Maurer's universal statistical test can widely detect non-randomness of given sequences. Coron proposed an improved test, and further Yamamoto and Liu proposed a new test based on Coron's test. These tests use normal distributions as their reference distributions, but the soundness has not been theoretically discussed so far. Additionally, Yamamoto and Liu's test uses an experimental value as the variance of its reference distribution. In this paper, we theoretically derive the variance of the reference distribution of Yamamoto and Liu's test and prove that the true reference distribution of Coron's test converges to a normal distribution in some sense. We can apply the proof to the other tests with small changes.
In this paper, we propose a strategy to construct a multi-objective optimization algorithm from a single-objective optimization algorithm by using the Bézier simplex model. Also, we extend the stability of optimization algorithms in the sense of Probability Approximately Correct (PAC) learning and define the PAC stability. We prove that it leads to an upper bound on the generalization with high probability. Furthermore, we show that multi-objective optimization algorithms derived from a gradient descent-based single-objective optimization algorithm are PAC stable. We conducted numerical experiments and demonstrated that our method achieved lower generalization errors than the existing multi-objective optimization algorithm.Preprint. Under review.
Maurer's universal statistical test can widely detect non-randomness of given sequences. Coron proposed an improved test, and further Yamamoto and Liu proposed a new test based on Coron's test. These tests use normal distributions as their reference distributions, but the soundness has not been theoretically discussed so far. Additionally, Yamamoto and Liu's test uses an experimental value as the variance of its reference distribution. In this paper, we theoretically derive the variance of the reference distribution of Yamamoto and Liu's test and prove that the true reference distribution of Coron's test converges to a normal distribution in some sense. We can apply the proof to the other tests with small changes.
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