Disentangling feature and lazy training in deep neural networks

Two distinct limits for deep learning have been derived as the network width h -> infinity, depending on how the weights of the last layer scale with h. In the neural tangent Kernel (NTK) limit, the dynamics becomes linear in the weights and is described by a frozen kernel Theta (the NTK). By contrast, in the mean-field limit, the dynamics can be expressed in terms of the distribution of the parameters associated with a neuron, that follows a partial differential equation. In this work we consider deep networks where the weights in the last layer scale as alpha h(-1/2) at initialization. By varying alpha and h, we probe the crossover between the two limits. We observe two the previously identified regimes of 'lazy training' and 'feature training'. In the lazy-training regime, the dynamics is almost linear and the NTK barely changes after initialization. The feature-training regime includes the mean-field formulation as a limiting case and is characterized by a kernel that evolves in time, and thus learns some features. We perform numerical experiments on MNIST, Fashion-MNIST, EMNIST and CIFAR10 and consider various architectures. We find that: (i) the two regimes are separated by an alpha* that scales as 1/root h. (ii) Network architecture and data structure play an important role in determining which regime is better: in our tests, fully-connected networks perform generally better in the lazy-training regime, unlike convolutional networks. (iii) In both regimes, the fluctuations delta F induced on the learned function by initial conditions decay as delta F similar to 1/root h, leading to a performance that increases with h. The same improvement can also be obtained at an intermediate width by ensemble-averaging several networks that are trained independently. (iv) In the feature-training regime we identify a time scale t(1) similar to root h alpha, such that for t < t(1) the dynamics is linear. At t similar to t(1), the output has grown by a magnitude root h and the changes of the tangent kernel parallel to Delta Theta parallel to become significant. Ultimately, it follows parallel to Delta Theta parallel to similar to(root h alpha)(-a) for ReLU and Softplus activation functions, with a < 2 and a -> 2 as depth grows. We provide scaling arguments supporting these findings.

On Vacuous and Non-Vacuous Generalization Bounds for Deep Neural Networks.

Konstantinos Pitas

Deep neural networks have been empirically successful in a variety of tasks, however their theoretical understanding is still poor. In particular, modern deep neural networks have many more parameters than training data. Thus, in principle they should overfit the training samples and exhibit poor generalization to the complete data distribution. Counter intuitively however, they manage to achieve both high training accuracy and high testing accuracy. One can prove generalization using a validation set, however this can be difficult when training samples are limited and at the same time we do not obtain any information about why deep neural networks generalize well. Another approach is to estimate the complexity of the deep neural network. The hypothesis is that if a network with high training accuracy has high complexity it will have memorized the data, while if it has low complexity it will have learned generalizable patterns. In the first part of this thesis we explore Spectral Complexity, a measure of complexity that depends on combinations of norms of the weight matrices of the deep neural network. For a dataset that is difficult to classify, with no underlying model and/or no recurring pattern, for example one where the labels have been chosen randomly, spectral complexity has a large value, reflecting that the network needs to memorize the labels, and will not generalize well. Putting back the real labels, the spectral complexity becomes lower reflecting that some structure is present and the network has learned patterns that might generalize to unseen data. Spectral complexity results in vacuous estimates of the generalization error (the difference between the training and testing accuracy), and we show that it can lead to counterintuitive results when comparing the generalization error of different architectures. In the second part of the thesis we explore non-vacuous estimates of the generalization error. In Chapter 2 we analyze the case of PAC-Bayes where a posterior distribution over the weights of a deep neural network is learned using stochastic variational inference, and the generalization error is the KL divergence between this posterior and a prior distribution. We find that a common approximation where the posterior is constrained to be Gaussian with diagonal covariance, known as the mean-field approximation, limits significantly any gains in bound tightness. We find that, if we choose the prior mean to be the random network initialization, the generalization error estimate tightens significantly. In Chapter 3 we explore an existing approach to learning the prior mean, in PAC-Bayes, from the training set. Specifically, we explore differential privacy, which ensures that the training samples contribute only a limited amount of information to the prior, making it distribution and not training set dependent. In this way the prior should generalize well to unseen data (as it hasn't memorized individual samples) and at the same time any posterior distribution that is close to it in terms of the KL divergence will also exhibit good generalization.

