Optimizers¶

Overview¶

The scikit-learn library provides functionality for training linear models and a large number of related tools. The present module provides simplified interfaces for various linear model regression methods. These methods are set up in a way that work out of the box for typical problems in cluster expansion and force constant potential construction, including slight adjustments to scikit-learn default values. If you need more flexibility, extended functionality or the ability to fine-tune parameters that are not included in this interface, it is possible to use scikit-learn directly.

The most commonly used fit methods in the present context are LASSO, automatic relevance determination regression (ARDR), recursive feature elimination with $$\ell_2$$-fitting (RFE-L2) as well as ordinary least-squares optimization (OLS). Below follows a short summary of the main algorithms. More information about the available linear models can be found in the scikit-learn documentation.

Least-squares¶

Ordinary least-squares (OLS) optimization is providing a solution to the linear problem

$\boldsymbol{A}\boldsymbol{x} = \boldsymbol{y},$

where $$\boldsymbol{A}$$ is the sensing matrix, $$\boldsymbol{y}$$ is the vector of target values, and $$\boldsymbol{x}$$ is the solution (parameter vector) that one seeks to obtain. The objective is given by

$\left\Vert\boldsymbol{A}\boldsymbol{x} - \boldsymbol{y}\right\Vert^2_2$

The OLS method is chosen by setting the fit_method keyword to least-squares.

LASSO¶

The least absolute shrinkage and selection operator (LASSO) is a method for performing variable selection and regularization in problems in statistics and machine learning. The optimization objective is given by

$\frac{1}{2 n_\text{samples}} \left\Vert\boldsymbol{A}\boldsymbol{x} - \boldsymbol{y}\right\Vert^2_2 + \alpha \Vert\boldsymbol{x}\Vert_1.$

While the first term ensures that $$\boldsymbol{x}$$ is a solution to the linear problem at hand, the second term introduces regularization and guides the algorithm toward finding sparse solutions, in the spirit of compressive sensing. In general, LASSO is suited for solving strongly underdetermined problems.

The LASSO optimizer is chosen by setting the fit_method keyword to lasso. The $$\alpha$$ parameter is set via the alpha keyword. If no value is specified a line scan will be carried out automatically to determine the optimal value.

Parameter Type Description Default
alpha float controls the sparsity of the solution vector None

Automatic relevance determination regression (ARDR)¶

Automatic relevance determination regression (ARDR) is an optimization algorithm provided by scikit-learn that is similar to Bayesian Ridge Regression, which provides a probabilistic model of the regression problem at hand. The method is also known as Sparse Bayesian Learning and Relevance Vector Machine.

The ARDR optimizer is chosen by setting the fit_method keyword to ardr. The threshold lambda parameter, which controls the sparsity of the solution vector, is set via the threshold_lambda keyword (default: 1e6).

Parameter Type Description Default
threshold_lambda float controls the sparsity of the solution vector 1e6

split-Bregman¶

The split-Bregman method [GolOsh09] is designed to solve a broad class of $$\ell_1$$-regularized problems. The solution vector $$\boldsymbol{x}$$ is given by

$\boldsymbol{x} = \arg\min_{\boldsymbol{x}, \boldsymbol{d}} \left\Vert\boldsymbol{d}\right\Vert_1 + \frac{1}{2} \left\Vert\boldsymbol{A}\boldsymbol{x} - \boldsymbol{y}\right\Vert^2 + \frac{\lambda}{2} \left\Vert\boldsymbol{d} - \mu \boldsymbol{x} \right\Vert^2,$

where $$\boldsymbol{d}$$ is an auxiliary quantity, while $$\mu$$ and $$\lambda$$ are hyperparameters that control the sparseness of the solution and the efficiency of the algorithm.

The split-Bregman implementation supports the following additional keywords.

Parameter Type Description Default
mu float sparseness parameter 1e-3
lmbda float weight of additional L2-norm in split-Bregman 100
n_iters int maximal number of split-Bregman iterations 1000
tol float convergence criterion iterative minimization 1e-6
verbose bool print additional information to stdout False

Recursive feature elimination¶

Recursive feature elimination (RFE) is a feature selection algorithm that obtains the optimal features by carrying out a series of fits, starting with the full set of parameters and then iteratively eliminating the less important ones. RFE needs to be combined with a specific fit method. Since RFE may require many hundreds of single fits its often advisable to use ordinary least-squares as training method, which is the default behavior. The present implementation is based on the implementation of feature selection in scikit-learn.

The RFE optimizer is chosen by setting the fit_method keyword to rfe. The n_features keyword allows one to specify the number of features to select. If this parameter is left unspecified RFE with cross-validation will be used to determine the optimal number of features.

