|
| 1 | +/// Momentum Optimization |
| 2 | +/// |
| 3 | +/// Momentum is an extension of gradient descent that accelerates convergence by accumulating |
| 4 | +/// a velocity vector in directions of persistent reduction in the objective function. |
| 5 | +/// This helps the optimizer navigate ravines and avoid getting stuck in local minima. |
| 6 | +/// |
| 7 | +/// The algorithm maintains a velocity vector that accumulates exponentially decaying moving |
| 8 | +/// averages of past gradients. This allows the optimizer to build up speed in consistent |
| 9 | +/// directions while dampening oscillations. |
| 10 | +/// |
| 11 | +/// The update equations are: |
| 12 | +/// velocity_{k+1} = beta * velocity_k + gradient_of_function(x_k) |
| 13 | +/// x_{k+1} = x_k - learning_rate * velocity_{k+1} |
| 14 | +/// |
| 15 | +/// where beta (typically 0.9) controls how much past gradients influence the current update. |
| 16 | +/// |
| 17 | +/// # Arguments |
| 18 | +/// |
| 19 | +/// * `derivative_fn` - The function that calculates the gradient of the objective function at a given point. |
| 20 | +/// * `x` - The initial parameter vector to be optimized. |
| 21 | +/// * `learning_rate` - Step size for each iteration. |
| 22 | +/// * `beta` - Momentum coefficient (typically 0.9). Higher values give more weight to past gradients. |
| 23 | +/// * `num_iterations` - The number of iterations to run the optimization. |
| 24 | +/// |
| 25 | +/// # Returns |
| 26 | +/// |
| 27 | +/// A reference to the optimized parameter vector `x`. |
| 28 | +#[allow(dead_code)] |
| 29 | +pub fn momentum( |
| 30 | + derivative: impl Fn(&[f64]) -> Vec<f64>, |
| 31 | + x: &mut Vec<f64>, |
| 32 | + learning_rate: f64, |
| 33 | + beta: f64, |
| 34 | + num_iterations: i32, |
| 35 | +) -> &mut Vec<f64> { |
| 36 | + // Initialize velocity vector to zero |
| 37 | + let mut velocity: Vec<f64> = vec![0.0; x.len()]; |
| 38 | + |
| 39 | + for _ in 0..num_iterations { |
| 40 | + let gradient = derivative(x); |
| 41 | + |
| 42 | + // Update velocity and parameters |
| 43 | + for ((x_k, vel), grad) in x.iter_mut().zip(velocity.iter_mut()).zip(gradient.iter()) { |
| 44 | + *vel = beta * *vel + grad; |
| 45 | + *x_k -= learning_rate * *vel; |
| 46 | + } |
| 47 | + } |
| 48 | + x |
| 49 | +} |
| 50 | + |
| 51 | +#[cfg(test)] |
| 52 | +mod test { |
| 53 | + use super::*; |
| 54 | + |
| 55 | + #[test] |
| 56 | + fn test_momentum_optimized() { |
| 57 | + fn derivative_of_square(params: &[f64]) -> Vec<f64> { |
| 58 | + params.iter().map(|x| 2.0 * x).collect() |
| 59 | + } |
| 60 | + |
| 61 | + let mut x: Vec<f64> = vec![5.0, 6.0]; |
| 62 | + let learning_rate: f64 = 0.01; |
| 63 | + let beta: f64 = 0.9; |
| 64 | + let num_iterations: i32 = 1000; |
| 65 | + |
| 66 | + let minimized_vector = momentum( |
| 67 | + derivative_of_square, |
| 68 | + &mut x, |
| 69 | + learning_rate, |
| 70 | + beta, |
| 71 | + num_iterations, |
| 72 | + ); |
| 73 | + |
| 74 | + let test_vector = [0.0, 0.0]; |
| 75 | + let tolerance = 1e-6; |
| 76 | + |
| 77 | + for (minimized_value, test_value) in minimized_vector.iter().zip(test_vector.iter()) { |
| 78 | + assert!((minimized_value - test_value).abs() < tolerance); |
| 79 | + } |
| 80 | + } |
| 81 | + |
| 82 | + #[test] |
| 83 | + fn test_momentum_unoptimized() { |
| 84 | + fn derivative_of_square(params: &[f64]) -> Vec<f64> { |
| 85 | + params.iter().map(|x| 2.0 * x).collect() |
| 86 | + } |
| 87 | + |
| 88 | + let mut x: Vec<f64> = vec![5.0, 6.0]; |
| 89 | + let learning_rate: f64 = 0.01; |
| 90 | + let beta: f64 = 0.9; |
| 91 | + let num_iterations: i32 = 10; |
| 92 | + |
| 93 | + let minimized_vector = momentum( |
| 94 | + derivative_of_square, |
| 95 | + &mut x, |
| 96 | + learning_rate, |
| 97 | + beta, |
| 98 | + num_iterations, |
| 99 | + ); |
| 100 | + |
| 101 | + let test_vector = [0.0, 0.0]; |
| 102 | + let tolerance = 1e-6; |
| 103 | + |
| 104 | + for (minimized_value, test_value) in minimized_vector.iter().zip(test_vector.iter()) { |
| 105 | + assert!((minimized_value - test_value).abs() >= tolerance); |
| 106 | + } |
| 107 | + } |
| 108 | + |
| 109 | + #[test] |
| 110 | + fn test_momentum_faster_than_gd() { |
| 111 | + fn derivative_of_square(params: &[f64]) -> Vec<f64> { |
| 112 | + params.iter().map(|x| 2.0 * x).collect() |
| 113 | + } |
| 114 | + |
| 115 | + // Test that momentum converges faster than gradient descent |
| 116 | + let mut x_momentum: Vec<f64> = vec![5.0, 6.0]; |
| 117 | + let mut x_gd: Vec<f64> = vec![5.0, 6.0]; |
| 118 | + let learning_rate: f64 = 0.01; |
| 119 | + let beta: f64 = 0.9; |
| 120 | + let num_iterations: i32 = 50; |
| 121 | + |
| 122 | + momentum( |
| 123 | + derivative_of_square, |
| 124 | + &mut x_momentum, |
| 125 | + learning_rate, |
| 126 | + beta, |
| 127 | + num_iterations, |
| 128 | + ); |
| 129 | + |
| 130 | + // Gradient descent from your original implementation |
| 131 | + for _ in 0..num_iterations { |
| 132 | + let gradient = derivative_of_square(&x_gd); |
| 133 | + for (x_k, grad) in x_gd.iter_mut().zip(gradient.iter()) { |
| 134 | + *x_k -= learning_rate * grad; |
| 135 | + } |
| 136 | + } |
| 137 | + |
| 138 | + // Momentum should be closer to zero |
| 139 | + let momentum_distance: f64 = x_momentum.iter().map(|x| x * x).sum(); |
| 140 | + let gd_distance: f64 = x_gd.iter().map(|x| x * x).sum(); |
| 141 | + |
| 142 | + assert!(momentum_distance < gd_distance); |
| 143 | + } |
| 144 | +} |
0 commit comments