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A scalable package for fitting sparse linear regression models with pairwise feature interactions, under strong hierarchy.

Project description


Hussein Hazimeh and Rahul Mazumder

Massachusetts Institute of Technology


hierScale is scalable toolkit for fitting sparse linear regression models with pairwise feature interactions. The optimization is done under the strong hierarchy (SH) constraint: an interaction coefficient is non-zero only if its associated main feature coefficients are non-zero. This constraint can enhance the interpretability of sparse interaction models and also reduce the future data collection costs; see the discussion in (Hazimeh and Mazumder, 2020).

More formally, given a data matrix X of main features and a response vector y, the toolkit fits a convex relaxation of the following model:

where X_i denotes the ith column (feature) of X and * refers to element-wise multiplication. The L0 norms impose sparsity on the coefficients and the constraints enforce SH. See (Hazimeh and Mazumder, 2020) for details on how the convex relaxation of the above problem is derived. The optimization is done for a regularization path (i.e., over a grid of lambda_1's and lambda_2's). We use proximal gradient descent (PGD) for optimization, along with novel proximal screening and gradient screening rules, which speed up PGD by over 4900x.


hierScale is written in Python 3. It uses Gurobi internally (for solving the LPs required for checking the optimality conditions). Before installing hierScale, please make sure that Gurobi and its Python interface (gurobipy) are installed.

To install hierScale, run the following command:

pip install hierScale

Quick Start

In Python, assuming you have the data X and y stored as numpy arrays, run the following to fit a regularization path:

from hierScale import hier_fit, hier_predict

# Set the parameters of the algorithm.
params = {
    "nLambda": 100, # Number of lambda_1's in the path.
    "maxSuppSize": 500, # Max support size to terminate the path at.

# Fit a path.
solutions, lambdas = hier_fit(X, y, params)

# solutions is a list of all the solutions in the path.
# To access the ith solution, say i=10, use the following:
current_solution = solutions[10]
current_solution.B # A dictionary of the non-zero coefficients in beta.
current_solution.T # A dictionary of the non-zero coefficients in theta.
current_solution.intercept # The intercept term.

# To predict the response given a matrix X, run the following:
hier_predict(current_solution, X)

# For more advanced usage and parameters, please check the documentation:


Learning Hierarchical Interactions at Scale: A Convex Optimization Approach.

Bibtex citation below:

  author    = {Hussein Hazimeh and
               Rahul Mazumder},
  title     = {Learning Hierarchical Interactions at Scale: A Convex Optimization Approach},
  booktitle = {The 23nd International Conference on Artificial Intelligence and Statistics,
               {AISTATS} 2020, 3-5 June 2020, Palermo, Sicily, Italy},
  series    = {Proceedings of Machine Learning Research},
  volume    = {108},
  publisher = {{PMLR}},
  year      = {2020},
  url       = {},

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