Uncertainty about Uncertainty: Optimal Adaptive Algorithms for Estimating Mixtures of Unknown Coins (1904.09228v3)

Abstract: Given a mixture between two populations of coins, "positive" coins that each have -- unknown and potentially different -- bias $\geq\frac{1}{2}+\Delta$ and "negative" coins with bias $\leq\frac{1}{2}-\Delta$, we consider the task of estimating the fraction $\rho$ of positive coins to within additive error $\epsilon$. We achieve an upper and lower bound of $\Theta(\frac{\rho}{\epsilon^2\Delta^2}\log\frac{1}{\delta})$ samples for a $1-\delta$ probability of success, where crucially, our lower bound applies to all fully-adaptive algorithms. Thus, our sample complexity bounds have tight dependence for every relevant problem parameter. A crucial component of our lower bound proof is a decomposition lemma (see Lemmas 17 and 18) showing how to assemble partially-adaptive bounds into a fully-adaptive bound, which may be of independent interest: though we invoke it for the special case of Bernoulli random variables (coins), it applies to general distributions. We present simulation results to demonstrate the practical efficacy of our approach for realistic problem parameters for crowdsourcing applications, focusing on the "rare events" regime where $\rho$ is small. The fine-grained adaptive flavor of both our algorithm and lower bound contrasts with much previous work in distributional testing and learning.

• The paper develops a novel adaptive sampling algorithm for estimating the fraction of biased coins with optimal sample complexity.
• It leverages both single-coin and cross-coin adaptivity to minimize coin flips needed for precise error control.
• Simulation results validate the theoretical guarantees, demonstrating significant improvements over non-adaptive methods.

Overview of "Uncertainty about Uncertainty: Optimal Adaptive Algorithms for Estimating Mixtures of Unknown Coins"

This essay examines the findings in "Uncertainty about Uncertainty: Optimal Adaptive Algorithms for Estimating Mixtures of Unknown Coins" by Jasper C.H. Lee and Paul Valiant. The paper addresses a statistical estimation problem involving mixtures of two types of coins—each coin with an unknown bias that is either $\geq \frac{1}{2}+\Delta$ or $\leq \frac{1}{2}-\Delta$. The objective is to estimate the fraction $\rho$ of positive-biased coins within a desired error margin using minimal coin flips.

Problem Setting and Relevance

The problem is framed in the context of practical applications such as crowdsourcing. Given a set of data, a relevant task is to approximate the fraction of data meeting a specific quality criterion. This approximation is equivalent to classifying crowdsourced judgments with high variability in accuracy. By leveraging a nuanced adaptive sampling approach, the paper aims to optimize this estimation task.

Methodological Approach

The authors propose an algorithmic framework using adaptive methods, exploring both "single-coin adaptivity"—which governs decisions on further sampling of a particular coin—and "cross-coin adaptivity"—which determines subsequent coin flips based on observations up to that point. Lemma-driven proofs establish the theoretical guarantees of the approach, culminating in the critical result that the sample complexity is $$\Theta(\frac{\rho}{^2\Delta^2}\log\frac{1}{\delta})$$. This bound is rigorously demonstrated to be tight, providing key insights into the problem's parameters: $\rho$, $$.

Results

Simulation experiments corroborate the paper's theoretically grounded claims. The Triangular Walk Algorithm, developed within the work, effectively balances exploration across both high-quality and low-quality coins to estimate $\rho$ rapidly and efficiently. As shown, the number of samples required by the algorithm compared to non-adaptive methods is minimized while maintaining error thresholds.

Contributions and Implications

The paper contributes significantly to the field through a thorough theoretical foundation balanced with practical heuristics. Importantly, it underscores adaptivity's role in high-uncertainty estimation contexts, expanding the repository of tools for statistical estimation and machine learning applications where true distributions are unknown or noisy.

Examining how these methods might pertain to other areas of AI—such as reinforcement learning settings or sensor fusion in autonomous systems—shows the broader utility of adaptive sampling strategies. Future work could explore the convergence properties of these methods under dynamic environments, thus generalizing from static task settings to live, evolving datasets.

Overall, "Uncertainty about Uncertainty" enriches our understanding of optimal decision making in environments rife with incomplete information. Through specialized algorithms and rigorous mathematical treatments, it bridges theoretical insight with applicable strategies to meet the demands of contemporary data-driven fields.