--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/subproblem/l1squared_unconstrained.rs Thu Aug 29 00:00:00 2024 -0500
@@ -0,0 +1,195 @@
+/*!
+Iterative algorithms for solving the finite-dimensional subproblem without constraints.
+*/
+
+use nalgebra::DVector;
+use numeric_literals::replace_float_literals;
+use itertools::izip;
+use std::cmp::Ordering::*;
+
+use std::iter::zip;
+use alg_tools::iterate::{
+ AlgIteratorFactory,
+ AlgIteratorState,
+};
+use alg_tools::nalgebra_support::ToNalgebraRealField;
+use alg_tools::nanleast::NaNLeast;
+use alg_tools::norms::{Dist, L1};
+
+use crate::types::*;
+use super::{
+ InnerMethod,
+ InnerSettings
+};
+use super::unconstrained::soft_thresholding;
+use super::l1squared_nonneg::max_interval_dist_to_zero;
+
+/// Calculate $prox_f(x)$ for $f(x)=\frac{β}{2}\norm{x-y}_1^2$.
+///
+/// To derive an algorithm for this, we can assume that $y=0$, as
+/// $prox_f(x) = prox_{f_0}(x - y) - y$ for $f_0=\frac{β}{2}\norm{x}_1^2$.
+/// Now, the optimality conditions for $w = prox_f(x)$ are
+/// $$\tag{*}
+/// 0 ∈ w-x + β\norm{w}_1\sign w.
+/// $$
+/// Clearly then $w = \soft_{β\norm{w}_1}(x)$.
+/// Thus the components of $x$ with smallest absolute value will be zeroed out.
+/// Denoting by $w'$ the non-zero components, and by $x'$ the corresponding components
+/// of $x$, and by $m$ their count, multipying the corresponding lines of (*) by $\sign x'$,
+/// we obtain
+/// $$
+/// \norm{x'}_1 = (1+βm)\norm{w'}_1.
+/// $$
+/// That is, $\norm{w}_1=\norm{w'}_1=\norm{x'}_1/(1+βm)$.
+/// Thus, sorting $x$ by absolute value, and sequentially in order eliminating the smallest
+/// elements, we can easily calculate what $\norm{w}_1$ should be for that choice, and
+/// then easily calculate $w = \soft_{β\norm{w}_1}(x)$. We just have to verify that
+/// the resulting $w$ has the same norm. There's a shortcut to this, as we work
+/// sequentially: just check that the smallest assumed-nonzero component $i$ satisfies the
+/// condition of soft-thresholding to remain non-zero: $|x_i|>τ\norm{x'}/(1+τm)$.
+/// Clearly, if this condition fails for x_i, it will fail for all the components
+/// already exluced. While, if it holds, it will hold for all components not excluded.
+#[replace_float_literals(F::cast_from(literal))]
+pub(super) fn l1squared_prox<F :Float + nalgebra::RealField>(
+ sorted_abs : &mut DVector<F>,
+ x : &mut DVector<F>,
+ y : &DVector<F>,
+ β : F
+) {
+ sorted_abs.copy_from(x);
+ sorted_abs.axpy(-1.0, y, 1.0);
+ sorted_abs.apply(|z_i| *z_i = num_traits::abs(*z_i));
+ sorted_abs.as_mut_slice().sort_unstable_by(|a, b| NaNLeast(*a).cmp(&NaNLeast(*b)));
+
+ let mut n = sorted_abs.sum();
+ for (m, az_i) in zip((1..=x.len() as u32).rev(), sorted_abs) {
+ // test first
+ let tmp = β*n/(1.0 + β*F::cast_from(m));
+ if *az_i <= tmp {
+ // Fail
+ n -= *az_i;
+ } else {
+ // Success
+ x.zip_apply(y, |x_i, y_i| *x_i = y_i + soft_thresholding(*x_i-y_i, tmp));
+ return
+ }
+ }
+ // m = 0 should always work, but x is zero.
+ x.fill(0.0);
+}
+
+/// Returns the ∞-norm minimal subdifferential of $x ↦ (β/2)|x-y|_1^2 - g^⊤ x + λ\|x\|₁$ at $x$.
+///
+/// `v` will be modified and cannot be trusted to contain useful values afterwards.
