Descenso de gradiente

Optimization algorithm

Para conocer el método analítico llamado "descenso más pronunciado", consulte Método de descenso más pronunciado .

El descenso de gradiente es un algoritmo de optimización iterativo de primer orden para encontrar un mínimo local de una función diferenciable . La idea es dar pasos repetidos en la dirección opuesta al gradiente (o gradiente aproximado) de la función en el punto actual, porque esta es la dirección de descenso más pronunciado. Por el contrario, dar un paso en la dirección del gradiente conducirá a un máximo local de esa función; el procedimiento se conoce como ascenso en pendiente .

El descenso de gradiente se atribuye generalmente a Cauchy , quien lo sugirió por primera vez en 1847. ^[1] Hadamard propuso de forma independiente un método similar en 1907. ^{[2] [3]} Sus propiedades de convergencia para problemas de optimización no lineal fueron estudiadas por primera vez por Haskell Curry en 1944. , ^[4] con el método cada vez más estudiado y utilizado en las décadas siguientes, ^{[5] [6]} también llamado a menudo descenso más empinado.

Descripción

Ilustración de descenso de gradiente en una serie de conjuntos de niveles

El descenso de gradiente se basa en la observación de que si la función multivariable está definida y es diferenciable en una vecindad de un punto , entonces disminuye más rápido si se va en la dirección del gradiente negativo de en . De ello se deduce que, si${\displaystyle F(\mathbf {x} )}$${\displaystyle \mathbf {a} }$${\displaystyle F(\mathbf {x} )}$${\displaystyle \mathbf {a} }$${\displaystyle F}$${\displaystyle \mathbf {a} ,-\nabla F(\mathbf {a} )}$

${\displaystyle \mathbf {a} _{n+1}=\mathbf {a} _{n}-\gamma \nabla F(\mathbf {a} _{n})}$

por lo suficientemente pequeño, entonces . En otras palabras, el término se resta porque queremos movernos contra el gradiente, hacia el mínimo local. Con esta observación en mente, uno comienza con una suposición de un mínimo local de y considera la secuencia tal que${\displaystyle \gamma \in \mathbb {R} _{+}}$${\displaystyle F(\mathbf {a_{n}} )\geq F(\mathbf {a_{n+1}} )}$${\displaystyle \gamma \nabla F(\mathbf {a} )}$${\displaystyle \mathbf {a} }$${\displaystyle \mathbf {x} _{0}}$${\displaystyle F}$${\displaystyle \mathbf {x} _{0},\mathbf {x} _{1},\mathbf {x} _{2},\ldots }$

${\displaystyle \mathbf {x} _{n+1}=\mathbf {x} _{n}-\gamma _{n}\nabla F(\mathbf {x} _{n}),\ n\geq 0.}$

Tenemos una secuencia monótona

${\displaystyle F(\mathbf {x} _{0})\geq F(\mathbf {x} _{1})\geq F(\mathbf {x} _{2})\geq \cdots ,}$

por lo que es de esperar que la secuencia converja al mínimo local deseado. Tenga en cuenta que el valor del tamaño del paso puede cambiar en cada iteración. Con ciertas suposiciones sobre la función (por ejemplo, convexa y Lipschitz ) y opciones particulares de (por ejemplo, elegidas mediante una búsqueda de línea que satisfaga las condiciones de Wolfe , o el método de Barzilai-Borwein ^{[7] [8] que se} muestra a continuación),${\displaystyle (\mathbf {x} _{n})}$ ${\displaystyle \gamma }$${\displaystyle F}$${\displaystyle F}$ ${\displaystyle \nabla F}$ ${\displaystyle \gamma }$

${\displaystyle \gamma _{n}={\frac {\left|\left(\mathbf {x} _{n}-\mathbf {x} _{n-1}\right)^{T}\left[\nabla F(\mathbf {x} _{n})-\nabla F(\mathbf {x} _{n-1})\right]\right|}{\left\|\nabla F(\mathbf {x} _{n})-\nabla F(\mathbf {x} _{n-1})\right\|^{2}}}}$

se puede garantizar la convergencia a un mínimo local. Cuando la función es convexa , todos los mínimos locales también son mínimos globales, por lo que en este caso el descenso del gradiente puede converger a la solución global.${\displaystyle F}$