After the optimal number of features has been determined the final model is trained. The fit method for the final fit can be controlled via final_estimator. Here, estimator and final_estimator can be set to any of the fit methods described in this section. For example, estimator='lasso' implies that a LASSO-CV scan is carried out for each fit in the RFE algorithm.

Parameter Type Description Default
n_features int number of features to select None
step int number parameters to eliminate
float percentage of parameters to eliminate 0.04
cv_splits int number of CV splits (90/10) used when optimizing n_features 5
estimator str fit method to be used in RFE algorithm 'least-squares'
final_estimator str fit method to be used in the final fit = estimator
estimator_kwargs dict keyword arguments for fit method defined by estimator {}
final_estimator_kwargs dict keyword arguments for fit method defined by final_estimator {}

Note

When running on multi-core systems please be mindful of memory consumption. By default all CPUs will be used (n_jobs=-1), which will duplicate data and can require a lot of memory, potentially giving rise to errors. To prevent this behavior you can set the [n_jobs parameter](https://scikit-learn.org/stable/glossary.html#term-n-jobs) explicitly, which is handed over directly to scikit-learn.

Other methods¶

The optimizers furthermore support the ridge method (ridge), the elastic net method (elasticnet) as well as Bayesian ridge regression (bayesian-ridge).

Optimizer¶

class icet.fitting.Optimizer(fit_data, fit_method='least-squares', standardize=True, train_size=0.75, test_size=None, train_set=None, test_set=None, check_condition=True, seed=42, **kwargs)[source]

This optimizer finds a solution to the linear $$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{y}$$ problem.

One has to specify either train_size/test_size or train_set/test_set If either train_set or test_set (or both) is specified the fractions will be ignored.

Warning

Repeatedly setting up a Optimizer and training without changing the seed for the random number generator will yield identical or correlated results, to avoid this please specify a different seed when setting up multiple Optimizer instances.

Parameters: fit_data (tuple(numpy.ndarray, numpy.ndarray)) – the first element of the tuple represents the fit matrix A (N, M array) while the second element represents the vector of target values y (N array); here N (=rows of A, elements of y) equals the number of target values and M (=columns of A) equals the number of parameters fit_method (str) – method to be used for training; possible choice are “least-squares”, “lasso”, “elasticnet”, “bayesian-ridge”, “ardr”, “rfe-l2”, “split-bregman” standardize (bool) – if True the fit matrix is standardized before fitting train_size (float or int) – If float represents the fraction of fit_data (rows) to be used for training. If int, represents the absolute number of rows to be used for training. test_size (float or int) – If float represents the fraction of fit_data (rows) to be used for testing. If int, represents the absolute number of rows to be used for testing. train_set (tuple or list(int)) – indices of rows of A/y to be used for training test_set (tuple or list(int)) – indices of rows of A/y to be used for testing check_condition (bool) – if True the condition number will be checked (this can be sligthly more time consuming for larger matrices) seed (int) – seed for pseudo random number generator
train_scatter_data

target and predicted value for each row in the training set

Type: ScatterData
test_scatter_data

target and predicted value for each row in the test set

Type: ScatterData
compute_rmse(A, y)

Returns the root mean squared error (RMSE) using $$\boldsymbol{A}$$, $$\boldsymbol{y}$$, and the vector of fitted parameters $$\boldsymbol{x}$$, corresponding to $$\|\boldsymbol{A}\boldsymbol{x}-\boldsymbol{y}\|_2$$.

Parameters: A (ndarray) – fit matrix (N,M array) where N (=rows of A, elements of y) equals the number of target values and M (=columns of A) equals the number of parameters (=elements of x) y (ndarray) – vector of target values float
contributions_test

average contribution to the predicted values for the test set from each parameter

Return type: ndarray
contributions_train

average contribution to the predicted values for the train set from each parameter

Return type: ndarray
fit_method

fit method

Return type: str
get_contributions(A)

Returns the average contribution for each row of A to the predicted values from each element of the parameter vector.

Parameters: A (ndarray) – fit matrix where N (=rows of A, elements of y) equals the number of target values and M (=columns of A) equals the number of parameters ndarray
n_nonzero_parameters

number of non-zero parameters

Return type: int
n_parameters

number of parameters (=columns in A matrix)

Return type: int
n_target_values

number of target values (=rows in A matrix)

Return type: int
parameters

copy of parameter vector

Return type: ndarray
predict(A)

Predicts data given an input matrix $$\boldsymbol{A}$$, i.e., $$\boldsymbol{A}\boldsymbol{x}$$, where $$\boldsymbol{x}$$ is the vector of the fitted parameters. The method returns the vector of predicted values or a float if a single row provided as input.

