A Formalization of The Natural Gradient Method for General Similarity Measures

In optimization, the natural gradient method is well-known for likelihood maximization. The method uses the Kullback-Leibler divergence, corresponding infinitesimally to the Fisher-Rao metric, which is pulled back to the parameter space of a family of probability distributions. This way, gradients with respect to the parameters respect the Fisher-Rao geometry of the space of distributions, which might differ vastly from the standard Euclidean geometry of the parameter space, often leading to faster convergence. However, when minimizing an arbitrary similarity measure between distributions, it is generally unclear which metric to use. We provide a general framework that, given a similarity measure, derives a metric for the natural gradient. We then discuss connections between the natural gradient method and multiple other optimization techniques in the literature. Finally, we provide computations of the formal natural gradient to show overlap with well-known cases and to compute natural gradients in novel frameworks

A group-theoretic approach to formalizing bootstrapping problems

The bootstrapping problem consists in designing agents that learn a model of themselves and the world, and utilize it to achieve useful tasks. It is different from other learning problems as the agent starts with uninterpreted observations and commands, and with minimal prior information about the world. In this paper, we give a mathematical formalization of this aspect of the problem. We argue that the vague constraint of having "no prior information" can be recast as a precise algebraic condition on the agent: that its behavior is invariant to particular classes of nuisances on the world, which we show can be well represented by actions of groups (diffeomorphisms, permutations, linear transformations) on observations and commands. We then introduce the class of bilinear gradient dynamics sensors (BGDS) as a candidate for learning generic robotic sensorimotor cascades. We show how framing the problem as rejection of group nuisances allows a compact and modular analysis of typical preprocessing stages, such as learning the topology of the sensors. We demonstrate learning and using such models on real-world range-finder and camera data from publicly available datasets

Supervised and unsupervised methods for learning representations of linguistic units

Word representations, also called word embeddings, are generic representations, often high-dimensional vectors. They map the discrete space of words into a continuous vector space, which allows us to handle rare or even unseen events, e.g. by considering the nearest neighbors. Many Natural Language Processing tasks can be improved by word representations if we extend the task specific training data by the general knowledge incorporated in the word representations. The first publication investigates a supervised, graph-based method to create word representations. This method leads to a graph-theoretic similarity measure, CoSimRank, with equivalent formalizations that show CoSimRank’s close relationship to Personalized Page-Rank and SimRank. The new formalization is efficient because it can use the graph-based word representation to compute a single node similarity without having to compute the similarities of the entire graph. We also show how we can take advantage of fast matrix multiplication algorithms. In the second publication, we use existing unsupervised methods for word representation learning and combine these with semantic resources by learning representations for non-word objects like synsets and entities. We also investigate improved word representations which incorporate the semantic information from the resource. The method is flexible in that it can take any word representations as input and does not need an additional training corpus. A sparse tensor formalization guarantees efficiency and parallelizability. In the third publication, we introduce a method that learns an orthogonal transformation of the word representation space that focuses the information relevant for a task in an ultradense subspace of a dimensionality that is smaller by a factor of 100 than the original space. We use ultradense representations for a Lexicon Creation task in which words are annotated with three types of lexical information – sentiment, concreteness and frequency. The final publication introduces a new calculus for the interpretable ultradense subspaces, including polarity, concreteness, frequency and part-of-speech (POS). The calculus supports operations like “−1 × hate = love” and “give me a neutral word for greasy” (i.e., oleaginous) and extends existing analogy computations like “king − man + woman = queen”.Wortrepräsentationen, sogenannte Word Embeddings, sind generische Repräsentationen, meist hochdimensionale Vektoren. Sie bilden den diskreten Raum der Wörter in einen stetigen Vektorraum ab und erlauben uns, seltene oder ungesehene Ereignisse zu behandeln -- zum Beispiel durch die Betrachtung der nächsten Nachbarn. Viele Probleme der Computerlinguistik können durch Wortrepräsentationen gelöst werden, indem wir spezifische Trainingsdaten um die allgemeinen Informationen erweitern, welche in den Wortrepräsentationen enthalten sind. In der ersten Publikation untersuchen wir überwachte, graphenbasierte Methodenn um Wortrepräsentationen zu erzeugen. Diese Methoden führen zu einem graphenbasierten Ähnlichkeitsmaß, CoSimRank, für welches zwei äquivalente Formulierungen existieren, die sowohl die enge Beziehung zum personalisierten PageRank als auch zum SimRank zeigen. Die neue Formulierung kann einzelne Knotenähnlichkeiten effektiv berechnen, da graphenbasierte Wortrepräsentationen benutzt werden können. In der zweiten Publikation verwenden wir existierende Wortrepräsentationen und kombinieren diese mit semantischen Ressourcen, indem wir Repräsentationen für Objekte lernen, welche keine Wörter sind, wie zum Beispiel Synsets und Entitäten. Die Flexibilität unserer Methode zeichnet sich dadurch aus, dass wir beliebige Wortrepräsentationen als Eingabe verwenden können und keinen zusätzlichen Trainingskorpus benötigen. In der dritten Publikation stellen wir eine Methode vor, die eine Orthogonaltransformation des Vektorraums der Wortrepräsentationen lernt. Diese Transformation fokussiert relevante Informationen in einen ultra-kompakten Untervektorraum. Wir benutzen die ultra-kompakten Repräsentationen zur Erstellung von Wörterbüchern mit drei verschiedene Angaben -- Stimmung, Konkretheit und Häufigkeit. Die letzte Publikation präsentiert eine neue Rechenmethode für die interpretierbaren ultra-kompakten Untervektorräume -- Stimmung, Konkretheit, Häufigkeit und Wortart. Diese Rechenmethode beinhaltet Operationen wie ”−1 × Hass = Liebe” und ”neutrales Wort für Winkeladvokat” (d.h., Anwalt) und erweitert existierende Rechenmethoden, wie ”Onkel − Mann + Frau = Tante”

On the adequacy of current empirical evaluations of formal models of categorization

Categorization is one of the fundamental building blocks of cognition, and the study of categorization is notable for the extent to which formal modeling has been a central and influential component of research. However, the field has seen a proliferation of noncomplementary models with little consensus on the relative adequacy of these accounts. Progress in assessing the relative adequacy of formal categorization models has, to date, been limited because (a) formal model comparisons are narrow in the number of models and phenomena considered and (b) models do not often clearly define their explanatory scope. Progress is further hampered by the practice of fitting models with arbitrarily variable parameters to each data set independently. Reviewing examples of good practice in the literature, we conclude that model comparisons are most fruitful when relative adequacy is assessed by comparing well-defined models on the basis of the number and proportion of irreversible, ordinal, penetrable successes (principles of minimal flexibility, breadth, good-enough precision, maximal simplicity, and psychological focus)