Gradient descent

 ( Mathematical Optimization ) 

Gradient descent is a method for unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate function.

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. It is particularly useful in machine learning for minimizing the cost or loss function.

Gradient descent should not be confused with local search algorithms, although both are Iterative method for optimization.

Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847.

Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades.

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

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Description

Gradient descent is based on the observation that if the multi-variable function $f(\backslash mathbf\{x\})$ is defined and differentiable in a neighborhood of a point $\backslash mathbf\{a\}$, then $f(\backslash mathbf\{x\})$ decreases fastest if one goes from $\backslash mathbf\{a\}$ in the direction of the negative gradient of $f$ at $\backslash mathbf\{a\},\; -\backslash nabla\; f(\backslash mathbf\{a\})$. It follows that, if

$\backslash mathbf\{a\}\_\{n+1\}\; =\; \backslash mathbf\{a\}\_n-\backslash eta\; \backslash nabla\; f(\backslash mathbf\{a\}\_n)$

for a small enough step size or learning rate $\backslash eta\backslash in\; \backslash R\_\{+\}$, then $f(\backslash mathbf\{a\_n\})\backslash geq\; f(\backslash mathbf\{a\_\{n+1\}\})$. In other words, the term $\backslash eta\backslash nabla\; f(\backslash mathbf\{a\})$ is subtracted from $\backslash mathbf\{a\}$ because we want to move against the gradient, toward the local minimum. With this observation in mind, one starts with a guess $\backslash mathbf\{x\}\_0$ for a local minimum of $f$, and considers the sequence $\backslash mathbf\{x\}\_0,\; \backslash mathbf\{x\}\_1,\; \backslash mathbf\{x\}\_2,\; \backslash ldots$ such that

$\backslash mathbf\{x\}\_\{n+1\}=\backslash mathbf\{x\}\_n-\backslash eta\_n\backslash nabla\; f(\backslash mathbf\{x\}\_n),\backslash \; n\; \backslash ge\; 0.$

We have a monotonic sequence

$f(\backslash mathbf\{x\}\_0)\backslash ge\; f(\backslash mathbf\{x\}\_1)\backslash ge\; f(\backslash mathbf\{x\}\_2)\backslash ge\; \backslash cdots,$

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

It is possible to guarantee the convergence to a local minimum under certain assumptions on the function $f$ (for example, $f$ Convex function and $\backslash nabla\; f$ Lipschitz) and particular choices of $\backslash eta$. Those include the sequence

$\backslash eta\_\{n\}\; =\; \backslash frac\{\; \backslash left\; |\; \backslash left\; (\backslash mathbf\; x\_\{n\}\; -\; \backslash mathbf\; x\_\{n-1\}\; \backslash right\; )^\backslash top\; \backslash left\; \backslash nabla\; \backslash right\; |\}\{\backslash left\; \backslash |\backslash nabla\; f(\backslash mathbf\{x\}\_\{n\})\; -\; \backslash nabla\; f(\backslash mathbf\{x\}\_\{n-1\})\; \backslash right\; \backslash |^2\}$

as in the Barzilai-Borwein method,

R. Fletcher (2025). 9780387242545, Springer. ISBN 9780387242545

or a sequence $\backslash eta\_n$ satisfying the Wolfe conditions (which can be found by using line search). When the function $f$ is Convex function, all local minima are also global minima, so in this case gradient descent can converge to the global solution.

This process is illustrated in the adjacent picture. Here, $f$ is assumed to be defined on the plane, and that its graph has a bowl shape. The blue curves are the , that is, the regions on which the value of $f$ is constant. A red arrow originating at a point shows the direction of the negative gradient at that point. Note that the (negative) gradient at a point is orthogonal to the contour line going through that point. We see that gradient descent leads us to the bottom of the bowl, that is, to the point where the value of the function $f$ is minimal.