EPFL2020

Slow dynamics in structured neural network models

Samuel Pavio Muscinelli

Humans and some other animals are able to perform tasks that require coordination of movements across multiple temporal scales, ranging from hundreds of milliseconds to several seconds. The fast timescale at which neurons naturally operate, on the order of tens of milliseconds, is well-suited to support motor control of rapid movements. In contrast, to coordinate movements on the order of seconds, a neural network should produce reliable dynamics on a similarly âslowâ timescale. Neurons and synapses exhibit biophysical mechanisms whose timescales range from tens of milliseconds to hours, which suggests a possible role of these mechanisms in producing slow reliable dynamics. However, how such mechanisms influence network dynamics is not yet understood. An alternative approach to achieve slow dynamics in a neural network consists in modifying its connectivity structure. Still, the limitations of this approach and in particular to what degree the weights require fine-tuning, remain unclear. Understanding how both the single neuron mechanisms and the connectivity structure might influence the network dynamics to produce slow timescales is the main goal of this thesis. We first consider the possibility of obtaining slow dynamics in binary networks by tuning their connectivity. It is known that binary networks can produce sequential dynamics. However, if the sequences consist of random patterns, the typical length of the longest sequence that can be produced grows linearly with the number of units. Here, we show that we can overcome this limitation by carefully designing the sequence structure. More precisely, we obtain a constructive proof that allows to obtain sequences whose length scales exponentially with the number of units. To achieve this however, one needs to exponentially fine-tune the connectivity matrix. Next, we focus on the interaction between single neuron mechanisms and recurrent dynamics. Particular attention is dedicated to adaptation, which is known to have a broad range of timescales and is therefore particularly interesting for the subject of this thesis. We study the dynamics of a random network with adaptation using mean-field techniques, and we show that the network can enter a state of resonant chaos. Interestingly, the resonance frequency of this state is independent of the connectivity strength and depends only on the properties of the single neuron model. The approach used to study networks with adaptation can also be applied when considering linear rate units with an arbitrary number of auxiliary variables. Based on a qualitative analysis of the mean-field theory for a random network whose neurons are described by a D -dimensional rate model, we conclude that the statistics of the chaotic dynamics are strongly influenced by the single neuron model under investigation. Using a reservoir computing approach, we show preliminary evidence that slow adaptation can be beneficial when performing tasks that require slow timescales. The positive impact of adaptation on the network performance is particularly strong in the presence of noise. Finally, we propose a network architecture in which the slowing-down effect due to adaptation is combined with a hierarchical structure, with the purpose of efficiently generate sequences that require multiple, hierarchically organized timescales.

EPFL2018

Visual Analysis of Maya Glyphs via Crowdsourcing and Deep Learning

Gülcan Can

In this dissertation, we study visual analysis methods for complex ancient Maya writings. The unit sign of a Maya text is called glyph, and may have either semantic or syllabic significance. There are over 800 identified glyph categories, and over 1400 variations across these categories. To enable fast manipulation of data by scholars in Humanities, it is desirable to have automatic visual analysis tools such as glyph categorization, localization, and visualization. Analysis and recognition of glyphs are challenging problems. The same patterns may be observed in different signs but with different compositions. The inter-class variance can thus be significantly low. On the opposite, the intra-class variance can be high, as the visual variants within the same semantic category may differ to a large extent except for some patterns specific to the category. Another related challenge of Maya writings is the lack of a large dataset to study the glyph patterns. Consequently, we study local shape representations, both knowledge-driven and data-driven, over a set of frequent syllabic glyphs as well as other binary shapes, i.e. sketches. This comparative study indicates that a large data corpus and a deep network architecture are needed to learn data-driven representations that can capture the complex compositions of local patterns. To build a large glyph dataset in a short period of time, we study a crowdsourcing approach as an alternative to time-consuming data preparation of experts. Specifically, we work on individual glyph segmentation out of glyph-blocks from the three remaining codices (i.e. folded bark pages painted with a brush). With gradual steps in our crowdsourcing approach, we observe that providing supervision and careful task design are key aspects for non-experts to generate high-quality annotations. This way, we obtain a large dataset (over 9000) of individual Maya glyphs. We analyze this crowdsourced glyph dataset with both knowledge-driven and data-driven visual representations. First, we evaluate two competitive knowledge-driven representations, namely Histogram of Oriented Shape Context and Histogram of Oriented Gradients. Secondly, thanks to the large size of the crowdsourced dataset, we study visual representation learning with deep Convolutional Neural Networks. We adopt three data-driven approaches: assess- ing representations from pretrained networks, fine-tuning the last convolutional block of a pretrained network, and training a network from scratch. Finally, we investigate different glyph visualization tasks based on the studied representations. First, we explore the visual structure of several glyph corpora by applying a non-linear dimensionality reduction method, namely t-distributed Stochastic Neighborhood Embedding, Secondly, we propose a way to inspect the discriminative parts of individual glyphs according to the trained deep networks. For this purpose, we use the Gradient-weighted Class Activation Mapping method and highlight the network activations as a heatmap visualization over an input image. We assess whether the highlighted parts correspond to distinguishing parts of glyphs in a perceptual crowdsourcing study. Overall, this thesis presents a promising crowdsourcing approach, competitive data-driven visual representations, and interpretable visualization methods that can be applied to explore various other Digital Humanities datasets.

EPFL2017