finitediff/vec/
hessian.rs

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// Copyright 2018-2024 argmin developers
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// http://apache.org/licenses/LICENSE-2.0> or the MIT license <LICENSE-MIT or
// http://opensource.org/licenses/MIT>, at your option. This file may not be
// copied, modified, or distributed except according to those terms.

use std::ops::AddAssign;

use anyhow::{anyhow, Error};
use num::{Float, FromPrimitive};

use crate::utils::{mod_and_calc, restore_symmetry_vec, KV};

use super::{CostFn, GradientFn};

pub fn forward_hessian_vec<F>(x: &Vec<F>, grad: GradientFn<'_, F>) -> Result<Vec<Vec<F>>, Error>
where
    F: Float + FromPrimitive,
{
    let eps_sqrt = F::epsilon().sqrt();
    let fx = (grad)(x)?;
    let mut xt = x.clone();
    let out: Vec<Vec<F>> = (0..x.len())
        .map(|i| {
            let fx1 = mod_and_calc(&mut xt, grad, i, eps_sqrt)?;
            Ok(fx1
                .iter()
                .zip(fx.iter())
                .map(|(&a, &b)| (a - b) / eps_sqrt)
                .collect::<Vec<F>>())
        })
        .collect::<Result<_, Error>>()?;

    // restore symmetry
    Ok(restore_symmetry_vec(out))
}

pub fn central_hessian_vec<F>(x: &[F], grad: GradientFn<'_, F>) -> Result<Vec<Vec<F>>, Error>
where
    F: Float + FromPrimitive,
{
    let eps_cbrt = F::epsilon().cbrt();
    let mut xt = x.to_owned();
    let out: Vec<Vec<F>> = (0..x.len())
        .map(|i| {
            let fx1 = mod_and_calc(&mut xt, grad, i, eps_cbrt)?;
            let fx2 = mod_and_calc(&mut xt, grad, i, -eps_cbrt)?;
            Ok(fx1
                .iter()
                .zip(fx2.iter())
                .map(|(&a, &b)| (a - b) / (F::from_f64(2.0).unwrap() * eps_cbrt))
                .collect::<Vec<F>>())
        })
        .collect::<Result<_, Error>>()?;

    // restore symmetry
    Ok(restore_symmetry_vec(out))
}

pub fn forward_hessian_vec_prod_vec<F>(
    x: &Vec<F>,
    grad: GradientFn<'_, F>,
    p: &[F],
) -> Result<Vec<F>, Error>
where
    F: Float,
{
    let eps_sqrt = F::epsilon().sqrt();
    let fx = (grad)(x)?;
    let out: Vec<F> = {
        let x1 = x
            .iter()
            .zip(p.iter())
            .map(|(&xi, &pi)| xi + pi * eps_sqrt)
            .collect();
        let fx1 = (grad)(&x1)?;
        fx1.iter()
            .zip(fx.iter())
            .map(|(&a, &b)| (a - b) / eps_sqrt)
            .collect::<Vec<F>>()
    };
    Ok(out)
}

pub fn central_hessian_vec_prod_vec<F>(
    x: &[F],
    grad: GradientFn<'_, F>,
    p: &[F],
) -> Result<Vec<F>, Error>
where
    F: Float + FromPrimitive,
{
    let eps_cbrt = F::epsilon().cbrt();
    let out: Vec<F> = {
        // TODO: Do this in single array
        let x1 = x
            .iter()
            .zip(p.iter())
            .map(|(&xi, &pi)| xi + pi * eps_cbrt)
            .collect();
        let x2 = x
            .iter()
            .zip(p.iter())
            .map(|(&xi, &pi)| xi - pi * eps_cbrt)
            .collect();
        let fx1 = (grad)(&x1)?;
        let fx2 = (grad)(&x2)?;
        fx1.iter()
            .zip(fx2.iter())
            .map(|(&a, &b)| (a - b) / (F::from_f64(2.0).unwrap() * eps_cbrt))
            .collect::<Vec<F>>()
    };
    Ok(out)
}

pub fn forward_hessian_nograd_vec<F>(x: &Vec<F>, f: CostFn<'_, F>) -> Result<Vec<Vec<F>>, Error>
where
    F: Float + FromPrimitive + AddAssign,
{
    // TODO: Check why this is necessary
    let eps_nograd = F::from_f64(2.0).unwrap() * F::epsilon();
    let eps_sqrt_nograd = eps_nograd.sqrt();

    let fx = (f)(x)?;
    let n = x.len();
    let mut xt = x.clone();

    // Precompute f(x + sqrt(EPS) * e_i) for all i
    let fxei: Vec<F> = (0..n)
        .map(|i| mod_and_calc(&mut xt, f, i, eps_sqrt_nograd))
        .collect::<Result<_, Error>>()?;

    let mut out: Vec<Vec<F>> = vec![vec![F::from_f64(0.0).unwrap(); n]; n];

    for i in 0..n {
        for j in 0..=i {
            let t = {
                let xti = xt[i];
                let xtj = xt[j];
                xt[i] += eps_sqrt_nograd;
                xt[j] += eps_sqrt_nograd;
                let fxij = (f)(&xt)?;
                xt[i] = xti;
                xt[j] = xtj;
                (fxij - fxei[i] - fxei[j] + fx) / eps_nograd
            };
            out[i][j] = t;
            out[j][i] = t;
        }
    }
    Ok(out)
}

