Lazy Sampling Strategies

• Lazy sampling is a deferred computation strategy that delays evaluation until necessary, reducing memory and processing overhead.
• It is applied across domains like submodular maximization, kernel approximation, and Bayesian control to improve efficiency.
• Practical implementations use methods such as stochastic-greedy and trie-based sampling to achieve scalable, incremental performance.

Lazy sampling is a general computational strategy in which random values, combinatorial decisions, or expensive computations are deferred and executed only when explicitly required by the algorithm or system. This paradigm targets scalability, efficiency, and memory savings in contexts ranging from submodular optimization and probabilistic programming to cryptographic proofs and statistical inference. While implementations and analytical guarantees vary by domain, the unifying feature is that only a dynamically determined subset of operations or samples are actually instantiated during execution.

1. Principles and Formal Definitions

Lazy sampling is characterized by on-demand realization of random, deterministic, or computational resources, avoiding up-front computation or evaluation across the entire input or state space. The canonical example in randomized algorithms involves maintaining a table of drawn samples and generating new ones only as queries arise. In formal logic or numerical linear algebra, lazy variants update computational state incrementally, avoiding global recomputation or full evaluation.

Submodular maximization provides a precise template: for a monotone submodular function $f:2^V\to\mathbb{R}_+$ over a finite ground set $V$, classical algorithms instantiate every candidate inclusion to $S$ at each iteration, whereas lazy (stochastic-greedy) approaches query only a random subset $R\subset V\setminus S$ in each step, and possibly further restrict evaluation within $R$ via partial upper bounds (Mirzasoleiman et al., 2014).

2. Representative Algorithms and Data Structures

Submodular Function Maximization: STOCHASTIC-GREEDY

STOCHASTIC-GREEDY replaces exhaustive greedy scans by random sampling. In iteration $i$, it draws a sample $R\subset V\setminus A$ of cardinality

$s = \lceil(n/k) \ln(1/\varepsilon)\rceil,$

maximizes $\Delta(a|A)$ over $a\in R$, and augments $A$ by the best candidate. Performance is dramatically improved by combining this step with lazy marginal evaluation—only the highest-potential elements (under upper bound $\rho(e)$) in $R$ are certified via explicit computation.

Trie-based Incremental Sampling

Sequence-model lazy sampling instantiates a trie of prefixes, each node tracking unsampled probability mass. Random choices traverse this trie, expanding children only as needed, and updating mass after each termination; this realizes both sampling without replacement and efficient expectation estimation for exponentially large discrete spaces (Shi et al., 2020).

Lazy Pivoted Cholesky/Farthest-Point Sampling

In kernel approximation, the lazy pivoted Cholesky algorithm maintains diagonal residuals $d[i]$ and selects the farthest point at each iteration using only the required kernel column. Updates are strictly local; the Schur complement is never globally computed, and memory requirements do not scale quadratically with $N$ (Shabat, 7 Jan 2026).

Lazy Posterior Sampling for Bayesian Control

Lazy PSRL defers expensive policy recomputation: new posterior samples and associated optimal policies are drawn only when the information matrix determinant passes a fixed multiplicative threshold, yielding an $O(\log T)$ bound on recomputation steps with $\widetilde O(\sqrt T)$ regret (Abbasi-Yadkori et al., 2014).

Sampling for Data Freshness: Lazy Threshold Policies

In sampling for age-of-information optimization, optimal causal sampling policies take the form of deterministic or randomized threshold rules. Samples are generated only when the receiver age crosses a computed threshold, with bisection techniques used to establish the precise value. The threshold and sampling rates are derived directly from constrained Markov decision process analysis (Sun et al., 2018).

Lazy ABC for Likelihood-Free Inference

In computational statistics, lazy ABC divides simulation into partial (cheap) and full (expensive) stages, and employs a random continuation rule $\alpha(\theta, X)$. The algorithm may terminate early with probability $1-\alpha$, ensuring unbiasedness via a correction to sample weights (Prangle, 2014).

3. Theoretical Guarantees and Analytical Tools

Lazy sampling strategies typically offer strong theoretical guarantees with respect to approximation error, computational complexity, and resource usage.

