We provide a refined characterization of the super-Turing computational power of analog, evolving, and stochastic neural networks based on the Kolmogorov complexity of their real weights, evolving weights, and real probabilities, respectively. First, we retrieve an infinite hierarchy of classes of analog networks defined in terms of the Kolmogorov complexity of their underlying real weights. This hierarchy is located between the complexity classes $\mathbf{P}$ and $\mathbf{P/poly}$. Then, we generalize this result to the case of evolving networks. A similar hierarchy of Kolomogorov-based complexity classes of evolving networks is obtained. This hierarchy also lies between $\mathbf{P}$ and $\mathbf{P/poly}$. Finally, we extend these results to the case of stochastic networks employing real probabilities as source of randomness. An infinite hierarchy of stochastic networks based on the Kolmogorov complexity of their probabilities is therefore achieved. In this case, the hierarchy bridges the gap between $\mathbf{BPP}$ and $\mathbf{BPP/log^*}$. Beyond proving the existence and providing examples of such hierarchies, we describe a generic way of constructing them based on classes of functions of increasing complexity. For the sake of clarity, this study is formulated within the framework of echo state networks. Overall, this paper intends to fill the missing results and provide a unified view about the refined capabilities of analog, evolving and stochastic neural networks.

Nowadays, increasingly more data are available as knowledge graphs (KGs). While this data model supports advanced reasoning and querying, they remain difficult to mine due to their size and complexity. Graph mining approaches can be used to extract patterns from KGs. However this presents two main issues. First, graph mining approaches tend to extract too many patterns for a human analyst to interpret (pattern explosion). Second, real-life KGs tend to differ from the graphs usually treated in graph mining: they are multigraphs, their vertex degrees tend to follow a power-law, and the way in which they model knowledge can produce spurious patterns. Recently, a graph mining approach named GraphMDL+ has been proposed to tackle the problem of pattern explosion, using the Minimum Description Length (MDL) principle. However, GraphMDL+, like other graph mining approaches, is not suited for KGs without adaptations. In this paper we propose KG-MDL, a graph pattern mining approach based on the MDL principle that, given a KG, generates a human-sized and descriptive set of graph patterns, and so in a parameter-less and anytime way. We report on experiments on medium-sized KGs showing that our approach generates sets of patterns that are both small enough to be interpreted by humans and descriptive of the KG. We show that the extracted patterns highlight relevant characteristics of the data: both of the schema used to create the data, and of the concrete facts it contains. We also discuss the issues related to mining graph patterns on knowledge graphs, as opposed to other types of graph data.

Multivariate Hawkes processes (MHPs) are versatile probabilistic tools used to model various real-life phenomena: earthquakes, operations on stock markets, neuronal activity, virus propagation and many others. In this paper, we focus on MHPs with exponential decay kernels and estimate connectivity graphs, which represent the Granger causal relations between their components. We approach this inference problem by proposing an optimization criterion and model selection algorithm based on the minimum message length (MML) principle. MML compares Granger causal models using the Occam's razor principle in the following way: even when models have a comparable goodness-of-fit to the observed data, the one generating the most concise explanation of the data is preferred. While most of the state-of-art methods using lasso-type penalization tend to overfitting in scenarios with short time horizons, the proposed MML-based method achieves high F1 scores in these settings. We conduct a numerical study comparing the proposed algorithm to other related classical and state-of-art methods, where we achieve the highest F1 scores in specific sparse graph settings. We illustrate the proposed method also on G7 sovereign bond data and obtain causal connections, which are in agreement with the expert knowledge available in the literature.

Deep Neural Networks (DNNs) deployed to the real world are regularly subject to out-of-distribution (OoD) data, various types of noise, and shifting conceptual objectives. This paper proposes a framework for adapting to data distribution drift modeled by benchmark Continual Learning datasets. We develop and evaluate a method of Continual Learning that leverages uncertainty quantification from Bayesian Inference to mitigate catastrophic forgetting. We expand on previous approaches by removing the need for Monte Carlo sampling of the model weights to sample the predictive distribution. We optimize a closed-form Evidence Lower Bound (ELBO) objective approximating the predictive distribution by propagating the first two moments of a distribution, i.e. mean and covariance, through all network layers. Catastrophic forgetting is mitigated by using the closed-form ELBO to approximate the Minimum Description Length (MDL) Principle, inherently penalizing changes in the model likelihood by minimizing the KL Divergence between the variational posterior for the current task and the previous task's variational posterior acting as the prior. Leveraging the approximation of the MDL principle, we aim to initially learn a sparse variational posterior and then minimize additional model complexity learned for subsequent tasks. Our approach is evaluated for the task incremental learning scenario using density propagated versions of fully-connected and convolutional neural networks across multiple sequential benchmark datasets with varying task sequence lengths. Ultimately, this procedure produces a minimally complex network over a series of tasks mitigating catastrophic forgetting.

