256 lines
8.0 KiB
Rust
256 lines
8.0 KiB
Rust
//! An Interior-Point Method for Large-Scale l1-Regularized Least Squares
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//!
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//! This is a specialized interior-point method for solving large-scale 1-regularized LSPs that uses the
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//! preconditioned conjugate gradients algorithm to compute the search direction.
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//!
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//! The interior-point method can solve large sparse problems, with a million variables and observations, in a few tens of minutes on a PC.
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//! It can efficiently solve large dense problems, that arise in sparse signal recovery with orthogonal transforms, by exploiting fast algorithms for these transforms.
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//!
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//! ## References:
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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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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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pub 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, M: Matrix<T>> InteriorPointOptimizer<T, M> {
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pub 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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pub 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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lambda: T,
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max_iter: usize,
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tol: 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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let lambda = lambda.max(T::epsilon());
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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() / lambda)
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.min(T::two() * p_f64 / T::from(1e-3).unwrap());
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for ntiter in 0..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 > lambda {
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let lnu = lambda / max_xnu;
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nu.mul_scalar_mut(lnu);
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}
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let pobj = z.dot(&z) + lambda * 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 < 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 = lambda - (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) + lambda * 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) + lambda * newu.sum() - 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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