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A Library for Scaling Mixed-Integer Optimization-Based Machine Learning.

Project description


BackboneLearn is an open-source software package and framework for scaling mixed integer optimization (MIO) problems with indicator variables to high-dimensional problems. This optimization paradigm can naturally be used to formulate fundamental problems in interpretable supervised learning (e.g., sparse regression and decision trees), in unsupervised learning (e.g., clustering), and beyond; BackboneLearn solves the aforementioned problems faster than exact methods and with higher accuracy than commonly used heuristics, while also scaling to large problem sizes. The package is built in Python and is user-friendly and easily extendible: users can directly implement a backbone algorithm for their MIO problem at hand.

What do we mean by indicators?

Indicators are binary variables that are part of the MIO problem we use to train the aforementioned models.

  • Sparse regression: Each regression coefficient (and the corresponding feature) is paired with an indicator, which is 1 if the coefficient is nonzero and 0 otherwise.
  • Decision trees: An indicator corresponds to a feature in a decision tree node, being nonzero if that feature is chosen for branching at that node.
  • Clustering: An indicator represents whether a pair of data points are in the same cluster, being nonzero if they are clustered together.


The backbone framework, upon which BackboneLearn is built, operates in two phases: we first extract a “backbone set” of potentially ``relevant indicators'' (i.e., indicators that are nonzero in the optimal solution) by solving a number of specially chosen, tractable subproblems; we then use traditional techniques to solve a reduced problem to optimality or near-optimality, considering only the backbone indicators. A screening step often proceeds the first phase, to discard indicators that are almost surely irrelevant. For more details, check the paper by Bertsimas and Digalakis Jr (2022)

Getting Started


Install BackboneLearn using pip:

pip install backbone-learn

Note: For ODT implementations, please follow the instructions described in the libary to install compile CBC from source using coinbrew:

Example Usage

Here are some example usages of BackboneLearn for different tasks that you could find in 'notebooks':

Sparse Regression with L0BnB Model

from backbone_learn.backbone.backbone_sparse_regression import BackboneSparseRegression
# Initialize BackboneSparseRegression
backbone = BackboneSparseRegression(alpha=0.5, beta=0.5, num_subproblems=5, num_iterations=1, lambda_2=0.001, max_nonzeros=10)
# Fit the model, y)
# Make predictions
y_pred = backbone.predict(X)

Decision Trees with BendersOCT Model

from backbone_learn.backbone.backbone_decision_tree import BackboneDecisionTree
# Initialize BackboneDecisionTree
backbone = BackboneDecisionTree(alpha=0.5, beta=0.5, num_subproblems=5, num_iterations=1, depth=3, _lambda=0.5)
# Fit the model, y)
# Make predictions
y_pred = backbone.predict(X)

Clustering with MIO Formulation Model

from backbone_learn.backbone.backbone_clustering import BackboneClustering
# Initialize BackboneClustering
backbone = BackboneClustering(beta=1.0, num_subproblems=5, num_iterations=1, n_clusters=5)
# Fit the model
# Make predictions
y_pred = backbone.predict(X)

Running Simulations

To run benchmark simulations, please follow these commands and change the hyper-parameters for each script in 'experiments':

# Install Poetry dependencies
poetry install
# Activate Poetry shell (optional but recommended)
poetry shell
# Run benchmark simulations for decision trees, sparse regression, and clustering
python experiments/
python experiments/
python experiments/

Custom Implementation Guide

Follow these steps to implement your backbone algorithms in BackboneLearn:

Custom Screening Method

  • Extend ScreenSelectorBase in 'backbone_learn/screen_selector'.
  • Implement the calculate_utilities method, which computes utilities (or importances) for each feature based on your screening criteria. Features with the lowest scores may be candidates for elimination. The number of features to keep is defined as a fraction of the total features (alpha * n_features).

Custom Heuristic Method

  • Extend HeuristicSolverBase in 'backbone_learn/heuristic_solvers'.
  • Implement fit and get_relevant_features. The fit method trains a model within each subproblem, which needs to be highly efficient. This function fits a sparse model (regression, trees, etc.) to the data inputs of the subproblem. The get_relevant_features method identifies and extracts all features deemed relevant by the model. Ensure to add any necessary parameters in the init or fit method and pass them to the backbone. The fraction of features to include in the feature set for each subproblem is defined by 'beta'. The number of subproblems and iterations are defined by 'num_subproblems' and 'n_iterations', accordingly.

Custom Exact Solver

  • Extend ExactSolverBase in 'backbone_learn/exact_solvers'.
  • Implement fit and predict methods. The fit function is applied to the reduced backbone set and does not need to be efficient; instead, it should use a method with optimality guarantees. The model parameters can again be defined in init and passed in the backbone class. The predict function ensures that the model can be used for prediction.

Custom Backbone Algorithm

  • Extend BackboneSupervised for supervised algorithms or BackboneUnsupervised for unsupervised algorithms in 'backbone_learn/backbone'. The supervised algorithms perform feature selection, while the unsupervised algorithms perform data selection.
  • Implement set_solvers. Add your customized screen selector, heuristic solver, and exact solver as class instances. It's optional to define a screen selector and heuristic solver. Parameters for each solver are recommended to be passed manually. See examples in backbone_learn/backbone/backbone_decision_tree.

Example Usage for Customized Backbone Algorithm

Here's an example of how you can crate a custom Backbone algorithm for supervised method:

class CustomBackboneAlgorithm(BackboneSupervised):
    def set_solvers(self, **kwargs):
        # Init screen selector or set None to skip
        self.screen_selector = CustomScreenSelector(**kwarg)
        # Init heuristic solver or set None to skip
        self.heuristic_solver = CustomHeuristicSolver(**kwarg)
         # Init exact solver
        self.exact_solver = CustomExactSolver(**kwarg)

Here's how you can use the custom Backbone algorithm:

# Initialize with custom parameters
backbone_algorithm = CustomBackboneAlgorithm(alpha=0.5, beta=0.3, num_subproblems=3,**kwarg)

# Fit the model, y)

# Make predictions
predictions = backbone_algorithm.predict(X_new)

Citing BackboneLearn

If you find BackboneLearn useful in your research, please consider citing the following papers.

Paper 1 (Toolkit):

Coming soon

Paper 2 Methodology:

@article{bertsimas2022backbone, title={The backbone method for ultra-high dimensional sparse machine learning}, author={Bertsimas, Dimitris and Digalakis Jr, Vassilis}, journal={Machine Learning}, volume={111}, number={6}, pages={2161--2212}, year={2022}, publisher={Springer} }


Apache License 2.0

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