arXiv subjects

Thomas Hofmann

Publications and source records attributed to Thomas Hofmann.

At least 109 records · Page 6Linked to original sources

arXiv2021

Precise characterization of the prior predictive distribution of deep ReLU networks

Recent works on Bayesian neural networks (BNNs) have highlighted the need to better understand the implications of using Gaussian priors in combination with the compositional structure of the network architecture. Similar in spirit to the kind of analysis that has been developed to devise better initialization schemes for neural networks (cf. He- or Xavier initialization), we derive a precise characterization of the prior predictive distribution of finite-width ReLU networks with Gaussian weights. While theoretical results have been obtained for their heavy-tailedness, the full characterization of the prior predictive distribution (i.e. its density, CDF and moments), remained unknown prior to this work. Our analysis, based on the Meijer-G function, allows us to quantify the influence of architectural choices such as the width or depth of the network on the resulting shape of the prior predictive distribution. We also formally connect our results to previous work in the infinite width setting, demonstrating that the moments of the distribution converge to those of a normal log-normal mixture in the infinite depth limit. Finally, our results provide valuable guidance on prior design: for instance, controlling the predictive variance with depth- and width-informed priors on the weights of the network.

cs.LG↗

arXiv2021

Uniform Convergence, Adversarial Spheres and a Simple Remedy

Previous work has cast doubt on the general framework of uniform convergence and its ability to explain generalization in neural networks. By considering a specific dataset, it was observed that a neural network completely misclassifies a projection of the training data (adversarial set), rendering any existing generalization bound based on uniform convergence vacuous. We provide an extensive theoretical investigation of the previously studied data setting through the lens of infinitely-wide models. We prove that the Neural Tangent Kernel (NTK) also suffers from the same phenomenon and we uncover its origin. We highlight the important role of the output bias and show theoretically as well as empirically how a sensible choice completely mitigates the problem. We identify sharp phase transitions in the accuracy on the adversarial set and study its dependency on the training sample size. As a result, we are able to characterize critical sample sizes beyond which the effect disappears. Moreover, we study decompositions of a neural network into a clean and noisy part by considering its canonical decomposition into its different eigenfunctions and show empirically that for too small bias the adversarial phenomenon still persists.

cs.LG↗

arXiv2021

Learning Generative Models of Textured 3D Meshes from Real-World Images

Recent advances in differentiable rendering have sparked an interest in learning generative models of textured 3D meshes from image collections. These models natively disentangle pose and appearance, enable downstream applications in computer graphics, and improve the ability of generative models to understand the concept of image formation. Although there has been prior work on learning such models from collections of 2D images, these approaches require a delicate pose estimation step that exploits annotated keypoints, thereby restricting their applicability to a few specific datasets. In this work, we propose a GAN framework for generating textured triangle meshes without relying on such annotations. We show that the performance of our approach is on par with prior work that relies on ground-truth keypoints, and more importantly, we demonstrate the generality of our method by setting new baselines on a larger set of categories from ImageNet - for which keypoints are not available - without any class-specific hyperparameter tuning. We release our code at https://github.com/dariopavllo/textured-3d-gan

cs.CV↗

arXiv2021

How to Query Language Models?

Large pre-trained language models (LMs) are capable of not only recovering linguistic but also factual and commonsense knowledge. To access the knowledge stored in mask-based LMs, we can use cloze-style questions and let the model fill in the blank. The flexibility advantage over structured knowledge bases comes with the drawback of finding the right query for a certain information need. Inspired by human behavior to disambiguate a question, we propose to query LMs by example. To clarify the ambivalent question "Who does Neuer play for?", a successful strategy is to demonstrate the relation using another subject, e.g., "Ronaldo plays for Portugal. Who does Neuer play for?". We apply this approach of querying by example to the LAMA probe and obtain substantial improvements of up to 37.8% for BERT-large on the T-REx data when providing only 10 demonstrations--even outperforming a baseline that queries the model with up to 40 paraphrases of the question. The examples are provided through the model's context and thus require neither fine-tuning nor an additional forward pass. This suggests that LMs contain more factual and commonsense knowledge than previously assumed--if we query the model in the right way.