Este proceso se ilustra en la imagen adyacente. Aquí se supone que está definido en el plano y que su gráfico tiene forma de cuenco . Las curvas azules son las curvas de nivel , es decir, las regiones en las que el valor de es constante. Una flecha roja que se origina en un punto muestra la dirección del gradiente negativo en ese punto. Tenga en cuenta que el gradiente (negativo) en un punto es ortogonal a la línea de contorno que pasa por ese punto. Vemos que el descenso de gradiente nos lleva al fondo del cuenco, es decir, al punto donde el valor de la función es mínimo.${\displaystyle F}$${\displaystyle F}$${\displaystyle F}$

Una analogía para entender el descenso de gradientes

Niebla en las montañas

La intuición básica detrás del descenso de gradientes puede ilustrarse mediante un escenario hipotético. Una persona está atrapada en las montañas y está tratando de bajar (es decir, tratando de encontrar el mínimo global). Hay una densa niebla que hace que la visibilidad sea extremadamente baja. Por lo tanto, el camino de bajada de la montaña no es visible, por lo que deben usar la información local para encontrar el mínimo. Pueden usar el método de descenso en gradiente, que implica mirar la pendiente de la colina en su posición actual y luego proceder en la dirección con el descenso más empinado (es decir, cuesta abajo). Si estuvieran tratando de encontrar la cima de la montaña (es decir, el máximo), entonces continuarían en la dirección del ascenso más empinado (es decir, cuesta arriba). Usando este método, eventualmente encontrarían su camino hacia abajo de la montaña o posiblemente se quedarían atrapados en algún agujero (es decir, mínimo local opunto de silla de montar ), como un lago de montaña. Sin embargo, suponga también que la pendiente de la colina no es inmediatamente obvia con una simple observación, sino que requiere un instrumento sofisticado para medir, que la persona tiene en ese momento. Se necesita bastante tiempo para medir la pendiente de la colina con el instrumento, por lo que deben minimizar el uso del instrumento si quieren bajar de la montaña antes de la puesta del sol. La dificultad entonces es elegir la frecuencia con la que deben medir la pendiente del cerro para no salirse de la pista.

En esta analogía, la persona representa el algoritmo y el camino recorrido por la montaña representa la secuencia de ajustes de parámetros que explorará el algoritmo. La pendiente de la colina representa la pendiente de la superficie de error en ese punto. El instrumento utilizado para medir la pendiente es la diferenciación (la pendiente de la superficie de error se puede calcular tomando la derivada de la función de error al cuadrado en ese punto). La dirección en la que eligen viajar se alinea con el gradiente de la superficie de error en ese punto. La cantidad de tiempo que viajan antes de tomar otra medida es el tamaño del paso.

Ejemplos

El gradiente de descenso tiene problemas con funciones patológicas como la función de Rosenbrock que se muestra aquí.

${\displaystyle f(x_{1},x_{2})=(1-x_{1})^{2}+100(x_{2}-x_{1}^{2})^{2}.}$

La función Rosenbrock tiene un estrecho valle curvo que contiene el mínimo. El fondo del valle es muy plano. Debido al valle plano curvo, la optimización se realiza en zigzag lentamente con pasos pequeños hacia el mínimo.

La naturaleza en zigzag del método también es evidente a continuación, donde el método de descenso de gradiente se aplica a

${\displaystyle F(x,y)=\sin \left({\frac {1}{2}}x^{2}-{\frac {1}{4}}y^{2}+3\right)\cos \left(2x+1-e^{y}\right).}$

Elegir el tamaño del paso y la dirección de descenso

Dado que el uso de un tamaño de paso demasiado pequeño ralentizaría la convergencia y uno demasiado grande conduciría a la divergencia, encontrar un buen ajuste de es un problema práctico importante. Philip Wolfe también abogó por utilizar "elecciones inteligentes de la dirección [de descenso]" en la práctica. ^[9] Si bien el uso de una dirección que se desvía de la dirección de descenso más empinada puede parecer contrario a la intuición, la idea es que la pendiente más pequeña se puede compensar manteniéndose en una distancia mucho más larga.${\displaystyle \gamma }$${\displaystyle \gamma }$${\displaystyle \gamma }$

Para razonar sobre esto matemáticamente, usemos una dirección y un tamaño de paso y consideremos la actualización más general:${\displaystyle \mathbf {p} _{n}}$${\displaystyle \gamma _{n}}$

${\displaystyle \mathbf {a} _{n+1}=\mathbf {a} _{n}-\gamma _{n}\,\mathbf {p} _{n}}$.

