Gradient descent

From Rosetta Code
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.

Task

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[edit]

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[0]-1)*(x[0]-1)*
math.Exp(-x[1]*x[1]) + x[1]*(x[1]+2)*
math.Exp(-2*x[0]*x[0])
}
 
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[0] =", x[0], "\b, x[1] =", x[1])
}
 
Output:
Testing steepest descent method:
The minimum is at x[0] = 0.10764302056464771, x[1] = -1.223351901171944

Julia[edit]

using Optim, Base.MathConstants
 
f(x) = (x[1] - 1) * (x[1] - 1) * e^(-x[2]^2) + x[2] * (x[2] + 2) * e^(-2 * x[1]^2)
 
println(optimize(f, [0.1, -1.0], GradientDescent()))
 
Output:
Results of Optimization Algorithm
 * Algorithm: Gradient Descent
 * 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
 * Gradient Calls: 35

Phix[edit]

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 z
end 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 x
end function
 
constant tolerance = 0.0000006, alpha = 0.1
sequence x = steepestDescent({0.1,-1}, alpha, tolerance)
printf(1,"Testing steepest descent method:\n")
printf(1,"The minimum is at x[1] = %.16f, x[1] = %.16f\n", x)
Output:
Testing steepest descent method:
The minimum is at x[1] = 0.1076572080934996,    x[1] = -1.2232976080475890  -- (64 bit)
The minimum is at x[1] = 0.1073980565405569,    x[1] = -1.2233251778997771  -- (32 bit)

TypeScript[edit]

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 function
export 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[0] - 1) * (x[0] - 1)
* Math.exp(-x[1] * x[1]) + x[1] * (x[1] + 2)
* Math.exp(-2 * x[0] * x[0]);
}
 
export const gradientDescentMain = () => {
let tolerance: number = 0.0000006;
let alpha: number = 0.1;
let x: number[] = [2];
 
//Initial guesses
x[0] = 0.1;
//of location of minimums
x[1] = -1;
steepestDescent(x, alpha, tolerance);
 
console.log("Testing steepest descent method");
console.log("The minimum is at x[0] = " + x[0]
+ ", x[1] = " + x[1]);
// console.log("");
}
 
gradientDescentMain();
 
 
Output:
Testing steepest descent method
The minimum is at x[0] = 0.10768224291553158, x[1] = -1.2233090211217854

Linear Regression[edit]

Translation of 
  •   [Linear Regression using Gradient Descent by Adarsh Menon]



 
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[0]; });
Y_arr = data.map(function (value, index) { return value[1]; });
 
// 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[0];
D_c = all[1];
 
m = m - L * D_m; // Update m
c = c - L * D_c; // Update c
}
 
console.log("m: " + m + " c: " + c);
}
 
gradientDescentMain();