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Importance Sampling for Nonlinear Models

Prakash Palanivelu Rajmohan, Fred Roosta

TL;DR

This work addresses the challenge of extending importance sampling to nonlinear models by introducing the nonlinear adjoint operator, enabling an inner-product-like representation that yields nonlinear leverage and norm scores. By formulating a nonlinear dual matrix $\widehat{\mathbf{F}}^{\star}(\bm\theta)$, the authors derive sampling schemes with guarantees analogous to linear subspace embeddings for nonlinear losses, including squared loss. They provide parameter-independent score estimation, an $\varepsilon$-net based lower bound, and conditions under which the full-data loss can be approximated from sampled data with small error. Empirical results on regression and binary classification tasks demonstrate that nonlinear scores identify important samples and aid outlier detection, while reducing training costs compared to linear or uniform sampling. The framework promises practical impact for efficient training and model diagnostics in neural networks and generalized linear predictors, with potential extensions to active and transfer learning.

Abstract

While norm-based and leverage-score-based methods have been extensively studied for identifying "important" data points in linear models, analogous tools for nonlinear models remain significantly underdeveloped. By introducing the concept of the adjoint operator of a nonlinear map, we address this gap and generalize norm-based and leverage-score-based importance sampling to nonlinear settings. We demonstrate that sampling based on these generalized notions of norm and leverage scores provides approximation guarantees for the underlying nonlinear mapping, similar to linear subspace embeddings. As direct applications, these nonlinear scores not only reduce the computational complexity of training nonlinear models by enabling efficient sampling over large datasets but also offer a novel mechanism for model explainability and outlier detection. Our contributions are supported by both theoretical analyses and experimental results across a variety of supervised learning scenarios.

Importance Sampling for Nonlinear Models

TL;DR

This work addresses the challenge of extending importance sampling to nonlinear models by introducing the nonlinear adjoint operator, enabling an inner-product-like representation that yields nonlinear leverage and norm scores. By formulating a nonlinear dual matrix , the authors derive sampling schemes with guarantees analogous to linear subspace embeddings for nonlinear losses, including squared loss. They provide parameter-independent score estimation, an -net based lower bound, and conditions under which the full-data loss can be approximated from sampled data with small error. Empirical results on regression and binary classification tasks demonstrate that nonlinear scores identify important samples and aid outlier detection, while reducing training costs compared to linear or uniform sampling. The framework promises practical impact for efficient training and model diagnostics in neural networks and generalized linear predictors, with potential extensions to active and transfer learning.

Abstract

While norm-based and leverage-score-based methods have been extensively studied for identifying "important" data points in linear models, analogous tools for nonlinear models remain significantly underdeveloped. By introducing the concept of the adjoint operator of a nonlinear map, we address this gap and generalize norm-based and leverage-score-based importance sampling to nonlinear settings. We demonstrate that sampling based on these generalized notions of norm and leverage scores provides approximation guarantees for the underlying nonlinear mapping, similar to linear subspace embeddings. As direct applications, these nonlinear scores not only reduce the computational complexity of training nonlinear models by enabling efficient sampling over large datasets but also offer a novel mechanism for model explainability and outlier detection. Our contributions are supported by both theoretical analyses and experimental results across a variety of supervised learning scenarios.
Paper Structure (22 sections, 2 theorems, 63 equations, 10 figures)

This paper contains 22 sections, 2 theorems, 63 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Importance Sampling in Linear Models.
  4. Importance Sampling in Nonlinear Models.
  5. Nonlinear Approximation: Our Approach
  6. Nonlinear Adjoint Operator
  7. Parameter-dependent Approximations
  8. Parameter-independent Approximation
  9. Estimating Sampling Scores
  10. Lower bounding $\mathcal{L}_{\mathcal{S}}(\bm \theta^{\star}_{\mathcal{S}})$
  11. Experiments
  12. Conclusions and Further Thoughts
  13. Appendix
  14. Proof of \ref{['prop:f*_composit']}
  15. Norm Score Approximation for \ref{['ex:nn_estimate']}
  16. ...and 7 more sections

Key Result

Proposition 3.1

Let $f = g \circ h = g(h)$ where $g:\mathbb{R} \to \mathbb{R}$ and $h:\mathbb{R}^{p} \to \mathbb{R}$. Also, assume that $h$ is positively homogeneous of degree $\alpha \in \mathbb{R}$, i.e., $h(t \bm \theta) = t^{\alpha} h(\bm \theta)$ for any $t > 0$. Then

Figures (10)

  • Figure 1: Comparison of sampling strategies. The Y-axis shows $\log\left[( \mathcal{L}(\bm {\theta}^{\star}_{\mathcal{S}}) - \mathcal{L}(\bm {\theta}^{\star}))/\mathcal{L}(\bm {\theta}^{\star})\right]$ against sample size (as a percentage of total data), where $\bm {\theta}^{\star}_{\mathcal{S}}$ is the optimal parameter from training on sampled data. "RN", "LS", and "UN" denote Row Norm, Leverage Scores, and Uniform Sampling, respectively, with "L" and "A" indicating linear and adjoint-based nonlinear variants. Nonlinear importance scores consistently outperform all other alternatives.
  • Figure 2: Comparisons of high and low linear/nonlinear leverage scores across multiple datasets. "High" and "Low" refer to images with the highest and lowest scores, respectively. In subfigures (a)-(j), the top row shows images selected using nonlinear leverage scores (\ref{['def:lev_score']}), while the bottom row uses linear leverage scores. Samples with higher nonlinear scores contain distinct patterns and are harder to classify, while lower scores correspond to straightforward samples. In contrast, linear scores select less insightful samples. Notably, our method identifies mislabeled and noisy samples (outliers) with high scores. In FER, the top three rows represent the scores calculated at $\approx$75% (final), $\approx$65%, and $\approx$55% (initial) training accuracy, while the bottom row uses linear scores. Under-trained models highlight extreme expressions, while fully trained models detect subtler emotions and facial features like accessories, tears, and aging.
  • Figure 3: Illustration of an $\varepsilon$-net covering $\mathcal{B}^{\star}_{R}$. The larger circle has radius $(1+\varepsilon/2)R$, while $\mathcal{B}^{\star}_{R}$ has radius $R$. Blue dots denote the $\varepsilon$-net, and dashed circles of radius $\varepsilon R/2$ cover all points.
  • Figure 4: Illustration of quantitative results on datasets used in \ref{['fig:all_datasets']}, SVHN & QD. The Y-axis shows Log(MSE) on training data against sample size (in percentage of total data). "LS' and "UN" denote Leverage Scores and Uniform Sampling schemas, respectively, with "L" and "A" indicating linear and adjoint-based nonlinear variants.
  • Figure 5: Top 50 images with the highest and lowest nonlinear leverage scores in each grouping for the SVHN dataset.
  • ...and 5 more figures

Theorems & Definitions (15)

  • Definition 3.1: Adjoint Operator
  • Proposition 3.1
  • proof
  • Example 3.1: Generalized Linear Predictors
  • Example 3.2: ReLU Neural Networks
  • Definition 3.2: Nonlinear Dual Matrix
  • Definition 3.3: Nonlinear Leverage Scores
  • Definition 3.4: Nonlinear Norm Scores
  • Remark 3.1
  • Remark 3.2
  • ...and 5 more