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Theoretical Analysis of Measure Consistency Regularization for Partially Observed Data

Yinsong Wang, Shahin Shahrampour

TL;DR

The paper tackles the theoretical underpinnings of Measure Consistency Regularization (MCR) for imputation with partially observed data, introducing Neural Net Distance (NND) as a flexible IPM-based regularizer that aligns imputed data with fully observed data. It provides estimation error bounds under both perfect and imperfect training, showing that MCR can reduce generalization error but that robustness in the imperfect setting depends on controllable quantities such as the duality gap and distribution shift, which the authors address with a duality-gap stopping protocol. The work combines rigorous theory with extensive simulations and multi-domain real-data experiments (images, sensors, and single-cell data) to demonstrate MCR’s practical benefits and to guide robust training. The protocol and bounds offer a principled framework for deploying MCR in real-world imputation tasks, while suggesting broader implications for two-stage training and post-training in complex models such as LLMs. Overall, the paper advances a domain-agnostic, theory-backed approach to improving imputations under partial observability with concrete procedures to preserve generalization.

Abstract

The problem of corrupted data, missing features, or missing modalities continues to plague the modern machine learning landscape. To address this issue, a class of regularization methods that enforce consistency between imputed and fully observed data has emerged as a promising approach for improving model generalization, particularly in partially observed settings. We refer to this class of methods as Measure Consistency Regularization (MCR). Despite its empirical success in various applications, such as image inpainting, data imputation and semi-supervised learning, a fundamental understanding of the theoretical underpinnings of MCR remains limited. This paper bridges this gap by offering theoretical insights into why, when, and how MCR enhances imputation quality under partial observability, viewed through the lens of neural network distance. Our theoretical analysis identifies the term responsible for MCR's generalization advantage and extends to the imperfect training regime, demonstrating that this advantage is not always guaranteed. Guided by these insights, we propose a novel training protocol that monitors the duality gap to determine an early stopping point that preserves the generalization benefit. We then provide detailed empirical evidence to support our theoretical claims and to show the effectiveness and accuracy of our proposed stopping condition. We further provide a set of real-world data simulations to show the versatility of MCR under different model architectures designed for different data sources.

Theoretical Analysis of Measure Consistency Regularization for Partially Observed Data

TL;DR

The paper tackles the theoretical underpinnings of Measure Consistency Regularization (MCR) for imputation with partially observed data, introducing Neural Net Distance (NND) as a flexible IPM-based regularizer that aligns imputed data with fully observed data. It provides estimation error bounds under both perfect and imperfect training, showing that MCR can reduce generalization error but that robustness in the imperfect setting depends on controllable quantities such as the duality gap and distribution shift, which the authors address with a duality-gap stopping protocol. The work combines rigorous theory with extensive simulations and multi-domain real-data experiments (images, sensors, and single-cell data) to demonstrate MCR’s practical benefits and to guide robust training. The protocol and bounds offer a principled framework for deploying MCR in real-world imputation tasks, while suggesting broader implications for two-stage training and post-training in complex models such as LLMs. Overall, the paper advances a domain-agnostic, theory-backed approach to improving imputations under partial observability with concrete procedures to preserve generalization.

Abstract

The problem of corrupted data, missing features, or missing modalities continues to plague the modern machine learning landscape. To address this issue, a class of regularization methods that enforce consistency between imputed and fully observed data has emerged as a promising approach for improving model generalization, particularly in partially observed settings. We refer to this class of methods as Measure Consistency Regularization (MCR). Despite its empirical success in various applications, such as image inpainting, data imputation and semi-supervised learning, a fundamental understanding of the theoretical underpinnings of MCR remains limited. This paper bridges this gap by offering theoretical insights into why, when, and how MCR enhances imputation quality under partial observability, viewed through the lens of neural network distance. Our theoretical analysis identifies the term responsible for MCR's generalization advantage and extends to the imperfect training regime, demonstrating that this advantage is not always guaranteed. Guided by these insights, we propose a novel training protocol that monitors the duality gap to determine an early stopping point that preserves the generalization benefit. We then provide detailed empirical evidence to support our theoretical claims and to show the effectiveness and accuracy of our proposed stopping condition. We further provide a set of real-world data simulations to show the versatility of MCR under different model architectures designed for different data sources.
Paper Structure (15 sections, 6 theorems, 67 equations, 7 figures, 12 tables, 1 algorithm)

This paper contains 15 sections, 6 theorems, 67 equations, 7 figures, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Measure Consistency Regularization with Neural Net Distance
  3. Imputation without Regularization
  4. Measure Consistency Regularization
  5. Neural Net Distance for MCR
  6. Literature Review
  7. On the Generalization Benefit of MCR
  8. Perfect Training
  9. Imperfect Training
  10. Handling the Optimality Gap
  11. Simulations
  12. Verifying Theoretical Claims
  13. Evaluating the Duality Gap Stopping Condition
  14. Versatility of MCR
  15. Conclusion

Key Result

Theorem 1

Suppose Assumptions assump:1--assump:3 hold. Let $\hat{\mathfrak{R}}_n(\mathcal{G}_{nn})$ denote the Rademacher complexity of $\mathcal{G}_{nn}$ on $n$ samples, and let $\mathcal{H} := \{h(\mathbf{x})=g(\mathbf{x},f(\mathbf{x})): g \in \mathcal{G}_{nn}, f \in \mathcal{F}\}$. Then,

Figures (7)

  • Figure 1: An intuitive illustration of the effect of measure consistency regularization (MCR). Without MCR, minimizing the supervised loss on the fully observed empirical measure $\boldsymbol{\pi}_l$ admits a large set of solutions, making the imputed measure induced by the learned function likely to deviate from the true measure $\boldsymbol{\pi}$. By enforcing MCR, the learning process encourages the imputed measure to align with $\boldsymbol{\pi}_l$, thereby shrinking the solution set and increasing the likelihood that the imputed measure remains close to the true measure.
  • Figure 2: Reconstruction error reduction in percentage with MCR. Left: The error reduction percentage over varying model width. Right: The error reduction percentage over varying model depth.
  • Figure 3: Error reduction vs $\log \lambda_d$. We do not observe a statistically significant trend across different $\lambda_d$.
  • Figure 4: Error reduction vs $\hat{\xi}$. The condition threshold in Eq. \ref{['eq:dg_bound']} based on our $\hat{\xi}$ estimates in Eq. \ref{['eq:xi']} accurately reflects the tipping point where MCR's benefit disappears.
  • Figure 5: The test error curve comparison between vanilla training and MCR training. Left: Test error curve over $2000$ epochs. Right: Test error curve comparison in the first $100$ epochs.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • proof
  • Corollary 4
  • Remark 1
  • Lemma 5
  • Lemma 6