feat: adds LASSO
This commit is contained in:
Volodymyr Orlov
@@ -0,0 +1,509 @@
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//! # Lasso
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//!
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//! [Linear regression](../linear_regression/index.html) is the standard algorithm for predicting a quantitative response \\(y\\) on the basis of a linear combination of explanatory variables \\(X\\)
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//! that assumes that there is approximately a linear relationship between \\(X\\) and \\(y\\).
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//! Lasso is an extension to linear regression that adds L1 regularization term to the loss function during training.
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//!
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//! Similar to [ridge regression](../ridge_regression/index.html), the lasso shrinks the coefficient estimates towards zero when. However, in the case of the lasso, the l1 penalty has the effect of
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//! forcing some of the coefficient estimates to be exactly equal to zero when the tuning parameter \\(\alpha\\) is sufficiently large.
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//!
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//! Lasso coefficient estimates solve the problem:
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//!
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//! \\[\underset{\beta}{minimize} \space \space \sum_{i=1}^n \left( y_i - \beta_0 - \sum_{j=1}^p \beta_jx_{ij} \right)^2 + \alpha \sum_{j=1}^p \lVert \beta_j \rVert_1\\]
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//!
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//! This problem is solved with an interior-point method that is comparable to coordinate descent in solving large problems with modest accuracy,
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//! but is able to solve them with high accuracy with relatively small additional computational cost.
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//!
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//! ## References:
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//!
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//! * ["An Introduction to Statistical Learning", James G., Witten D., Hastie T., Tibshirani R., 6.2. Shrinkage Methods](http://faculty.marshall.usc.edu/gareth-james/ISL/)
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//! * ["An Interior-Point Method for Large-Scale l1-Regularized Least Squares", K. Koh, M. Lustig, S. Boyd, D. Gorinevsky](https://web.stanford.edu/~boyd/papers/pdf/l1_ls.pdf)
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//! * [Simple Matlab Solver for l1-regularized Least Squares Problems](https://web.stanford.edu/~boyd/l1_ls/)
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//!
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//! <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
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//! <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
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use std::fmt::Debug;
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use serde::{Deserialize, Serialize};
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use crate::error::Failed;
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use crate::linalg::BaseVector;
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use crate::linalg::Matrix;
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use crate::linear::bg_solver::BiconjugateGradientSolver;
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use crate::math::num::RealNumber;
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/// Lasso regression parameters
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#[derive(Serialize, Deserialize, Debug)]
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pub struct LassoParameters<T: RealNumber> {
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/// Controls the strength of the penalty to the loss function.
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pub alpha: T,
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/// If true the regressors X will be normalized before regression
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/// by subtracting the mean and dividing by the standard deviation.
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pub normalize: bool,
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/// The tolerance for the optimization
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pub tol: T,
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/// The maximum number of iterations
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pub max_iter: usize,
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}
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#[derive(Serialize, Deserialize, Debug)]
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/// Lasso regressor
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pub struct Lasso<T: RealNumber, M: Matrix<T>> {
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coefficients: M,
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intercept: T,
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}
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struct InteriorPointOptimizer<T: RealNumber, M: Matrix<T>> {
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ata: M,
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d1: Vec<T>,
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d2: Vec<T>,
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prb: Vec<T>,
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prs: Vec<T>,
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}
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impl<T: RealNumber> Default for LassoParameters<T> {
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fn default() -> Self {
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LassoParameters {
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alpha: T::one(),
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normalize: true,
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tol: T::from_f64(1e-4).unwrap(),
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max_iter: 1000,
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}
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}
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}
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impl<T: RealNumber, M: Matrix<T>> PartialEq for Lasso<T, M> {
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fn eq(&self, other: &Self) -> bool {
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self.coefficients == other.coefficients
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&& (self.intercept - other.intercept).abs() <= T::epsilon()
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}
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}
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impl<T: RealNumber, M: Matrix<T>> Lasso<T, M> {
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/// Fits Lasso regression to your data.
