Gradient descent
In mathematics gradient descent (also often called steepest descent) is a iterative optimization algorithm for finding a local minimum of a differentiable 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 local maximum of that function; the procedure is then known as gradient ascent. Gradient descent is generally attributed to Cauchy, who first suggested it in 1847. 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. Description. Gradient descent is based on the observation that if the multi-variable function formula_1 is defined and differentiable in a neighborhood of a point formula_2, then formula_1 decreases "fastest" if one goes from formula_2 in the direction of the negative gradient of formula_5 at formula_6. It follows that, if for a small enough step size or learning rate formula_8, then formula_9. In other words, the term formula_10  is subtracted from formula_2 because we want to move against the gradient, toward the local minimum. With this observation in mind, one starts with a guess formula_12 for a local minimum of formula_5, and considers the sequence formula_14 such that We have a monotonic sequence so, hopefully, the sequence formula_17 converges to the desired local minimum. Note that the value of the "step size" formula_18 is allowed to change at every iteration. With certain assumptions on the function formula_5 (for example, formula_5 convex and formula_21 Lipschitz) and particular choices of formula_18 (e.g., chosen either via a line search that satisfies  the Wolfe conditions, or the Barzilai–Borwein method shown as following), convergence to a local minimum can be guaranteed. When the function formula_5 is 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, formula_5 is assumed to be defined on the plane, and that its graph has a bowl shape. The blue curves are the contour lines, that is, the regions on which the value of formula_5 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 formula_5 is minimal. An analogy for understanding gradient descent. The basic intuition behind gradient descent can be illustrated by a hypothetical scenario. A person is stuck in the mountains and is 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 person happens 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 person represents 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 error surface at that point. The instrument used to measure steepness is differentiation (the slope of the error surface can be calculated by taking the derivative of the squared error function at that point). The direction they choose to travel in aligns with the gradient of the error surface at that point. The amount of time they travel before taking another measurement is the step size. Examples . Gradient descent has problems with pathological functions such as the Rosenbrock function shown here. The Rosenbrock function has a narrow-curved valley that contains the minimum. The bottom of the valley is very flat. Because of the curved flat valley, the optimization is zigzagging slowly with small step sizes towards the minimum. The Whiplash gradient descent, which is a closed-loop algorithm, successfully minimizes this function. The zigzagging nature of the method is also evident below, where the gradient ascent method is applied to Choosing the step size and descent direction. Since using a step size formula_18 that is too  small would slow convergence, and a formula_18 too large would lead to divergence, finding a good setting of formula_18 is an important practical problem. Philip Wolfe also advocated for using "clever choices of the [descent] direction" in practice. Whilst 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, let's use a direction formula_33 and step size formula_34 and consider the more general update: Finding good settings of formula_33 and formula_34  requires a little thought. First of all, we would like the update direction to point downhill. Mathematically, letting formula_38 denote the angle between formula_39 and formula_33, this requires that formula_41 To say more, we need more information about the objective function that we are optimising. Under the fairly weak assumption that formula_5 is continuously differentiable, we may prove that: This inequality implies that the amount by which we can be sure the function formula_5 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 formula_33 and formula_34 to choose an optimal step size and direction. The problem is that evaluating the second term in square brackets requires evaluating formula_46, and extra gradient evaluations are generally expensive and undesirable. Some ways around this problem are: Usually by following one of the recipes above, convergence to a local minimum can be guaranteed. When the function formula_5 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. Gradient descent can be used to solve a system of linear equations reformulated as a quadratic minimization problem. If the system matrix formula_67 is real symmetric and positive-definite, an objective function is defined as the quadratic function, with minimization of so that For a general real matrix formula_67, linear least squares define In traditional linear least squares for real formula_67 and formula_73 the Euclidean norm is used, in which case The line search minimization, finding the locally optimal step size formula_18 on every iteration, can be performed analytically  for quadratic functions, and explicit formulas for the locally optimal formula_18 are known. For example, for real symmetric and positive-definite matrix formula_67, a simple algorithm can be as follows, To avoid multiplying by formula_67 twice per iteration, we note that formula_80 implies formula_81, which gives the traditional algorithm, 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 formula_83 of the system matrix formula_67 (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 preconditioning, where gradient descent may require less assumptions on the preconditioner. 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, "x", "x", and "x". This example shows one iteration of the gradient descent. Consider the nonlinear system of equations Let us introduce the associated function where  One might now define the objective function which we will attempt to minimize. As an initial guess, let us use We know that where the Jacobian matrix formula_91 is given by We calculate: Thus and Now, a suitable formula_96 must be found such that This can be done with any of a variety of line search algorithms. One might also simply guess formula_98 which gives Evaluating the objective function at this value, yields The decrease from formula_101 to the next step's value of 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. 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 formula_18 on every iteration. Performing the line search can be time-consuming. Conversely, using a fixed small formula_18 can yield poor convergence. Methods based on Newton's method and inversion of the Hessian using conjugate gradient techniques can be better alternatives. 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 formula_105 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 formula_106 to a gradient flow. In turn, this equation may be derived as an optimal controller for the control system formula_107 with formula_108 given in feedback form formula_109. However, don't let this analogy to mean that the convergence of the gradient descent in the discrete setting follows from results known for the gradient flow : while the gradient flow has very good convergence guarantees, the situation in the discrete setting is more complicated and one has to be careful in choosing a good learning rate. More generally, it is misleading to think that the good properties of some ODE will automatically guarantee good properties of a corresponding discrete version. 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 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 formula_5 is convex and formula_21 is Lipschitz, and it is not assumed that formula_5 is strongly convex, then the error in the objective value generated at each step formula_113 by the gradient descent method will be bounded by formula_114. Using the Nesterov acceleration technique, the error  decreases at formula_115. It is known that the rate formula_116 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) 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. 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. 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 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). Gradient  descent is a special case of mirror descent using the squared Euclidean distance as the given Bregman divergence.