On the impossibility of discovering a formula for primes using AI

The present work explores the theoretical limits of Machine Learning (ML) within the framework of Kolmogorov's theory of Algorithmic Probability, which clarifies the notion of entropy as Expected Kolmogorov Complexity and formalizes other fundamental concepts such as Occam's razor via Levin's Universal Distribution. As a fundamental application, we develop Maximum Entropy methods that allow us to derive the Erd\H{o}s--Kac Law in Probabilistic Number Theory, and establish the impossibility of discovering a formula for primes using Machine Learning via the Prime Coding Theorem.Comment: 23 page

Application of information theory and statistical learning to anomaly detection

In today\u27s highly networked world, computer intrusions and other attacks area constant threat. The detection of such attacks, especially attacks that are new or previously unknown, is important to secure networks and computers. A major focus of current research efforts in this area is on anomaly detection.;In this dissertation, we explore applications of information theory and statistical learning to anomaly detection. Specifically, we look at two difficult detection problems in network and system security, (1) detecting covert channels, and (2) determining if a user is a human or bot. We link both of these problems to entropy, a measure of randomness information content, or complexity, a concept that is central to information theory. The behavior of bots is low in entropy when tasks are rigidly repeated or high in entropy when behavior is pseudo-random. In contrast, human behavior is complex and medium in entropy. Similarly, covert channels either create regularity, resulting in low entropy, or encode extra information, resulting in high entropy. Meanwhile, legitimate traffic is characterized by complex interdependencies and moderate entropy. In addition, we utilize statistical learning algorithms, Bayesian learning, neural networks, and maximum likelihood estimation, in both modeling and detecting of covert channels and bots.;Our results using entropy and statistical learning techniques are excellent. By using entropy to detect covert channels, we detected three different covert timing channels that were not detected by previous detection methods. Then, using entropy and Bayesian learning to detect chat bots, we detected 100% of chat bots with a false positive rate of only 0.05% in over 1400 hours of chat traces. Lastly, using neural networks and the idea of human observational proofs to detect game bots, we detected 99.8% of game bots with no false positives in 95 hours of traces. Our work shows that a combination of entropy measures and statistical learning algorithms is a powerful and highly effective tool for anomaly detection

A VLSI-design of the minimum entropy neuron

One of the most interesting domains of feedforward networks is the processing of sensor signals. There do exist some networks which extract most of the information by implementing the maximum entropy principle for Gaussian sources. This is done by transforming input patterns to the base of eigenvectors of the input autocorrelation matrix with the biggest eigenvalues. The basic building block of these networks is the linear neuron, learning with the Oja learning rule. Nevertheless, some researchers in pattern recognition theory claim that for pattern recognition and classification clustering transformations are needed which reduce the intra-class entropy. This leads to stable, reliable features and is implemented for Gaussian sources by a linear transformation using the eigenvectors with the smallest eigenvalues. In another paper (Brause 1992) it is shown that the basic building block for such a transformation can be implemented by a linear neuron using an Anti-Hebb rule and restricted weights. This paper shows the analog VLSI design for such a building block, using standard modules of multiplication and addition. The most tedious problem in this VLSI-application is the design of an analog vector normalization circuitry. It can be shown that the standard approaches of weight summation will not give the convergence to the eigenvectors for a proper feature transformation. To avoid this problem, our design differs significantly from the standard approaches by computing the real Euclidean norm. Keywords: minimum entropy, principal component analysis, VLSI, neural networks, surface approximation, cluster transformation, weight normalization circuit

A Tight Excess Risk Bound via a Unified PAC-Bayesian-Rademacher-Shtarkov-MDL Complexity

We present a novel notion of complexity that interpolates between and generalizes some classic existing complexity notions in learning theory: for estimators like empirical risk minimization (ERM) with arbitrary bounded losses, it is upper bounded in terms of data-independent Rademacher complexity; for generalized Bayesian estimators, it is upper bounded by the data-dependent information complexity (also known as stochastic or PAC-Bayesian, $\mathrm{KL}(\text{posterior} \operatorname{\|} \text{prior})$ complexity. For (penalized) ERM, the new complexity reduces to (generalized) normalized maximum likelihood (NML) complexity, i.e. a minimax log-loss individual-sequence regret. Our first main result bounds excess risk in terms of the new complexity. Our second main result links the new complexity via Rademacher complexity to $L_2(P)$ entropy, thereby generalizing earlier results of Opper, Haussler, Lugosi, and Cesa-Bianchi who did the log-loss case with $L_\infty$. Together, these results recover optimal bounds for VC- and large (polynomial entropy) classes, replacing localized Rademacher complexity by a simpler analysis which almost completely separates the two aspects that determine the achievable rates: 'easiness' (Bernstein) conditions and model complexity.Comment: 38 page