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[[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>\gammaeta</math> is allowed to change at every iteration.
It is possible to guarantee the [[Convergent series|convergence]] to a local minimum under 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>. Those include the 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> \gamma_neta_n</math> satisfying the [[Wolfe conditions]] (which can be found by using [[line search]]). 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.
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 ===
=== 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> 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.
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 inner 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==
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 ===
In steepest descent applied to solving <math> A \vecmathbf{A x} = \vecmathbf{b} </math>, where <math> \mathbf{A} </math> is symmetric positive-definite, the residual vectors <math> \vecmathbf{r}_k = \vecmathbf{b} - \mathbf{A}\vecmathbf{x}_k </math> are orthogonal across iterations:
:<math>
\vecmathbf{r}_{k+1} ^\cdottop \vecmathbf{r}_k = 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==
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>
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
Thus
:<math>\mathbf{x}^{(1)}= \mathbf{0}-\gamma_0eta_0 \begin{bmatrix}
-7.5\\
-2\\
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}
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.
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.
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]].
===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({k^{-1}}\right)</math>. Using the Nesterov acceleration technique, the error decreases at <math display="inline">\mathcal{O}\left({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]] |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]].
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