non-linear Minimization Problems |journal=Quart. Appl. Math. |volume=2 |year=1944 |issue=3 |pages=258–261 |doi=10.1090/qam/10667 |doi-access=free }}</ref> with the method becoming increasingly well-studied and used in the following decades.<ref name="BP" /><ref name="AK82" />

It is particularly useful in machine learning for minimizing the cost or loss function.<ref name="auto">{{citeCite webbook |last1=Boyd |first1=Stephen |url=httpshttp://keepnotesdx.comdoi.org/document10.1017/stanford-university/319-gradient-descent-intuitioncbo9780511804441 | title=GradientConvex descentOptimization intuition|last2=Vandenberghe &#124;|first2=Lieven Stanford|date=2004-03-08 |publisher=Cambridge University -Press KeepNotes|doi=10.1017/cbo9780511804441 |isbn=978-0-521-83378-3}}</ref> Gradient descent should not be confused with [[Local search (optimization)|local search]] algorithms, although allboth are [[Iterative method|iterative methods]] for [[Global optimization|optimization]].

Gradient descent is generally attributed to [[Augustin-Louis Cauchy]], who first suggested it in 1847.<ref>{{cite journal book|firstvauthors=((Lemaréchal, C.)) |last title=LemaréchalOptimization Stories |author-link veditors=Claude((Grötschel, LemaréchalM.)) |title date=1 January 2012 | chapter=Cauchy and the Gradientgradient Methodmethod |journal series=DocDocumenta MathMathematica ExtraSeries | publisher=EMS Press | volume=6 | edition=1st | pages=251–254 |year chapter-url=2012https://www.math.uni-bielefeld.de/documenta/vol-ismp/40_lemarechal-claude.pdf| doi=10.4171/dms/6/27 | doi-access=free | isbn=978-3-936609-58-5|access-date=2020-01-26 |archive-date=2018-12-29 |archive-url=https://web.archive.org/web/20181229073335/https://www.math.uni-bielefeld.de/documenta/vol-ismp/40_lemarechal-claude.pdf |url-status=dead}}</ref> [[Jacques Hadamard]] independently proposed a similar method in 1907.<ref>{{Cite journal|last=Hadamard|first=Jacques|date=1908|title=Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées|journal=Mémoires présentés par divers savants éstrangers à l'Académie des Sciences de l'Institut de France|volume=33}}</ref><ref>{{cite journal |last1=Courant |first1=R. |title=Variational methods for the solution of problems of equilibrium and vibrations |journal=Bulletin of the American Mathematical Society |date=1943 |volume=49 |issue=1 |pages=1–23 |doi=10.1090/S0002-9904-1943-07818-4 |doi-access=free }}</ref> Its convergence properties for non-linear optimization problems were first studied by [[Haskell Curry]] in 1944,<ref>{{cite journal |first=Haskell B. |last=Curry |title=The Method of Steepest Descent for Non-linear Minimization Problems |journal=Quart. Appl. Math. |volume=2 |year=1944 |issue=3 |pages=258–261 |doi=10.1090/qam/10667 |doi-access=free }}</ref> with the method becoming increasingly well-studied and used in the following decades.<ref name="BP" /><ref name="AK82" />

A simple extension of gradient descent, [[stochastic gradient descent]], serves as the most basic algorithm used for training most [[deep network]]s today.

A simple extension of gradient descent, [[stochastic gradient descent]], serves as the most basic algorithm used for training most [[deep neural network|deep network]]s today.

