[Paper Review] Mitigating Bias in Adaptive Data Gathering via Differential Privacy

Seth Neel, Aaron Roth|arXiv (Cornell University)|Jun 6, 2018

Advanced Bandit Algorithms ResearchDecision Sciences28 references4 citations

TL;DR

This paper proposes using differential privacy in adaptive data gathering—such as bandit algorithms—to mitigate systematic bias in empirical estimates and invalid statistical inference. By ensuring selection decisions are differentially private, the method bounds bias in mean estimates and enables valid p-value corrections, achieving near-optimal regret with strong theoretical guarantees across stochastic and linear contextual bandit settings.

ABSTRACT

Data that is gathered adaptively --- via bandit algorithms, for example --- exhibits bias. This is true both when gathering simple numeric valued data --- the empirical means kept track of by stochastic bandit algorithms are biased downwards --- and when gathering more complicated data --- running hypothesis tests on complex data gathered via contextual bandit algorithms leads to false discovery. In this paper, we show that this problem is mitigated if the data collection procedure is differentially private. This lets us both bound the bias of simple numeric valued quantities (like the empirical means of stochastic bandit algorithms), and correct the p-values of hypothesis tests run on the adaptively gathered data. Moreover, there exist differentially private bandit algorithms with near optimal regret bounds: we apply existing theorems in the simple stochastic case, and give a new analysis for linear contextual bandits. We complement our theoretical results with experiments validating our theory.

Motivation & Objective

• Address the problem of systematic bias in adaptive data gathering, where selection of data points depends on prior observations, leading to distorted empirical estimates and false discoveries.
• Overcome limitations of existing de-biasing methods that require complex post-processing or are restricted to specific models, by introducing a general, principled approach using differential privacy.
• Provide theoretical bounds on bias for both simple stochastic bandits and linear contextual bandits, ensuring robustness without additional de-biasing procedures.
• Enable valid statistical inference—particularly corrected p-values—on adaptively gathered data, crucial for hypothesis testing in sequential experimentation.
• Develop differentially private bandit algorithms with near-optimal regret and strong bias guarantees, applicable to real-world adaptive learning systems.

Proposed method

• Apply differential privacy to the selection mechanism of bandit algorithms, ensuring that decisions to explore or exploit arms depend only weakly on observed rewards, thus limiting information leakage.
• Use the connection between differential privacy and adaptive data analysis to bound the bias of empirical means in stochastic bandits, leveraging max-information techniques from Dwork et al. (2015).
• Design a differentially private variant of the UCB algorithm (private UCB) that maintains empirical means with bias on the order of $ O(1/\sqrt{K \cdot T}) $, significantly lower than non-private UCB.
• Extend the framework to linear contextual bandits by ensuring reward-based decisions are differentially private, while allowing contexts to be public, improving accuracy over prior private contextual bandit methods.
• Apply existing privacy-based adaptive analysis tools (e.g., max-information bounds) to correct p-values for hypothesis tests on adaptively gathered data, ensuring valid Type I error control.
• Use Monte Carlo simulations to validate theoretical claims, comparing private and non-private bandit algorithms across bias, regret, and p-value distribution metrics.

Experimental results

Research questions

• RQ1 Can differential privacy be used to bound the bias in empirical mean estimates produced by adaptive data gathering in stochastic bandit settings?
• RQ2 Does differential privacy enable valid statistical inference—specifically corrected p-values—on data gathered via adaptive procedures like contextual bandits?
• RQ3 Can differentially private bandit algorithms achieve near-optimal regret while simultaneously ensuring low bias in estimated arm means?
• RQ4 How does the bias in empirical means and p-values differ between private and non-private bandit algorithms in both stochastic and linear contextual settings?
• RQ5 To what extent do theoretical bounds on bias from differential privacy hold in practice, and how do they compare to empirical observations in simulations?

Key findings

• Private UCB reduced average absolute bias to 0.00176 across 20 arms over 500 rounds, compared to 0.0698 for non-private UCB—over 40 times higher bias.
• In a second experiment with $ K=5 $, $ T=100,000 $, and $ \epsilon=400 $, private UCB achieved average bias of 0.0015 per arm (indistinguishable from zero), while non-private UCB had 0.011, a 7.5-fold increase.
• The most biased arm under non-private UCB had a measured bias of approximately 0.14, indicating strong negative skew, whereas private UCB showed no such skew and bias was statistically indistinguishable from zero.
• In linear contextual bandits, non-private UCB led to a 76% false discovery rate in z-tests for zero regression coefficients (when the null was true), showing severe p-value inflation due to adaptivity.
• The p-value distribution under non-private UCB was strongly downward skewed, with 76% of p-values below the 0.05 threshold, while the theoretical expectation under the null is 5%, confirming invalid inference.
• Private UCB maintained sublinear regret comparable to non-private UCB, demonstrating that privacy does not come at the cost of learning efficiency in the tested settings.