We provide a function space characterization of the inductive bias resulting from minimizing the $\ell_2$ norm of the weights in multi-channel convolutional neural networks with linear activations and empirically test our resulting hypothesis on ReLU networks trained using gradient descent. We define an induced regularizer in the function space as the minimum $\ell_2$ norm of weights of a network required to realize a function. For two layer linear convolutional networks with $C$ output channels and kernel size $K$, we show the following: (a) If the inputs to the network are single channeled, the induced regularizer for any $K$ is independent of the number of output channels $C$. Furthermore, we derive the regularizer is a norm given by a semidefinite program (SDP). (b) In contrast, for multi-channel inputs, multiple output channels can be necessary to merely realize all matrix-valued linear functions and thus the inductive bias does depend on $C$. However, for sufficiently large $C$, the induced regularizer is again given by an SDP that is independent of $C$. In particular, the induced regularizer for $K=1$ and $K=D$ (input dimension) is given in closed form as the nuclear norm and the $\ell_{2,1}$ group-sparse norm, respectively, of the Fourier coefficients of the linear predictor. We investigate the broader applicability of our theoretical results to implicit regularization from gradient descent on linear and ReLU networks through experiments on MNIST and CIFAR-10 datasets.
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
We analyze the inductive bias of gradient descent for weight normalized smooth homogeneous neural nets, when trained on exponential or cross-entropy loss. We analyse both standard weight normalization (SWN) and exponential weight normalization (EWN), and show that the gradient flow path with EWN is equivalent to gradient flow on standard networks with an adaptive learning rate. We extend these results to gradient descent, and establish asymptotic relations between weights and gradients for both SWN and EWN. We also show that EWN causes weights to be updated in a way that prefers asymptotic rel…
State-of-the-art neural networks are heavily over-parameterized, making the optimization algorithm a crucial ingredient for learning predictive models with good generalization properties. A recent line of work has shown that in a certain over-parameterized regime, the learning dynamics of gradient descent are governed by a certain kernel obtained at initialization, called the neural tangent kernel. We study the inductive bias of learning in such a regime by analyzing this kernel and the corresponding function space (RKHS). In particular, we study smoothness, approximation, and stability proper…
Georgia Institute of Technology（佐治亚理工学院）；Pennsylvania State University（宾夕法尼亚州立大学）
Adversarial training is a principled approach for training robust neural networks. Despite of tremendous successes in practice, its theoretical properties still remain largely unexplored. In this paper, we provide new theoretical insights of gradient descent based adversarial training by studying its computational properties, specifically on its inductive bias. We take the binary classification task on linearly separable data as an illustrative example, where the loss asymptotically attains its infimum as the parameter diverges to infinity along certain directions. Specifically, we show that w…
The empirical success of deep convolutional networks on tasks involving high-dimensional data such as images or audio suggests that they can efficiently approximate certain functions that are well-suited for such tasks. In this paper, we study this through the lens of kernel methods, by considering simple hierarchical kernels with two or three convolution and pooling layers, inspired by convolutional kernel networks. These achieve good empirical performance on standard vision datasets, while providing a precise description of their functional space that yields new insights on their inductive b…
Convolutional neural networks often dominate fully-connected counterparts in generalization performance, especially on image classification tasks. This is often explained in terms of 'better inductive bias'. However, this has not been made mathematically rigorous, and the hurdle is that the fully connected net can always simulate the convolutional net (for a fixed task). Thus the training algorithm plays a role. The current work describes a natural task on which a provable sample complexity gap can be shown, for standard training algorithms. We construct a single natural distribution on $\math…
Allen Institute for AI（艾伦人工智能研究所）；New York University（纽约大学）；University of Washington（华盛顿大学）；Bar Ilan University（巴伊兰大学）；Hebrew University of Jerusalem（希伯来大学）