cs.CL↗

arXiv2021

Analytic Insights into Structure and Rank of Neural Network Hessian Maps

The Hessian of a neural network captures parameter interactions through second-order derivatives of the loss. It is a fundamental object of study, closely tied to various problems in deep learning, including model design, optimization, and generalization. Most prior work has been empirical, typically focusing on low-rank approximations and heuristics that are blind to the network structure. In contrast, we develop theoretical tools to analyze the range of the Hessian map, providing us with a precise understanding of its rank deficiency as well as the structural reasons behind it. This yields exact formulas and tight upper bounds for the Hessian rank of deep linear networks, allowing for an elegant interpretation in terms of rank deficiency. Moreover, we demonstrate that our bounds remain faithful as an estimate of the numerical Hessian rank, for a larger class of models such as rectified and hyperbolic tangent networks. Further, we also investigate the implications of model architecture (e.g.~width, depth, bias) on the rank deficiency. Overall, our work provides novel insights into the source and extent of redundancy in overparameterized networks.

cs.LG↗

arXiv2021

Vanishing Curvature and the Power of Adaptive Methods in Randomly Initialized Deep Networks

This paper revisits the so-called vanishing gradient phenomenon, which commonly occurs in deep randomly initialized neural networks. Leveraging an in-depth analysis of neural chains, we first show that vanishing gradients cannot be circumvented when the network width scales with less than O(depth), even when initialized with the popular Xavier and He initializations. Second, we extend the analysis to second-order derivatives and show that random i.i.d. initialization also gives rise to Hessian matrices with eigenspectra that vanish as networks grow in depth. Whenever this happens, optimizers are initialized in a very flat, saddle point-like plateau, which is particularly hard to escape with stochastic gradient descent (SGD) as its escaping time is inversely related to curvature. We believe that this observation is crucial for fully understanding (a) historical difficulties of training deep nets with vanilla SGD, (b) the success of adaptive gradient methods (which naturally adapt to curvature and thus quickly escape flat plateaus) and (c) the effectiveness of modern architectural components like residual connections and normalization layers.

cs.LG↗

arXiv2021

Generative Minimization Networks: Training GANs Without Competition

Many applications in machine learning can be framed as minimization problems and solved efficiently using gradient-based techniques. However, recent applications of generative models, particularly GANs, have triggered interest in solving min-max games for which standard optimization techniques are often not suitable. Among known problems experienced by practitioners is the lack of convergence guarantees or convergence to a non-optimum cycle. At the heart of these problems is the min-max structure of the GAN objective which creates non-trivial dependencies between the players. We propose to address this problem by optimizing a different objective that circumvents the min-max structure using the notion of duality gap from game theory. We provide novel convergence guarantees on this objective and demonstrate why the obtained limit point solves the problem better than known techniques.

cs.LG↗

arXiv2021

SwissDial: Parallel Multidialectal Corpus of Spoken Swiss German

Swiss German is a dialect continuum whose natively acquired dialects significantly differ from the formal variety of the language. These dialects are mostly used for verbal communication and do not have standard orthography. This has led to a lack of annotated datasets, rendering the use of many NLP methods infeasible. In this paper, we introduce the first annotated parallel corpus of spoken Swiss German across 8 major dialects, plus a Standard German reference. Our goal has been to create and to make available a basic dataset for employing data-driven NLP applications in Swiss German. We present our data collection procedure in detail and validate the quality of our corpus by conducting experiments with the recent neural models for speech synthesis.

cs.CL↗

arXiv2021

Revisiting the Role of Euler Numerical Integration on Acceleration and Stability in Convex Optimization

Viewing optimization methods as numerical integrators for ordinary differential equations (ODEs) provides a thought-provoking modern framework for studying accelerated first-order optimizers. In this literature, acceleration is often supposed to be linked to the quality of the integrator (accuracy, energy preservation, symplecticity). In this work, we propose a novel ordinary differential equation that questions this connection: both the explicit and the semi-implicit (a.k.a symplectic) Euler discretizations on this ODE lead to an accelerated algorithm for convex programming. Although semi-implicit methods are well-known in numerical analysis to enjoy many desirable features for the integration of physical systems, our findings show that these properties do not necessarily relate to acceleration.

math.OC↗

arXiv2020

Adversarial Training is a Form of Data-dependent Operator Norm Regularization

We establish a theoretical link between adversarial training and operator norm regularization for deep neural networks. Specifically, we prove that $\ell_p$-norm constrained projected gradient ascent based adversarial training with an $\ell_q$-norm loss on the logits of clean and perturbed inputs is equivalent to data-dependent (p, q) operator norm regularization. This fundamental connection confirms the long-standing argument that a network's sensitivity to adversarial examples is tied to its spectral properties and hints at novel ways to robustify and defend against adversarial attacks. We provide extensive empirical evidence on state-of-the-art network architectures to support our theoretical results.