Encontrar una buena configuración de y requiere un poco de reflexión. En primer lugar, nos gustaría que la dirección de actualización apunte cuesta abajo. Matemáticamente, dejar denotar el ángulo entre y , esto requiere que Para decir más, necesitamos más información sobre la función objetivo que estamos optimizando. Bajo el supuesto bastante débil de que es continuamente diferenciable, podemos probar que: ^[10]${\displaystyle \mathbf {p} _{n}}$${\displaystyle \gamma _{n}}$${\displaystyle \theta _{n}}$${\displaystyle \nabla F(\mathbf {a_{n}} )}$${\displaystyle \mathbf {p} _{n}}$${\displaystyle \cos \theta _{n}>0.}$${\displaystyle F}$

${\displaystyle F(\mathbf {a} _{n+1})\leq F(\mathbf {a} _{n})-\gamma _{n}\|\nabla F(\mathbf {a} _{n})\|_{2}\|\mathbf {p} _{n}\|_{2}\left[\cos \theta _{n}-\max _{t\in [0,1]}{\frac {\|\nabla F(\mathbf {a} _{n}-t\gamma _{n}\mathbf {p} _{n})-\nabla F(\mathbf {a} _{n})\|_{2}}{\|\nabla F(\mathbf {a} _{n})\|_{2}}}\right]}$

( 1 )

This inequality implies that the amount by which we can be sure the function ${\displaystyle F}$ 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 (1) could be optimized over ${\displaystyle \mathbf {p} _{n}}$ and ${\displaystyle \gamma _{n}}$ to choose an optimal step size and direction. The problem is that evaluating the second term in square brackets requires evaluating ${\displaystyle \nabla F(\mathbf {a} _{n}-t\gamma _{n}\mathbf {p} _{n})}$, 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 ${\displaystyle \mathbf {p} _{n}=\nabla F(\mathbf {a_{n}} )}$, and use line search to find a suitable step-size ${\displaystyle \gamma _{n}}$, such as one that satisfies the Wolfe conditions.
• Assuming that ${\displaystyle F}$ is twice-differentiable, use its Hessian ${\displaystyle \nabla ^{2}F}$ to estimate ${\displaystyle \|\nabla F(\mathbf {a} _{n}-t\gamma _{n}\mathbf {p} _{n})-\nabla F(\mathbf {a} _{n})\|_{2}\approx \|t\gamma _{n}\nabla ^{2}F(\mathbf {a} _{n})\mathbf {p} _{n}\|.}$Then choose ${\displaystyle \mathbf {p} _{n}}$ and ${\displaystyle \gamma _{n}}$ by optimising inequality (1).
• Assuming that ${\displaystyle \nabla F}$ is Lipschitz, use its Lipschitz constant ${\displaystyle L}$ to bound ${\displaystyle \|\nabla F(\mathbf {a} _{n}-t\gamma _{n}\mathbf {p} _{n})-\nabla F(\mathbf {a} _{n})\|_{2}\leq Lt\gamma _{n}\|\mathbf {p} _{n}\|.}$ Then choose ${\displaystyle \mathbf {p} _{n}}$ and ${\displaystyle \gamma _{n}}$ by optimising inequality (1).
• Build a custom model of ${\displaystyle \max _{t\in [0,1]}{\frac {\|\nabla F(\mathbf {a} _{n}-t\gamma _{n}\mathbf {p} _{n})-\nabla F(\mathbf {a} _{n})\|_{2}}{\|\nabla F(\mathbf {a} _{n})\|_{2}}}}$ for ${\displaystyle F}$. Then choose ${\displaystyle \mathbf {p} _{n}}$ and ${\displaystyle \gamma _{n}}$ by optimising inequality (1).
• Under stronger assumptions on the function ${\displaystyle F}$ such as convexity, more advanced techniques may be possible.