==Description==

[[File:Gradient descent.svg|thumb|350px|Illustration of gradient descent on a series of [[level set]]s]]

Gradient descent is based on the observation that if the [[multi-variable function]] <math>Ff(\mathbf{x})</math> is [[Defined and undefined|defined]] and [[Differentiable function|differentiable]] in a neighborhood of a point <math>\mathbf{a}</math>, then <math>Ff(\mathbf{x})</math> decreases ''fastest'' if one goes from <math>\mathbf{a}</math> in the direction of the negative [[gradient]] of <math>Ff</math> at <math>\mathbf{a}, -\nabla Ff(\mathbf{a})</math>. It follows that, if

:<math> \mathbf{a}_{n+1} = \mathbf{a}_n-\gammaeta \nabla Ff(\mathbf{a}_n)</math>

for a small enough step size or [[learning rate]] <math>\gamma eta\in \R_{+}</math>, then <math>Ff(\mathbf{a_n})\geq Ff(\mathbf{a_{n+1}})</math>. In other words, the term <math>\gammaeta\nabla Ff(\mathbf{a})</math> is subtracted from <math>\mathbf{a}</math> because we want to move against the gradient, toward the local minimum. With this observation in mind, one starts with a guess <math>\mathbf{x}_0</math> for a local minimum of <math>Ff</math>, and considers the sequence <math>\mathbf{x}_0, \mathbf{x}_1, \mathbf{x}_2, \ldots</math> such that

:<math>\mathbf{x}_{n+1}=\mathbf{x}_n-\gamma_n eta_n\nabla Ff(\mathbf{x}_n),\ n \ge 0.</math>

We have a [[Monotonic function|monotonic]] sequence

:<math>Ff(\mathbf{x}_0)\ge Ff(\mathbf{x}_1)\ge Ff(\mathbf{x}_2)\ge \cdots,</math>

so the sequence <math>(\mathbf{x}_n)</math> converges to the desired local minimum. Note that the value of the ''step size'' <math>\eta</math> is allowed to change at every iteration.

so,It hopefully,is the sequence <math>(\mathbf{x}_n)</math> convergespossible to the desired local minimum. Note thatguarantee the value[[Convergent of the ''step size'' <math>\gamma</math> is allowedseries|convergence]] to changea atlocal every iteration.minimum Withunder certain assumptions on the function <math>Ff</math> (for example, <math>Ff</math> [[Convex function|convex]] and <math>\nabla Ff</math> [[Lipschitz continuity|Lipschitz]]) and particular choices of <math>\gammaeta</math> (e.g., chosenThose either via a [[line search]] that satisfiesinclude the [[Wolfe conditions]], or the [[Barzilai-Borwein method]]<ref>{{cite journal|last1=Barzilai|first1=Jonathan|last2=Borwein|first2=Jonathan M.|year=1988|title=Two-Point Step Size Gradient Methods|journal=IMA Journal of Numerical Analysis|volume=8|issue=1|pages=141–148|doi=10.1093/imanum/8.1.141}}</ref><ref>{{cite book|last=Fletcher|first=R.|title=Optimization and Control with Applications|publisher=Springer|year=2005|isbn=0-387-24254-6|editor-last=Qi|editor-first=L.|series=Applied Optimization|volume=96|___location=Boston|pages=235–256|chapter=On the Barzilai–Borwein Method|editor2-last=Teo|editor2-first=K.|editor3-last=Yang|editor3-first=X.}}</ref> shown as following),sequence

: <math>\gamma_eta_{n} = \frac{ \left | \left (\mathbf x_{n} - \mathbf x_{n-1} \right )^T\top \left [\nabla Ff (\mathbf x_{n}) - \nabla Ff (\mathbf x_{n-1}) \right ] \right |}{\left \|\nabla Ff(\mathbf{x}_{n}) - \nabla Ff(\mathbf{x}_{n-1}) \right \|^2}</math>

as in the [[Barzilai-Borwein method]],<ref>{{cite journal |last1=Barzilai |first1=Jonathan |last2=Borwein |first2=Jonathan M. |year=1988 |title=Two-Point Step Size Gradient Methods |journal=IMA Journal of Numerical Analysis |volume=8 |issue=1 |pages=141–148 |doi=10.1093/imanum/8.1.141}}</ref><ref>{{cite book |last=Fletcher |first=R. |title=Optimization and Control with Applications |publisher=Springer |year=2005 |isbn=0-387-24254-6 |editor-last=Qi |editor-first=L. |series=Applied Optimization |volume=96 |___location=Boston |pages=235–256 |chapter=On the Barzilai–Borwein Method |editor2-last=Teo |editor2-first=K. |editor3-last=Yang |editor3-first=X.}}</ref> or a sequence <math> \eta_n</math> satisfying the [[Wolfe conditions]] (which can be found by using [[line search]]). When the function <math>f</math> is [[Convex function|convex]], all local minima are also global minima, so in this case gradient descent can converge to the global solution.