cs.LG↗

arXiv2020

Convolutional Generation of Textured 3D Meshes

While recent generative models for 2D images achieve impressive visual results, they clearly lack the ability to perform 3D reasoning. This heavily restricts the degree of control over generated objects as well as the possible applications of such models. In this work, we bridge this gap by leveraging recent advances in differentiable rendering. We design a framework that can generate triangle meshes and associated high-resolution texture maps, using only 2D supervision from single-view natural images. A key contribution of our work is the encoding of the mesh and texture as 2D representations, which are semantically aligned and can be easily modeled by a 2D convolutional GAN. We demonstrate the efficacy of our method on Pascal3D+ Cars and CUB, both in an unconditional setting and in settings where the model is conditioned on class labels, attributes, and text. Finally, we propose an evaluation methodology that assesses the mesh and texture quality separately.

cs.CV↗

arXiv2020

Controlling Style and Semantics in Weakly-Supervised Image Generation

We propose a weakly-supervised approach for conditional image generation of complex scenes where a user has fine control over objects appearing in the scene. We exploit sparse semantic maps to control object shapes and classes, as well as textual descriptions or attributes to control both local and global style. In order to condition our model on textual descriptions, we introduce a semantic attention module whose computational cost is independent of the image resolution. To further augment the controllability of the scene, we propose a two-step generation scheme that decomposes background and foreground. The label maps used to train our model are produced by a large-vocabulary object detector, which enables access to unlabeled data and provides structured instance information. In such a setting, we report better FID scores compared to fully-supervised settings where the model is trained on ground-truth semantic maps. We also showcase the ability of our model to manipulate a scene on complex datasets such as COCO and Visual Genome.

cs.CV↗

arXiv2020

A domain agnostic measure for monitoring and evaluating GANs

Generative Adversarial Networks (GANs) have shown remarkable results in modeling complex distributions, but their evaluation remains an unsettled issue. Evaluations are essential for: (i) relative assessment of different models and (ii) monitoring the progress of a single model throughout training. The latter cannot be determined by simply inspecting the generator and discriminator loss curves as they behave non-intuitively. We leverage the notion of duality gap from game theory to propose a measure that addresses both (i) and (ii) at a low computational cost. Extensive experiments show the effectiveness of this measure to rank different GAN models and capture the typical GAN failure scenarios, including mode collapse and non-convergent behaviours. This evaluation metric also provides meaningful monitoring on the progression of the loss during training. It highly correlates with FID on natural image datasets, and with domain specific scores for text, sound and cosmology data where FID is not directly suitable. In particular, our proposed metric requires no labels or a pretrained classifier, making it domain agnostic.

cs.LG↗

arXiv2020

Batch Normalization Provably Avoids Rank Collapse for Randomly Initialised Deep Networks

Randomly initialized neural networks are known to become harder to train with increasing depth, unless architectural enhancements like residual connections and batch normalization are used. We here investigate this phenomenon by revisiting the connection between random initialization in deep networks and spectral instabilities in products of random matrices. Given the rich literature on random matrices, it is not surprising to find that the rank of the intermediate representations in unnormalized networks collapses quickly with depth. In this work we highlight the fact that batch normalization is an effective strategy to avoid rank collapse for both linear and ReLU networks. Leveraging tools from Markov chain theory, we derive a meaningful lower rank bound in deep linear networks. Empirically, we also demonstrate that this rank robustness generalizes to ReLU nets. Finally, we conduct an extensive set of experiments on real-world data sets, which confirm that rank stability is indeed a crucial condition for training modern-day deep neural architectures.

stat.ML↗

arXiv2020

BERT as a Teacher: Contextual Embeddings for Sequence-Level Reward

Measuring the quality of a generated sequence against a set of references is a central problem in many learning frameworks, be it to compute a score, to assign a reward, or to perform discrimination. Despite great advances in model architectures, metrics that scale independently of the number of references are still based on n-gram estimates. We show that the underlying operations, counting words and comparing counts, can be lifted to embedding words and comparing embeddings. An in-depth analysis of BERT embeddings shows empirically that contextual embeddings can be employed to capture the required dependencies while maintaining the necessary scalability through appropriate pruning and smoothing techniques. We cast unconditional generation as a reinforcement learning problem and show that our reward function indeed provides a more effective learning signal than n-gram reward in this challenging setting.

