[pull] master from TheAlgorithms:master

DIRECTORY.md

@@ -174,6 +174,7 @@
     * Optimization
       * [Adam](https://github.com/TheAlgorithms/Rust/blob/master/src/machine_learning/optimization/adam.rs)
       * [Gradient Descent](https://github.com/TheAlgorithms/Rust/blob/master/src/machine_learning/optimization/gradient_descent.rs)
+      * [Momentum](https://github.com/TheAlgorithms/Rust/blob/master/src/machine_learning/optimization/momentum.rs)
   * Math
     * [Abs](https://github.com/TheAlgorithms/Rust/blob/master/src/math/abs.rs)
     * [Aliquot Sum](https://github.com/TheAlgorithms/Rust/blob/master/src/math/aliquot_sum.rs)

src/machine_learning/optimization/mod.rs

@@ -1,5 +1,6 @@
 mod adam;
 mod gradient_descent;
+mod momentum;
 
 pub use self::adam::Adam;
 pub use self::gradient_descent::gradient_descent;

src/machine_learning/optimization/momentum.rs

@@ -0,0 +1,144 @@
+/// Momentum Optimization
+///
+/// Momentum is an extension of gradient descent that accelerates convergence by accumulating
+/// a velocity vector in directions of persistent reduction in the objective function.
+/// This helps the optimizer navigate ravines and avoid getting stuck in local minima.
+///
+/// The algorithm maintains a velocity vector that accumulates exponentially decaying moving
+/// averages of past gradients. This allows the optimizer to build up speed in consistent
+/// directions while dampening oscillations.
+///
+/// The update equations are:
+/// velocity_{k+1} = beta * velocity_k + gradient_of_function(x_k)
+/// x_{k+1} = x_k - learning_rate * velocity_{k+1}
+///
+/// where beta (typically 0.9) controls how much past gradients influence the current update.
+///
+/// # Arguments
+///
+/// * `derivative_fn` - The function that calculates the gradient of the objective function at a given point.
+/// * `x` - The initial parameter vector to be optimized.
+/// * `learning_rate` - Step size for each iteration.
+/// * `beta` - Momentum coefficient (typically 0.9). Higher values give more weight to past gradients.
+/// * `num_iterations` - The number of iterations to run the optimization.
+///
+/// # Returns
+///
+/// A reference to the optimized parameter vector `x`.
+#[allow(dead_code)]
+pub fn momentum(
+    derivative: impl Fn(&[f64]) -> Vec<f64>,
+    x: &mut Vec<f64>,
+    learning_rate: f64,
+    beta: f64,
+    num_iterations: i32,
+) -> &mut Vec<f64> {
+    // Initialize velocity vector to zero
+    let mut velocity: Vec<f64> = vec![0.0; x.len()];
+
+    for _ in 0..num_iterations {
+        let gradient = derivative(x);
+
+        // Update velocity and parameters
+        for ((x_k, vel), grad) in x.iter_mut().zip(velocity.iter_mut()).zip(gradient.iter()) {
+            *vel = beta * *vel + grad;
+            *x_k -= learning_rate * *vel;
+        }
+    }
+    x
+}
+
+#[cfg(test)]
+mod test {
+    use super::*;
+
+    #[test]
+    fn test_momentum_optimized() {
+        fn derivative_of_square(params: &[f64]) -> Vec<f64> {
+            params.iter().map(|x| 2.0 * x).collect()
+        }
+
+        let mut x: Vec<f64> = vec![5.0, 6.0];
+        let learning_rate: f64 = 0.01;
+        let beta: f64 = 0.9;
+        let num_iterations: i32 = 1000;
+
+        let minimized_vector = momentum(
+            derivative_of_square,
+            &mut x,
+            learning_rate,
+            beta,
+            num_iterations,
+        );
+
+        let test_vector = [0.0, 0.0];
+        let tolerance = 1e-6;
+
+        for (minimized_value, test_value) in minimized_vector.iter().zip(test_vector.iter()) {
+            assert!((minimized_value - test_value).abs() < tolerance);
+        }
+    }
+
+    #[test]
+    fn test_momentum_unoptimized() {
+        fn derivative_of_square(params: &[f64]) -> Vec<f64> {
+            params.iter().map(|x| 2.0 * x).collect()
+        }
+
+        let mut x: Vec<f64> = vec![5.0, 6.0];
+        let learning_rate: f64 = 0.01;
+        let beta: f64 = 0.9;
+        let num_iterations: i32 = 10;
+
+        let minimized_vector = momentum(
+            derivative_of_square,
+            &mut x,
+            learning_rate,
+            beta,
+            num_iterations,
+        );
+
+        let test_vector = [0.0, 0.0];
+        let tolerance = 1e-6;
+
+        for (minimized_value, test_value) in minimized_vector.iter().zip(test_vector.iter()) {
+            assert!((minimized_value - test_value).abs() >= tolerance);
+        }
+    }
+
+    #[test]
+    fn test_momentum_faster_than_gd() {
+        fn derivative_of_square(params: &[f64]) -> Vec<f64> {
+            params.iter().map(|x| 2.0 * x).collect()
+        }
+
+        // Test that momentum converges faster than gradient descent
+        let mut x_momentum: Vec<f64> = vec![5.0, 6.0];
+        let mut x_gd: Vec<f64> = vec![5.0, 6.0];
+        let learning_rate: f64 = 0.01;
+        let beta: f64 = 0.9;
+        let num_iterations: i32 = 50;
+
+        momentum(
+            derivative_of_square,
+            &mut x_momentum,
+            learning_rate,
+            beta,
+            num_iterations,
+        );
+
+        // Gradient descent from your original implementation
+        for _ in 0..num_iterations {
+            let gradient = derivative_of_square(&x_gd);
+            for (x_k, grad) in x_gd.iter_mut().zip(gradient.iter()) {
+                *x_k -= learning_rate * grad;
+            }
+        }
+
+        // Momentum should be closer to zero
+        let momentum_distance: f64 = x_momentum.iter().map(|x| x * x).sum();
+        let gd_distance: f64 = x_gd.iter().map(|x| x * x).sum();
+
+        assert!(momentum_distance < gd_distance);
+    }
+}