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/// * `x` - _NxM_ matrix with _N_ observations and _M_ features in each observation.
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/// * `y` - target values
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/// * `parameters` - other parameters, use `Default::default()` to set parameters to default values.
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pub fn fit(
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x: &M,
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y: &M::RowVector,
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parameters: LassoParameters<T>,
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) -> Result<Lasso<T, M>, Failed> {
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let (n, p) = x.shape();
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if n <= p {
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return Err(Failed::fit(
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"Number of rows in X should be >= number of columns in X",
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));
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}
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if parameters.alpha < T::zero() {
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return Err(Failed::fit("alpha should be >= 0"));
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}
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if parameters.tol <= T::zero() {
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return Err(Failed::fit("tol should be > 0"));
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}
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if parameters.max_iter == 0 {
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return Err(Failed::fit("max_iter should be > 0"));
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}
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if y.len() != n {
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return Err(Failed::fit("Number of rows in X should = len(y)"));
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}
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let (w, b) = if parameters.normalize {
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let (scaled_x, col_mean, col_std) = Self::rescale_x(x)?;
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let mut optimizer = InteriorPointOptimizer::new(&scaled_x, p);
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let mut w = optimizer.optimize(&scaled_x, y, ¶meters)?;
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for j in 0..p {
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w.set(j, 0, w.get(j, 0) / col_std[j]);
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}
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let mut b = T::zero();
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for i in 0..p {
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b += w.get(i, 0) * col_mean[i];
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}
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b = y.mean() - b;
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(w, b)
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} else {
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let mut optimizer = InteriorPointOptimizer::new(x, p);
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let w = optimizer.optimize(x, y, ¶meters)?;
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(w, y.mean())
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};
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Ok(Lasso {
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intercept: b,
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coefficients: w,
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})
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}
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/// Predict target values from `x`
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/// * `x` - _KxM_ data where _K_ is number of observations and _M_ is number of features.
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pub fn predict(&self, x: &M) -> Result<M::RowVector, Failed> {
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let (nrows, _) = x.shape();
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let mut y_hat = x.matmul(&self.coefficients);
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y_hat.add_mut(&M::fill(nrows, 1, self.intercept));
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Ok(y_hat.transpose().to_row_vector())
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}
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/// Get estimates regression coefficients
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pub fn coefficients(&self) -> &M {
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&self.coefficients
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}
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/// Get estimate of intercept
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pub fn intercept(&self) -> T {
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self.intercept
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}
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fn rescale_x(x: &M) -> Result<(M, Vec<T>, Vec<T>), Failed> {
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let col_mean = x.mean(0);
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let col_std = x.std(0);
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for i in 0..col_std.len() {
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if (col_std[i] - T::zero()).abs() < T::epsilon() {
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return Err(Failed::fit(&format!(
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"Cannot rescale constant column {}",
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i
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)));
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}
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}
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let mut scaled_x = x.clone();
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scaled_x.scale_mut(&col_mean, &col_std, 0);
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Ok((scaled_x, col_mean, col_std))
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}