cs.LG↗

arXiv2019

Cosmological N-body simulations: a challenge for scalable generative models

Deep generative models, such as Generative Adversarial Networks (GANs) or Variational Autoencoders (VAs) have been demonstrated to produce images of high visual quality. However, the existing hardware severely limits the size of the images that can be generated. The rapid growth of high dimensional data in many fields of science therefore poses a significant challenge for generative models. In cosmology, the large-scale, three-dimensional matter distribution, modeled with N-body simulations, plays a crucial role in understanding the evolution of the universe. As these simulations are computationally very expensive, GANs have recently generated interest as a possible method to emulate these datasets, but they have been, so far, mostly limited to two dimensional data. In this work, we introduce a new benchmark for the generation of three dimensional N-body simulations, in order to stimulate new ideas in the machine learning community and move closer to the practical use of generative models in cosmology. As a first benchmark result, we propose a scalable GAN approach for training a generator of N-body three-dimensional cubes. Our technique relies on two key building blocks, (i) splitting the generation of the high-dimensional data into smaller parts, and (ii) using a multi-scale approach that efficiently captures global image features that might otherwise be lost in the splitting process. We evaluate the performance of our model for the generation of N-body samples using various statistical measures commonly used in cosmology. Our results show that the proposed model produces samples of high visual quality, although the statistical analysis reveals that capturing rare features in the data poses significant problems for the generative models. We make the data, quality evaluation routines, and the proposed GAN architecture publicly available at https://github.com/nperraud/3DcosmoGAN

physics.comp-ph↗

arXiv2019

Mixing of Stochastic Accelerated Gradient Descent

We study the mixing properties for stochastic accelerated gradient descent (SAGD) on least-squares regression. First, we show that stochastic gradient descent (SGD) and SAGD are simulating the same invariant distribution. Motivated by this, we then establish mixing rate for SAGD-iterates and compare it with those of SGD-iterates. Theoretically, we prove that the chain of SAGD iterates is geometrically ergodic --using a proper choice of parameters and under regularity assumptions on the input distribution. More specifically, we derive an explicit mixing rate depending on the first 4 moments of the data distribution. By means of illustrative examples, we prove that SAGD-iterate chain mixes faster than the chain of iterates obtained by SGD. Furthermore, we highlight applications of the established mixing rate in the convergence analysis of SAGD on realizable objectives. The proposed analysis is based on a novel non-asymptotic analysis of products of random matrices. This theoretical result is substantiated and validated by experiments.

math.OC↗

arXiv2019

Cosmological constraints with deep learning from KiDS-450 weak lensing maps

Convolutional Neural Networks (CNN) have recently been demonstrated on synthetic data to improve upon the precision of cosmological inference. In particular they have the potential to yield more precise cosmological constraints from weak lensing mass maps than the two-point functions. We present the cosmological results with a CNN from the KiDS-450 tomographic weak lensing dataset, constraining the total matter density $Ω_m$, the fluctuation amplitude $σ_8$, and the intrinsic alignment amplitude $A_{\rm{IA}}$. We use a grid of N-body simulations to generate a training set of tomographic weak lensing maps. We test the robustness of the expected constraints to various effects, such as baryonic feedback, simulation accuracy, different value of $H_0$, or the lightcone projection technique. We train a set of ResNet-based CNNs with varying depths to analyze sets of tomographic KiDS mass maps divided into 20 flat regions, with applied Gaussian smoothing of $σ=2.34$ arcmin. The uncertainties on shear calibration and $n(z)$ error are marginalized in the likelihood pipeline. Following a blinding scheme, we derive constraints of $S_8 = σ_8 (Ω_m/0.3)^{0.5} = 0.777^{+0.038}_{-0.036}$ with our CNN analysis, with $A_{\rm{IA}}=1.398^{+0.779}_{-0.724}$. We compare this result to the power spectrum analysis on the same maps and likelihood pipeline and find an improvement of about $30\%$ for the CNN. We discuss how our results offer excellent prospects for the use of deep learning in future cosmological data analysis.

astro-ph.CO↗