[[Convergent series|convergence]] to a local minimum can be guaranteed. When the function <math>F</math> is [[Convex function|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, <math>Ff</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>Ff</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>Ff</math> is minimal.

=== An analogy for understanding gradient descent ===

[[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. APeople person isare stuck in the mountains and isare 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 personpersons happenshappen 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 personpersons representsrepresent 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.

=== Choosing the step size and descent direction ===

Since using a step size <math>\gammaeta</math> that is too small would slow convergence, and a <math>\gammaeta</math> too large would lead to overshoot and divergence, finding a good setting of <math>\gammaeta</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> WhilstWhile 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, consider a direction <math> \mathbf{p}_n</math> and step size <math> \gamma_neta_n</math> and consider the more general update:

:<math> \mathbf{a}_{n+1} = \mathbf{a}_n-\gamma_neta_n\,\mathbf{p}_n</math>.

Finding good settings of <math> \mathbf{p}_n</math> and <math> \gamma_neta_n</math> requires some thought. First of all, we would like the update direction to point downhill. Mathematically, letting <math> \theta_n</math> denote the angle between <math>-\nabla Ff(\mathbf{a_n})</math> and <math> \mathbf{p}_n</math>, this requires that <math> \cos \theta_n > 0.</math> To say more, we need more information about the objective function that we are optimising. Under the fairly weak assumption that <math>Ff</math> is continuously differentiable, we may prove that:<ref>{{cite arXiv|last1=Bernstein|first1=Jeremy|last2=Vahdat|first2=Arash|last3=Yue|first3=Yisong|last4=Liu|first4=Ming-Yu|date=2020-06-12|title=On the distance between two neural networks and the stability of learning|class=cs.LG|eprint=2002.03432}}</ref>

{{NumBlk|:|<math> Ff(\mathbf{a}_{n+1}) \leq Ff(\mathbf{a}_n) - \gamma_neta_n \|\nabla Ff(\mathbf{a}_n)\|_2 \|\mathbf{p}_n\|_2 \left[(\cos \theta_n - \max_{t\in[0,1]} \frac{\|\nabla Ff(\mathbf{a}_n - t \gamma_neta_n \mathbf{p}_n) - \nabla Ff(\mathbf{a}_n)\|_2}{\| \nabla Ff(\mathbf{a}_n) \|_2}\right])</math>|{{EquationRef|1}}}}

This inequality implies that the amount by which we can be sure the function <math>Ff</math> 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 ({{EquationNote|1}}) could be optimized over <math> \mathbf{p}_n</math> and <math> \gamma_neta_n</math> to choose an optimal step size and direction. The problem is that evaluating the second term in square brackets requires evaluating <math> \nabla Ff(\mathbf{a}_n - t \gamma_neta_n \mathbf{p}_n)</math>, 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 <math>\mathbf{p}_n = \nabla Ff(\mathbf{a_n})</math>, and use [[line search]] to find a suitable step-size <math> \gamma_n</math>, such as one that satisfies the [[Wolfe conditions]]. A more economic way of choosing learning rates is [[backtracking line search]], a method that has both good theoretical guarantees and experimental results. Note that one does not need to choose <math>\mathbf{p}_n </math> to be the gradient; any direction that has positive intersectioninner product with the gradient will result in a reduction of the function value (for a sufficiently small value of <math> \gamma_neta_n</math>).

