Matteo Marsili

Rongrong Xie, Matteo Marsili

We discuss probabilistic neural networks with a fixed internal representation as models for machine understanding. Here understanding is intended as mapping data to an already existing representation which encodes an {\em a priori} organisation of the feature space. We derive the internal representation by requiring that it satisfies the principles of maximal relevance and of maximal ignorance about how different features are combined. We show that, when hidden units are binary variables, these two principles identify a unique model -- the Hierarchical Feature Model (HFM) -- which is fully solvable and provides a natural interpretation in terms of features. We argue that learning machines with this architecture enjoy a number of interesting properties, like the continuity of the representation with respect to changes in parameters and data, the possibility to control the level of compression and the ability to support functions that go beyond generalisation. We explore the behaviour of the model with extensive numerical experiments and argue that models where the internal representation is fixed reproduce a learning modality which is qualitatively different from that of traditional models such as Restricted Boltzmann Machines.

Carlo Orientale Caputo, Elias Seiffert, Enrico Frausin et al.

Abstraction is the process of extracting the essential features from raw data while ignoring irrelevant details. It is well known that abstraction emerges with depth in neural networks, where deep layers capture abstract characteristics of data by combining lower level features encoded in shallow layers (e.g. edges). Yet we argue that depth alone is not enough to develop truly abstract representations. We advocate that the level of abstraction crucially depends on how broad the training set is. We address the issue within a renormalisation group approach where a representation is expanded to encompass a broader set of data. We take the unique fixed point of this transformation -- the Hierarchical Feature Model -- as a candidate for a representation which is absolutely abstract. This theoretical picture is tested in numerical experiments based on Deep Belief Networks and auto-encoders trained on data of different breadth. These show that representations in neural networks approach the Hierarchical Feature Model as the data get broader and as depth increases, in agreement with theoretical predictions.

Fariba Jangjoo, Matteo Marsili, Yasser Roudi

Closed-loop learning is the process of repeatedly estimating a model from data generated from the model itself. It is receiving great attention due to the possibility that large neural network models may, in the future, be primarily trained with data generated by artificial neural networks themselves. We study this process for models that belong to exponential families, deriving equations of motions that govern the dynamics of the parameters. We show that maximum likelihood estimation of the parameters endows sufficient statistics with the martingale property and that as a result the process converges to absorbing states that amplify initial biases present in the data. However, we show that this outcome may be prevented if the data contains at least one data point generated from a ground truth model, by relying on maximum a posteriori estimation or by introducing regularisation.

Matteo Marsili, Yasser Roudi

Learning is a distinctive feature of intelligent behaviour. High-throughput experimental data and Big Data promise to open new windows on complex systems such as cells, the brain or our societies. Yet, the puzzling success of Artificial Intelligence and Machine Learning shows that we still have a poor conceptual understanding of learning. These applications push statistical inference into uncharted territories where data is high-dimensional and scarce, and prior information on "true" models is scant if not totally absent. Here we review recent progress on understanding learning, based on the notion of "relevance". The relevance, as we define it here, quantifies the amount of information that a dataset or the internal representation of a learning machine contains on the generative model of the data. This allows us to define maximally informative samples, on one hand, and optimal learning machines on the other. These are ideal limits of samples and of machines, that contain the maximal amount of information about the unknown generative process, at a given resolution (or level of compression). Both ideal limits exhibit critical features in the statistical sense: Maximally informative samples are characterised by a power-law frequency distribution (statistical criticality) and optimal learning machines by an anomalously large susceptibility. The trade-off between resolution (i.e. compression) and relevance distinguishes the regime of noisy representations from that of lossy compression. These are separated by a special point characterised by Zipf's law statistics. This identifies samples obeying Zipf's law as the most compressed loss-less representations that are optimal in the sense of maximal relevance. Criticality in optimal learning machines manifests in an exponential degeneracy of energy levels, that leads to unusual thermodynamic properties.

Rongrong Xie, Matteo Marsili

We study a generic ensemble of deep belief networks which is parametrized by the distribution of energy levels of the hidden states of each layer. We show that, within a random energy approach, statistical dependence can propagate from the visible to deep layers only if each layer is tuned close to the critical point during learning. As a consequence, efficiently trained learning machines are characterised by a broad distribution of energy levels. The analysis of Deep Belief Networks and Restricted Boltzmann Machines on different datasets confirms these conclusions.