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

Solution of a linear system

The steepest descent algorithm applied to the Wiener filter^[11]

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

${\displaystyle A\mathbf {x} -\mathbf {b} =0}$

reformulated as a quadratic minimization problem. If the system matrix ${\displaystyle A}$ is real symmetric and positive-definite, the quadratic function to minimize commonly is

${\displaystyle F(\mathbf {x} )=\mathbf {x} ^{T}A\mathbf {x} -2\mathbf {x} ^{T}\mathbf {b} ,}$

so that

${\displaystyle \nabla F(\mathbf {x} )=2(A\mathbf {x} -\mathbf {b} ).}$

For a general real matrix ${\displaystyle A}$, linear least squares define

${\displaystyle F(\mathbf {x} )=\left\|A\mathbf {x} -\mathbf {b} \right\|^{2}.}$

In traditional linear least squares for real ${\displaystyle A}$ and ${\displaystyle \mathbf {b} }$ the Euclidean norm is used, in which case

${\displaystyle \nabla F(\mathbf {x} )=2A^{T}(A\mathbf {x} -\mathbf {b} ).}$

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

For example, for real symmetric and positive-definite matrix ${\displaystyle A}$, a simple algorithm can be as follows,^[5]

${\displaystyle {\begin{aligned}&{\text{repeat in the loop:}}\\&\qquad \mathbf {r} :=\mathbf {b} -\mathbf {Ax} \\&\qquad \gamma :={\mathbf {r} ^{\mathsf {T}}\mathbf {r} }/{\mathbf {r} ^{\mathsf {T}}\mathbf {Ar} }\\&\qquad \mathbf {x} :=\mathbf {x} +\gamma \mathbf {r} \\&\qquad {\hbox{if }}\mathbf {r} ^{\mathsf {T}}\mathbf {r} {\text{ is sufficiently small, then exit loop}}\\&{\text{end repeat loop}}\\&{\text{return }}\mathbf {x} {\text{ as the result}}\end{aligned}}}$

To avoid multiplying by ${\displaystyle A}$ twice per iteration, we note that ${\displaystyle \mathbf {x} :=\mathbf {x} +\gamma \mathbf {r} }$ implies ${\displaystyle \mathbf {r} :=\mathbf {r} -\gamma \mathbf {Ar} }$, which gives the traditional algorithm,^[13]

${\displaystyle {\begin{aligned}&\mathbf {r} :=\mathbf {b} -\mathbf {Ax} \\&{\text{repeat in the loop:}}\\&\qquad \gamma :={\mathbf {r} ^{\mathsf {T}}\mathbf {r} }/{\mathbf {r} ^{\mathsf {T}}\mathbf {Ar} }\\&\qquad \mathbf {x} :=\mathbf {x} +\gamma \mathbf {r} \\&\qquad {\hbox{if }}\mathbf {r} ^{\mathsf {T}}\mathbf {r} {\text{ is sufficiently small, then exit loop}}\\&\qquad \mathbf {r} :=\mathbf {r} -\gamma \mathbf {Ar} \\&{\text{end repeat loop}}\\&{\text{return }}\mathbf {x} {\text{ as the result}}\end{aligned}}}$

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 ${\displaystyle \kappa (A)}$ of the system matrix ${\displaystyle A}$ (the ratio of the maximum to minimum eigenvalues of ${\displaystyle A^{T}A}$), while the convergence of conjugate gradient method is typically determined by a square root of the condition number, i.e., is much faster. Both methods can benefit from preconditioning, where gradient descent may require less assumptions on the preconditioner.^[13]

Solution of a non-linear system

Gradient descent can also be used to solve a system of nonlinear equations. Below is an example that shows how to use the gradient descent to solve for three unknown variables, x₁, x₂, and x₃. This example shows one iteration of the gradient descent.

Consider the nonlinear system of equations

${\displaystyle {\begin{cases}3x_{1}-\cos(x_{2}x_{3})-{\tfrac {3}{2}}=0\\4x_{1}^{2}-625x_{2}^{2}+2x_{2}-1=0\\\exp(-x_{1}x_{2})+20x_{3}+{\tfrac {10\pi -3}{3}}=0\end{cases}}}$