The capacity of neural networks like the widely adopted transformer is known to be very high. Evidence is emerging that they learn successfully due to inductive bias in the training routine, typically a variant of gradient descent (GD). To better understand this bias, we study the tendency for transformer parameters to grow in magnitude ($\ell_2$ norm) during training, and its implications for the emergent representations within self attention layers. Empirically, we document norm growth in the training of transformer language models, including T5 during its pretraining. As the parameters grow…
Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Bias
Department of Computer Science and Engineering（计算机科学与工程系）；The Hong Kong University of Science and Technology（香港科学与技术大学）；School of Mathematical Sciences（数学科学学院）；Peking University（北京大学）；Department of Computer Science（计算机科学系）；University of Warwick（沃里克大学）；Industrial Engineering & Management Sciences（工业工程与管理科学）；Northwestern University（西北大学）
Ecole Normale Supérieure（巴黎高等师范学院）；Facebook AI Research（脸书人工智能研究院）
Convolutional architectures have proven extremely successful for vision tasks. Their hard inductive biases enable sample-efficient learning, but come at the cost of a potentially lower performance ceiling. Vision Transformers (ViTs) rely on more flexible self-attention layers, and have recently outperformed CNNs for image classification. However, they require costly pre-training on large external datasets or distillation from pre-trained convolutional networks. In this paper, we ask the following question: is it possible to combine the strengths of these two architectures while avoiding their …
Sorbonne universités（索邦大学）；Université de technologie de Compiègne（贡比涅技术大学）；CNRS（法国国家科学研究中心）；Heudiasyc（人机交互与自动化实验室）
In inductive transfer learning, fine-tuning pre-trained convolutional networks substantially outperforms training from scratch. When using fine-tuning, the underlying assumption is that the pre-trained model extracts generic features, which are at least partially relevant for solving the target task, but would be difficult to extract from the limited amount of data available on the target task. However, besides the initialization with the pre-trained model and the early stopping, there is no mechanism in fine-tuning for retaining the features learned on the source task. In this paper, we inves…
Our formal understanding of the inductive bias that drives the success of convolutional networks on computer vision tasks is limited. In particular, it is unclear what makes hypotheses spaces born from convolution and pooling operations so suitable for natural images. In this paper we study the ability of convolutional networks to model correlations among regions of their input. We theoretically analyze convolutional arithmetic circuits, and empirically validate our findings on other types of convolutional networks as well. Correlations are formalized through the notion of separation rank, whi…
We propose a Healthcare Graph Convolutional Network (HealGCN) to offer disease self-diagnosis service for online users based on Electronic Healthcare Records (EHRs). Two main challenges are focused in this paper for online disease diagnosis: (1) serving cold-start users via graph convolutional networks and (2) handling scarce clinical description via a symptom retrieval system. To this end, we first organize the EHR data into a heterogeneous graph that is capable of modeling complex interactions among users, symptoms and diseases, and tailor the graph representation learning towards disease di…
The driving force behind convolutional networks - the most successful deep learning architecture to date, is their expressive power. Despite its wide acceptance and vast empirical evidence, formal analyses supporting this belief are scarce. The primary notions for formally reasoning about expressiveness are efficiency and inductive bias. Expressive efficiency refers to the ability of a network architecture to realize functions that require an alternative architecture to be much larger. Inductive bias refers to the prioritization of some functions over others given prior knowledge regarding a t…
We consider the problem of learning efficient and inductive graph convolutional networks for text classification with a large number of examples and features. Existing state-of-the-art graph embedding based methods such as predictive text embedding (PTE) and TextGCN have shortcomings in terms of predictive performance, scalability and inductive capability. To address these limitations, we propose a heterogeneous graph convolutional network (HeteGCN) modeling approach that unites the best aspects of PTE and TextGCN together. The main idea is to learn feature embeddings and derive document embed…
Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. However, extending GCNs to hyperbolic geometry presents several unique challenges because it is not clear how to define neural network operations, such as feature transformation and aggregation, in hyperbolic space. Furthermore, since input features are often Euclidean, it is unc…