Parameters: A (ndarray) – fit matrix where N (=rows of A, elements of y) equals the number of target values and M (=columns of A) equals the number of parameters Union[ndarray, float]
rmse_test

root mean squared error for test set

Return type: float
rmse_train

root mean squared error for training set

Return type: float
seed

seed used to initialize pseudo random number generator

Return type: int
standardize

if True standardize the fit matrix before fitting

Return type: bool
summary

Return type: Dict[str, Any]
test_fraction

fraction of rows included in test set

Return type: float
test_set

indices of rows included in the test set

Return type: List[int]
test_size

number of rows included in test set

Return type: int
train()[source]

Carries out training.

Return type: None
train_fraction

fraction of rows included in training set

Return type: float
train_set

indices of rows included in the training set

Return type: List[int]
train_size

number of rows included in training set

Return type: int

EnsembleOptimizer¶

class icet.fitting.EnsembleOptimizer(fit_data, fit_method='least-squares', standardize=True, ensemble_size=50, train_size=1.0, bootstrap=True, check_condition=True, seed=42, **kwargs)[source]

The ensemble optimizer carries out a series of single optimization runs using the Optimizer class in order to solve the linear $$\boldsymbol{A}\boldsymbol{x} = \boldsymbol{y}$$ problem. Subsequently, it provides access to various ensemble averaged quantities such as errors and parameters.

Warning

Repeatedly setting up a EnsembleOptimizer and training without changing the seed for the random number generator will yield identical or correlated results, to avoid this please specify a different seed when setting up multiple EnsembleOptimizer instances.

Parameters: fit_data (tuple(numpy.ndarray, numpy.ndarray)) – the first element of the tuple represents the fit matrix A (N, M array) while the second element represents the vector of target values y (N array); here N (=rows of A, elements of y) equals the number of target values and M (=columns of A) equals the number of parameters fit_method (str) – method to be used for training; possible choice are “least-squares”, “lasso”, “elasticnet”, “bayesian-ridge”, “ardr”, “rfe-l2”, “split-bregman” standardize (bool) – if True the fit matrix is standardized before fitting ensemble_size (int) – number of fits in the ensemble train_size (float or int) – if float represents the fraction of fit_data (rows) to be used for training; if int, represents the absolute number of rows to be used for training bootstrap (bool) – if True sampling will be carried out with replacement check_condition (bool) – if True the condition number will be checked (this can be sligthly more time consuming for larger matrices) seed (int) – seed for pseudo random number generator
bootstrap

True if sampling is carried out with replacement

Return type: bool
compute_rmse(A, y)

Returns the root mean squared error (RMSE) using $$\boldsymbol{A}$$, $$\boldsymbol{y}$$, and the vector of fitted parameters $$\boldsymbol{x}$$, corresponding to $$\|\boldsymbol{A}\boldsymbol{x}-\boldsymbol{y}\|_2$$.

Parameters: A (ndarray) – fit matrix (N,M array) where N (=rows of A, elements of y) equals the number of target values and M (=columns of A) equals the number of parameters (=elements of x) y (ndarray) – vector of target values float
ensemble_size

number of train rounds

Return type: int
error_matrix

matrix of fit errors where N is the number of target values and M is the number of fits (i.e., the size of the ensemble)

Return type: ndarray
fit_method

fit method

Return type: str
get_contributions(A)

Returns the average contribution for each row of A to the predicted values from each element of the parameter vector.

Parameters: A (ndarray) – fit matrix where N (=rows of A, elements of y) equals the number of target values and M (=columns of A) equals the number of parameters ndarray
n_nonzero_parameters

number of non-zero parameters

Return type: int
n_parameters

number of parameters (=columns in A matrix)

Return type: int
n_target_values

number of target values (=rows in A matrix)

Return type: int
parameter_vectors

all parameter vectors in the ensemble

Return type: List[ndarray]
parameters

copy of parameter vector

Return type: ndarray
parameters_std

standard deviation for each parameter

Return type: ndarray
predict(A, return_std=False)[source]

Predicts data given an input matrix $$oldsymbol{A}$$, i.e., $$\boldsymbol{A}\boldsymbol{x}$$, where $$\boldsymbol{x}$$ is the vector of the fitted parameters. The method returns the vector of predicted values and optionally also the vector of standard deviations.

By using all parameter vectors in the ensemble a standard deviation of the prediction can be obtained.

Parameters: A (ndarray) – fit matrix where N (=rows of A, elements of y) equals the number of target values and M (=columns of A) equals the number of parameters return_std (bool) – whether or not to return the standard deviation of the prediction Union[ndarray, Tuple[ndarray, ndarray]]
rmse_test

ensemble average of root mean squared error over test sets

Return type: float
rmse_test_ensemble

root mean squared test errors obtained during for each fit in ensemble

Return type: ndarray
rmse_train

ensemble average of root mean squared error over train sets

Return type: float
rmse_train_ensemble

root mean squared train errors obtained during for each fit in ensemble

Return type: ndarray
seed

seed used to initialize pseudo random number generator

Return type: int
standardize

if True standardize the fit matrix before fitting

Return type: bool
summary

Return type: Dict[str, Any]
train()[source]

Carries out ensemble training and construct the final model by averaging over all models in the ensemble.