* Assuming that <math>Ff</math> is twice-differentiable, use its Hessian <math>\nabla^2 Ff</math> to estimate <math> \|\nabla Ff(\mathbf{a}_n - t \gamma_neta_n \mathbf{p}_n) - \nabla Ff(\mathbf{a}_n)\|_2 \approx \| t \gamma_neta_n \nabla^2 Ff(\mathbf{a}_n) \mathbf{p}_n\|.</math>Then choose <math> \mathbf{p}_n</math> and <math> \gamma_neta_n</math> by optimising inequality ({{EquationNote|1}}).

* Assuming that <math>\nabla Ff</math> is [[Lipschitz continuity|Lipschitz]], use its Lipschitz constant <math> L</math> to bound <math> \|\nabla Ff(\mathbf{a}_n - t \gamma_neta_n \mathbf{p}_n) - \nabla Ff(\mathbf{a}_n)\|_2 \leq L t \gamma_neta_n \|\mathbf{p}_n\|.</math> Then choose <math> \mathbf{p}_n</math> and <math> \gamma_neta_n</math> by optimising inequality ({{EquationNote|1}}).

* Build a custom model of <math> \max_{t\in[0,1]} \frac{\|\nabla Ff(\mathbf{a}_n - t \gamma_neta_n \mathbf{p}_n) - \nabla Ff(\mathbf{a}_n)\|_2}{\| \nabla Ff(\mathbf{a}_n) \|_2}</math> for <math>Ff</math>. Then choose <math> \mathbf{p}_n</math> and <math> \gamma_neta_n</math> by optimising inequality ({{EquationNote|1}}).

* Under stronger assumptions on the function <math>Ff</math> 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 <math>Ff</math> is [[Convex function|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==

[[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>]]

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

:<math>\mathbf{A}\mathbf{x}-\mathbf{b}=0</math>

reformulated as a quadratic minimization problem.

If the system matrix <math>\mathbf{A}</math> is real [[Symmetric matrix|symmetric]] and [[positive-definite matrix|positive-definite]], an objective function is defined as the quadratic function, with minimization of

:<math>Ff(\mathbf{x})=\mathbf{x}^T\top \mathbf{A}\mathbf{x}-2\mathbf{x}^T\top \mathbf{b},</math>

so that

:<math>\nabla Ff(\mathbf{x})=2(\mathbf{A}\mathbf{x}-\mathbf{b}).</math>

For a general real matrix <math>\mathbf{A}</math>, [[linear least squares]] define

:<math>Ff(\mathbf{x})=\left\|\mathbf{A}\mathbf{x}-\mathbf{b}\right\|^2.</math>

In traditional linear least squares for real <math>\mathbf{A}</math> and <math>\mathbf{b}</math> the [[Euclidean norm]] is used, in which case

:<math>\nabla Ff(\mathbf{x})=2A2\mathbf{A}^T\top(\mathbf{A}\mathbf{x}-\mathbf{b}).</math>

The [[line search]] minimization, finding the locally optimal step size <math>\gammaeta</math> on every iteration, can be performed analytically for quadratic functions, and explicit formulas for the locally optimal <math>\gammaeta</math> are known.<ref name="BP">{{cite book |author-link=Boris T. Polyak |last=Polyak |first=Boris |title=Introduction to Optimization |year=1987 |language=en |url=https://www.researchgate.net/publication/342978480 }}</ref><ref name=saad1996iterative>{{cite book|last=Saad|first=Yousef|title=Iterative methods for sparse linear systems|year=2003|publisher=Society for Industrial and Applied Mathematics|___location=Philadelphia, Pa.|isbn=978-0-89871-534-7|pages=[https://archive.org/details/iterativemethods0000saad/page/195 195]|edition=2nd|url=https://archive.org/details/iterativemethods0000saad/page/195}}</ref>

For example, for real [[Symmetric matrix|symmetric]] and [[positive-definite matrix|positive-definite]] matrix <math>\mathbf{A}</math>, a simple algorithm can be as follows,<ref name="BP" />