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An analogy for understanding gradient descent

The basic intuition behind gradient descent can be illustrated by a hypothetical scenario. People are stuck in the mountains and are trying to get down (i.e., trying to find the global minimum). There is heavy fog such that visibility is extremely low. Therefore, the path down the mountain is not visible, so they must use local information to find the minimum. They can use the method of gradient descent, which involves looking at the steepness of the hill at their current position, then proceeding in the direction with the steepest descent (i.e., downhill). If they were trying to find the top of the mountain (i.e., the maximum), then they would proceed in the direction of steepest ascent (i.e., uphill). Using this method, they would eventually find their way down the mountain or possibly get stuck in some hole (i.e., local minimum or saddle point), like a mountain lake. However, assume also that the steepness of the hill is not immediately obvious with simple observation, but rather it requires a sophisticated instrument to measure, which the persons happen to have at the moment. It takes quite some time to measure the steepness of the hill with the instrument, thus they should minimize their use of the instrument if they wanted to get down the mountain before sunset. The difficulty then is choosing the frequency at which they should measure the steepness of the hill so not to go off track.

In this analogy, the persons represent the algorithm, and the path taken down the mountain represents the sequence of parameter settings that the algorithm will explore. The steepness of the hill represents the slope of the function at that point. The instrument used to measure steepness is differentiation. The direction they choose to travel in aligns with the gradient of the function at that point. The amount of time they travel before taking another measurement is the step size.

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Choosing the step size and descent direction

Since using a step size $\backslash eta$ that is too small would slow convergence, and a $\backslash eta$ too large would lead to overshoot and divergence, finding a good setting of $\backslash eta$ is an important practical problem. Philip Wolfe also advocated using "clever choices of the descent direction" in practice. 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 $\backslash mathbf\{p\}\_n$ and step size $\backslash eta\_n$ and consider the more general update:

$\backslash mathbf\{a\}\_\{n+1\}\; =\; \backslash mathbf\{a\}\_n-\backslash eta\_n\backslash ,\backslash mathbf\{p\}\_n$.

Finding good settings of $\backslash mathbf\{p\}\_n$ and $\backslash eta\_n$ requires some thought. First of all, we would like the update direction to point downhill. Mathematically, letting $\backslash theta\_n$ denote the angle between $-\backslash nabla\; f(\backslash mathbf\{a\_n\})$ and $\backslash mathbf\{p\}\_n$, this requires that $\backslash cos\; \backslash theta\_n\; >\; 0.$ To say more, we need more information about the objective function that we are optimising. Under the fairly weak assumption that $f$ is continuously differentiable, we may prove that:

This inequality implies that the amount by which we can be sure the function $f$ is decreased depends on a trade off between the two terms in square brackets. The first term in square brackets measures the angle between the descent direction and the negative gradient. The second term measures how quickly the gradient changes along the descent direction.

