Gradient descent is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Gradient descent (also known as steepest descent) is a first-order iterative optimization algorithm for finding the minimum of a function which is described in this Wikipedia article.

Use this algorithm to search for minimum values of the bi-variate function:

```  f(x, y) = (x - 1)(x - 1)e^(-y^2) + y(y+2)e^(-2x^2)
```

around x = 0.1 and y = -1.

This book excerpt shows sample C# code for solving this task.

## Go

This is a translation of the C# code in the book excerpt linked to above and hence also of the first Typescript example below.

For some unknown reason the results differ from the other solutions after the first 4 decimal places but are near enough for an approximate method such as this.

`package main import (    "fmt"    "math") func steepestDescent(x []float64, alpha, tolerance float64) {    n := len(x)    h := tolerance    g0 := g(x) // Initial estimate of result.     // Calculate initial gradient.    fi := gradG(x, h)     // Calculate initial norm.    delG := 0.0    for i := 0; i < n; i++ {        delG += fi[i] * fi[i]    }    delG = math.Sqrt(delG)    b := alpha / delG     // Iterate until value is <= tolerance.    for delG > tolerance {        // Calculate next value.        for i := 0; i < n; i++ {            x[i] -= b * fi[i]        }        h /= 2         // Calculate next gradient.        fi = gradG(x, h)         // Calculate next norm.        delG = 0        for i := 0; i < n; i++ {            delG += fi[i] * fi[i]        }        delG = math.Sqrt(delG)        b = alpha / delG         // Calculate next value.        g1 := g(x)         // Adjust parameter.        if g1 > g0 {            alpha /= 2        } else {            g0 = g1        }    }} // Provides a rough calculation of gradient g(x).func gradG(x []float64, h float64) []float64 {    n := len(x)    z := make([]float64, n)    y := make([]float64, n)    copy(y, x)    g0 := g(x)     for i := 0; i < n; i++ {        y[i] += h        z[i] = (g(y) - g0) / h    }    return z} // Function for which minimum is to be found.func g(x []float64) float64 {    return (x-1)*(x-1)*        math.Exp(-x*x) + x*(x+2)*        math.Exp(-2*x*x)} func main() {    tolerance := 0.0000006    alpha := 0.1    x := []float64{0.1, -1} // Initial guess of location of minimum.     steepestDescent(x, alpha, tolerance)    fmt.Println("Testing steepest descent method:")    fmt.Println("The minimum is at x =", x, "\b, x =", x)} `
Output:
```Testing steepest descent method:
The minimum is at x = 0.10764302056464771, x = -1.223351901171944
```

## Julia

`using Optim, Base.MathConstants f(x) = (x - 1) * (x - 1) * e^(-x^2) + x * (x + 2) * e^(-2 * x^2) println(optimize(f, [0.1, -1.0], GradientDescent())) `
Output:
```Results of Optimization Algorithm
* Starting Point: [0.1,-1.0]
* Minimizer: [0.107626844383003,-1.2232596628723371]
* Minimum: -7.500634e-01
* Iterations: 14
* Convergence: true
* |x - x'| ≤ 0.0e+00: false
|x - x'| = 2.97e-09
* |f(x) - f(x')| ≤ 0.0e+00 |f(x)|: true
|f(x) - f(x')| = 0.00e+00 |f(x)|
* |g(x)| ≤ 1.0e-08: true
|g(x)| = 2.54e-09
* Stopped by an increasing objective: false
* Reached Maximum Number of Iterations: false
* Objective Calls: 35
```

## Perl

Calculate with `bignum` for numerical stability.

