Abstract
Sampling is a key step of stochastic methods. In some contexts, minimizing the sample size can be of critical importance and determine by itself the viability of various approaches. Rethinking the way samples are generated can even lead to new algorithms, better suited than their alternatives to tackle specific real-world problems for which the sampling task is a costly step of the estimation process. For instance, strategies of formal verification and statistical model checking can be greatly improved by the use of adaptive stopping algorithms. Instead of initially computing the necessary sample size, those algorithms generate the samples progressively while continuously monitoring their progress along the way, allowing them to stop the sampling process as soon as possible. We present a generalization of two existing adaptive stopping algorithms for statistical model checking, and we show how this generalization can be exploited to derive tailor-made variations for specific use cases.
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Acknowledgements
M. Parmentier is funded by a FNRS PhD Grant and by the UCLouvain.
A. Legay is funded by a FNRS PDR - T013721.
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Appendix: Concentration Inequalities
Appendix: Concentration Inequalities
This is a nonexhaustive list of some of the most well-known and useful concentration inequalities. Most of them are regarded as relatively elementary results in statistics [6, 10, 38].
Let X be a random variable.
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Markov’s inequality (X must be almost surely nonnegative):
$$\forall \epsilon > 0:P(X \ge \epsilon ) \le \frac{E(X)}{\epsilon }$$ -
Chebyshev’s inequality (E(X) and V(X) must be finite):
$$\forall \epsilon > 0:P(|X-E(X)| \ge \epsilon ) \le \frac{V(X)}{{\epsilon }^2}$$ -
Chernoff bound (the moment generating function of X, \(M_X(t) = E(e^{tX})\), must be well-defined and finite):
$$\forall \epsilon > 0:P(X \ge \epsilon ) \le \frac{E(e^{tX})}{e^{t\epsilon }}$$
Now let \(X = S_n = X_1 + ... + X_n\) be the sum of n independent random variables, with each \(X_i\) almost surely bounded within \([a_i,b_i]\). Let \(R_i = b_i - a_i\) be the range of \(X_i\). Let be \(R = \underset{1\le i\le n}{\max }(R_i)\).
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Hoeffding’s inequality:
$$\forall \epsilon > 0:P(|S_n - E(S_n)| \ge \epsilon ) \le 2\exp \left( -\frac{2{\epsilon }^2}{\sum _{i=1}^{n} (b_i - a_i)^2}\right) $$ -
Bernstein’s inequality:
$$\forall \epsilon > 0:{P(|S_n - E(S_n)| \ge \epsilon ) \le 2\exp \left( -\frac{{\epsilon }^2}{2V(S_n)+(2/3)\epsilon R}\right) }$$ -
Both Azuma’s Inequality and McDiamid’s inequality are equivalent to Hoeffding’s inequality with those assumptions:
$$\forall \epsilon > 0:P(|S_n - E(S_n)| \ge \epsilon ) \le 2\exp \left( -\frac{2{\epsilon }^2}{\sum _{i=1}^{n} (b_i - a_i)^2}\right) $$ -
Benett’s inequality (with \(f(x) = (1+x)\log (1+x)-x\)):
$$\forall \epsilon > 0:{P(|S_n - E(S_n)| \ge \epsilon ) \le 2\exp \left( -\frac{V(S_n)}{R^2} f\left( \frac{\epsilon R}{V(S_n)}\right) \right) }$$
Additionally, if the random variables \(X_1\), ..., \(X_n\) are independent and identically distributed (i.i.d.), the central limit theorem itself can be interpreted as a concentration inequality: as n grows, \(S_n - E(S_n)\) can be approximated with \(Z \sim \mathcal {N}(0,V(S_n))\), which implies:
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Parmentier, M., Legay, A. (2025). Adaptive Stopping Algorithms Based on Concentration Inequalities. In: Steffen, B. (eds) Bridging the Gap Between AI and Reality. AISoLA 2024. Lecture Notes in Computer Science, vol 15217. Springer, Cham. https://doi.org/10.1007/978-3-031-75434-0_23
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