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The Big Data Bootstrap
Ariel Kleiner, Ameet Talwalkar, Purnamrita Sarkar, Michael Jordan
TL;DR
Large datasets make bootstrap-based estimator-quality assessment computationally demanding. BLB combines bootstrapping small subsets with subsampling to provide a scalable, automatic alternative, and the paper reports favorable computational behavior, statistical properties, and robustness to subset size. Its scope includes large-scale distributed settings, where full bootstrap resamples can impose prohibitive computation and storage costs.
Problem
The bootstrap provides generic estimator-quality assessment but can be prohibitively expensive for large datasets, while alternatives can be sensitive to hyperparameters and require convergence-rate knowledge.
Method
BLB averages bootstrap-based quality estimates computed from multiple small subsets of the original dataset.
Results
BLB retains bootstrap applicability and statistical properties while offering a more favorable computational profile and greater robustness to subset-size choice than m out of n bootstrap and subsampling.
Takeaways & Limitations
BLB is presented as an automatic estimator-quality assessment procedure suited to large-scale data and modern parallel and distributed computing architectures.
Takeaways & Limitations
Efficient automatic selection of BLB's subset size b remains difficult, and devising a computationally efficient stabilization-based method is left for future work.
Abstract
from arXiv · showhide
The bootstrap provides a simple and powerful means of assessing the quality of estimators. However, in settings involving large datasets, the computation of bootstrap-based quantities can be prohibitively demanding. As an alternative, we present the Bag of Little Bootstraps (BLB), a new procedure which incorporates features of both the bootstrap and subsampling to obtain a robust, computationally efficient means of assessing estimator quality. BLB is well suited to modern parallel and distributed computing architectures and retains the generic applicability, statistical efficiency, and favorable theoretical properties of the bootstrap. We provide the results of an extensive empirical and theoretical investigation of BLB's behavior, including a study of its statistical correctness, its large-scale implementation and performance, selection of hyperparameters, and performance on real data.
1. Introduction
Estimator-quality assessment is important for quantifying uncertainty, bias, and risk, but large datasets make automatic methods computationally demanding. BLB addresses this gap by combining bootstrap and subsampling features while preserving broad applicability and improving scalability.
- 1. Introduction: Estimator-quality measures such as confidence regions, bias, and risk provide information beyond point estimates and support inference and data-use decisions.Applications include hypothesis testing, bias correction, resource allocation, active learning, and feature selection.
- 1. Introduction: Classical asymptotic analysis can be statistically correct and analytically computable, but applies only when the relevant estimator-specific analysis is tractable and available.Automatic methods reduce reliance on such specialized analysis, although large-scale computation remains a challenge.
- 1. Introduction: The bootstrap has strong statistical properties but can become prohibitively expensive on large datasets, while m out of n bootstrap and subsampling depend sensitively on hyperparameters and convergence-rate knowledge.These limitations motivate addressing computational cost, scalability, statistical correctness, and automation together.
- 1. Introduction: BLB combines bootstrapping multiple small subsets to assess estimator quality with substantially lower computational cost than the bootstrap.It is designed for large datasets and parallel or distributed architectures.
- 1. Introduction: The paper studies BLB's statistical correctness, distributed implementation, hyperparameter selection, and real-data performance.These investigations cover both theoretical and empirical behavior.
2. Setting and Notation
The paper models data as an i.i.d. sample from an unknown population and defines estimator quality through the distribution of an estimator induced by that population. The goal is to estimate quality metrics from observed data alone.
- 2. Setting and Notation: The observed sample is modeled as i.i.d. draws from an unknown distribution P, and the estimator is obtained by applying θ to the empirical distribution P_n.The estimate is written as θ̂_n = θ(P_n).
- 2. Setting and Notation: Estimator quality is characterized by the distribution Q_n(P) of the random estimate θ̂_n, which depends on both the population distribution and estimator mapping.Examples include estimates of correlation, regression parameters, and classification accuracy.