Translation of: Perl 6
`use strict;use warnings;use bignum; sub steepestDescent {    my(\$alpha, \$tolerance, @x) = @_;    my \$N = @x;    my \$h = \$tolerance;    my \$g0 = g(@x) ;    # Initial estimate of result.     my @fi = gradG(\$h, @x) ;    #  Calculate initial gradient     # Calculate initial norm.    my \$delG = 0;    for (0..\$N-1) { \$delG += \$fi[\$_]**2 }    my \$b = \$alpha / sqrt(\$delG);     while ( \$delG > \$tolerance ) {   # Iterate until value is <= tolerance.       #  Calculate next value.       for (0..\$N-1) { \$x[\$_] -= \$b * \$fi[\$_] }       \$h /= 2;        @fi = gradG(\$h, @x);    # Calculate next gradient.       # Calculate next norm.       \$delG = 0;       for (0..\$N-1) { \$delG += \$fi[\$_]**2 }       \$b = \$alpha / sqrt(\$delG);        my \$g1 = g(@x);   # Calculate next value.        \$g1 > \$g0 ? (\$alpha /= 2) : (\$g0 = \$g1);  # Adjust parameter.    }    @x} # Provides a rough calculation of gradient g(x).sub gradG {    my(\$h, @x) = @_;    my \$N = @x;    my @y = @x;    my \$g0 = g(@x);    my @z;    for (0..\$N-1) { \$y[\$_] += \$h ; \$z[\$_] = (g(@y) - \$g0) / \$h }    return @z} # Function for which minimum is to be found.sub g { my(@x) = @_; (\$x-1)**2 * exp(-\$x**2) + \$x*(\$x+2) * exp(-2*\$x**2) }; my \$tolerance = 0.0000001;my \$alpha     = 0.01;my @x = <0.1 -1>; # Initial guess of location of minimum. printf "The minimum is at x = %.6f, x = %.6f", steepestDescent(\$alpha, \$tolerance, @x);`
Output:
`The minimum is at x = 0.107653, x = -1.223370`

## Perl 6

Translation of: Go
`#!/usr/bin/env perl6 use v6.d; sub steepestDescent(@x, \$alpha is copy, \$h is copy) {    my \$g0 = g(@x) ;    # Initial estimate of result.    my @fi = gradG(@x, \$h, \$g0) ;    #  Calculate initial gradient    # Calculate initial norm.   my \$b = \$alpha / sqrt(my \$delG = sum(map {\$_²}, @fi));    while ( \$delG > \$h ) {   # Iterate until value is <= tolerance.       for @fi.kv -> \$i, \$j { @x[\$i] -= \$b * \$j } #  Calculate next value.       # Calculate next gradient and next value      @fi = gradG(@x, \$h /= 2, my \$g1 = g(@x));         \$b = \$alpha / sqrt(\$delG = sum(map {\$_²}, @fi) );  # Calculate next norm.       \$g1 > \$g0 ?? ( \$alpha /= 2 ) !! ( \$g0 = \$g1 )   # Adjust parameter.   }} sub gradG(@x is copy, \$h, \$g0) { # gives a rough calculation of gradient g(x).   return map { \$_ += \$h ; (g(@x) - \$g0) / \$h }, @x} # Function for which minimum is to be found.sub g(\x) { (x-1)² * exp(-x²) + x*(x+2) * exp(-2*x²) }  my \$tolerance = 0.0000006 ; my \$alpha = 0.1; my @x = 0.1, -1; # Initial guess of location of minimum. steepestDescent(@x, \$alpha, \$tolerance); say "Testing steepest descent method:";say "The minimum is at x = ", @x, ", x = ", @x; `
Output:
```Testing steepest descent method:
The minimum is at x = 0.10743450794656964, x = -1.2233956711774543
```

## Phix

Translation of: Go

... and just like Go, the results don't quite match anything else.

