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When are Unbiased Monte Carlo Estimators More Preferable than Biased Ones?

Guanyang Wang, Jose Blanchet, Peter W. Glynn

TL;DR

This paper formulates a rigorous framework to compare unbiased and biased Monte Carlo estimators under massively parallel computation. It shows that unbiased methods tend to have favorable completion times, tightly linked to the tail behavior of their per-replication running times, but do not automatically reduce total computational cost relative to biased methods. The framework is applied to multilevel and Markov chain Monte Carlo, revealing that unbiased randomized MLMC (rMLMC) can achieve the optimal $O(\epsilon^{-2})$ cost with favorable completion-time properties, while standard MLMC remains optimal for total cost in non-parallel settings. In MCMC, debiasing via coupling yields unbiased estimators with competitive cost and substantially improved completion-time performance in parallel environments. The numerical case study on Gaussian mean estimation demonstrates the practical benefits of unbiased MCMC under heavy parallelization, offering guidance on when to adopt unbiased versus biased strategies and highlighting opportunities for hybrids and variance reduction techniques.

Abstract

Due to the potential benefits of parallelization, designing unbiased Monte Carlo estimators, primarily in the setting of randomized multilevel Monte Carlo, has recently become very popular in operations research and computational statistics. However, existing work primarily substantiates the benefits of unbiased estimators at an intuitive level or using empirical evaluations. The intuition being that unbiased estimators can be replicated in parallel enabling fast estimation in terms of wall-clock time. This intuition ignores that, typically, bias will be introduced due to impatience because most unbiased estimators necesitate random completion times. This paper provides a mathematical framework for comparing these methods under various metrics, such as completion time and overall computational cost. Under practical assumptions, our findings reveal that unbiased methods typically have superior completion times - the degree of superiority being quantifiable through the tail behavior of their running time distribution - but they may not automatically provide substantial savings in overall computational costs. We apply our findings to Markov Chain Monte Carlo and Multilevel Monte Carlo methods to identify the conditions and scenarios where unbiased methods have an advantage, thus assisting practitioners in making informed choices between unbiased and biased methods.

When are Unbiased Monte Carlo Estimators More Preferable than Biased Ones?

TL;DR

This paper formulates a rigorous framework to compare unbiased and biased Monte Carlo estimators under massively parallel computation. It shows that unbiased methods tend to have favorable completion times, tightly linked to the tail behavior of their per-replication running times, but do not automatically reduce total computational cost relative to biased methods. The framework is applied to multilevel and Markov chain Monte Carlo, revealing that unbiased randomized MLMC (rMLMC) can achieve the optimal cost with favorable completion-time properties, while standard MLMC remains optimal for total cost in non-parallel settings. In MCMC, debiasing via coupling yields unbiased estimators with competitive cost and substantially improved completion-time performance in parallel environments. The numerical case study on Gaussian mean estimation demonstrates the practical benefits of unbiased MCMC under heavy parallelization, offering guidance on when to adopt unbiased versus biased strategies and highlighting opportunities for hybrids and variance reduction techniques.

Abstract

Due to the potential benefits of parallelization, designing unbiased Monte Carlo estimators, primarily in the setting of randomized multilevel Monte Carlo, has recently become very popular in operations research and computational statistics. However, existing work primarily substantiates the benefits of unbiased estimators at an intuitive level or using empirical evaluations. The intuition being that unbiased estimators can be replicated in parallel enabling fast estimation in terms of wall-clock time. This intuition ignores that, typically, bias will be introduced due to impatience because most unbiased estimators necesitate random completion times. This paper provides a mathematical framework for comparing these methods under various metrics, such as completion time and overall computational cost. Under practical assumptions, our findings reveal that unbiased methods typically have superior completion times - the degree of superiority being quantifiable through the tail behavior of their running time distribution - but they may not automatically provide substantial savings in overall computational costs. We apply our findings to Markov Chain Monte Carlo and Multilevel Monte Carlo methods to identify the conditions and scenarios where unbiased methods have an advantage, thus assisting practitioners in making informed choices between unbiased and biased methods.
Paper Structure (17 sections, 16 theorems, 33 equations, 2 figures, 2 algorithms)

This paper contains 17 sections, 16 theorems, 33 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Cost Analysis under different metrics
  3. Problem Formulation
  4. Analysis of unbiased algorithms
  5. A limit theorem for the completion time
  6. Analysis of biased algorithms
  7. A remark on other metrics
  8. Applications on Multilevel Monte Carlo
  9. Non-randomized Multilevel Monte Carlo: Optimal total cost and completion time analysis
  10. Randomized Multilevel Monte Carlo
  11. Remarks on practical implementation and further extensions
  12. Applications on Markov Chain Monte Carlo
  13. Standard MCMC: cost and completion time analysis
  14. Unbiased estimator by coupling
  15. Remarks on practical implementation and further extensions
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose ${\mathcal{A}}$ is an algorithm satisfying $b_{\mathcal{A}} = 0$, $v_{\mathcal{A}} < \infty$, and $\mathbb{E}[C({\mathcal{A}})]< \infty$. Then the expected total computation cost for $\cal A$ to achieve $\epsilon^2$-MSE equals Meanwhile, if $m$ processors are available, then users can allocate the calculations suitably such that the expected worst-case computational cost is no more than

Figures (2)

  • Figure 1: Performance evaluation of the implemented algorithm across $10^5$ processors. Each plot is based on the average results of 100 independent repetitions of the same experiment. (a) The relationship between the number of processors and completion time, demonstrating how parallelization scales. (b) A plot of completion time against the logarithm of squared error. (c) The relationship between the number of processors and the logarithm of squared error. (d) A log-log plot of the number of processors against squared error, with a fitted regression line.
  • Figure 2: Comparative Analysis of Unbiased and Biased MCMC Methods. Each plot is based on the average results of 100 independent repetitions of the same experiment.(a) The plot shows how squared error diminishes as the number of processors increases for various methods. The steelblue curve represents the unbiased MCMC method, which has a maximum completion time of $28.71$. Curves in other colors represent standard MCMC methods conducted for {10, 20, 30, 50, 100} iterations. (b) This plot displays the relationship between completion time and squared error for both standard MCMC (in red) and unbiased MCMC (in blue). (c) This plot displays the relationship between the logarithm of completion time and the logarithm of squared error for standard MCMC (in orange) and unbiased MCMC (in steelblue).

Theorems & Definitions (26)

  • Example 1
  • Example 2
  • Example 3
  • Theorem 1
  • proof : Proof of Theorem \ref{['thm:cost-unbiased']}
  • Remark 1
  • Definition 1: Sub-gaussian
  • Definition 2: Sub-exponential
  • Theorem 2
  • proof
  • ...and 16 more