argmin

A pure Rust optimization framework

This crate offers a (work in progress) numerical optimization toolbox/framework written entirely in Rust. It is at the moment potentially very buggy. Please use with care and report any bugs you encounter. This crate is looking for contributors!

Documentation of most recent release

Documentation of master

Design goals

This crate's intention is to be useful to users as well as developers of optimization algorithms, meaning that it should be both easy to apply and easy to implement algorithms. In particular, as a developer of optimization algorithms you should not need to worry about usability features (such as logging, dealing with different types, setters and getters for certain common parameters, counting cost function and gradient evaluations, termination, and so on). Instead you can focus on implementing your algorithm.

• Easy framework for the implementation of optimization algorithms: Implement a single iteration of your method and let the framework do the rest. This leads to similar interfaces for different solvers, making it easy for users.
• Pure Rust implementations of a wide range of optimization methods: This avoids the need to compile and interface C/C++/Fortran code.
• Type-agnostic: Many problems require data structures that go beyond simple vectors to represent the parameters. In argmin, everything is generic: All that needs to be done is implementing certain traits on your data type. For common types, these traits are already implemented.
• Convenient: Easy and consistent logging of anything that may be important. Log to the terminal, to a file or implement your own observers. Future plans include sending metrics to databases and connecting to big data piplines.
• Algorithm evaluation: Methods to assess the performance of an algorithm for different parameter settings, problem classes, ...

Since this crate is in a very early stage, so far most points are only partially implemented or remain future plans.

Algorithms

• Line searches
  • Backtracking line search
  • More-Thuente line search
  • Hager-Zhang line search
• Trust region method
  • Cauchy point method
  • Dogleg method
  • Steihaug method
• Steepest descent
• Conjugate gradient method
• Nonlinear conjugate gradient method
• Newton methods
  • Newton's method
  • Newton-CG
• Quasi-Newton methods
  • BFGS
  • L-BFGS
  • DFP
  • SR1
  • SR1-TrustRegion
• Gauss-Newton method
• Gauss-Newton method with linesearch
• Landweber iteration
• Nelder-Mead method
• Simulated Annealing
• Particle Swarm Optimization

Usage

Add this to your Cargo.toml:

toml [dependencies] argmin = "0.2.2"

Optional features (recommended)

There are additional features which can be activated in Cargo.toml:

toml [dependencies] argmin = { version = "0.2.2", features = ["ctrlc", "ndarrayl"] }

These may become default features in the future. Without these features compilation to wasm32-unknown-unkown seems to be possible.

• ctrlc: Uses the ctrlc crate to properly stop the optimization (and return the current best result) after pressing Ctrl+C.
• ndarrayl: Support for ndarray, ndarray-linalg and ndarray-rand.

Defining a problem

A problem can be defined by implementing the ArgminOp trait which comes with the associated types Param, Output and Hessian. Param is the type of your parameter vector (i.e. the input to your cost function), Output is the type returned by the cost function and Hessian is the type of the Hessian. The trait provides the following methods:

• apply(&self, p: &Self::Param) -> Result<Self::Output, Error>: Applys the cost function to parameters p of type Self::Param and returns the cost function value.
• gradient(&self, p: &Self::Param) -> Result<Self::Param, Error>: Computes the gradient at p.
• hessian(&self, p: &Self::Param) -> Result<Self::Hessian, Error>: Computes the Hessian at p.

The following code snippet shows an example of how to use the Rosenbrock test functions from argmin-testfunctions in argmin:

```rust use argmin::testfunctions::{rosenbrock2d, rosenbrock2dderivative, rosenbrock2d_hessian}; use argmin::prelude::*; use serde::{Serialize, Deserialize};

/// First, create a struct for your problem

[derive(Clone, Default, Serialize, Deserialize)]

struct Rosenbrock { a: f64, b: f64, }

/// Implement ArgminOp for Rosenbrock impl ArgminOp for Rosenbrock { /// Type of the parameter vector type Param = Vec; /// Type of the return value computed by the cost function type Output = f64; /// Type of the Hessian. Can be () if not needed. type Hessian = Vec>;

/// Apply the cost function to a parameter `p`
fn apply(&self, p: &Self::Param) -> Result<Self::Output, Error> {
    Ok(rosenbrock_2d(p, self.a, self.b))
}

/// Compute the gradient at parameter `p`.
fn gradient(&self, p: &Self::Param) -> Result<Self::Param, Error> {
    Ok(rosenbrock_2d_derivative(p, self.a, self.b))
}

/// Compute the Hessian at parameter `p`.
fn hessian(&self, p: &Self::Param) -> Result<Self::Hessian, Error> {
    let t = rosenbrock_2d_hessian(p, self.a, self.b);
    Ok(vec![vec![t[0], t[1]], vec![t[2], t[3]]])
}

} ```

It is optional to implement any of these methods, as there are default implementations which will return an Err when called. What needs to be implemented is defined by the requirements of the solver that is to be used.

Running a solver

The following example shows how to use the previously shown definition of a problem in a Steepest Descent (Gradient Descent) solver.