`-- Function for which minimum is to be found.function g(sequence x)    atom {x0,x1} = x    return (x0-1)*(x0-1)*exp(-x1*x1) +                x1*(x1+2)*exp(-2*x0*x0)end function -- Provides a rough calculation of gradient g(x).function gradG(sequence x, atom h)    integer n = length(x)    sequence z = repeat(0, n)    atom g0 := g(x)    for i=1 to n do        x[i] += h        z[i] = (g(x) - g0) / h    end for    return zend function function steepestDescent(sequence x, atom alpha, tolerance)    integer n = length(x)    atom h = tolerance,         g0 = g(x) -- Initial estimate of result.     -- Calculate initial gradient.    sequence fi = gradG(x, h)     -- Calculate initial norm.    atom delG = sqrt(sum(sq_mul(fi,fi))),         b = alpha / delG     -- Iterate until value is <= tolerance.    while delG>tolerance do        -- Calculate next value.        x = sq_sub(x,sq_mul(b,fi))        h /= 2         -- Calculate next gradient.        fi = gradG(x, h)         -- Calculate next norm.        delG = sqrt(sum(sq_mul(fi,fi)))        b = alpha / delG         -- Calculate next value.        atom g1 = g(x)         -- Adjust parameter.        if g1>g0 then            alpha /= 2        else            g0 = g1        end if    end while    return xend function constant tolerance = 0.0000006, alpha = 0.1sequence x = steepestDescent({0.1,-1}, alpha, tolerance)printf(1,"Testing steepest descent method:\n")printf(1,"The minimum is at x = %.16f,  x = %.16f\n", x)`
Output:
```Testing steepest descent method:
The minimum is at x = 0.1076572080934996,    x = -1.2232976080475890  -- (64 bit)
The minimum is at x = 0.1073980565405569,    x = -1.2233251778997771  -- (32 bit)
```

## Racket

Translation of: Go

Note the different implementation of `grad`. I believe that the vector should be reset and only the partial derivative in a particular dimension is to be used. For this reason, I've _yet another_ result!

I could have used ∇ and Δ in the variable names, but it looked too confusing, so I've gone with grad- and del-

`#lang racket (define (apply-vector f v)  (apply f (vector->list v))) ;; Provides a rough calculation of gradient g(v).(define ((grad/del f) v δ #:fv (fv (apply-vector f v)))  (define dim (vector-length v))  (define tmp (vector-copy v))  (define grad (for/vector #:length dim ((i dim)                            (v_i v))              (vector-set! tmp i (+ v_i δ))              (define ∂f/∂v_i (/ (- (apply-vector f tmp) fv) δ))              (vector-set! tmp i v_i)              ∂f/∂v_i))  (values grad (sqrt (for/sum ((∂_i grad)) (sqr ∂_i))))) (define (steepest-descent g x α tolerance)  (define grad/del-g (grad/del g))   (define (loop x δ α gx grad-gx del-gx b)    (cond      [(<= del-gx tolerance) x]      [else        (define δ´ (/ δ 2))        (define x´ (vector-map + (vector-map (curry * (- b)) grad-gx) x))        (define gx´ (apply-vector g x´))        (define-values (grad-gx´ del-gx´) (grad/del-g x´ δ´ #:fv gx´))        (define b´ (/ α del-gx´))        (if (> gx´ gx)            (loop x´ δ´ (/ α 2) gx  grad-gx´ del-gx´ b´)            (loop x´ δ´ α       gx´ grad-gx´ del-gx´ b´))]))   (define gx (apply-vector g x))  (define δ tolerance)  (define-values (grad-gx del-gx) (grad/del-g x δ #:fv gx))  (loop x δ α gx grad-gx del-gx (/ α del-gx))) (define (Gradient-descent)  (steepest-descent    (λ (x y)       (+ (* (- x 1) (- x 1) (exp (- (sqr y))))        (* y (+ y 2) (exp (- (* 2 (sqr x)))))))    #(0.1 -1.) 0.1 0.0000006)) (module+ main  (Gradient-descent)) `
Output:
`'#(0.10760797905122492 -1.2232993981966753)`

## TypeScript

Translation of
•   [Numerical Methods, Algorithms and Tools in C# by Waldemar Dos Passos (18.2 Gradient Descent Method]

