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Improving Decision Sparsity

Yiyang Sun, Tong Wang, Cynthia Rudin

TL;DR

This work dramatically expands a notion of decision sparsity called the Sparse Explanation Value (SEV) so that its explanations are more meaningful, and proposes algorithms that optimize decision sparsity in machine learning models.

Abstract

Sparsity is a central aspect of interpretability in machine learning. Typically, sparsity is measured in terms of the size of a model globally, such as the number of variables it uses. However, this notion of sparsity is not particularly relevant for decision-making; someone subjected to a decision does not care about variables that do not contribute to the decision. In this work, we dramatically expand a notion of decision sparsity called the Sparse Explanation Value(SEV) so that its explanations are more meaningful. SEV considers movement along a hypercube towards a reference point. By allowing flexibility in that reference and by considering how distances along the hypercube translate to distances in feature space, we can derive sparser and more meaningful explanations for various types of function classes. We present cluster-based SEV and its variant tree-based SEV, introduce a method that improves credibility of explanations, and propose algorithms that optimize decision sparsity in machine learning models.

Improving Decision Sparsity

TL;DR

This work dramatically expands a notion of decision sparsity called the Sparse Explanation Value (SEV) so that its explanations are more meaningful, and proposes algorithms that optimize decision sparsity in machine learning models.

Abstract

Sparsity is a central aspect of interpretability in machine learning. Typically, sparsity is measured in terms of the size of a model globally, such as the number of variables it uses. However, this notion of sparsity is not particularly relevant for decision-making; someone subjected to a decision does not care about variables that do not contribute to the decision. In this work, we dramatically expand a notion of decision sparsity called the Sparse Explanation Value(SEV) so that its explanations are more meaningful. SEV considers movement along a hypercube towards a reference point. By allowing flexibility in that reference and by considering how distances along the hypercube translate to distances in feature space, we can derive sparser and more meaningful explanations for various types of function classes. We present cluster-based SEV and its variant tree-based SEV, introduce a method that improves credibility of explanations, and propose algorithms that optimize decision sparsity in machine learning models.

Paper Structure

This paper contains 37 sections, 5 theorems, 16 equations, 13 figures, 15 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Local Sparsity Optimization Models
  4. Preliminaries and Motivation
  5. Recap of Sparse Explanation Values
  6. Motivation of Our Work: Sensitivity to Reference Points
  7. Selecting References.
  8. Meaningful and Credible SEV
  9. Cluster-based SEV: Improving Closeness
  10. Tree-based SEV: SEV$^{\copyright}$ Variant with Useful Properties and Computational Benefits
  11. Improving Credibility for All SEV Calculations
  12. Flexible Reference SEV: Improving Sparsity
  13. Optimizing Models for SEV$^{-}$
  14. Gradient-based SEV Optimization
  15. Search-based SEV Optimization
  16. ...and 22 more sections

Key Result

Theorem 4.1

With a single decision classifier DT and a positively-predicted query $\boldsymbol{x}_i$, define $N_i$ as the leaf that captures it. If $N_i$ has a sibling leaf, or any internal node in its decision path has a negatively-predicted child leaf, then SEV$^{T}$ is equal to 1.

Figures (13)

  • Figure 1: SEV Hypercube
  • Figure 2: Cluster-based SEV
  • Figure 3: SEV$^{T}$ Preprocessing
  • Figure 4: Efficient SEV$^{T}$ calculation: Query (node 7) has SEV$^{T}$=1, which goes to node 10. The path to this node is recorded as LL at node 3, which is along the decision path to node 7.
  • Figure 5: Explanation performance under different models and metrics. We desire lower SEV$^{-}$ for sparsity, lower $\ell_\infty$ for closeness and higher log likelihood for credibility (shaded regions)
  • ...and 8 more figures

Theorems & Definitions (7)

  • Definition 3.1: SEV$^{}$ hypercube
  • Definition 3.2: SEV$^{-}$
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem L.1
  • Theorem M.1