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Mixtures of Transparent Local Models

Niffa Cheick Oumar Diaby, Thierry Duchesne, Mario Marchand

TL;DR

This work introduces Mixtures of Transparent Local Models (MoTLM), a framework for interpretable prediction that partitions the input space around points of interest and learns simple local linear predictors within each locality, with an external predictor for regions outside all localities. Grounded in the PAC-Bayesian theory, MoTLM optimizes a loss that blends local predictions with locality-aware gating and provides rigorous risk bounds via KL-divergence between posterior and prior distributions. The authors derive closed-form loss and KL terms for binary classification and linear regression with known PoIs, and extend to unknown PoIs using non-central chi-square and related functions, employing Quasi-Monte Carlo to approximate complex terms. Empirically, MoTLM achieves competitive accuracy on synthetic and real datasets while maintaining interpretability, outperforming or matching linear baselines and approaching kernel-based opaque models in some settings. The approach offers a principled, interpretable alternative to post-hoc explanations and suggests promising directions for extending to other transparent models and distance metrics.

Abstract

The predominance of machine learning models in many spheres of human activity has led to a growing demand for their transparency. The transparency of models makes it possible to discern some factors, such as security or non-discrimination. In this paper, we propose a mixture of transparent local models as an alternative solution for designing interpretable (or transparent) models. Our approach is designed for the situations where a simple and transparent function is suitable for modeling the label of instances in some localities/regions of the input space, but may change abruptly as we move from one locality to another. Consequently, the proposed algorithm is to learn both the transparent labeling function and the locality of the input space where the labeling function achieves a small risk in its assigned locality. By using a new multi-predictor (and multi-locality) loss function, we established rigorous PAC-Bayesian risk bounds for the case of binary linear classification problem and that of linear regression. In both cases, synthetic data sets were used to illustrate how the learning algorithms work. The results obtained from real data sets highlight the competitiveness of our approach compared to other existing methods as well as certain opaque models. Keywords: PAC-Bayes, risk bounds, local models, transparent models, mixtures of local transparent models.

Mixtures of Transparent Local Models

TL;DR

This work introduces Mixtures of Transparent Local Models (MoTLM), a framework for interpretable prediction that partitions the input space around points of interest and learns simple local linear predictors within each locality, with an external predictor for regions outside all localities. Grounded in the PAC-Bayesian theory, MoTLM optimizes a loss that blends local predictions with locality-aware gating and provides rigorous risk bounds via KL-divergence between posterior and prior distributions. The authors derive closed-form loss and KL terms for binary classification and linear regression with known PoIs, and extend to unknown PoIs using non-central chi-square and related functions, employing Quasi-Monte Carlo to approximate complex terms. Empirically, MoTLM achieves competitive accuracy on synthetic and real datasets while maintaining interpretability, outperforming or matching linear baselines and approaching kernel-based opaque models in some settings. The approach offers a principled, interpretable alternative to post-hoc explanations and suggests promising directions for extending to other transparent models and distance metrics.

Abstract

The predominance of machine learning models in many spheres of human activity has led to a growing demand for their transparency. The transparency of models makes it possible to discern some factors, such as security or non-discrimination. In this paper, we propose a mixture of transparent local models as an alternative solution for designing interpretable (or transparent) models. Our approach is designed for the situations where a simple and transparent function is suitable for modeling the label of instances in some localities/regions of the input space, but may change abruptly as we move from one locality to another. Consequently, the proposed algorithm is to learn both the transparent labeling function and the locality of the input space where the labeling function achieves a small risk in its assigned locality. By using a new multi-predictor (and multi-locality) loss function, we established rigorous PAC-Bayesian risk bounds for the case of binary linear classification problem and that of linear regression. In both cases, synthetic data sets were used to illustrate how the learning algorithms work. The results obtained from real data sets highlight the competitiveness of our approach compared to other existing methods as well as certain opaque models. Keywords: PAC-Bayes, risk bounds, local models, transparent models, mixtures of local transparent models.
Paper Structure (38 sections, 2 theorems, 91 equations, 11 figures, 10 tables)

This paper contains 38 sections, 2 theorems, 91 equations, 11 figures, 10 tables.

Table of Contents

  1. Introduction
  2. PAC-Bayesian Machine Learning
  3. Basic Notions of Machine Learning
  4. PAC-Bayesian Risk Bounds
  5. The PAC-Bayesian Learning Algorithm
  6. Basic Elements for Mixtures of Transparent Local Models
  7. Vicinity Function
  8. A Loss Function Defined on a Set of Models and Localities
  9. Risk Function
  10. The Kullback-Leibler (KL) Divergence Between Q and P
  11. Hypothesis Classes
  12. Classes of Distributions
  13. Mixtures of Transparent Local Models with Known Points of Interest
  14. Binary Linear Classification
  15. Linear Regression
  16. ...and 23 more sections

Key Result

Theorem 2

MM-LN-24 Consider some distribution ${\mathcal{D}}$ over the space ${\mathcal{X}} \times {\mathcal{Y}}$, a class of predictors ${\mathcal{H}}: {\mathcal{X}} \rightarrow {\mathcal{Y}}$, and a prior distribution $P$ on ${\mathcal{H}}$. Let $\phi : {\mathds{R}}_{+} \rightarrow {\mathds{R}}_{+}$, and $f Then, for any $\lambda > 0$ and for any $\delta \in (0, 1)$, with probability at least $1 - \delta$

Figures (11)

  • Figure 1: Example of global models.
  • Figure 2: Example of mixture of transparent local models.
  • Figure 3: Synthetic data sets for binary linear classification with known points of interest.
  • Figure 4: Separating hyperplanes obtained using the linear SVM on the training set.
  • Figure 5: Results of mixtures of transparent local linear classifiers on the training set, with known points of interest.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Definition 1
  • Theorem 2
  • Corollary 3
  • Remark 4