` // Using the steepest-descent method to search// for minimum values of a multi-variable functionexport const steepestDescent = (x: number[], alpha: number, tolerance: number) => {     let n: number = x.length; // size of input array    let h: number = 0.0000006; //Tolerance factor    let g0: number = g(x); //Initial estimate of result     //Calculate initial gradient    let fi: number[] = [n];     //Calculate initial norm    fi = GradG(x, h);    // console.log("fi:"+fi);     //Calculate initial norm    let DelG: number = 0.0;     for (let i: number = 0; i < n; ++i) {        DelG += fi[i] * fi[i];    }    DelG = Math.sqrt(DelG);    let b: number = alpha / DelG;     //Iterate until value is <= tolerance limit    while (DelG > tolerance) {        //Calculate next value        for (let i = 0; i < n; ++i) {            x[i] -= b * fi[i];        }        h /= 2;         //Calculate next gradient        fi = GradG(x, h);        //Calculate next norm        DelG = 0;        for (let i: number = 0; i < n; ++i) {            DelG += fi[i] * fi[i];        }         DelG = Math.sqrt(DelG);        b = alpha / DelG;         //Calculate next value        let g1: number = g(x);         //Adjust parameter        if (g1 > g0) alpha /= 2;        else g0 = g1;    }} // Provides a rough calculation of gradient g(x).export const GradG = (x: number[], h: number) => {     let n: number = x.length;    let z: number[] = [n];    let y: number[] = x;    let g0: number = g(x);     // console.log("y:" + y);     for (let i = 0; i < n; ++i) {        y[i] += h;        z[i] = (g(y) - g0) / h;    }    // console.log("z:"+z);    return z;} // Method to provide function g(x).export const g = (x: number[]) => {    return (x - 1) * (x - 1)        * Math.exp(-x * x) + x * (x + 2)        * Math.exp(-2 * x * x);} export const gradientDescentMain = () => {    let tolerance: number = 0.0000006;    let alpha: number = 0.1;    let x: number[] = ;     //Initial guesses    x = 0.1;    //of location of minimums     x = -1;    steepestDescent(x, alpha, tolerance);     console.log("Testing steepest descent method");    console.log("The minimum is at x = " + x        + ", x = " + x);    // console.log("");} gradientDescentMain();  `
Output:
```Testing steepest descent method
The minimum is at x = 0.10768224291553158, x = -1.2233090211217854
```