- 2. Setting and Notation: The target is a metric ξ(Q_n(P)) that summarizes estimator behavior, such as a confidence region, standard error, or bias.Because P and Q_n(P) are unknown, this metric must be estimated from observed data and knowledge of θ.
3. Related Work
The bootstrap offers a generic plug-in approach to estimator-quality assessment but requires repeated computation on full-sized resamples. Smaller-resample alternatives reduce computation, yet require convergence-rate knowledge and are sensitive to the chosen resample size.
- 3. Related Work: The bootstrap estimates estimator-quality metrics by repeatedly resampling n observations from the empirical distribution and computing the estimator on each resample.The resulting empirical distribution of estimates provides a Monte Carlo approximation to the target quality metric.
- 3. Related Work: Bootstrap efficiency improvements mainly reduce the number of Monte Carlo resamples, leaving repeated estimator computation on datasets comparable in size to the original.They also introduce additional implementation complexity.
- 3. Related Work: The m out of n bootstrap and subsampling compute estimates on m<n observations and apply an analytical correction to approximate the n-sample quality metric.The correction requires prior knowledge of the estimator's convergence rate.
- 3. Related Work: Although smaller resamples improve computational cost, these alternatives require convergence-rate knowledge and are sensitive to the choice of m.Automatic m-selection schemes can require enough additional computation to eliminate the savings.
4. Bag of Little Bootstraps (BLB)
BLB averages bootstrap-based quality estimates computed from multiple small subsets of the full dataset. Because each resample has at most b distinct points, BLB reduces computation and storage while retaining bootstrap-style estimation and enabling distributed execution.
- 4. Bag of Little Bootstraps (BLB): BLB samples s subsets of size b<n and averages the quality estimates obtained by bootstrapping within those subsets.The subsets are sampled uniformly from the original data, with disjoint subsets also allowed.
- 4. Bag of Little Bootstraps (BLB): Within each subset, BLB repeatedly resamples n points, computes the estimator, forms the empirical distribution of estimates, and approximates the subset-specific quality metric by Monte Carlo.This preserves the bootstrap's inner resampling structure while using smaller stored subsets.
- 4. Bag of Little Bootstraps (BLB): Each BLB resample contains at most b distinct data points, so weighted representations can make time and storage scale with b rather than n.The resample can be represented by distinct points and their sampled counts.
- 4. Bag of Little Bootstraps (BLB): For n=1,000,000 and b=n^0.6, BLB subsets and resamples contain at most 3,981 distinct points, compared with approximately 632,000 in a bootstrap resample.With 1 MB per data point, the corresponding storage is at most 4 GB for BLB versus approximately 632 GB for a bootstrap resample.
- 4. Bag of Little Bootstraps (BLB): BLB has a more favorable computational profile, supports parallel and distributed architectures, retains bootstrap statistical properties, and is more robust to subset-size choice than m out of n bootstrap and subsampling.The paper reports that modest numbers of subsets and Monte Carlo iterations typically suffice for high accuracy.
5. Statistical Correctness
BLB matches the bootstrap’s statistical performance while being more robust to the choice of subset size b across simulated estimation tasks. Experiments and theory support its correctness, including higher-order convergence under conditions that allow b to remain much smaller than n.
- BLB shares the bootstrap’s favorable statistical performance and is more robust than the m out of n bootstrap and subsampling to the choice of b.This conclusion is supported by both simulation and theoretical analysis.
- BLB converges to low relative error faster than the bootstrap for b > n^0.5, while b = n^0.5 yields somewhat higher relative error.The comparison uses representative classification experiments with StudentT-distributed covariates.
- For the b out of n bootstrap, b ≤ n^0.6 fails to converge to low relative error, whereas BLB substantially outperforms it even at b = n^0.5.