:<math>\begin{align}

& \text{repeat in the loop:} \\

& \qquad \mathbf{r} := \mathbf{b} - \mathbf{A x} \\

& \qquad \gammaeta := {\mathbf{r}^\mathsf{T}top \mathbf{r}}/{\mathbf{r}^\mathsf{T}top \mathbf{A r}} \\

& \qquad \mathbf{x} := \mathbf{x} + \gamma eta\mathbf{r} \\

& \qquad \hbox{if } \mathbf{r}^\mathsf{T}top \mathbf{r} \text{ is sufficiently small, then exit loop} \\

& \text{end repeat loop} \\

& \text{return } \mathbf{x} \text{ as the result}

\end{align}</math>

To avoid multiplying by <math>\mathbf{A}</math> twice per iteration,

we note that <math>\mathbf{x} := \mathbf{x} + \gamma eta\mathbf{r}</math> implies <math>\mathbf{r} := \mathbf{r} - \gamma eta\mathbf{A r}</math>, which gives the traditional algorithm,<ref name=":0">{{cite journal |first1=Henricus |last1=Bouwmeester |first2=Andrew |last2=Dougherty |first3=Andrew V. |last3=Knyazev |title=Nonsymmetric Preconditioning for Conjugate Gradient and Steepest Descent Methods |journal=Procedia Computer Science |volume=51 |pages=276–285 |year=2015 |doi=10.1016/j.procs.2015.05.241 |doi-access=free |arxiv=1212.6680 }}</ref>

:<math>\begin{align}

& \mathbf{r} := \mathbf{b} - \mathbf{A x} \\

& \text{repeat in the loop:} \\

& \qquad \gammaeta := {\mathbf{r}^\mathsf{T}top \mathbf{r}}/{\mathbf{r}^\mathsf{T}top \mathbf{A r}} \\

& \qquad \mathbf{x} := \mathbf{x} + \gamma eta\mathbf{r} \\

& \qquad \hbox{if } \mathbf{r}^\mathsf{T}top \mathbf{r} \text{ is sufficiently small, then exit loop} \\

& \qquad \mathbf{r} := \mathbf{r} - \gamma eta\mathbf{A r} \\

& \text{end repeat loop} \\

& \text{return } \mathbf{x} \text{ as the result}

\end{align}</math>

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}^TA\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" />

===Geometric behavior and residual orthogonality===

==Solution of a non-linear system==

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}, \mathbf{r}_k\rangle = 0.

</math>

Because each step is taken in the steepest direction, steepest-descent steps alternate between directions aligned with the extreme axes of the elongated level sets. When <math>\kappa(\mathbf{A})</math> is large, this produces a 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]]]]

As shown in the image on the right, steepest descent converges slowly due to the high condition number of <math> \mathbf{A} </math>, and the orthogonality of residuals forces each new direction to undo the overshoot from the previous step. The result is a path that zigzags toward the solution. This inefficiency is one reason conjugate gradient or preconditioning methods are preferred.<ref>{{Cite book | author1=Holmes, M. | title=Introduction to Scientific Computing and Data Analysis, 2nd Ed | year=2023 | publisher=Springer | isbn=978-3-031-22429-4 }}</ref>

==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''₁, ''x''₂, and ''x''₃. This example shows one iteration of the gradient descent.

Line 141 ⟶ 156:

One might now define the objective function

:<math>\begin{align}Ff(\mathbf{x}) &= \frac{1}{2} G^\mathrm{T}top(\mathbf{x}) G(\mathbf{x}) \\&=\frac{1}{2} \left[ \left (3x_1-\cos(x_2x_3)-\frac{3}{2} \right)^2 + \left(4x_1^2-625x_2^2+2x_2-1 \right)^2 +\right.\\

&{}\qquad\left. \left(\exp(-x_1x_2) + 20x_3 + \frac{10\pi-3}{3} \right)^2 \right],\end{align}</math>

Line 153 ⟶ 168:

