Crate ganesh

ganesh, (/ɡəˈneɪʃ/) named after the Hindu god of wisdom, provides several common minimization algorithms as well as a straightforward, trait-based interface to create your own extensions. This crate is intended to be as simple as possible. The user needs to implement the Function trait on some struct which will take a vector of parameters and return a single-valued Result ($f(\mathbb{R}^n) \to \mathbb{R}$). Users can optionally provide a gradient function to speed up some algorithms, but a default central finite-difference implementation is provided so that all algorithms will work out of the box.

This crate is still in an early development phase, and the API is not stable. It can (and likely will) be subject to breaking changes before the 1.0.0 version release (and hopefully not many after that).

Table of Contents

• Key Features
• Quick Start
• Bounds
• Future Plans

Key Features

• Simple but powerful trait-oriented library which tries to follow the Unix philosophy of “do one thing and do it well”.
• Generics to allow for different numeric types to be used in the provided algorithms.
• Algorithms that are simple to use with sensible defaults.
• Traits which make developing future algorithms simple and consistent.

Quick Start

This crate provides some common test functions in the test_functions module. Consider the following implementation of the Rosenbrock function:

use std::convert::Infallible;
use ganesh::prelude::*;

pub struct Rosenbrock {
    pub n: usize,
}
impl Function<f64, (), Infallible> for Rosenbrock {
    fn evaluate(&self, x: &[f64], _user_data: &mut ()) -> Result<f64, Infallible> {
        Ok((0..(self.n - 1))
            .map(|i| 100.0 * (x[i + 1] - x[i].powi(2)).powi(2) + (1.0 - x[i]).powi(2))
            .sum())
    }
}

To minimize this function, we could consider using the Nelder-Mead algorithm:

use ganesh::prelude::*;
use ganesh::algorithms::NelderMead;


let problem = Rosenbrock { n: 2 };
let nm = NelderMead::default();
let mut m = Minimizer::new(nm, 2);
let x0 = &[2.0, 2.0];
m.minimize(&problem, x0, &mut ()).unwrap();
println!("{}", m.status);

This should output

MSG:       term_f = STDDEV
X:         [0.9999999946231828, 0.9999999884539057]
F(X):      0.00000000000000009170942877687133
N_EVALS:   160
CONVERGED: true

Bounds

All minimizers in ganesh have access to a feature which allows algorithms which usually function in unbounded parameter spaces to only return results inside a bounding box. This is done via a parameter transformation, the same one used by LMFIT and MINUIT. While the user inputs parameters within the bounds, unbounded algorithms can (and in practice will) convert those values to a set of unbounded “internal” parameters. When functions are called, however, these internal parameters are converted back into bounded “external” parameters, via the following transformations:

Upper and lower bounds:

x_\text{int} = \arcsin\left(2\frac{x_\text{ext} - x_\text{min}}{x_\text{max} - x_\text{min}} - 1\right)

x_\text{ext} = x_\text{min} + \left(\sin(x_\text{int}) + 1\right)\frac{x_\text{max} - x_\text{min}}{2}

Upper bound only:

x_\text{int} = \sqrt{(x_\text{max} - x_\text{ext} + 1)^2 - 1}

x_\text{ext} = x_\text{max} + 1 - \sqrt{x_\text{int}^2 + 1}

Lower bound only:

x_\text{int} = \sqrt{(x_\text{ext} - x_\text{min} + 1)^2 - 1}

x_\text{ext} = x_\text{min} - 1 + \sqrt{x_\text{int}^2 + 1}

As noted in the documentation for both LMFIT and MINUIT, these bounds should be used with caution. They turn linear problems into nonlinear ones, which can mess with error propagation and even fit convergence, not to mention increase function complexity. Methods which output covariance matrices need to be adjusted if bounded, and MINUIT recommends fitting a second time near a minimum without bounds to ensure proper error propagation.

Future Plans

• Eventually, I would like to implement BGFS and variants, MCMC algorithms, and some more modern gradient-free optimization techniques.
• There are probably many optimizations and algorithm extensions that I’m missing right now because I just wanted to get it working first.
• A test suite

Modules

• algorithms
  Module containing minimization algorithms
• observers
  Module containing Observers
• prelude
  Prelude module containing everything someone should need to use this crate for non-development purposes
• test_functions
  Module containing standard functions for testing algorithms

Macros

• convert
  Convenience macro for converting raw numeric values to a generic

Structs

• Minimizer
  The main struct used for running Algorithms on Functions.
• Status
  A status message struct containing all information about a minimization result.

Enums

• Bound
  An enum that describes a bound/limit on a parameter in a minimization.

Traits

• Algorithm
  A trait representing a minimization algorithm.
• Function
  A trait which describes a function $f(\mathbb{R}^n) \to \mathbb{R}$
• Observer
  A trait which holds a callback function that can be used to check an Algorithm’s Status during a minimization.