In principle inequality () could be optimized over $\backslash mathbf\{p\}\_n$ and $\backslash eta\_n$ to choose an optimal step size and direction. The problem is that evaluating the second term in square brackets requires evaluating $\backslash nabla\; f(\backslash mathbf\{a\}\_n\; -\; t\; \backslash eta\_n\; \backslash mathbf\{p\}\_n)$, 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 $\backslash mathbf\{p\}\_n\; =\; \backslash nabla\; f(\backslash mathbf\{a\_n\})$, and use line search to find a suitable step-size $\backslash gamma\_n$, 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 $\backslash mathbf\{p\}\_n$ 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 $\backslash eta\_n$).
• Assuming that $f$ is twice-differentiable, use its Hessian $\backslash nabla^2\; f$ to estimate $\backslash |\backslash nabla\; f(\backslash mathbf\{a\}\_n\; -\; t\; \backslash eta\_n\; \backslash mathbf\{p\}\_n)\; -\; \backslash nabla\; f(\backslash mathbf\{a\}\_n)\backslash |\_2\; \backslash approx\; \backslash |\; t\; \backslash eta\_n\; \backslash nabla^2\; f(\backslash mathbf\{a\}\_n)\; \backslash mathbf\{p\}\_n\backslash |.$Then choose $\backslash mathbf\{p\}\_n$ and $\backslash eta\_n$ by optimising inequality ().
• Assuming that $\backslash nabla\; f$ is Lipschitz, use its Lipschitz constant $L$ to bound $\backslash |\backslash nabla\; f(\backslash mathbf\{a\}\_n\; -\; t\; \backslash eta\_n\; \backslash mathbf\{p\}\_n)\; -\; \backslash nabla\; f(\backslash mathbf\{a\}\_n)\backslash |\_2\; \backslash leq\; L\; t\; \backslash eta\_n\; \backslash |\backslash mathbf\{p\}\_n\backslash |.$ Then choose $\backslash mathbf\{p\}\_n$ and $\backslash eta\_n$ by optimising inequality ().
• Build a custom model of $\backslash max\_\{t\backslash in0,1\}\; \backslash frac\{\backslash |\backslash nabla\; f(\backslash mathbf\{a\}\_n\; -\; t\; \backslash eta\_n\; \backslash mathbf\{p\}\_n)\; -\; \backslash nabla\; f(\backslash mathbf\{a\}\_n)\backslash |\_2\}\{\backslash |\; \backslash nabla\; f(\backslash mathbf\{a\}\_n)\; \backslash |\_2\}$ for $f$. Then choose $\backslash mathbf\{p\}\_n$ and $\backslash eta\_n$ by optimising inequality ().
• Under stronger assumptions on the function $f$ such as Convex function, 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 $f$ is Convex function, all local minima are also global minima, so in this case gradient descent can converge to the global solution.

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Solution of a linear system

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

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

reformulated as a quadratic minimization problem. If the system matrix $\backslash mathbf\{A\}$ is real Symmetric matrix and positive-definite, an objective function is defined as the quadratic function, with minimization of

$f(\backslash mathbf\{x\})=\backslash mathbf\{x\}^\backslash top\; \backslash mathbf\{A\}\backslash mathbf\{x\}-2\backslash mathbf\{x\}^\backslash top\; \backslash mathbf\{b\},$

so that

$\backslash nabla\; f(\backslash mathbf\{x\})=2(\backslash mathbf\{A\}\backslash mathbf\{x\}-\backslash mathbf\{b\}).$

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

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

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

$\backslash nabla\; f(\backslash mathbf\{x\})=2\backslash mathbf\{A\}^\backslash top(\backslash mathbf\{A\}\backslash mathbf\{x\}-\backslash mathbf\{b\}).$

The line search minimization, finding the locally optimal step size $\backslash eta$ on every iteration, can be performed analytically for quadratic functions, and explicit formulas for the locally optimal $\backslash eta$ are known.

Yousef Saad (2025). 9780898715347, Society for Industrial and Applied Mathematics. . ISBN 9780898715347

For example, for real Symmetric matrix and positive-definite matrix $\backslash mathbf\{A\}$, a simple algorithm can be as follows,

$\backslash begin\{align\}$

& \text{repeat in the loop:} \\ & \qquad \mathbf{r} := \mathbf{b} - \mathbf{A x} \\ & \qquad \eta := {\mathbf{r}^\top \mathbf{r}}/{\mathbf{r}^\top \mathbf{A r}} \\ & \qquad \mathbf{x} := \mathbf{x} + \eta\mathbf{r} \\ & \qquad \hbox{if } \mathbf{r}^\top \mathbf{r} \text{ is sufficiently small, then exit loop} \\ & \text{end repeat loop} \\ & \text{return } \mathbf{x} \text{ as the result} \end{align}

To avoid multiplying by $\backslash mathbf\{A\}$ twice per iteration, we note that $\backslash mathbf\{x\}\; :=\; \backslash mathbf\{x\}\; +\; \backslash eta\backslash mathbf\{r\}$ implies $\backslash mathbf\{r\}\; :=\; \backslash mathbf\{r\}\; -\; \backslash eta\backslash mathbf\{A\; r\}$, which gives the traditional algorithm,