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}
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impl<T: RealNumber, M: Matrix<T>> InteriorPointOptimizer<T, M> {
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fn new(a: &M, n: usize) -> InteriorPointOptimizer<T, M> {
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InteriorPointOptimizer {
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ata: a.ab(true, a, false),
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d1: vec![T::zero(); n],
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d2: vec![T::zero(); n],
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prb: vec![T::zero(); n],
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prs: vec![T::zero(); n],
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}
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}
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fn optimize(
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&mut self,
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x: &M,
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y: &M::RowVector,
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parameters: &LassoParameters<T>,
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) -> Result<M, Failed> {
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let (n, p) = x.shape();
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let p_f64 = T::from_usize(p).unwrap();
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//parameters
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let pcgmaxi = 5000;
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let min_pcgtol = T::from_f64(0.1).unwrap();
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let eta = T::from_f64(1E-3).unwrap();
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let alpha = T::from_f64(0.01).unwrap();
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let beta = T::from_f64(0.5).unwrap();
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let gamma = T::from_f64(-0.25).unwrap();
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let mu = T::two();
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let y = M::from_row_vector(y.sub_scalar(y.mean())).transpose();
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let mut max_ls_iter = 100;
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let mut pitr = 0;
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let mut w = M::zeros(p, 1);
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let mut neww = w.clone();
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let mut u = M::ones(p, 1);
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let mut newu = u.clone();
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let mut f = M::fill(p, 2, -T::one());
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let mut newf = f.clone();
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let mut q1 = vec![T::zero(); p];
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let mut q2 = vec![T::zero(); p];
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let mut dx = M::zeros(p, 1);
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let mut du = M::zeros(p, 1);
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let mut dxu = M::zeros(2 * p, 1);
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let mut grad = M::zeros(2 * p, 1);
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let mut nu = M::zeros(n, 1);
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let mut dobj = T::zero();
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let mut s = T::infinity();
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let mut t = T::one()
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.max(T::one() / parameters.alpha)
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.min(T::two() * p_f64 / T::from(1e-3).unwrap());
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for ntiter in 0..parameters.max_iter {
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let mut z = x.matmul(&w);
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for i in 0..n {
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z.set(i, 0, z.get(i, 0) - y.get(i, 0));
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nu.set(i, 0, T::two() * z.get(i, 0));
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}
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// CALCULATE DUALITY GAP
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let xnu = x.ab(true, &nu, false);
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let max_xnu = xnu.norm(T::infinity());
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if max_xnu > parameters.alpha {
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let lnu = parameters.alpha / max_xnu;
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nu.mul_scalar_mut(lnu);
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}
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let pobj = z.dot(&z) + parameters.alpha * w.norm(T::one());
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dobj = dobj.max(gamma * nu.dot(&nu) - nu.dot(&y));
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let gap = pobj - dobj;
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// STOPPING CRITERION
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if gap / dobj < parameters.tol {
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break;
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}
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// UPDATE t
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if s >= T::half() {
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t = t.max((T::two() * p_f64 * mu / gap).min(mu * t));
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}
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// CALCULATE NEWTON STEP
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for i in 0..p {
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let q1i = T::one() / (u.get(i, 0) + w.get(i, 0));
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let q2i = T::one() / (u.get(i, 0) - w.get(i, 0));
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q1[i] = q1i;
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q2[i] = q2i;
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self.d1[i] = (q1i * q1i + q2i * q2i) / t;
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self.d2[i] = (q1i * q1i - q2i * q2i) / t;
