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The study of Deep Network (DN) training dynamics has largely focused on the evolution of the loss function, evaluated on or around train and test set data points. In fact, many DN phenomenon were first introduced in literature with that respect, e.g., double descent, grokking. In this study, we look at the training dynamics of the input space partition or linear regions formed by continuous piecewise affine DNs, e.g., networks with (leaky)ReLU nonlinearities. First, we present a novel statistic that encompasses the local complexity (LC) of the DN based on the concentration of linear regions inside arbitrary dimensional neighborhoods around data points. We observe that during training, the LC around data points undergoes a number of phases, starting with a decreasing trend after initialization, followed by an ascent and ending with a final descending trend. Using exact visualization methods, we come across the perplexing observation that during the final LC descent phase of training, linear regions migrate away from training and test samples towards the decision boundary, making the DN input-output nearly linear everywhere else. We also observe that the different LC phases are closely related to the memorization and generalization performance of the DN, especially during grokking.

Byzantine agreement tasks have been studied extensively through many diverse frameworks ranging from epistemic modal logic to combinatorial, topological and even game theoretical approaches. Among byzantine agreement tasks, firing rebels with relay is of particular interest, since it is a basic primitive necessary for solving more complex tasks such as Consistent Broadcast. The epistemic logic approach has yielded both necessary conditions and sufficient knowledge conditions for solving firing rebels with relay. However, these conditions are stated in terms of knowledge and in principle do not explore the conditions on the communication structure which is often assumed to be complete. That is, any process is assumed to be capable of communicating with any other process at any time. In this paper, we characterize byzantine firing rebels solvability with and without relay in terms of the communication structure of the system. We define a relation between asynchronous message schedules and directed graph sequences, which we call network abstractions. This allows us to capture the message relay capabilities of the system into a combinatorial object. Although there are some similarities between network abstractions and causal cones, there is a fundamental difference. Namely, causal cones allow only to look at events in the past, while network abstractions allow us to reason about future possibilities. Thus enabling us to reason about liveness properties, which can not be expressed by looking only at past events. Furthermore, we formalize our characterization by using a temporal epistemic logic for byzantine systems. Such formulation constitutes the necessary and sufficient a-priori knowledge regarding network connectivity for solving firing rebels with relay.

Recent studies have shown that sequence-to-sequence (seq2seq) models struggle with compositional generalization (CG), i.e., the ability to systematically generalize to unseen compositions of seen components. There is mounting evidence that one of the reasons hindering CG is the representation of the encoder uppermost layer is entangled, i.e., the syntactic and semantic representations of sequences are entangled. However, we consider that the previously identified representation entanglement problem is not comprehensive enough. Additionally, we hypothesize that the source keys and values representations passing into different decoder layers are also entangled. Starting from this intuition, we propose \textsc{CompoSition} (\textbf{Compo}se \textbf{S}yntactic and Semant\textbf{i}c Representa\textbf{tion}s), an extension to seq2seq models which learns to compose representations of different encoder layers dynamically for different tasks, since recent studies reveal that the bottom layers of the Transformer encoder contain more syntactic information and the top ones contain more semantic information. Specifically, we introduce a \textit{composed layer} between the encoder and decoder to compose different encoder layers' representations to generate specific keys and values passing into different decoder layers. \textsc{CompoSition} achieves competitive results on two comprehensive and realistic benchmarks, which empirically demonstrates the effectiveness of our proposal. Codes are available at~\url{//github.com/thinkaboutzero/COMPOSITION}.