$\backslash begin\{align\}$

& \mathbf{r} := \mathbf{b} - \mathbf{A x} \\ & \text{repeat in the loop:} \\ & \qquad \eta := {\mathbf{r}^\top \mathbf{r}}/{\mathbf{r}^\top \mathbf{A r}} \\ & \qquad \mathbf{x} := \mathbf{x} + \eta\mathbf{r} \\ & \qquad \hbox{if } \mathbf{r}^\top \mathbf{r} \text{ is sufficiently small, then exit loop} \\ & \qquad \mathbf{r} := \mathbf{r} - \eta\mathbf{A r} \\ & \text{end repeat loop} \\ & \text{return } \mathbf{x} \text{ as the result} \end{align}

The method is rarely used for solving linear equations, with the conjugate gradient method being one of the most popular alternatives. The number of gradient descent iterations is commonly proportional to the spectral condition number $\backslash kappa(\backslash mathbf\{A\})$ of the system matrix $\backslash mathbf\{A\}$ (the ratio of the maximum to minimum eigenvalues of , while the convergence of conjugate gradient method is typically determined by a square root of the condition number, i.e., is much faster. Both methods can benefit from Preconditioner, where gradient descent may require less assumptions on the preconditioner.

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Geometric behavior and residual orthogonality In steepest descent applied to solving $\backslash mathbf\{A\; x\}\; =\; \backslash mathbf\{b\}$, where $\backslash mathbf\{A\}$ is symmetric positive-definite, the residual vectors $\backslash mathbf\{r\}\_k\; =\; \backslash mathbf\{b\}\; -\; \backslash mathbf\{A\}\backslash mathbf\{x\}\_k$ are orthogonal across iterations:

$$

\mathbf{r}_{k+1}^\top \mathbf{r}_k = 0.

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 $\backslash kappa(\backslash mathbf\{A\})$ is large, this produces a characteristic zig-zag path. The poor conditioning of $\backslash mathbf\{A\}$ is the primary cause of the slow convergence, and orthogonality of successive residuals reinforces this alternation.

As shown in the image on the right, steepest descent converges slowly due to the high condition number of $\backslash mathbf\{A\}$, 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.

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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

$\backslash begin\{cases\}$

3x_1-\cos(x_2x_3)-\tfrac{3}{2} =0 \\ 4x_1^2-625x_2^2+2x_2-1 = 0 \\ \exp(-x_1x_2)+20x_3+\tfrac{10\pi-3}{3} =0 \end{cases}

Let us introduce the associated function

$G(\backslash mathbf\{x\})\; =\; \backslash begin\{bmatrix\}$

3x_1-\cos(x_2x_3)-\tfrac{3}{2} \\ 4x_1^2-625x_2^2+2x_2-1 \\ \exp(-x_1x_2)+20x_3+\tfrac{10\pi-3}{3} \\ \end{bmatrix},

where

$\backslash mathbf\{x\}\; =\backslash begin\{bmatrix\}$

 x_1 \\
 x_2 \\
 x_3 \\
     

\end{bmatrix}.

One might now define the objective function

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

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

$\backslash mathbf\{x\}^\{(0)\}=\; \backslash mathbf\{0\}\; =\; \backslash begin\{bmatrix\}$

 0 \\
 0 \\
 0 \\ \end{bmatrix}.
     

We know that

$\backslash mathbf\{x\}^\{(1)\}=\backslash mathbf\{0\}-\backslash eta\_0\; \backslash nabla\; f(\backslash mathbf\{0\})\; =\; \backslash mathbf\{0\}-\backslash eta\_0\; J\_G(\backslash mathbf\{0\})^\backslash top\; G(\backslash mathbf\{0\}),$

where the Jacobian matrix $J\_G$ is given by

$J\_G(\backslash mathbf\{x\})\; =\; \backslash begin\{bmatrix\}$

 3 & \sin(x_2x_3)x_3 & \sin(x_2x_3)x_2   \\
 8x_1 & -1250x_2+2 & 0 \\
 -x_2\exp{(-x_1x_2)} & -x_1\exp(-x_1x_2) & 20\\
     

\end{bmatrix}.

