src/subproblem/l1squared_unconstrained.rs

/*!
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 =>
            l1squared_unconstrained_pdps(y, g, λ, β, x, inner_τ, inner_σ, iterator),
        other => unimplemented!("${other:?} is unimplemented"),
    }
}