We know that

:<math>\mathbf{x}^{(1)}=\mathbf{0}-\gamma_0eta_0 \nabla Ff(\mathbf{0}) = \mathbf{0}-\gamma_0eta_0 J_G(\mathbf{0})^\mathrm{T}top G(\mathbf{0}),</math>

where the [[Jacobian matrix]] <math>J_G</math> is given by

Line 177 ⟶ 192:

Thus

:<math>\mathbf{x}^{(1)}= \mathbf{0}-\gamma_0eta_0 \begin{bmatrix}

-7.5\\

-2\\

Line 185 ⟶ 200:

and

:<math>Ff(\mathbf{0}) = 0.5 \left( (-2.5)^2 + (-1)^2 + (10.472)^2 \right) = 58.456.</math>

[[File:Gradient Descent Example Nonlinear Equations.gif|thumb|right|350px|An animation showing the first 83 iterations of gradient descent applied to this example. Surfaces are [[isosurface]]s of <math>Ff(\mathbf{x}^{(n)})</math> at current guess <math>\mathbf{x}^{(n)}</math>, and arrows show the direction of descent. Due to a small and constant step size, the convergence is slow.]]

Now, a suitable <math>\gamma_0eta_0</math> must be found such that

:<math>Ff\left (\mathbf{x}^{(1)}\right ) \le Ff\left (\mathbf{x}^{(0)}\right ) = Ff(\mathbf{0}).</math>

This can be done with any of a variety of [[line search]] algorithms. One might also simply guess <math>\gamma_0eta_0=0.001,</math> which gives

:<math> \mathbf{x}^{(1)}=\begin{bmatrix}

Line 203 ⟶ 218:

Evaluating the objective function at this value, yields

:<math>Ff \left (\mathbf{x}^{(1)}\right ) = 0.5 \left ((-2.48)^2 + (-1.00)^2 + (6.28)^2 \right ) = 23.306.</math>

The decrease from <math>Ff(\mathbf{0})=58.456</math> to the next step's value of

:<math> Ff \left (\mathbf{x}^{(1)}\right ) =23.306 </math>

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.<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 [[Cauchy-SchwarzCauchy–Schwarz inequality]], i.e. 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 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 Frank–Wolfe<ref>{{Cite journal |last=Meyer |first=Gerard G. L. |date=November 1974 |title=Accelerated Frank–Wolfe Algorithms |url=http://epubs.siam.org/doi/10.1137/0312050 |journal=SIAM Journal on Control |language=en |volume=12 |issue=4 |pages=655–663 |doi=10.1137/0312050 |issn=0036-1402|url-access=subscription }}</ref>) and heavy-ball parameters (exponential moving averages<ref>{{Citation |last1=Kingma |first1=Diederik P. |title=Adam: A Method for Stochastic Optimization |date=2017-01-29 |last2=Ba |first2=Jimmy|arxiv=1412.6980 }}</ref> and positive-negative momentum<ref>{{Cite journal |last1=Xie |first1=Zeke |last2=Yuan |first2=Li |last3=Zhu |first3=Zhanxing |last4=Sugiyama |first4=Masashi |date=2021-07-01 |title=Positive-Negative Momentum: Manipulating Stochastic Gradient Noise to Improve Generalization |url=https://proceedings.mlr.press/v139/xie21h.html |journal=Proceedings of the 38th International Conference on Machine Learning |language=en |publisher=PMLR |pages=11448–11458|arxiv=2103.17182 }}</ref>). The main examples of such optimizers are Adam, DiffGrad, Yogi, AdaBelief, etc.

The gradient descent can be combined with a [[line search]], finding the locally optimal step size <math>\gamma</math> on every iteration. Performing the line search can be time-consuming. Conversely, using a fixed small <math>\gamma</math> 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.<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>\gammaeta.</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.