We calculate:

$J\_G(\backslash mathbf\{0\})\; =\; \backslash begin\{bmatrix\}$

 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

$\backslash mathbf\{x\}^\{(1)\}=\; \backslash mathbf\{0\}-\backslash eta\_0\; \backslash begin\{bmatrix\}$

 -7.5\\
 -2\\
 209.44
     

\end{bmatrix},

and

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

Now, a suitable $\backslash eta\_0$ must be found such that

$f\backslash left\; (\backslash mathbf\{x\}^\{(1)\}\backslash right\; )\; \backslash le\; f\backslash left\; (\backslash mathbf\{x\}^\{(0)\}\backslash right\; )\; =\; f(\backslash mathbf\{0\}).$

This can be done with any of a variety of line search algorithms. One might also simply guess $\backslash eta\_0=0.001,$ which gives

$\backslash mathbf\{x\}^\{(1)\}=\backslash begin\{bmatrix\}$

  0.0075  \\
  0.002   \\
 -0.20944 \\
     

\end{bmatrix}.

Evaluating the objective function at this value, yields

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

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

$f\; \backslash left\; (\backslash mathbf\{x\}^\{(1)\}\backslash right\; )\; =23.306$

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

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

(1982). 9780080230368, Pergamon Press. ISBN 9780080230368

That gradient descent works in any number of dimensions (finite number at least) can be seen as a consequence of the Cauchy-Schwarz inequality, i.e. 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 (Nesterov, Polyak, and Frank-Wolfe) and heavy-ball parameters (exponential moving averages and positive-negative momentum). The main examples of such optimizers are Adam, DiffGrad, Yogi, AdaBelief, etc.

Methods based on Newton's method and inversion of the Hessian matrix using conjugate gradient techniques can be better alternatives.

T. Strutz (2025). 9783658114558, Springer Vieweg. ISBN 9783658114558

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 $\backslash eta.$ 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.

While it is sometimes possible to substitute gradient descent for a local search algorithm, gradient descent is not in the same family: although it is an iterative method for local optimization, it relies on an loss function rather than an explicit exploration of a Feasible region.

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

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Modifications Gradient descent can converge to a local minimum and slow down in a neighborhood of a saddle point. Even for unconstrained quadratic minimization, gradient descent develops a 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.

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Fast gradient methods Yurii Nesterov has proposed

Yurii Nesterov (2025). 9781402075537, Springer. ISBN 9781402075537

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

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

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

This technique is used in stochastic gradient descent and as an extension to the backpropagation algorithms used to train artificial neural networks. Part of a lecture series for the Coursera online course Neural Networks for Machine Learning . In the direction of updating, stochastic gradient descent adds a stochastic property. The weights can be used to calculate the derivatives.

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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).

(2025). 9781441995681, Springer. ISBN 9781441995681

Gradient descent is a special case of mirror descent using the squared Euclidean distance as the given Bregman divergence.

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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. For example, if the objective is assumed to be strongly convex and lipschitz smooth, then gradient descent converges linearly with a fixed step size. Looser assumptions lead to either weaker convergence guarantees or require a more sophisticated step size selection.

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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
• CLS (continuous local search)
• Neuroevolution

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Further reading

• (2025). 9780521833783, Cambridge University Press. ISBN 9780521833783
• (2025). 9781118279014, Wiley. ISBN 9781118279014
• David M. Himmelblau (1972). 9780070289215, McGraw-Hill. ISBN 9780070289215

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External links

• 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
• Archived at Ghostarchive and the target="_blank" rel="nofollow"> Wayback Machine:

Categories: Mathematical Optimization, First Order Methods, Optimization Algorithms And Methods, Gradient Methods

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