### Linear Regression

Translation of
` let data: number[][] =    [[32.5023452694530, 31.70700584656990],    [53.4268040332750, 68.77759598163890],    [61.5303580256364, 62.56238229794580],    [47.4756396347860, 71.54663223356770],    [59.8132078695123, 87.23092513368730],    [55.1421884139438, 78.21151827079920],    [52.2117966922140, 79.64197304980870],    [39.2995666943170, 59.17148932186950],    [48.1050416917682, 75.33124229706300],    [52.5500144427338, 71.30087988685030],    [45.4197301449737, 55.16567714595910],    [54.3516348812289, 82.47884675749790],    [44.1640494967733, 62.00892324572580],    [58.1684707168577, 75.39287042599490],    [56.7272080570966, 81.43619215887860],    [48.9558885660937, 60.72360244067390],    [44.6871962314809, 82.89250373145370],    [60.2973268513334, 97.37989686216600],    [45.6186437729558, 48.84715331735500],    [38.8168175374456, 56.87721318626850],    [66.1898166067526, 83.87856466460270],    [65.4160517451340, 118.59121730252200],    [47.4812086078678, 57.25181946226890],    [41.5756426174870, 51.39174407983230],    [51.8451869056394, 75.38065166531230],    [59.3708220110895, 74.76556403215130],    [57.3100034383480, 95.45505292257470],    [63.6155612514533, 95.22936601755530],    [46.7376194079769, 79.05240616956550],    [50.5567601485477, 83.43207142132370],    [52.2239960855530, 63.35879031749780],    [35.5678300477466, 41.41288530370050],    [42.4364769440556, 76.61734128007400],    [58.1645401101928, 96.76956642610810],    [57.5044476153417, 74.08413011660250],    [45.4405307253199, 66.58814441422850],    [61.8962226802912, 77.76848241779300],    [33.0938317361639, 50.71958891231200],    [36.4360095113868, 62.12457081807170],    [37.6756548608507, 60.81024664990220],    [44.5556083832753, 52.68298336638770],    [43.3182826318657, 58.56982471769280],    [50.0731456322890, 82.90598148507050],    [43.8706126452183, 61.42470980433910],    [62.9974807475530, 115.24415280079500],    [32.6690437634671, 45.57058882337600],    [40.1668990087037, 54.08405479622360],    [53.5750775316736, 87.99445275811040],    [33.8642149717782, 52.72549437590040],    [64.7071386661212, 93.57611869265820],    [38.1198240268228, 80.16627544737090],    [44.5025380646451, 65.10171157056030],    [40.5995383845523, 65.56230126040030],    [41.7206763563412, 65.28088692082280],    [51.0886346783367, 73.43464154632430],    [55.0780959049232, 71.13972785861890],    [41.3777265348952, 79.10282968354980],    [62.4946974272697, 86.52053844034710],    [49.2038875408260, 84.74269780782620],    [41.1026851873496, 59.35885024862490],    [41.1820161051698, 61.68403752483360],    [50.1863894948806, 69.84760415824910],    [52.3784462192362, 86.09829120577410],    [50.1354854862861, 59.10883926769960],    [33.6447060061917, 69.89968164362760],    [39.5579012229068, 44.86249071116430],    [56.1303888168754, 85.49806777884020],    [57.3620521332382, 95.53668684646720],    [60.2692143939979, 70.25193441977150],    [35.6780938894107, 52.72173496477490],    [31.5881169981328, 50.39267013507980],    [53.6609322616730, 63.64239877565770],    [46.6822286494719, 72.24725106866230],    [43.1078202191024, 57.81251297618140],    [70.3460756150493, 104.25710158543800],    [44.4928558808540, 86.64202031882200],    [57.5045333032684, 91.48677800011010],    [36.9300766091918, 55.23166088621280],    [55.8057333579427, 79.55043667850760],    [38.9547690733770, 44.84712424246760],    [56.9012147022470, 80.20752313968270],    [56.8689006613840, 83.14274979204340],    [34.3331247042160, 55.72348926054390],    [59.0497412146668, 77.63418251167780],    [57.7882239932306, 99.05141484174820],    [54.2823287059674, 79.12064627468000],    [51.0887198989791, 69.58889785111840],    [50.2828363482307, 69.51050331149430],    [44.2117417520901, 73.68756431831720],    [38.0054880080606, 61.36690453724010],    [32.9404799426182, 67.17065576899510],    [53.6916395710700, 85.66820314500150],    [68.7657342696216, 114.85387123391300],    [46.2309664983102, 90.12357206996740],    [68.3193608182553, 97.91982103524280],    [50.0301743403121, 81.53699078301500],    [49.2397653427537, 72.11183246961560],    [50.0395759398759, 85.23200734232560],    [48.1498588910288, 66.22495788805460],    [25.1284846477723, 53.45439421485050]]; function lossFunction(arr0: number[], arr1: number[], arr2: number[]) {     let n: number = arr0.length; // Number of elements in X     //D_m = (-2/n) * sum(X * (Y - Y_pred))  # Derivative wrt m    let a: number = (-2 / n) * (arr0.map((a, i) => a * (arr1[i] - arr2[i]))).reduce((sum, current) => sum + current);    //D_c = (-2/n) * sum(Y - Y_pred)  # Derivative wrt c    let b: number = (-2 / n) * (arr1.map((a, i) => (a - arr2[i]))).reduce((sum, current) => sum + current);    return [a, b];} export const gradientDescentMain = () => {     // Building the model    let m: number = 0;    let c: number = 0;    let X_arr: number[];    let Y_arr: number[];    let Y_pred_arr: number[];    let D_m: number = 0;    let D_c: number = 0;     let L: number = 0.00000001;  // The learning Rate    let epochs: number = 10000000;  // The number of iterations to perform gradient descent     //Initial guesses    for (let i = 0; i < epochs; i++) {        X_arr = data.map(function (value, index) { return value; });        Y_arr = data.map(function (value, index) { return value; });         // The current predicted value of Y        Y_pred_arr = X_arr.map((a) => ((m * a) + c));         let all = lossFunction(X_arr, Y_arr, Y_pred_arr);        D_m = all;        D_c = all;         m = m - L * D_m;  // Update m        c = c - L * D_c;  // Update c    }     console.log("m: " + m + " c: " + c);} gradientDescentMain(); `