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}
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let mut gradphi = x.ab(true, &z, false);
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for i in 0..p {
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let g1 = T::two() * gradphi.get(i, 0) - (q1[i] - q2[i]) / t;
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let g2 = parameters.alpha - (q1[i] + q2[i]) / t;
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gradphi.set(i, 0, g1);
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grad.set(i, 0, -g1);
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grad.set(i + p, 0, -g2);
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}
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for i in 0..p {
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self.prb[i] = T::two() + self.d1[i];
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self.prs[i] = self.prb[i] * self.d1[i] - self.d2[i] * self.d2[i];
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}
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let normg = grad.norm2();
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let mut pcgtol = min_pcgtol.min(eta * gap / T::one().min(normg));
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if ntiter != 0 && pitr == 0 {
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pcgtol *= min_pcgtol;
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}
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let error = self.solve_mut(x, &grad, &mut dxu, pcgtol, pcgmaxi)?;
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if error > pcgtol {
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pitr = pcgmaxi;
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}
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for i in 0..p {
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dx.set(i, 0, dxu.get(i, 0));
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du.set(i, 0, dxu.get(i + p, 0));
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}
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// BACKTRACKING LINE SEARCH
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let phi = z.dot(&z) + parameters.alpha * u.sum() - Self::sumlogneg(&f) / t;
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s = T::one();
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let gdx = grad.dot(&dxu);
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let lsiter = 0;
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while lsiter < max_ls_iter {
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for i in 0..p {
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neww.set(i, 0, w.get(i, 0) + s * dx.get(i, 0));
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newu.set(i, 0, u.get(i, 0) + s * du.get(i, 0));
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newf.set(i, 0, neww.get(i, 0) - newu.get(i, 0));
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newf.set(i, 1, -neww.get(i, 0) - newu.get(i, 0));
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}
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if newf.max() < T::zero() {
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let mut newz = x.matmul(&neww);
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for i in 0..n {
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newz.set(i, 0, newz.get(i, 0) - y.get(i, 0));
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}
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let newphi = newz.dot(&newz) + parameters.alpha * newu.sum()
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- Self::sumlogneg(&newf) / t;
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if newphi - phi <= alpha * s * gdx {
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break;
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}
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}
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s = beta * s;
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max_ls_iter += 1;
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}
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if lsiter == max_ls_iter {
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return Err(Failed::fit(
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"Exceeded maximum number of iteration for interior point optimizer",
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));
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}
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w.copy_from(&neww);
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u.copy_from(&newu);
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f.copy_from(&newf);
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}
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Ok(w)
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}
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fn sumlogneg(f: &M) -> T {
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let (n, _) = f.shape();
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let mut sum = T::zero();
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for i in 0..n {
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sum += (-f.get(i, 0)).ln();
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sum += (-f.get(i, 1)).ln();
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}
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sum
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}
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}
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impl<'a, T: RealNumber, M: Matrix<T>> BiconjugateGradientSolver<T, M>
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for InteriorPointOptimizer<T, M>
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{
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fn solve_preconditioner(&self, a: &M, b: &M, x: &mut M) {
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let (_, p) = a.shape();
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for i in 0..p {
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x.set(
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i,
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0,
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(self.d1[i] * b.get(i, 0) - self.d2[i] * b.get(i + p, 0)) / self.prs[i],
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);