+#[replace_float_literals(F::cast_from(literal))]
+fn min_subdifferential<F : Float + nalgebra::RealField>(
+ y : &DVector<F>,
+ x : &DVector<F>,
+ g : &DVector<F>,
+ λ : F,
+ β : F
+) -> F {
+ let mut val = 0.0;
+ let tmp = β*y.dist(x, L1);
+ for (&g_i, &x_i, y_i) in izip!(g.iter(), x.iter(), y.iter()) {
+ let (mut lb, mut ub) = (-g_i, -g_i);
+ match x_i.partial_cmp(y_i) {
+ Some(Greater) => { lb += tmp; ub += tmp },
+ Some(Less) => { lb -= tmp; ub -= tmp },
+ Some(Equal) => { lb -= tmp; ub += tmp },
+ None => {},
+ }
+ match x_i.partial_cmp(&0.0) {
+ Some(Greater) => { lb += λ; ub += λ },
+ Some(Less) => { lb -= λ; ub -= λ },
+ Some(Equal) => { lb -= λ; ub += λ },
+ None => {},
+ };
+ val = max_interval_dist_to_zero(val, lb, ub);
+ }
+ val
+}
+
+
+/// PDPS implementation of [`l1squared_unconstrained`].
+/// For detailed documentation of the inputs and outputs, refer to there.
+///
+/// The `λ` component of the model is handled in the proximal step instead of the gradient step
+/// for potential performance improvements.
+#[replace_float_literals(F::cast_from(literal).to_nalgebra_mixed())]
+pub fn l1squared_unconstrained_pdps<F, I>(
+ y : &DVector<F::MixedType>,
+ g : &DVector<F::MixedType>,
+ λ_ : F,
+ β_ : F,
+ x : &mut DVector<F::MixedType>,
+ τ_ : F,
+ σ_ : F,
+ iterator : I
+) -> usize
+where F : Float + ToNalgebraRealField,
+ I : AlgIteratorFactory<F>
+{
+ let λ = λ_.to_nalgebra_mixed();
+ let β = β_.to_nalgebra_mixed();
+ let τ = τ_.to_nalgebra_mixed();
+ let σ = σ_.to_nalgebra_mixed();
+ let mut w = DVector::zeros(x.len());
+ let mut tmp = DVector::zeros(x.len());
+ let mut xprev = x.clone();
+ let mut iters = 0;
+
+ iterator.iterate(|state| {
+ // Primal step: x^{k+1} = prox_{τ|.-y|_1^2}(x^k - τ (w^k - g))
+ x.axpy(-τ, &w, 1.0);
+ x.axpy(τ, g, 1.0);
+ l1squared_prox(&mut tmp, x, y, τ*β);
+
+ // Dual step: w^{k+1} = proj_{[-λ,λ]}(w^k + σ(2x^{k+1}-x^k))
+ w.axpy(2.0*σ, x, 1.0);
+ w.axpy(-σ, &xprev, 1.0);
+ w.apply(|w_i| *w_i = num_traits::clamp(*w_i, -λ, λ));
+ xprev.copy_from(x);
+
+ iters +=1;
+
+ state.if_verbose(|| {
+ F::from_nalgebra_mixed(min_subdifferential(y, x, g, λ, β))
+ })
+ });
+
+ iters
+}
+
+
+/// This function applies an iterative method for the solution of the problem
+/// <div>$$
+/// \min_{x ∈ ℝ^n} \frac{β}{2} |x-y|_1^2 - g^⊤ x + λ\|x\|₁.
+/// $$</div>
+/// Only PDPS is supported.
+///
+/// This function returns the number of iterations taken.
+#[replace_float_literals(F::cast_from(literal))]
+pub fn l1squared_unconstrained<F, I>(
+ y : &DVector<F::MixedType>,
+ g : &DVector<F::MixedType>,
+ λ : F,
+ β : F,
+ x : &mut DVector<F::MixedType>,
+ inner : &InnerSettings<F>,
+ iterator : I
+) -> usize
+where F : Float + ToNalgebraRealField,
+ I : AlgIteratorFactory<F>
+{
+ // Estimate of ‖K‖ for K=Id.
+ let normest = 1.0;
+
+ let (inner_τ, inner_σ) = (inner.pdps_τσ0.0 / normest, inner.pdps_τσ0.1 / normest);
+
+ match inner.method {
+ InnerMethod::PDPS | InnerMethod::Default =>
+ l1squared_unconstrained_pdps(y, g, λ, β, x, inner_τ, inner_σ, iterator),
+ _ => unimplemented!(),
+ }
+}