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Computationally efficient reductions between some statistical models

Mengqi Lou, Guy Bresler, Ashwin Pananjady

TL;DR

This work develops nonasymptotic, computationally efficient reductions between statistical experiments, enabling the transformation of samples from a source model to a target model without knowing the source parameters. It introduces two general reduction techniques and applies them to Laplace, Erlang, and Uniform location models to achieve structure-preserving, missing-data–robust reductions to broad target families, with explicit signed kernels and a rejection-sampling algorithm that bounds total-variation deficiency. The authors demonstrate concrete consequences for high-dimensional problems, including mixtures of experts and phase retrieval, denoising with log-concave noise, and transforming Laplace privacy mechanisms to Gaussian ones, enabling transferable risk and privacy guarantees. The results show TV deficiencies can decay exponentially in key parameters, offering practical, computationally feasible pathways to transfer learning and hardness results across models. Overall, the framework provides a versatile toolkit for transferring statistical risk bounds and algorithmic guarantees across a range of noise models and privacy settings in high dimensions.

Abstract

We study the problem of approximately transforming a sample from a source statistical model to a sample from a target statistical model without knowing the parameters of the source model, and construct several computationally efficient such reductions between canonical statistical experiments. In particular, we provide computationally efficient procedures that approximately reduce uniform, Erlang, and Laplace location models to general target families. We illustrate our methodology by establishing nonasymptotic reductions between some canonical high-dimensional problems, spanning mixtures of experts, phase retrieval, and signal denoising. Notably, the reductions are structure-preserving and can accommodate missing data. We also point to a possible application in transforming one differentially private mechanism to another.

Computationally efficient reductions between some statistical models

TL;DR

This work develops nonasymptotic, computationally efficient reductions between statistical experiments, enabling the transformation of samples from a source model to a target model without knowing the source parameters. It introduces two general reduction techniques and applies them to Laplace, Erlang, and Uniform location models to achieve structure-preserving, missing-data–robust reductions to broad target families, with explicit signed kernels and a rejection-sampling algorithm that bounds total-variation deficiency. The authors demonstrate concrete consequences for high-dimensional problems, including mixtures of experts and phase retrieval, denoising with log-concave noise, and transforming Laplace privacy mechanisms to Gaussian ones, enabling transferable risk and privacy guarantees. The results show TV deficiencies can decay exponentially in key parameters, offering practical, computationally feasible pathways to transfer learning and hardness results across models. Overall, the framework provides a versatile toolkit for transferring statistical risk bounds and algorithmic guarantees across a range of noise models and privacy settings in high dimensions.

Abstract

We study the problem of approximately transforming a sample from a source statistical model to a sample from a target statistical model without knowing the parameters of the source model, and construct several computationally efficient such reductions between canonical statistical experiments. In particular, we provide computationally efficient procedures that approximately reduce uniform, Erlang, and Laplace location models to general target families. We illustrate our methodology by establishing nonasymptotic reductions between some canonical high-dimensional problems, spanning mixtures of experts, phase retrieval, and signal denoising. Notably, the reductions are structure-preserving and can accommodate missing data. We also point to a possible application in transforming one differentially private mechanism to another.
Paper Structure (62 sections, 21 theorems, 283 equations, 2 figures)

This paper contains 62 sections, 21 theorems, 283 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Single-sample reductions:
  4. Asymptotic perspectives:
  5. Other related papers:
  6. Contributions, motivating examples, and organization
  7. Motivation 1
  8. Motivation 2
  9. Motivation 3
  10. Background and preliminary observations
  11. Definitions
  12. Objective
  13. Additional notation
  14. How good is a natural plug-in approach?
  15. An oracle rejection sampling algorithm
  16. ...and 47 more sections

Key Result

Proposition 1

There is a universal constant $c > 0$ such that the following is true. Suppose $\theta \in \Theta$ and the source location model generates the random variable $X_{\theta} = \theta + W$. Consider any of the following cases: Recall the plugin reduction from Eq. eq:reduction-plugin, and let $Y_{\theta} \sim \mathcal{N}(\theta, \sigma^2)$ denote the desired target for some $\sigma \geq 1$. Then in ca

Figures (2)

  • Figure 1: Simulation of $\mathsf{Lap}(\theta, 1)$ to $\mathcal{N}(\theta, \sigma^2)$ reduction, plotted for $\theta \in \{-5, -2.5, 0, 2.5, 5\}$ and $\sigma = 5$.
  • Figure 2: Simulation of $\mathsf{Unif}([\theta-1/2,\theta + 1/2])$ to $\mathcal{N}(10\cdot|\theta|,\sigma^2)$ reduction, plotted for parameters $\theta \in\{ -1/2, -1/4, 0, 1/4, 1/2\}$ and $\sigma = 10$.

Theorems & Definitions (32)

  • Proposition 1
  • Lemma 1
  • Proposition 2
  • Theorem 1
  • Proposition 3
  • Theorem 2
  • Example 1: Gaussian Target
  • Example 2: Logistic Target
  • Remark 1: Exact comparison of experiments: Exponential to logistic
  • Example 3
  • ...and 22 more