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Multilevel Control Functional

Kaiyu Li, Yiming Yang, Xiaoyuan Cheng, Yi He, Zhuo Sun

TL;DR

This work proposes an extension of control variates, multilevel control functional (MLCF), which uses non-parametric Stein-based control variate and multifidelity models with lower cost to gain better performance.

Abstract

Control variates are variance reduction techniques for Monte Carlo estimators. They play a critical role in improving Monte Carlo estimators in scientific and machine learning applications that involve computationally expensive integrals. We introduce multilevel control functionals (MLCFs), a novel and widely applicable extension of control variates that combines non-parametric Stein-based control variates with multi-fidelity methods. We show that when the integrand and the density are smooth, and when the dimensionality is not very high, MLCFs enjoy a faster convergence rate. We provide both theoretical analysis and empirical assessments on differential equation examples, including Bayesian inference for ecological models, to demonstrate the effectiveness of our proposed approach. Furthermore, we extend MLCFs for variational inference, and demonstrate improved performance empirically through Bayesian neural network examples.

Multilevel Control Functional

TL;DR

This work proposes an extension of control variates, multilevel control functional (MLCF), which uses non-parametric Stein-based control variate and multifidelity models with lower cost to gain better performance.

Abstract

Control variates are variance reduction techniques for Monte Carlo estimators. They play a critical role in improving Monte Carlo estimators in scientific and machine learning applications that involve computationally expensive integrals. We introduce multilevel control functionals (MLCFs), a novel and widely applicable extension of control variates that combines non-parametric Stein-based control variates with multi-fidelity methods. We show that when the integrand and the density are smooth, and when the dimensionality is not very high, MLCFs enjoy a faster convergence rate. We provide both theoretical analysis and empirical assessments on differential equation examples, including Bayesian inference for ecological models, to demonstrate the effectiveness of our proposed approach. Furthermore, we extend MLCFs for variational inference, and demonstrate improved performance empirically through Bayesian neural network examples.
Paper Structure (56 sections, 4 theorems, 57 equations, 12 figures, 6 tables)

This paper contains 56 sections, 4 theorems, 57 equations, 12 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Limitations of Previous Techniques
  4. Our Contributions
  5. Background
  6. Control Variates
  7. Monte Carlo and Control Variates
  8. General Recipe of Stein-based Control Variates
  9. Control Functionals
  10. Multi-fidelity Models and Multilevel Monte Carlo
  11. Methodology
  12. Assumptions
  13. Extensions: MLCFs for Variational Inference
  14. Experimental Results
  15. Synthetic Example
  16. ...and 41 more sections

Key Result

Proposition 3.1

Given $X_l$, the associated score evaluations $\{\nabla \log \pi(x_{(l,i)})\}_{l=1}^{n_l}$, and the function evaluations $\{f_l(x_{(l,i)})-f_{l-1}(x_{(l,i)})\}_{i=1}^{n_l}\}$ for $l\in\{0,\ldots,L\}$, we split it into two parts: $X^0_l=(x_{(l,1)},\ldots,x_{(l,m_l)})^\top$ and $X^1_l=(x_{(l,m_l+1)},\ where $a_l=\pmb{1}^{\top} k_0^l(X^0_l,X^0_l) ^{-1} (f_l(X^0_l)-f_{l-1}(X^0_l) )/\pmb{1}^{\top}k_0^

Figures (12)

  • Figure 1: Illustration Example. (a): $f_0$, $f_1$ and $f_2$ are coarse, medium and fine approximations to $f$. (b)(c)(d): Compared with MLMC, after applying MLCF, the green curves become much flatter and closer to $\Pi[f_l-f_{l-1}]$ (red dotted lines) than the original $f_l-f_{l-1}$ (blue curves) used at each MLMC level. (e): Compared with CF, although applying CF to $f_2$ (purple curve) already reduces variance, MLCF leverages the multilevel structure. Thus, fine levels, e.g. $f_2-f_1$ (in (d)), themselves show smaller variance than $f_2$ (in (e)) and MLCF further decreases this variance (in (d)). This demonstrates that MLCF reduces the variance significantly.
  • Figure 2: Synthetic Example: Absolute integration error under a budget constraint (Y-axis log-scale).
  • Figure 3: Boundary-value ODE: Absolute integration error under a budget constraint.
  • Figure 4: Bayesian Inference for Lotka-Volterra: Absolute integration error under a budget constraint.
  • Figure 5: MLCF for Variation Inference of Bayesian Neural Networks: Hidden Dimension Size is 15 (num. of param. is 392).
  • ...and 7 more figures

Theorems & Definitions (8)

  • Proposition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proposition 3.4
  • proof
  • proof
  • proof : Proof of \ref{['theorem: n']}
  • proof