We consider the problem of learning latent features (aka embedding) for users and items in a recommendation setting. Given only a user-item interaction graph, the goal is to recommend items for each user. Traditional approaches employ matrix factorization-based collaborative filtering methods. Recent methods using graph convolutional networks (e.g., LightGCN) achieve state-of-the-art performance. They learn both user and item embedding. One major drawback of most existing methods is that they are not inductive; they do not generalize for users and items unseen during training. Besides, existin…
The translation equivariance of convolutional layers enables convolutional neural networks to generalize well on image problems. While translation equivariance provides a powerful inductive bias for images, we often additionally desire equivariance to other transformations, such as rotations, especially for non-image data. We propose a general method to construct a convolutional layer that is equivariant to transformations from any specified Lie group with a surjective exponential map. Incorporating equivariance to a new group requires implementing only the group exponential and logarithm maps…
Laboratoire de Physique de l’École normale supérieure, ENS, Université PSL（巴黎高等师范学院物理实验室，巴黎高等师范学院，巴黎文理研究大学）；CNRS（法国国家科学研究中心）；Sorbonne Université（索邦大学）；Université de Paris（巴黎大学）
Pruning methods can considerably reduce the size of artificial neural networks without harming their performance. In some cases, they can even uncover sub-networks that, when trained in isolation, match or surpass the test accuracy of their dense counterparts. Here we study the inductive bias that pruning imprints in such "winning lottery tickets". Focusing on visual tasks, we analyze the architecture resulting from iterative magnitude pruning of a simple fully connected network (FCN). We show that the surviving node connectivity is local in input space, and organized in patterns reminiscent o…
ENS（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；EHESS（法国高等社会科学研究院）；CNRS（法国国家科学研究中心）；INRIA（法国国家信息与自动化研究所）；Meta AI Research（Meta人工智能研究院）
Parts of the brain that carry sensory tasks are organized topographically: nearby neurons are responsive to the same properties of input signals. Thus, in this work, inspired by the neuroscience literature, we proposed a new topographic inductive bias in Convolutional Neural Networks (CNNs). To achieve this, we introduced a new topographic loss and an efficient implementation to topographically organize each convolutional layer of any CNN. We benchmarked our new method on 4 datasets and 3 models in vision and audio tasks and showed equivalent performance to all benchmarks. Besides, we also sho…
Rice University（莱斯大学）；Technical University of Munich（慕尼黑工业大学）
Normalization layers are widely used in deep neural networks to stabilize training. In this paper, we consider the training of convolutional neural networks with gradient descent on a single training example. This optimization problem arises in recent approaches for solving inverse problems such as the deep image prior or the deep decoder. We show that for this setup, channel normalization, which centers and normalizes each channel individually, avoids vanishing gradients, whereas, without normalization, gradients vanish which prevents efficient optimization. This effect prevails in deep singl…
ETH Zurich（苏黎世联邦理工学院）；ETH AI Center（苏黎世联邦理工学院人工智能中心）；Max-Planck-Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Good generalization performance on high-dimensional data crucially hinges on a simple structure of the ground truth and a corresponding strong inductive bias of the estimator. Even though this intuition is valid for regularized models, in this paper we caution against a strong inductive bias for interpolation in the presence of noise: While a stronger inductive bias encourages a simpler structure that is more aligned with the ground truth, it also increases the detrimental effect of noise. Specifically, for both linear regression and classification with a sparse ground truth, we prove that min…
Harvard University（哈佛大学）；Microsoft Research NYC（微软研究院纽约）
Self-attention, an architectural motif designed to model long-range interactions in sequential data, has driven numerous recent breakthroughs in natural language processing and beyond. This work provides a theoretical analysis of the inductive biases of self-attention modules. Our focus is to rigorously establish which functions and long-range dependencies self-attention blocks prefer to represent. Our main result shows that bounded-norm Transformer networks "create sparse variables": a single self-attention head can represent a sparse function of the input sequence, with sample complexity sca…
