Threshold-Based Progressive Sampling

• Threshold-based progressive sampling is an adaptive method that dynamically adjusts sample sizes using quantifiable thresholds such as error bounds and confidence intervals.
• It leverages statistical concentration inequalities and resource-aware strategies to guarantee accuracy and computational efficiency in various analytical tasks.
• The algorithm provides scalable solutions across domains like itemset mining, streaming analytics, and stochastic optimization with rigorously defined stopping criteria.

A threshold-based progressive sampling algorithm is a methodological framework in which sampling is performed adaptively, with the sample size, inclusion probabilities, or subsampling regime dynamically governed by quantitative thresholds. Such thresholds are typically derived from statistical, optimization, or computational criteria—commonly confidence bounds, estimation error, optimization duality gaps, or stopping rules—which dictate when additional samples should be drawn or the process should terminate. This class of algorithms underpins a wide range of modern approaches for scalable inference, optimization, geometric model fitting, and combinatorial analytics, offering clear complexity guarantees and practical efficiency in both sequential and parallel contexts.

1. General Principles of Threshold-Based Progressive Sampling

Threshold-based progressive sampling algorithms operate by adaptively monitoring a formal statistical or computational quantity (such as error, confidence, duality gap, or empirical margin) and invoking sample augmentation or early stopping based on whether this quantity crosses a pre-defined threshold. This enables the algorithm to focus computation where it is most needed, halting when a user-prescribed accuracy or confidence parameter is achieved.

Classical examples include progressive itemset mining via sample-size bounds and error thresholds (Pietracaprina et al., 2010), PPS (probability proportional to size) streams with budget-based thresholds (Hentschel et al., 2021), adaptive optimization with trust-region or duality thresholds (Zhang et al., 30 Jul 2024), geometric fitting with RANSAC-style inlier thresholds (Barath et al., 2019), graph statistics approximators with confidence-interval based sampling (Pellegrina et al., 2021, Grinten et al., 2019), and adaptive threshold sketches for streaming analytics (Ting, 2017).

Central to all these methods is the explicit computation and updating of quantitative thresholds, which may be data-dependent, stochastic, or algorithmic.

2. Algorithmic Design Patterns and Stopping Criteria

A threshold-based progressive sampling algorithm typically proceeds in the following iterative loop:

1. Initialization: Set all statistical or algorithmic state, including initial thresholds that may depend on user parameters (e.g., $\varepsilon$ for accuracy, $\delta$ for confidence).
2. Sampling Step: Draw a batch of new samples, or process a new stream element (e.g., data point, scenario, or geometric correspondence), updating sample-related statistics.
3. Threshold Evaluation: Compute a criterion (often based on empirical error, frequency differences, or accumulated gradients).
4. Termination Check or Adaptation: If the criterion meets or falls below a specified threshold, halt and output the result; otherwise, adjust the threshold (tighten error, enlarge sample, lower duality gap) and repeat.

For example, the progressive itemset mining approach (Pietracaprina et al., 2010) evaluates after each sample increment whether the empirical gap between the estimated $K$-th and $(K+1)$-th most frequent itemsets in the sample exceeds a threshold margin $\varepsilon$, ensuring $(\varepsilon, \delta)$-approximation with high probability. Similarly, parallel adaptive sampling (Grinten et al., 2019) computes per-sample confidence intervals (e.g., Hoeffding bounds) and halts when all bounds fall below $\varepsilon$. In adaptive optimization (Zhang et al., 30 Jul 2024), a duality or trust-region gap decrements with iterations, and batch sizes are scaled dynamically according to a threshold formula derived from concentration inequalities.

3. Mathematical Foundations and Rigorous Guarantees

Threshold-based progressive sampling methods explicitly connect their sampling regime to formal probabilistic and optimization-theoretic guarantees. Key mathematical ingredients include:

• Concentration inequalities: Chernoff, Hoeffding, or McDiarmid bounds to control risk of estimation error, yielding explicit formulas for minimal sample sizes at given accuracy and confidence levels (Pietracaprina et al., 2010, Grinten et al., 2019, Pellegrina et al., 2021, Zhang et al., 30 Jul 2024).
• Adaptive confidence intervals: Statistical quantities (e.g., empirical Rademacher averages (Pellegrina et al., 2021)) are estimated at each step, ensuring a data-dependent and often non-uniform error control.
• Resource-aware sampling: Sample size or memory is bounded a priori or adaptively (e.g., by latent-size thresholds (Hentschel et al., 2021) or reservoir/bottom-$k$ logic (Ting, 2017)), maintaining algorithmic tractability and efficiency.
• Optimization duality and trust regions: In stochastic optimization, thresholds on functional gaps within trust-regions yield rigorous stopping conditions and sample size schedules (Zhang et al., 30 Jul 2024).