While it is sometimes possible to substitute gradient descent for a [[Local search (optimization)|local search]] algorithm, gradient descent is not in the same family: although it is an [[iterative method]] for [[Global optimization|local optimization]], it relies on an [[loss function|objective function’s gradient]] rather than an explicit exploration of a [[Feasible region|solution space]].

Gradient descent can be viewed as applying [[Euler's method]] for solving [[ordinary differential equations]] <math>x'(t)=-\nabla f(x(t))</math> to a [[gradient flow]]. In turn, this equation may be derived as an optimal controller<ref>{{cite journal |last1=Ross |first1=I.M. |title=An optimal control theory for nonlinear optimization |journal=Journal of Computational and Applied Mathematics |date=July 2019 |volume=354 |pages=39–51 |doi=10.1016/j.cam.2018.12.044 |s2cid=127649426 |doi-access=free }}</ref> for the control system <math>x'(t) = u(t)</math> with <math>u(t)</math> given in feedback form <math>u(t) = -\nabla f(x(t))</math>.

It can be shown that there is a correspondence between [[neuroevolution]] and gradient descent.<ref>{{cite journal |last1=Whitelam |first1=Stephen |last2=Selin |first2=Viktor |last3=Park |first3=Sang-Won |last4=Tamblyn |first4=Isaac |title=Correspondence between neuroevolution and gradient descent |journal=Nature Communications |date=2 November 2021 |volume=12 |issue=1 |pages=6317 |doi=10.1038/s41467-021-26568-2 |pmid=34728632 |pmc=8563972 |arxiv=2008.06643 |bibcode=2021NatCo..12.6317W }}</ref>

==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-zagzig–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<ref>{{cite book |first=Yurii |last=Nesterov |author-link=Yurii Nesterov |title=Introductory Lectures on Convex Optimization : A Basic Course |publisher=Springer |year=2004 |isbn=1-4020-7553-7 }}</ref> 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 <math>Ff</math> is convex and <math>\nabla Ff</math> is [[Lipschitz continuity|Lipschitz]], and it is not assumed that <math>Ff</math> is [[Convex function#Strongly convex functions|strongly convex]], then the error in the objective value generated at each step <math>k</math> by the gradient descent method will be [[Big O notation|bounded by]] <math display="inline">\mathcal{O}\left(\tfrac{k^{-1}{k}\right)</math>. Using the Nesterov acceleration technique, the error decreases at <math display="inline">\mathcal{O}\left(\tfrac{1}{k^{-2}}\right)</math>.<ref>{{cite web |url=https://www.seas.ucla.edu/~vandenbe/236C/lectures/fgrad.pdf |title=Fast Gradient Methods |work=Lecture notes for EE236C at UCLA |first=Lieven |last=Vandenberghe |date=2019 }}</ref><ref>{{Cite journal |last=Walkington |first=Noel J. |date=2023 |title=Nesterov's Method for Convex Optimization |url=https://epubs.siam.org/doi/10.1137/21M1390037 |journal=SIAM Review |language=en |volume=65 |issue=2 |pages=539–562 |doi=10.1137/21M1390037 |issn=0036-1445|url-access=subscription }}</ref> It is known that the rate <math>\mathcal{O}\left({k^{-2}}\right)</math> 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)<ref>{{cite journal |first1=D. |last1=Kim |first2=J. A. |last2=Fessler |title=Optimized First-order Methods for Smooth Convex Minimization |journal=[[Mathematical Programming|Math. Prog.]] |volume=151 |issue=1–2 |pages=81–107 |year=2016 |doi=10.1007/s10107-015-0949-3 |pmid=27765996 |pmc=5067109 |arxiv=1406.5468 |s2cid=207055414 }}</ref> reduces that constant by a factor of two and is an optimal first-order method for large-scale problems.<ref>{{cite journal |first=Yoel |last=Drori |date=2017 |title=The Exact Information-based Complexity of Smooth Convex Minimization |journal=Journal of Complexity |volume=39 |pages=1–16 |doi=10.1016/j.jco.2016.11.001 |arxiv=1606.01424 |s2cid=205861966 }}</ref>