• Submodular stochastic-greedy yields, in expectation, a $(1-1/e-\varepsilon)$ approximation to the optimum value using $O(n \log(1/\varepsilon))$ function evaluations, independent of $k$ (Mirzasoleiman et al., 2014).
• Lazy posterior sampling provably achieves $\widetilde O(\sqrt T)$ regret for average-cost MDPs, with no more than $O(\log T)$ policy updates (Abbasi-Yadkori et al., 2014).
• Trie-based incremental sampling ensures correct sampling without replacement and enables unbiased expectation estimation via the hindsight Gumbel estimator (Shi et al., 2020).
• Lazy ABC preserves the target posterior via unbiased reweighting, with practical efficiency gains proportional to the early stopping rate, subject to inflated estimator variance from the $1/\alpha$ factor (Prangle, 2014).
• Threshold policies in data freshness maximize general nonlinear age freshness objectives under sampling-rate constraints, with bisection search guaranteeing optimality in continuous and discrete time (Sun et al., 2018).
• Lazy pivoted Cholesky reduces memory and computational requirements from $O(N^2)$ and $O(N^3)$ to $O(NM)$ and $O(NM^2)$ for $M\ll N$, with deterministic output and spectral/tracial accuracy sufficient for large-scale kernel problems (Shabat, 7 Jan 2026).

4. Applications and Domains

Lazy sampling principles permeate diverse algorithmic areas:

• Data summarization: Active set selection in Gaussian processes, submodular sensor placement, and clustering exploit stochastic-greedy and lazy marginal evaluation for scalable optimization (Mirzasoleiman et al., 2014).
• Statistical inference: Lazy ABC algorithms make approximate Bayesian computation tractable for spatial extremes or epidemic models where full simulation is prohibitive (Prangle, 2014).
• Probabilistic programming: Trie-based incremental sampling enables sampling without replacement in program synthesis and combinatorial search, supporting expectation estimation and diversity (Shi et al., 2020).
• Robotics and motion planning: Asymptotically optimal lifelong planners utilize lazy sampling and search to dramatically cut collision and edge evaluation costs, outperforming eager planners in both static and dynamic environments (Huang et al., 2024).
• Complexity-theoretic and cryptographic proofs: Game-playing proofs and random oracle models adopt lazy sampling for indistinguishability results—the compressed quantum oracle technique extends this to coherent superposition queries (Czajkowski et al., 2019, Metere et al., 2023).

5. Comparative Analysis, Implementation, and Practical Guidance

Lazy sampling typically outperforms classic eager or full enumeration approaches in scenarios where evaluation cost, number of candidate points, or space complexity is dominant.

Context	Classical/Eager Method	Lazy Sampling Variant
Submodular maximization	Exhaustive greedy, $O(nk)$	STOCHASTIC-GREEDY, $O(n \log(1/\varepsilon))$
Kernel approx.	Full pivoted Cholesky, $O(N^2)$	Lazy FPS, local columns $O(NM^2)$
ABC inference	All simulations complete	Early stop with weight reweighting ($\alpha$)
Sequence sampling	Global heap or batch sampling	Trie-based one-at-a-time, no extra heap
Bayesian control	Policy recomputed per step	Policy recomputed only on sufficient variance
Motion planning	All edges evaluated	Only edges on candidate paths lazily checked
Random oracles (crypto)	Eager table, finite keys	On-demand memoization, compressed oracle

Recommended practices include tuning batch sizes or threshold parameters ($\varepsilon$ in submodular, $\alpha$ in control, $\lambda$ in ABC), further combining lazy sampling with lazy evaluation or incremental repair, and exploiting algorithmic structure (e.g., trie reuse in sequence models or informed rewiring in motion planning). In distributed or streaming settings, lazy sampling often yields direct linear-time speedups by replacing traditional greedy subroutines.

6. Extensions and Contemporary Developments

Recent advances generalize lazy sampling to quantum algorithms (compressed quantum oracles), probabilistic relational Hoare logic (direct lazy sampling proofs in EasyCrypt) and highly dynamic, lifelong learning contexts. In quantum cryptography, lazy sampling guarantees indistinguishability to any $q$-query adversary using only compressed databases and hybrid measurement arguments (Czajkowski et al., 2019). In formal verification, the direct lazy-sampling rule eliminates necessity for eager conversion, finite-memory bounds, and code-motion reasoning, significantly streamlining proofs in the random oracle model (Metere et al., 2023).

Lazy sampling methodology continues to evolve across computational science, combining randomized methods, incremental repair, threshold policies, and deferred evaluation, with substantial impact on the scalability and tractability of inference, optimization, search, and verification.