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x.set(
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i + p,
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0,
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(-self.d2[i] * b.get(i, 0) + self.prb[i] * b.get(i + p, 0)) / self.prs[i],
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);
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}
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}
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fn mat_vec_mul(&self, _: &M, x: &M, y: &mut M) {
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let (_, p) = self.ata.shape();
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let atax = self.ata.matmul(&x.slice(0..p, 0..1));
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for i in 0..p {
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y.set(
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i,
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0,
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T::two() * atax.get(i, 0) + self.d1[i] * x.get(i, 0) + self.d2[i] * x.get(i + p, 0),
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);
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y.set(
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i + p,
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0,
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self.d2[i] * x.get(i, 0) + self.d1[i] * x.get(i + p, 0),
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);
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}
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}
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fn mat_t_vec_mul(&self, a: &M, x: &M, y: &mut M) {
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self.mat_vec_mul(a, x, y);
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}
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}
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#[cfg(test)]
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mod tests {
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use super::*;
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use crate::linalg::naive::dense_matrix::*;
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use crate::metrics::mean_absolute_error;
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#[test]
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fn lasso_fit_predict() {
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let x = DenseMatrix::from_2d_array(&[
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&[234.289, 235.6, 159.0, 107.608, 1947., 60.323],
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&[259.426, 232.5, 145.6, 108.632, 1948., 61.122],
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&[258.054, 368.2, 161.6, 109.773, 1949., 60.171],
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&[284.599, 335.1, 165.0, 110.929, 1950., 61.187],
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&[328.975, 209.9, 309.9, 112.075, 1951., 63.221],
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&[346.999, 193.2, 359.4, 113.270, 1952., 63.639],
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&[365.385, 187.0, 354.7, 115.094, 1953., 64.989],
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&[363.112, 357.8, 335.0, 116.219, 1954., 63.761],
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&[397.469, 290.4, 304.8, 117.388, 1955., 66.019],
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&[419.180, 282.2, 285.7, 118.734, 1956., 67.857],
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&[442.769, 293.6, 279.8, 120.445, 1957., 68.169],
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&[444.546, 468.1, 263.7, 121.950, 1958., 66.513],
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&[482.704, 381.3, 255.2, 123.366, 1959., 68.655],
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&[502.601, 393.1, 251.4, 125.368, 1960., 69.564],
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&[518.173, 480.6, 257.2, 127.852, 1961., 69.331],
|
||||
&[554.894, 400.7, 282.7, 130.081, 1962., 70.551],
|
||||
]);
|
||||
|
||||
let y: Vec<f64> = vec![
|
||||
83.0, 88.5, 88.2, 89.5, 96.2, 98.1, 99.0, 100.0, 101.2, 104.6, 108.4, 110.8, 112.6,
|
||||
114.2, 115.7, 116.9,
|
||||
];
|
||||
|
||||
let y_hat = Lasso::fit(
|
||||
&x,
|
||||
&y,
|
||||
LassoParameters {
|
||||
alpha: 0.1,
|
||||
normalize: false,
|
||||
tol: 1e-4,
|
||||
max_iter: 1000,
|
||||
},
|
||||
)
|
||||
.and_then(|lr| lr.predict(&x))
|
||||
.unwrap();
|
||||
|
||||
assert!(mean_absolute_error(&y_hat, &y) < 2.0);
|
||||
|
||||
let y_hat = Lasso::fit(
|
||||
&x,
|
||||
&y,
|
||||
LassoParameters {
|
||||
alpha: 0.1,
|
||||
normalize: false,
|
||||
tol: 1e-4,
|
||||
max_iter: 1000,
|
||||
},
|
||||
)
|
||||
.and_then(|lr| lr.predict(&x))
|
||||
.unwrap();
|
||||
|
||||
assert!(mean_absolute_error(&y_hat, &y) < 2.0);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn serde() {
|
||||
let x = DenseMatrix::from_2d_array(&[
|
||||
&[234.289, 235.6, 159.0, 107.608, 1947., 60.323],
|
||||
&[259.426, 232.5, 145.6, 108.632, 1948., 61.122],
|
||||
&[258.054, 368.2, 161.6, 109.773, 1949., 60.171],
|
||||
&[284.599, 335.1, 165.0, 110.929, 1950., 61.187],
|
||||
&[328.975, 209.9, 309.9, 112.075, 1951., 63.221],
|
||||
&[346.999, 193.2, 359.4, 113.270, 1952., 63.639],
|
||||
&[365.385, 187.0, 354.7, 115.094, 1953., 64.989],
|
||||
&[363.112, 357.8, 335.0, 116.219, 1954., 63.761],
|
||||
&[397.469, 290.4, 304.8, 117.388, 1955., 66.019],
|
||||
&[419.180, 282.2, 285.7, 118.734, 1956., 67.857],
|
||||
&[442.769, 293.6, 279.8, 120.445, 1957., 68.169],
|
||||
&[444.546, 468.1, 263.7, 121.950, 1958., 66.513],
|
||||
&[482.704, 381.3, 255.2, 123.366, 1959., 68.655],
|
||||
&[502.601, 393.1, 251.4, 125.368, 1960., 69.564],
|
||||
&[518.173, 480.6, 257.2, 127.852, 1961., 69.331],
|
||||
&[554.894, 400.7, 282.7, 130.081, 1962., 70.551],
|
||||
]);
|
||||
|
||||
let y = vec![
|
||||
83.0, 88.5, 88.2, 89.5, 96.2, 98.1, 99.0, 100.0, 101.2, 104.6, 108.4, 110.8, 112.6,
|
||||
114.2, 115.7, 116.9,
|
||||
];
|
||||
|
||||
let lr = Lasso::fit(&x, &y, Default::default()).unwrap();
|
||||
|
||||
let deserialized_lr: Lasso<f64, DenseMatrix<f64>> =
|
||||
serde_json::from_str(&serde_json::to_string(&lr).unwrap()).unwrap();
|
||||
|
||||
assert_eq!(lr, deserialized_lr);
|
||||
}
|
||||
}
|
||||
Reference in New Issue
Block a user