Table 1 outlines representative threshold formulas in key domains:

Domain	Characteristic Threshold Formula	Reference
Top-$K$ itemset mining	$$N(\varepsilon,\delta,K,w)=\frac{2}{\varepsilon^2} \ln\left(\frac{2m+K(m-K)}{\delta}\right)$$	(Pietracaprina et al., 2010)
PPS streaming sampling (latent sample)	$$\rho_t = \min\left(\frac{1}{\max_{i\le t}w_i}, \frac{n}{\sum_{i=1}^t w_i}\right)$$	(Hentschel et al., 2021)
Parallel graph sampling (betweenness)	$$\tau \ge \frac{1}{2 \varepsilon^2} \ln\left(\frac{2n}{\delta}\right)$$	(Grinten et al., 2019)
Stochastic prog. hedging	$$|S_k| \ge (8 M_1^2/\kappa^2) (-\log(\varepsilon/2)) \delta_k^{-4}$$	(Zhang et al., 30 Jul 2024)
Betweenness (SILVAN Rademacher bound)	$\max_j \varepsilon_{F_j}(m) \leq \varepsilon$ (data-adaptive, non-uniform)	(Pellegrina et al., 2021)

4. Domain-Specific Instantiations

Threshold-based progressive sampling is realized differently across application domains:

• Frequent Itemset Mining: The algorithm tracks empirical frequencies as more transactions are sampled. Stopping occurs when the $K$-th largest frequency in the sample exceeds other observed frequencies by thresholds parameterized by $\varepsilon$, ensuring the output forms an $(\varepsilon,\delta)$-approximation to the global top-$K$ (Pietracaprina et al., 2010).
• Streaming PPS Sampling: The EB-PPS algorithm maintains a running threshold $\rho_t$. On arrival of new items, inclusion probabilities are scaled down by the updated threshold using a downsampling operator, constraining the latent sample size never to exceed $n$ while maintaining strict PPS property for every item (Hentschel et al., 2021).
• Stochastic Programming (Progressive Hedging): The adaptive PH scheme sets the scenario batch size at each iteration to satisfy a sample-size threshold derived from concentration inequalities, which guarantees sufficient approximation of the true dual objective on trust regions, thus contracting the duality gap efficiently (Zhang et al., 30 Jul 2024).
• Parallel Graph Analytics: Algorithms such as KADABRA and SILVAN use confidence interval thresholds to determine when to stop progressive sampling of shortest paths for betweenness centrality. In the parallel context, the epoch-based sampling framework ensures consistency with minimal synchronization, while maintaining global threshold criteria (Grinten et al., 2019, Pellegrina et al., 2021).
• Geometric Model Fitting: In robust estimation, RANSAC variants including Progressive-X (Barath et al., 2019) and P-NAPSAC (Barath et al., 2019) leverage threshold-based early stopping criteria, where thresholds function both in space (e.g., neighborhood radii) and in sample-score margins to separate inlier/outlier models or govern expansion from local to global samplers.

5. Data Structures and Computational Efficiency

Efficient threshold-based progressive samplers depend on carefully designed data structures and update mechanisms:

• Count-min and Bloom filter sketches: Facilitate scalable frequency tracking and threshold margin testing in itemset mining (Pietracaprina et al., 2010).
• Latent sample structures: Used in streaming PPS with bounded sample size, allowing amortized $O(1)$ time per update and robust threshold downsampling (Hentschel et al., 2021).
• Priority heaps and min-heaps: Underpin adaptive threshold sampling for top-$k$, heavy hitter, and sliding window contexts, enabling thresholds to monotonically decrease or adapt instantly to stream state (Ting, 2017).
• Parallel sample frames and atomic pointers: In high-dimensional or graph-sampling applications, per-thread sample frames with relaxed atomic operations realize threshold-based stopping with provably minimal synchronization overhead (Grinten et al., 2019).
• Rademacher matrix accumulators: Underlie sharp, non-uniform, data-dependent progressive thresholds in betweenness centrality estimation (Pellegrina et al., 2021).

6. Theoretical and Empirical Guarantees

Threshold-based progressive sampling algorithms provide explicit a priori and a posteriori guarantees:

• Sample size bounds: Derived from statistical concentration, these bounds guarantee that, with exponentially high probability, the output meets prescribed error/confidence parameters as soon as the stopping threshold is crossed (Pietracaprina et al., 2010, Zhang et al., 30 Jul 2024, Pellegrina et al., 2021).
• Unbiasedness and minimal variance: Adaptive threshold mechanisms maintain important statistical invariants, such as unbiasedness of Horvitz–Thompson estimators, and minimal variance of realized sample sizes under bounded PPS (Hentschel et al., 2021, Ting, 2017).
• Data-adaptive efficiency: In favorable regimes, progressive thresholding enables the algorithm to stop well before worst-case sample bounds are reached, as the empirical statistics typically separate earlier (Pietracaprina et al., 2010, Grinten et al., 2019, Pellegrina et al., 2021).
• Scalability and parallel speedup: Embeddings into parallel, epoch-based sampling architectures demonstrate nearly ideal speedup and negligible synchronization cost, confirmed by large empirical studies on up to 32-core machines (Grinten et al., 2019).

7. Variants and Generalizations

Numerous specialized variants extend the core framework:

• Adaptive thresholding in streaming: Integrates with reservoir, sliding window, stratified, and heavy hitter sampling, with fully substitutable threshold logic to guarantee statistical correctness in dynamic, streaming contexts (Ting, 2017).
• Geometric progression and local-global scheduling: Used in progressive geometric hypothesis generation, where thresholds dynamically adjust between local neighborhoods and global uniform sampling (Barath et al., 2019).
• Non-uniform, group-stratified error thresholds: Advanced by methods such as SILVAN, where empirical Rademacher complexity provides locally tight progressive bounds per stratum (Pellegrina et al., 2021).
• Optimization-aware batch sizing: Sample sizes are dynamically adjusted via trust-region radii or duality gaps, ensuring optimal contraction in stochastic optimization (Zhang et al., 30 Jul 2024).

As a unifying methodological paradigm, threshold-based progressive sampling underpins scalable, reliable algorithms for high-dimensional data analysis, combinatorial optimization, robust geometric estimation, and streaming analytics, providing formal quality guarantees that adapt to data, resource, and accuracy constraints.