pub fn forward_hessian_nograd_sparse_vec<F>(
    x: &Vec<F>,
    f: CostFn<'_, F>,
    indices: Vec<[usize; 2]>,
) -> Result<Vec<Vec<F>>, Error>
where
    F: Float + FromPrimitive + AddAssign,
{
    // TODO: Check why this is necessary
    let eps_nograd = F::from_f64(2.0).unwrap() * F::epsilon();
    let eps_sqrt_nograd = eps_nograd.sqrt();

    let fx = (f)(x)?;
    let n = x.len();
    let mut xt = x.clone();

    let mut idxs: Vec<usize> = indices
        .iter()
        .flat_map(|i| i.iter())
        .cloned()
        .collect::<Vec<usize>>();
    idxs.sort();
    idxs.dedup();

    let mut fxei = KV::new(idxs.len());

    for idx in idxs.iter() {
        fxei.set(*idx, mod_and_calc(&mut xt, f, *idx, eps_sqrt_nograd)?);
    }

    let mut out: Vec<Vec<F>> = vec![vec![F::from_f64(0.0).unwrap(); n]; n];
    for [i, j] in indices {
        let t = {
            let xti = xt[i];
            let xtj = xt[j];
            xt[i] += eps_sqrt_nograd;
            xt[j] += eps_sqrt_nograd;
            let fxij = (f)(&xt)?;
            xt[i] = xti;
            xt[j] = xtj;

            let fxi = fxei.get(i).ok_or(anyhow!("Bug"))?;
            let fxj = fxei.get(j).ok_or(anyhow!("Bug"))?;
            (fxij - fxi - fxj + fx) / eps_nograd
        };
        out[i][j] = t;
        out[j][i] = t;
    }
    Ok(out)
}

#[cfg(test)]
mod tests {
    use super::*;

    const COMP_ACC: f64 = 1e-6;

    fn f(x: &Vec<f64>) -> Result<f64, Error> {
        Ok(x[0] + x[1].powi(2) + x[2] * x[3].powi(2))
    }

    fn g(x: &Vec<f64>) -> Result<Vec<f64>, Error> {
        Ok(vec![1.0, 2.0 * x[1], x[3].powi(2), 2.0 * x[3] * x[2]])
    }

    fn x() -> Vec<f64> {
        vec![1.0f64, 1.0, 1.0, 1.0]
    }

    fn p() -> Vec<f64> {
        vec![2.0, 3.0, 4.0, 5.0]
    }

    fn res1() -> Vec<Vec<f64>> {
        vec![
            vec![0.0, 0.0, 0.0, 0.0],
            vec![0.0, 2.0, 0.0, 0.0],
            vec![0.0, 0.0, 0.0, 2.0],
            vec![0.0, 0.0, 2.0, 2.0],
        ]
    }

    fn res2() -> Vec<f64> {
        vec![0.0, 6.0, 10.0, 18.0]
    }

    #[test]
    fn test_forward_hessian_vec_f64() {
        let hessian = forward_hessian_vec(&x(), &g).unwrap();
        let res = res1();
        // println!("hessian:\n{:#?}", hessian);
        // println!("diff:\n{:#?}", diff);
        for i in 0..4 {
            for j in 0..4 {
                assert!((res[i][j] - hessian[i][j]).abs() < COMP_ACC)
            }
        }
    }

    #[test]
    fn test_central_hessian_vec_f64() {
        let hessian = central_hessian_vec(&x(), &g).unwrap();
        let res = res1();
        // println!("hessian:\n{:#?}", hessian);
        // println!("diff:\n{:#?}", diff);
        for i in 0..4 {
            for j in 0..4 {
                assert!((res[i][j] - hessian[i][j]).abs() < COMP_ACC)
            }
        }
    }

    #[test]
    fn test_forward_hessian_vec_prod_vec_f64() {
        let hessian = forward_hessian_vec_prod_vec(&x(), &g, &p()).unwrap();
        let res = res2();
        // println!("hessian:\n{:#?}", hessian);
        // println!("diff:\n{:#?}", diff);
        for i in 0..4 {
            assert!((res[i] - hessian[i]).abs() < COMP_ACC)
        }
    }

    #[test]
    fn test_central_hessian_vec_prod_vec_f64() {
        let hessian = central_hessian_vec_prod_vec(&x(), &g, &p()).unwrap();
        let res = res2();
        // println!("hessian:\n{:#?}", hessian);
        // println!("diff:\n{:#?}", diff);
        for i in 0..4 {
            assert!((res[i] - hessian[i]).abs() < COMP_ACC)
        }
    }

    #[test]
    fn test_forward_hessian_nograd_vec_f64() {
        let hessian = forward_hessian_nograd_vec(&x(), &f).unwrap();
        let res = res1();
        // println!("hessian:\n{:#?}", hessian);
        for i in 0..4 {
            for j in 0..4 {
                assert!((res[i][j] - hessian[i][j]).abs() < COMP_ACC)
            }
        }
    }

    #[test]
    fn test_forward_hessian_nograd_sparse_vec_f64() {
        let indices = vec![[1, 1], [2, 3], [3, 3]];
        let hessian = forward_hessian_nograd_sparse_vec(&x(), &f, indices).unwrap();
        let res = res1();
        // println!("hessian:\n{:#?}", hessian);
        // println!("diff:\n{:#?}", diff);
        for i in 0..4 {
            for j in 0..4 {
                assert!((res[i][j] - hessian[i][j]).abs() < COMP_ACC)
            }
        }
    }
}