Let us introduce the associated function

${\displaystyle G(\mathbf {x} )={\begin{bmatrix}3x_{1}-\cos(x_{2}x_{3})-{\tfrac {3}{2}}\\4x_{1}^{2}-625x_{2}^{2}+2x_{2}-1\\\exp(-x_{1}x_{2})+20x_{3}+{\tfrac {10\pi -3}{3}}\\\end{bmatrix}},}$

where

${\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\end{bmatrix}}.}$

One might now define the objective function

${\displaystyle F(\mathbf {x} )={\frac {1}{2}}G^{\mathrm {T} }(\mathbf {x} )G(\mathbf {x} )={\frac {1}{2}}\left[\left(3x_{1}-\cos(x_{2}x_{3})-{\frac {3}{2}}\right)^{2}+\left(4x_{1}^{2}-625x_{2}^{2}+2x_{2}-1\right)^{2}+\left(\exp(-x_{1}x_{2})+20x_{3}+{\frac {10\pi -3}{3}}\right)^{2}\right],}$

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

${\displaystyle \mathbf {x} ^{(0)}=\mathbf {0} ={\begin{bmatrix}0\\0\\0\\\end{bmatrix}}.}$

We know that

${\displaystyle \mathbf {x} ^{(1)}=\mathbf {0} -\gamma _{0}\nabla F(\mathbf {0} )=\mathbf {0} -\gamma _{0}J_{G}(\mathbf {0} )^{\mathrm {T} }G(\mathbf {0} ),}$

where the Jacobian matrix ${\displaystyle J_{G}}$ is given by

${\displaystyle J_{G}(\mathbf {x} )={\begin{bmatrix}3&\sin(x_{2}x_{3})x_{3}&\sin(x_{2}x_{3})x_{2}\\8x_{1}&-1250x_{2}+2&0\\-x_{2}\exp {(-x_{1}x_{2})}&-x_{1}\exp(-x_{1}x_{2})&20\\\end{bmatrix}}.}$

We calculate:

${\displaystyle J_{G}(\mathbf {0} )={\begin{bmatrix}3&0&0\\0&2&0\\0&0&20\end{bmatrix}},\qquad G(\mathbf {0} )={\begin{bmatrix}-2.5\\-1\\10.472\end{bmatrix}}.}$

Thus

${\displaystyle \mathbf {x} ^{(1)}=\mathbf {0} -\gamma _{0}{\begin{bmatrix}-7.5\\-2\\209.44\end{bmatrix}},}$

and

${\displaystyle F(\mathbf {0} )=0.5\left((-2.5)^{2}+(-1)^{2}+(10.472)^{2}\right)=58.456.}$

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

Now, a suitable ${\displaystyle \gamma _{0}}$ must be found such that

${\displaystyle F\left(\mathbf {x} ^{(1)}\right)\leq F\left(\mathbf {x} ^{(0)}\right)=F(\mathbf {0} ).}$

This can be done with any of a variety of line search algorithms. One might also simply guess ${\displaystyle \gamma _{0}=0.001,}$ which gives

${\displaystyle \mathbf {x} ^{(1)}={\begin{bmatrix}0.0075\\0.002\\-0.20944\\\end{bmatrix}}.}$

Evaluating the objective function at this value, yields

${\displaystyle F\left(\mathbf {x} ^{(1)}\right)=0.5\left((-2.48)^{2}+(-1.00)^{2}+(6.28)^{2}\right)=23.306.}$

The decrease from ${\displaystyle F(\mathbf {0} )=58.456}$ to the next step's value of

${\displaystyle F\left(\mathbf {x} ^{(1)}\right)=23.306}$

is a sizable decrease in the objective function. Further steps would reduce its value further, until an approximate solution to the system was found.

Comments

Gradient descent works in spaces of any number of dimensions, even in infinite-dimensional ones. In the latter case, the search space is typically a function space, and one calculates the Fréchet derivative of the functional to be minimized to determine the descent direction.^[6]

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

The gradient descent can take many iterations to compute a local minimum with a required accuracy, if the curvature in different directions is very different for the given function. For such functions, preconditioning, which changes the geometry of the space to shape the function level sets like concentric circles, cures the slow convergence. Constructing and applying preconditioning can be computationally expensive, however.

The gradient descent can be combined with a line search, finding the locally optimal step size ${\displaystyle \gamma }$ on every iteration. Performing the line search can be time-consuming. Conversely, using a fixed small ${\displaystyle \gamma }$ can yield poor convergence.