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,<ref name="BP" /> or to mass movement in [[Newtonian dynamics]] through a [[viscous]] medium in a [[conservative force]] field.<ref>{{cite journal|last1=Qian |first1=Ning |title=On the momentum term in gradient descent learning algorithms |journal=[[Neural Networks (journal)|Neural Networks]] |date=January 1999 |volume=12 |issue=1 |pages=145–151 |doi=10.1016/S0893-6080(98)00116-6 |pmid=12662723 |citeseerx=10.1.1.57.5612 |s2cid=2783597 }}</ref> 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]].<ref name="BP" />

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 network]]s.<ref>{{cite web|title=Momentum and Learning Rate Adaptation|url=http://www.willamette.edu/~gorr/classes/cs449/momrate.html|publisher=[[Willamette University]]|access-date=17 October 2014}}</ref><ref>{{cite web|author1=Geoffrey Hinton|author-link=Geoffrey Hinton|author2=Nitish Srivastava|author3=Kevin Swersky|title=The momentum method|url=https://www.coursera.org/lecture/neural-networks/the-momentum-method-Oya9a|website=[[Coursera]]|access-date=2 October 2018}} Part of a lecture series for the [[Coursera]] online course [https://www.coursera.org/learn/neural-networks Neural Networks for Machine Learning] {{Webarchive|url=https://web.archive.org/web/20161231174321/https://www.coursera.org/learn/neural-networks |date=2016-12-31 }}.</ref> In the direction of updating, stochastic gradient descent adds a stochastic property. The weights can be used to calculate the derivatives.

==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-backwardforward–backward algorithm]] for monotone inclusions (which includes [[convex programming]] and [[Variational inequality|variational inequalities]]).<ref>{{cite book |first1=P. L. |last1=Combettes |first2=J.-C. |last2=Pesquet |arxiv=0912.3522 |chapter=Proximal splitting methods in signal processing |title=Fixed-Point Algorithms for Inverse Problems in Science and Engineering |editor1-first=H. H. |editor1-last=Bauschke |editor2-link=Regina S. Burachik |editor2-first=R. S. |editor2-last=Burachik |editor3-first=P. L. |editor3-last=Combettes |editor4-first=V. |editor4-last=Elser |editor5-first=D. R. |editor5-last=Luke |editor6-first=H. |editor6-last=Wolkowicz |pages=185–212 |publisher=Springer |___location=New York |year=2011 |isbn=978-1-4419-9568-1 }}</ref>

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>

==Theoretical properties==

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==

Line 262 ⟶ 284:

* [[Hill climbing]]

* [[Quantum annealing]]

* [[TFNP#CLS|Continuous Local SearchCLS]] (continuous local search)

* [[Neuroevolution]]

{{Div col end}}

Line 268 ⟶ 291:

{{Reflist|30em}}

== Further reading ==

*{{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 }}

== External links ==

{{Commons category|Gradient descent}}

* [httphttps://codingplayground.blogspot.it/2013/05/learning-linear-regression-with.html Using gradient descent in C++, Boost, Ublas for linear regression]

* [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]

* Archived at [https://ghostarchive.org/varchive/youtube/20211211/IHZwWFHWa-w Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20171016173155/https://www.youtube.com/watch?v=IHZwWFHWa-w Wayback Machine]{{cbignore}}: {{cite web |work=3Blue1Brown |title=Gradient Descent, How Neural Networks Learn |date=October 16, 2017 |url=https://www.youtube.com/watch?v=IHZwWFHWa-w&list=PLZHQObOWTQDNU6R1_67000Dx_ZCJB-3pi&index=2 |via=[[YouTube]] }}{{cbignore}}

* [[arxiv:{{cite arXiv |eprint=2301.11235 |last1=Garrigos |first1=Guillaume |last2=Gower |first2=Robert M. |title=Handbook of Convergence Theorems for (Stochastic) Gradient Methods]] |date=2023 |class=math.OC }}

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