```rust use argmin::prelude::*; use argmin::solver::gradientdescent::SteepestDescent; use argmin::solver::linesearch::MoreThuenteLineSearch;

// Define cost function (must implement ArgminOperator) let cost = Rosenbrock { a: 1.0, b: 100.0 };

// Define initial parameter vector let init_param: Vec = vec![-1.2, 1.0];

// Set up line search let linesearch = MoreThuenteLineSearch::new();

// Set up solver let solver = SteepestDescent::new(linesearch);

// Run solver let res = Executor::new(cost, solver, initparam) // Add an observer which will log all iterations to the terminal .addobserver(ArgminSlogLogger::term(), ObserverMode::Always) // Set maximum iterations to 10 .max_iters(10) // run the solver on the defined problem .run()?;

// print result println!("{}", res); ```

Observing iterations

Argmin offers an interface to observe the state of the iteration at initialization as well as after every iteration. This includes the parameter vector, gradient, Hessian, iteration number, cost values and many more as well as solver-specific metrics. This interface can be used to implement loggers, send the information to a storage or to plot metrics. Observers need to implment the Observe trait. Argmin ships with a logger based on the slog crate. ArgminSlogLogger::term logs to the terminal and ArgminSlogLogger::file logs to a file in JSON format. Both loggers also come with a *_noblock version which does not block the execution of logging, but may drop some messages if the buffer is full. Parameter vectors can be written to disc using WriteToFile. For each observer it can be defined how often it will observe the progress of the solver. This is indicated via the enum ObserverMode which can be either Always, Never, NewBest (whenever a new best solution is found) or Every(i) which means every ith iteration.

rust let res = Executor::new(problem, solver, init_param) // Add an observer which will log all iterations to the terminal (without blocking) .add_observer(ArgminSlogLogger::term_noblock(), ObserverMode::Always) // Log to file whenever a new best solution is found .add_observer(ArgminSlogLogger::file("solver.log")?, ObserverMode::NewBest) // Write parameter vector to `params/param.arg` every 20th iteration .add_observer(WriteToFile::new("params", "param"), ObserverMode::Every(20)) // run the solver on the defined problem .run()?;

Checkpoints

The probability of crashes increases with runtime, therefore one may want to save checkpoints in order to be able to resume the optimization after a crash. The CheckpointMode defines how often checkpoints are saved and is either Never (default), Always (every iteration) or Every(u64) (every Nth iteration). It is set via the setter method checkpoint_mode of Executor. In addition, the directory where the checkpoints and a prefix for every file can be set via checkpoint_dir and checkpoint_name, respectively.

The following example shows how the from_checkpoint method can be used to resume from a checkpoint. In case this fails (for instance because the file does not exist, which could mean that this is the first run and there is nothing to resume from), it will resort to creating a new Executor, thus starting from scratch.

rust let res = Executor::from_checkpoint(".checkpoints/optim.arg") .unwrap_or(Executor::new(operator, solver, init_param)) .max_iters(iters) .checkpoint_dir(".checkpoints") .checkpoint_name("optim") .checkpoint_mode(CheckpointMode::Every(20)) .run()?;

Implementing an optimization algorithm

In this section we are going to implement the Landweber solver, which essentially is a special form of gradient descent. In iteration k, the new parameter vector x_{k+1} is calculated from the previous parameter vector x_k and the gradient at x_k according to the following update rule:

x_{k+1} = x_k - omega * \nabla f(x_k)

In order to implement this using the argmin framework, one first needs to define a struct which holds data specific to the solver. Then, the Solver trait needs to be implemented for the struct. This requires setting the associated constant NAME which gives your solver a name. The next_iter method defines the computations performed in a single iteration of the solver. Via the parameters op and state one has access to the operator (cost function, gradient computation, Hessian, ...) and to the current state of the optimization (parameter vectors, cost function values, iteration number, ...), respectively.

```rust use argmin::prelude::*; use serde::{Deserialize, Serialize};

// Define a struct which holds any parameters/data which are needed during the execution of the // solver. Note that this does not include parameter vectors, gradients, Hessians, cost // function values and so on, as those will be handled by the Executor.

[derive(Serialize, Deserialize)]

pub struct Landweber { /// omega omega: f64, }

impl Landweber { /// Constructor pub fn new(omega: f64) -> Self { Landweber { omega } } }

impl Solver for Landweber where // O always needs to implement ArgminOp O: ArgminOp, // O::Param needs to implement ArgminScaledSub because of the update formula O::Param: ArgminScaledSub, { // This gives the solver a name which will be used for logging const NAME: &'static str = "Landweber";

// Defines the computations performed in a single iteration.
fn next_iter(
    &mut self,
    // This gives access to the operator supplied to the `Executor`. `O` implements
    // `ArgminOp` and `OpWrapper` takes care of counting the calls to the respective
    // functions.
    op: &mut OpWrapper<O>,
    // Current state of the optimization. This gives access to the parameter vector,
    // gradient, Hessian and cost function value of the current, previous and best
    // iteration as well as current iteration number, and many more.
    state: &IterState<O>,
) -> Result<ArgminIterData<O>, Error> {
    // First we obtain the current parameter vector from the `state` struct (`x_k`).
    let xk = state.get_param();
    // Then we compute the gradient at `x_k` (`\nabla f(x_k)`)
    let grad = op.gradient(&xk)?;
    // Now subtract `\nabla f(x_k)` scaled by `omega` from `x_k` to compute `x_{k+1}`
    let xkp1 = xk.scaled_sub(&self.omega, &grad);
    // Return new paramter vector which will then be used by the `Executor` to update
    // `state`.
    Ok(ArgminIterData::new().param(xkp1))
}

} ```

TODOs

• More optimization methods
• Automatic differentiation
• Parallelization
• Tests
• Evaluation on real problems
• Evaluation framework
• Documentation & Tutorials
• C interface
• Python wrapper
• Solver and problem definition via a config file

Please open an issue if you want to contribute! Any help is appreciated!

License

Licensed under either of

• Apache License, Version 2.0, (LICENSE-APACHE or http://www.apache.org/licenses/LICENSE-2.0)
• MIT License (LICENSE-MIT or http://opensource.org/licenses/MIT)

at your option.

Contribution

Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache-2.0 license, shall be dual licensed as above, without any additional terms or conditions.

License: MIT OR Apache-2.0