Methods based on Newton's method and inversion of the Hessian using conjugate gradient techniques can be better alternatives.^[14][15] Generally, such methods converge in fewer iterations, but the cost of each iteration is higher. An example is the BFGS method which consists in calculating on every step a matrix by which the gradient vector is multiplied to go into a "better" direction, combined with a more sophisticated line search algorithm, to find the "best" value of ${\displaystyle \gamma .}$ For extremely large problems, where the computer-memory issues dominate, a limited-memory method such as L-BFGS should be used instead of BFGS or the steepest descent.

Gradient descent can be viewed as applying Euler's method for solving ordinary differential equations ${\displaystyle x'(t)=-\nabla f(x(t))}$ to a gradient flow. In turn, this equation may be derived as an optimal controller^[16] for the control system ${\displaystyle x'(t)=u(t)}$ with ${\displaystyle u(t)}$ given in feedback form ${\displaystyle u(t)=-\nabla f(x(t))}$.

Modifications

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

Fast gradient methods

Yurii Nesterov has proposed^[17] a simple modification that enables faster convergence for convex problems and has been since further generalized. For unconstrained smooth problems the method is called the Fast Gradient Method (FGM) or the Accelerated Gradient Method (AGM). Specifically, if the differentiable function ${\displaystyle F}$ is convex and ${\displaystyle \nabla F}$ is Lipschitz, and it is not assumed that ${\displaystyle F}$ is strongly convex, then the error in the objective value generated at each step ${\displaystyle k}$ by the gradient descent method will be bounded by ${\textstyle {\mathcal {O}}\left({\tfrac {1}{k}}\right)}$. Using the Nesterov acceleration technique, the error decreases at ${\textstyle {\mathcal {O}}\left({\tfrac {1}{k^{2}}}\right)}$.^[18] It is known that the rate ${\displaystyle {\mathcal {O}}\left({k^{-2}}\right)}$ for the decrease of the cost function is optimal for first-order optimization methods. Nevertheless, there is the opportunity to improve the algorithm by reducing the constant factor. The optimized gradient method (OGM)^[19] reduces that constant by a factor of two and is an optimal first-order method for large-scale problems.^[20]

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

Momentum or heavy ball method

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

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

Extensions

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

See also

• Backtracking line search
• Conjugate gradient method
• Stochastic gradient descent
• Rprop
• Delta rule
• Wolfe conditions
• Preconditioning
• Broyden–Fletcher–Goldfarb–Shanno algorithm
• Davidon–Fletcher–Powell formula
• Nelder–Mead method
• Gauss–Newton algorithm
• Hill climbing
• Quantum annealing