Return type: None
train_fraction

fraction of input data used for training; this value can differ slightly from the value set during initialization due to rounding

Return type: float
train_size

number of rows included in train sets; note that this will be different from the number of unique rows if boostrapping

Return type: int

CrossValidationEstimator¶

class icet.fitting.CrossValidationEstimator(fit_data, fit_method='least-squares', standardize=True, validation_method='k-fold', n_splits=10, check_condition=True, seed=42, **kwargs)[source]

This class provides an optimizer with cross validation for solving the linear $$\boldsymbol{A}\boldsymbol{x} = \boldsymbol{y}$$ problem. Cross-validation (CV) scores are calculated by splitting the available reference data in multiple different ways. It also produces the finalized model (using the full input data) for which the CV score is an estimation of its performance.

Warning

Repeatedly setting up a CrossValidationEstimator and training without changing the seed for the random number generator will yield identical or correlated results, to avoid this please specify a different seed when setting up multiple CrossValidationEstimator instances.

Parameters: fit_data (tupe(numpy.ndarray, numpy.ndarray)) – the first element of the tuple represents the fit matrix A (N, M array) while the second element represents the vector of target values y (N array); here N (=rows of A, elements of y) equals the number of target values and M (=columns of A) equals the number of parameters fit_method (str) – method to be used for training; possible choice are “least-squares”, “lasso”, “elasticnet”, “bayesian-ridge”, “ardr”, “rfe-l2”, “split-bregman” standardize (bool) – if True the fit matrix is standardized before fitting validation_method (str) – method to use for cross-validation; possible choices are “shuffle-split”, “k-fold” n_splits (int) – number of times the fit data set will be split for the cross-validation check_condition (bool) – if True the condition number will be checked (this can be sligthly more time consuming for larger matrices) seed (int) – seed for pseudo random number generator
train_scatter_data

contains target and predicted values from each individual traininig set in the cross-validation split; ScatterData is a namedtuple.

Type: ScatterData
validation_scatter_data

contains target and predicted values from each individual validation set in the cross-validation split; ScatterData is a namedtuple.

Type: ScatterData
compute_rmse(A, y)

Returns the root mean squared error (RMSE) using $$\boldsymbol{A}$$, $$\boldsymbol{y}$$, and the vector of fitted parameters $$\boldsymbol{x}$$, corresponding to $$\|\boldsymbol{A}\boldsymbol{x}-\boldsymbol{y}\|_2$$.

Parameters: A (ndarray) – fit matrix (N,M array) where N (=rows of A, elements of y) equals the number of target values and M (=columns of A) equals the number of parameters (=elements of x) y (ndarray) – vector of target values float
fit_method

fit method

Return type: str
get_contributions(A)

Returns the average contribution for each row of A to the predicted values from each element of the parameter vector.

Parameters: A (ndarray) – fit matrix where N (=rows of A, elements of y) equals the number of target values and M (=columns of A) equals the number of parameters ndarray
n_nonzero_parameters

number of non-zero parameters

Return type: int
n_nonzero_parameters_splits

number of non-zero parameters for each split

Return type: list
n_parameters

number of parameters (=columns in A matrix)

Return type: int
n_splits

number of splits (folds) used for cross-validation

Return type: int
n_target_values

number of target values (=rows in A matrix)

Return type: int
parameters

copy of parameter vector

Return type: ndarray
parameters_splits

all parameters obtained during cross-validation

Return type: ndarray
predict(A)

Predicts data given an input matrix $$\boldsymbol{A}$$, i.e., $$\boldsymbol{A}\boldsymbol{x}$$, where $$\boldsymbol{x}$$ is the vector of the fitted parameters. The method returns the vector of predicted values or a float if a single row provided as input.

Parameters: A (ndarray) – fit matrix where N (=rows of A, elements of y) equals the number of target values and M (=columns of A) equals the number of parameters Union[ndarray, float]
rmse_train

average root mean squared training error obtained during cross-validation

Return type: float
rmse_train_final

root mean squared error when using the full set of input data

Return type: float
rmse_train_splits

root mean squared training errors obtained during cross-validation

Return type: ndarray
rmse_validation

average root mean squared cross-validation error

Return type: float
rmse_validation_splits

root mean squared validation errors obtained during cross-validation

Return type: ndarray
seed

seed used to initialize pseudo random number generator

Return type: int
standardize

if True standardize the fit matrix before fitting

Return type: bool
summary

Return type: Dict[str, Any]
train()[source]

Constructs the final model using all input data available.

Return type: None
validate()[source]

Runs validation.

Return type: None
validation_method

validation method name

Return type: str