Clélia de Mulatier, Matteo Marsili

Finding the model that best describes a high-dimensional dataset is a daunting task, even more so if one aims to consider all possible high-order patterns of the data, going beyond pairwise models. For binary data, we show that this task becomes feasible when restricting the search to a family of simple models, that we call Minimally Complex Models (MCMs). MCMs are maximum entropy models that have interactions of arbitrarily high order grouped into independent components of minimal complexity. They are simple in information-theoretic terms, which means they can only fit well certain types of data patterns and are therefore easy to falsify. We show that Bayesian model selection restricted to these models is computationally feasible and has many advantages. First, the model evidence, which balances goodness-of-fit against complexity, can be computed efficiently without any parameter fitting, enabling very fast explorations of the space of MCMs. Second, the family of MCMs is invariant under gauge transformations, which can be used to develop a representation-independent approach to statistical modeling. For small systems (up to 15 variables), combining these two results allows us to select the best MCM among all, even though the number of models is already extremely large. For larger systems, we propose simple heuristics to find optimal MCMs in reasonable times. Besides, inference and sampling can be performed without any computational effort. Finally, because MCMs have interactions of any order, they can reveal the presence of important high-order dependencies in the data, providing a new approach to explore high-order dependencies in complex systems. We apply our method to synthetic data and real-world examples, illustrating how MCMs portray the structure of dependencies among variables in a simple manner, extracting falsifiable predictions on symmetries and invariance from the data.

Nicola Bulso, Matteo Marsili, Yasser Roudi

We investigate the complexity of logistic regression models which is defined by counting the number of indistinguishable distributions that the model can represent (Balasubramanian, 1997). We find that the complexity of logistic models with binary inputs does not only depend on the number of parameters but also on the distribution of inputs in a non-trivial way which standard treatments of complexity do not address. In particular, we observe that correlations among inputs induce effective dependencies among parameters thus constraining the model and, consequently, reducing its complexity. We derive simple relations for the upper and lower bounds of the complexity. Furthermore, we show analytically that, defining the model parameters on a finite support rather than the entire axis, decreases the complexity in a manner that critically depends on the size of the domain. Based on our findings, we propose a novel model selection criterion which takes into account the entropy of the input distribution. We test our proposal on the problem of selecting the input variables of a logistic regression model in a Bayesian Model Selection framework. In our numerical tests, we find that, while the reconstruction errors of standard model selection approaches (AIC, BIC, $\ell_1$ regularization) strongly depend on the sparsity of the ground truth, the reconstruction error of our method is always close to the minimum in all conditions of sparsity, data size and strength of input correlations. Finally, we observe that, when considering categorical instead of binary inputs, in a simple and mathematically tractable case, the contribution of the alphabet size to the complexity is very small compared to that of parameter space dimension. We further explore the issue by analysing the dataset of the "13 keys to the White House" which is a method for forecasting the outcomes of US presidential elections.

Juyong Song, Matteo Marsili, Junghyo Jo

Deep learning has been successfully applied to various tasks, but its underlying mechanism remains unclear. Neural networks associate similar inputs in the visible layer to the same state of hidden variables in deep layers. The fraction of inputs that are associated to the same state is a natural measure of similarity and is simply related to the cost in bits required to represent these inputs. The degeneracy of states with the same information cost provides instead a natural measure of noise and is simply related the entropy of the frequency of states, that we call relevance. Representations with minimal noise, at a given level of similarity (resolution), are those that maximise the relevance. A signature of such efficient representations is that frequency distributions follow power laws. We show, in extensive numerical experiments, that deep neural networks extract a hierarchy of efficient representations from data, because they i) achieve low levels of noise (i.e. high relevance) and ii) exhibit power law distributions. We also find that the layer that is most efficient to reliably generate patterns of training data is the one for which relevance and resolution are traded at the same price, which implies that frequency distribution follows Zipf's law.

Alberto Beretta, Claudia Battistin, Clélia de Mulatier et al.

Models can be simple for different reasons: because they yield a simple and computationally efficient interpretation of a generic dataset (e.g. in terms of pairwise dependences) - as in statistical learning - or because they capture the essential ingredients of a specific phenomenon - as e.g. in physics - leading to non-trivial falsifiable predictions. In information theory and Bayesian inference, the simplicity of a model is precisely quantified in the stochastic complexity, which measures the number of bits needed to encode its parameters. In order to understand how simple models look like, we study the stochastic complexity of spin models with interactions of arbitrary order. We highlight the existence of invariances with respect to bijections within the space of operators, which allow us to partition the space of all models into equivalence classes, in which models share the same complexity. We thus found that the complexity (or simplicity) of a model is not determined by the order of the interactions, but rather by their mutual arrangements. Models where statistical dependencies are localized on non-overlapping groups of few variables (and that afford predictions on independencies that are easy to falsify) are simple. On the contrary, fully connected pairwise models, which are often used in statistical learning, appear to be highly complex, because of their extended set of interactions.

Nicola Bulso, Matteo Marsili, Yasser Roudi

We propose a method for recovering the structure of a sparse undirected graphical model when very few samples are available. The method decides about the presence or absence of bonds between pairs of variable by considering one pair at a time and using a closed form formula, analytically derived by calculating the posterior probability for every possible model explaining a two body system using Jeffreys prior. The approach does not rely on the optimisation of any cost functions and consequently is much faster than existing algorithms. Despite this time and computational advantage, numerical results show that for several sparse topologies the algorithm is comparable to the best existing algorithms, and is more accurate in the presence of hidden variables. We apply this approach to the analysis of US stock market data and to neural data, in order to show its efficiency in recovering robust statistical dependencies in real data with non stationary correlations in time and space.