References

1. Lemaréchal, C. (2012). "Cauchy and the Gradient Method" (PDF). Doc Math Extra: 251–254.
2. Hadamard, Jacques (1908). "Mémoire sur le problème d'analyse relatif à Véquilibre des plaques élastiques encastrées,". Mémoires présentés par divers savants éstrangers à l'Académie des Sciences de l'Institut de France. 33.
3. Courant, Richard (January 1943). "Variational methods for the solution of problems of equilibrium and vibrations". Bull. Amer. Math. Soc. 49: 1–23 – via Project Euclid.
4. Curry, Haskell B. (1944). "The Method of Steepest Descent for Non-linear Minimization Problems". Quart. Appl. Math. 2 (3): 258–261. doi:10.1090/qam/10667.
5. Polyak, Boris (1987). Introduction to Optimization.
6. Akilov, G. P.; Kantorovich, L. V. (1982). Functional Analysis (2nd ed.). Pergamon Press. ISBN 0-08-023036-9.
7. Barzilai, Jonathan; Borwein, Jonathan M. (1988). "Two-Point Step Size Gradient Methods". IMA Journal of Numerical Analysis. 8 (1): 141–148. doi:10.1093/imanum/8.1.141.
8. Fletcher, R. (2005). "On the Barzilai–Borwein Method". In Qi, L.; Teo, K.; Yang, X. (eds.). Optimization and Control with Applications. Applied Optimization. 96. Boston: Springer. pp. 235–256. ISBN 0-387-24254-6.
9. Wolfe, Philip (1969-04-01). "Convergence Conditions for Ascent Methods". SIAM Review. 11 (2): 226–235. doi:10.1137/1011036. ISSN 0036-1445.
10. Bernstein, Jeremy; Vahdat, Arash; Yue, Yisong; Liu, Ming-Yu (2020-06-12). "On the distance between two neural networks and the stability of learning". arXiv:2002.03432 [cs.LG].
11. Haykin, Simon S. Adaptive filter theory. Pearson Education India, 2008. - p. 108-142, 217-242
12. Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed.). Philadelphia, Pa.: Society for Industrial and Applied Mathematics. pp. 195. ISBN 978-0-89871-534-7.
13. Bouwmeester, Henricus; Dougherty, Andrew; Knyazev, Andrew V. (2015). "Nonsymmetric Preconditioning for Conjugate Gradient and Steepest Descent Methods". Procedia Computer Science. 51: 276–285. doi:10.1016/j.procs.2015.05.241.
14. Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (1992). Numerical Recipes in C: The Art of Scientific Computing (2nd ed.). New York: Cambridge University Press. ISBN 0-521-43108-5.
15. Strutz, T. (2016). Data Fitting and Uncertainty: A Practical Introduction to Weighted Least Squares and Beyond (2nd ed.). Springer Vieweg. ISBN 978-3-658-11455-8.
16. Ross, I. M. (2019-07-01). "An optimal control theory for nonlinear optimization". Journal of Computational and Applied Mathematics. 354: 39–51. doi:10.1016/j.cam.2018.12.044. ISSN 0377-0427.
17. Nesterov, Yu. (2004). Introductory Lectures on Convex Optimization : A Basic Course. Springer. ISBN 1-4020-7553-7.
18. Vandenberghe, Lieven (2019). "Fast Gradient Methods" (PDF). Lecture notes for EE236C at UCLA.
19. Kim, D.; Fessler, J. A. (2016). "Optimized First-order Methods for Smooth Convex Minimization". Math. Prog. 151 (1–2): 81–107. arXiv:1406.5468. doi:10.1007/s10107-015-0949-3. PMC 5067109. PMID 27765996. S2CID 207055414.
20. Drori, Yoel (2017). "The Exact Information-based Complexity of Smooth Convex Minimization". Journal of Complexity. 39: 1–16. arXiv:1606.01424. doi:10.1016/j.jco.2016.11.001. S2CID 205861966.
21. Qian, Ning (January 1999). "On the momentum term in gradient descent learning algorithms" (PDF). Neural Networks. 12 (1): 145–151. CiteSeerX 10.1.1.57.5612. doi:10.1016/S0893-6080(98)00116-6. PMID 12662723. Archived from the original (PDF) on 8 May 2014. Retrieved 17 October 2014.
22. "Momentum and Learning Rate Adaptation". Willamette University. Retrieved 17 October 2014.
23. Geoffrey Hinton; Nitish Srivastava; Kevin Swersky. "The momentum method". Coursera. Retrieved 2 October 2018. Part of a lecture series for the Coursera online course Neural Networks for Machine Learning Archived 2016-12-31 at the Wayback Machine.
24. Combettes, P. L.; Pesquet, J.-C. (2011). "Proximal splitting methods in signal processing". In Bauschke, H. H.; Burachik, R. S.; Combettes, P. L.; Elser, V.; Luke, D. R.; Wolkowicz, H. (eds.). Fixed-Point Algorithms for Inverse Problems in Science and Engineering. New York: Springer. pp. 185–212. arXiv:0912.3522. ISBN 978-1-4419-9568-1.

Further reading

• Boyd, Stephen; Vandenberghe, Lieven (2004). "Unconstrained Minimization" (PDF). Convex Optimization. New York: Cambridge University Press. pp. 457–520. ISBN 0-521-83378-7.
• Chong, Edwin K. P.; Żak, Stanislaw H. (2013). "Gradient Methods". An Introduction to Optimization (Fourth ed.). Hoboken: Wiley. pp. 131–160. ISBN 978-1-118-27901-4.
• Himmelblau, David M. (1972). "Unconstrained Minimization Procedures Using Derivatives". Applied Nonlinear Programming. New York: McGraw-Hill. pp. 63–132. ISBN 0-07-028921-2.

External links

Wikimedia Commons has media related to Gradient descent.

• Using gradient descent in C++, Boost, Ublas for linear regression
• Series of Khan Academy videos discusses gradient ascent
• Online book teaching gradient descent in deep neural network context
• "Gradient Descent, How Neural Networks Learn". 3Blue1Brown. October 16, 2017 – via YouTube.