- BLB retains bootstrap-like statistical properties and is asymptotically consistent for a broad class of estimators and quality measures under the bootstrap’s prior analysis conditions.
- BLB can converge at O(1/n), rather than the O(1/√n) rate of asymptotic approximations, when b = Ω(√n) and s is sufficiently large.These conditions still permit b/n → 0 as n → ∞.
6. Scalability
BLB scales estimator-quality assessment more effectively than the bootstrap by distributing smaller subsamples and resamples, reducing repeated full-data computation and disk-access overhead.
- 6. Scalability: BLB reduces total computation for estimator-quality assessment and makes more natural use of parallel and distributed resources than the bootstrap.The paper also reports lower serial computation to reach comparably high accuracy.
- 6. Scalability: Bootstrap resamples contain approximately 63% of the original data points, so processing them repeatedly across a cluster incurs communication and coordination overhead.The overhead is especially problematic when cluster systems store data on disk rather than in memory.
- 6. Scalability: BLB enables simultaneous parallel computation across subsamples and resamples because its smaller data quantities can be transferred, stored, and processed on individual or small groups of nodes.This avoids repeatedly assigning an entire cluster to each large bootstrap resample.
- 6. Scalability: The large-scale experiment used 150 GB datasets, logistic regression, Poisson bootstrap resampling, and BLB settings r = 50, s = 5, and b = n^0.7.The disk-resident condition used 10 workers with 60 GB total memory; the cached condition used 20 workers with 240 GB.
- 6. Scalability: BLB produced a high-accuracy output on 150 GB of disk-resident data in less time than one bootstrap resample, while remaining substantially better when the data fit in memory.The experiments used Spark clusters with different storage configurations and compared relative error with total processing time.
7. Hyperparameter Selection
The paper studies how BLB’s subsample and resample counts affect accuracy and proposes adaptive stopping rules to avoid unnecessary computation. It finds that moderate settings can suffice, while automatic selection of subset size remains difficult.
- 7. Hyperparameter Selection: For b = n^0.7, choosing s ≥ 3 and r ≥ 50 is sufficient for BLB to achieve low relative error in the representative empirical setting.The minimum sufficient values can vary with the quality metric ξ and the underlying data distribution.
- 7. Hyperparameter Selection: BLB’s performance depends on r and s, but values that are not among the smallest can often be selected independently while retaining low relative error.The paper examines these effects empirically to provide practical guidance and adaptive selection methods.
- 7. Hyperparameter Selection: The resample-level estimates typically converge toward a constant target, often at rate O(1/√r), motivating convergence-based selection of r.The convergence constant is unknown, so the method monitors whether successive estimates fluctuate significantly.
- 7. Hyperparameter Selection: Adaptive selection of r and s lets BLB stop shortly after convergence, limiting unnecessary computation while using interpretable target-error and window-size parameters.The reported validation used ϵ = 0.05, with w = 20 for r and w = 3 for s.
- 7. Hyperparameter Selection: Efficient automatic selection of b is harder because computations for different subset sizes cannot be reused, although b = n^0.7 appears reasonable in many situations.The paper identifies efficient adaptive selection of b as future work.
8. Real Data
On real datasets, BLB produced confidence intervals comparable to the bootstrap while converging faster, whereas the b out of n bootstrap was sensitive to b.
- Real-data evaluation compared methods’ outputs using average absolute confidence-interval width because ground truth was unavailable.
- BLB and the bootstrap produced tightly clustered confidence-interval outputs, while BLB converged more quickly on UCI connect4.The dataset used logistic regression with d = 42 and n = 67,557.
- The b out of n bootstrap’s confidence-interval widths varied significantly as b changed, indicating less robust outputs.
9. Conclusion
BLB is presented as an alternative for automatic estimator-quality assessment that is accurate, scalable, and suited to parallel and distributed computing.
- BLB provides an automatic, accurate alternative for assessing estimator quality on large-scale data and modern parallel and distributed architectures.