Quantification represents the problem of predicting class distributions in a dataset. It also represents a growing research field in supervised machine learning, for which a large variety of different algorithms has been proposed in recent years. However, a comprehensive empirical comparison of quantification methods that supports algorithm selection is not available yet. In this work, we close this research gap by conducting a thorough empirical performance comparison of 24 different quantification methods on overall more than 40 data sets, considering binary as well as multiclass quantification settings. We observe that no single algorithm generally outperforms all competitors, but identify a group of methods including the threshold selection-based Median Sweep and TSMax methods, the DyS framework, and Friedman's method that performs best in the binary setting. For the multiclass setting, we observe that a different group of algorithms yields good performance, including the Generalized Probabilistic Adjusted Count, the readme method, the energy distance minimization method, the EM algorithm for quantification, and Friedman's method. We also find that tuning the underlying classifiers has in most cases only a limited impact on the quantification performance. More generally, we find that the performance on multiclass quantification is inferior to the results obtained in the binary setting. Our results can guide practitioners who intend to apply quantification algorithms and help researchers to identify opportunities for future research.

Tempered stable distributions are frequently used in financial applications (e.g., for option pricing) in which the tails of stable distributions would be too heavy. Given the non-explicit form of the probability density function, estimation relies on numerical algorithms which typically are time-consuming. We compare several parametric estimation methods such as the maximum likelihood method and different generalized method of moment approaches. We study large sample properties and derive consistency, asymptotic normality, and asymptotic efficiency results for our estimators. Additionally, we conduct simulation studies to analyze finite sample properties measured by the empirical bias, precision, and asymptotic confidence interval coverage rates and compare computational costs. We cover relevant subclasses of tempered stable distributions such as the classical tempered stable distribution and the tempered stable subordinator. Moreover, we discuss the normal tempered stable distribution which arises by subordinating a Brownian motion with a tempered stable subordinator. Our financial applications to log returns of asset indices and to energy spot prices illustrate the benefits of tempered stable models.

Natural gradient methods have been used to optimise the parameters of probability distributions in a variety of settings, often resulting in fast-converging procedures. Unfortunately, for many distributions of interest, computing the natural gradient has a number of challenges. In this work we propose a novel technique for tackling such issues, which involves reframing the optimisation as one with respect to the parameters of a surrogate distribution, for which computing the natural gradient is easy. We give several examples of existing methods that can be interpreted as applying this technique, and propose a new method for applying it to a wide variety of problems. Our method expands the set of distributions that can be efficiently targeted with natural gradients. Furthermore, it is fast, easy to understand, simple to implement using standard autodiff software, and does not require lengthy model-specific derivations. We demonstrate our method on maximum likelihood estimation and variational inference tasks.

Self-supervised learning, dubbed the dark matter of intelligence, is a promising path to advance machine learning. Yet, much like cooking, training SSL methods is a delicate art with a high barrier to entry. While many components are familiar, successfully training a SSL method involves a dizzying set of choices from the pretext tasks to training hyper-parameters. Our goal is to lower the barrier to entry into SSL research by laying the foundations and latest SSL recipes in the style of a cookbook. We hope to empower the curious researcher to navigate the terrain of methods, understand the role of the various knobs, and gain the know-how required to explore how delicious SSL can be.

Mathematical reasoning is a fundamental aspect of human intelligence and is applicable in various fields, including science, engineering, finance, and everyday life. The development of artificial intelligence (AI) systems capable of solving math problems and proving theorems has garnered significant interest in the fields of machine learning and natural language processing. For example, mathematics serves as a testbed for aspects of reasoning that are challenging for powerful deep learning models, driving new algorithmic and modeling advances. On the other hand, recent advances in large-scale neural language models have opened up new benchmarks and opportunities to use deep learning for mathematical reasoning. In this survey paper, we review the key tasks, datasets, and methods at the intersection of mathematical reasoning and deep learning over the past decade. We also evaluate existing benchmarks and methods, and discuss future research directions in this domain.

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.

We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding generalization power of overparametrized neural networks, the role of depth in deep architectures, the apparent absence of the curse of dimensionality, the surprisingly successful optimization performance despite the non-convexity of the problem, understanding what features are learned, why deep architectures perform exceptionally well in physical problems, and which fine aspects of an architecture affect the behavior of a learning task in which way. We present an overview of modern approaches that yield partial answers to these questions. For selected approaches, we describe the main ideas in more detail.