Inspired by Solomonoffs theory of inductive inference, we propose a prior based on circuit complexity. There are several advantages to this approach. First, it relies on a complexity measure that does not depend on the choice of UTM. There is one universal definition for Boolean circuits involving an universal operation such as nand with simple conversions to alternative definitions such as and, or, and not. Second, there is no analogue of the halting problem. The output value of a circuit can be calculated recursively by computer in time proportional to the number of gates, while a short program may run for a very long time. Our prior assumes that a Boolean function, or equivalently, Boolean string of fixed length, is generated by some Bayesian mixture of circuits. This model is appropriate for learning Boolean functions from partial information, a problem often encountered within machine learning as "binary classification." We argue that an inductive bias towards simple explanations as measured by circuit complexity is appropriate for this problem.

Histograms are among the most popular methods used in exploratory analysis to summarize univariate distributions. In particular, irregular histograms are good non-parametric density estimators that require very few parameters: the number of bins with their lengths and frequencies. Many approaches have been proposed in the literature to infer these parameters, either assuming hypotheses about the underlying data distributions or exploiting a model selection approach. In this paper, we focus on the G-Enum histogram method, which exploits the Minimum Description Length (MDL) principle to build histograms without any user parameter and achieves state-of-the art performance w.r.t accuracy; parsimony and computation time. We investigate on the limits of this method in the case of outliers or heavy-tailed distributions. We suggest a two-level heuristic to deal with such cases. The first level exploits a logarithmic transformation of the data to split the data set into a list of data subsets with a controlled range of values. The second level builds a sub-histogram for each data subset and aggregates them to obtain a complete histogram. Extensive experiments show the benefits of the approach.

When choosing between competing symbolic models for a data set, a human will naturally prefer the "simpler" expression or the one which more closely resembles equations previously seen in a similar context. This suggests a non-uniform prior on functions, which is, however, rarely considered within a symbolic regression (SR) framework. In this paper we develop methods to incorporate detailed prior information on both functions and their parameters into SR. Our prior on the structure of a function is based on a $n$-gram language model, which is sensitive to the arrangement of operators relative to one another in addition to the frequency of occurrence of each operator. We also develop a formalism based on the Fractional Bayes Factor to treat numerical parameter priors in such a way that models may be fairly compared though the Bayesian evidence, and explicitly compare Bayesian, Minimum Description Length and heuristic methods for model selection. We demonstrate the performance of our priors relative to literature standards on benchmarks and a real-world dataset from the field of cosmology.

Symbolic Regression (SR) algorithms attempt to learn analytic expressions which fit data accurately and in a highly interpretable manner. Conventional SR suffers from two fundamental issues which we address here. First, these methods search the space stochastically (typically using genetic programming) and hence do not necessarily find the best function. Second, the criteria used to select the equation optimally balancing accuracy with simplicity have been variable and subjective. To address these issues we introduce Exhaustive Symbolic Regression (ESR), which systematically and efficiently considers all possible equations -- made with a given basis set of operators and up to a specified maximum complexity -- and is therefore guaranteed to find the true optimum (if parameters are perfectly optimised) and a complete function ranking subject to these constraints. We implement the minimum description length principle as a rigorous method for combining these preferences into a single objective. To illustrate the power of ESR we apply it to a catalogue of cosmic chronometers and the Pantheon+ sample of supernovae to learn the Hubble rate as a function of redshift, finding $\sim$40 functions (out of 5.2 million trial functions) that fit the data more economically than the Friedmann equation. These low-redshift data therefore do not uniquely prefer the expansion history of the standard model of cosmology. We make our code and full equation sets publicly available.

Many organisations manage service quality and monitor a large set devices and servers where each entity is associated with telemetry or physical sensor data series. Recently, various methods have been proposed to detect behavioural anomalies, however existing approaches focus on multivariate time series and ignore communication between entities. Moreover, we aim to support end-users in not only in locating entities and sensors causing an anomaly at a certain period, but also explain this decision. We propose a scalable approach to detect anomalies using a two-step approach. First, we recover relations between entities in the network, since relations are often dynamic in nature and caused by an unknown underlying process. Next, we report anomalies based on an embedding of sequential patterns. Pattern mining is efficient and supports interpretation, i.e. patterns represent frequent occurring behaviour in time series. We extend pattern mining to filter sequential patterns based on frequency, temporal constraints and minimum description length. We collect and release two public datasets for international broadcasting and X from an Internet company. \textit{BAD} achieves an overall F1-Score of 0.78 on 9 benchmark datasets, significantly outperforming the best baseline by 3\%. Additionally, \textit{BAD} is also an order-of-magnitude faster than state-of-the-art anomaly detection methods.

The Game Developer's Conference, the largest event in the games industry, has hosted over 50 talks in the last decade about procedural generation, from small-scale independent speakers to large AAA companies, covering disciplines from programming to art to writing. Correspondingly, procedural generation has been an increasingly hot topic among game AI researchers in the last two decades. The Procedural Generation Workshop at FDG, now in its twelfth year, is one of the longest-running workshops in the field of game AI, and dedicated paper tracks at conferences are a regular occurrence. Despite the huge importance of content generation, and the wealth of time invested into developing practical techniques, the analysis of procedural generators is a relatively underdeveloped area of study. A few notable techniques have emerged over the last two decades of research [7, 8], as well as studies of efficacy [4, 9], but many of the techniques used by game researchers have changed little in that time. As a result, a lot of procedural generation work is done by'feel', with postmortems shared at events such as the Roguelike Celebration