As the deep neural networks are being applied to complex tasks, the size of the networks and architecture increases and their topology becomes more complicated too. At the same time, training becomes slow and at some instances inefficient. This motivated the introduction of various normalization techniques such as Batch Normalization and Layer Normalization. The aforementioned normalization methods use arithmetic operations to compute an approximation statistics (mainly the first and second moments) of the layer's data and use it to normalize it. The aforementioned methods use plain Monte Carl…
University of Stuttgart（斯图加特大学）；IIE, Chinese Academy of Sciences（中国科学院信息工程研究所）；Imperial College London（帝国理工学院）；Griffith University（格里菲斯大学）；AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Southampton（南安普顿大学）
Graph convolutional networks (GCNs) are powerful frameworks for learning embeddings of graph-structured data. GCNs are traditionally studied through the lens of Euclidean geometry. Recent works find that non-Euclidean Riemannian manifolds provide specific inductive biases for embedding hierarchical or spherical data. However, they cannot align well with data of mixed graph topologies. We consider a larger class of pseudo-Riemannian manifolds that generalize hyperboloid and sphere. We develop new geodesic tools that allow for extending neural network operations into geodesically disconnected ps…
Technical University of Munich（慕尼黑工业大学）；Whiting School of Engineering, Johns Hopkins University（约翰斯·霍普金斯大学怀廷工程学院）；Ca’ Foscari University of Venice（威尼斯大学）；University of Lugano（卢加诺大学）；NVIDIA（英伟达）；University of Oxford（牛津大学）；Twitter（推特）；Imperial College London（帝国理工学院）
Graph deep learning has recently emerged as a powerful ML concept allowing to generalize successful deep neural architectures to non-Euclidean structured data. Such methods have shown promising results on a broad spectrum of applications ranging from social science, biomedicine, and particle physics to computer vision, graphics, and chemistry. One of the limitations of the majority of current graph neural network architectures is that they are often restricted to the transductive setting and rely on the assumption that the underlying graph is {\em known} and {\em fixed}. Often, this assumption…
Convolutional neural network is a very important model of deep learning. It can help avoid the exploding/vanishing gradient problem and improve the generalizability of a neural network if the singular values of the Jacobian of a layer are bounded around $1$ in the training process. We propose a new penalty function for a convolutional kernel to let the singular values of the corresponding transformation matrix are bounded around $1$. We show how to carry out the gradient type methods. The penalty is about the structured transformation matrix corresponding to a convolutional kernel. This provid…
Borrowing from the transformer models that revolutionized the field of natural language processing, self-supervised feature learning for visual tasks has also seen state-of-the-art success using these extremely deep, isotropic networks. However, the typical AI researcher does not have the resources to evaluate, let alone train, a model with several billion parameters and quadratic self-attention activations. To facilitate further research, it is necessary to understand the features of these huge transformer models that can be adequately studied by the typical researcher. One interesting charac…
Deep Learning Inductive Biases for fMRI Time Series Classification during Resting-state and Movie-watching
School of Electrical and Computer Engineering, University of Tehran（埃特瓦尔大学电气与计算机工程学院）；School of Cognitive Sciences, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所（IPM）认知科学学院）
University of Rochester（罗切斯特大学）；University of Pennsylvania（宾夕法尼亚大学）
Machine learning frameworks such as graph neural networks typically rely on a given, fixed graph to exploit relational inductive biases and thus effectively learn from network data. However, when said graphs are (partially) unobserved, noisy, or dynamic, the problem of inferring graph structure from data becomes relevant. In this paper, we postulate a graph convolutional relationship between the observed and latent graphs, and formulate the graph learning task as a network inverse (deconvolution) problem. In lieu of eigendecomposition-based spectral methods or iterative optimization solutions,…
Tsinghua University（清华大学）；Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）；CIFAR AI Chair（CIFAR人工智能讲席）
Various forms of regularization in learning tasks strive for different notions of simplicity. This paper presents a spectral regularization technique, which attaches a unique inductive bias to sequence modeling based on an intuitive concept of simplicity defined in the Chomsky hierarchy. From fundamental connections between Hankel matrices and regular grammars, we propose to use the trace norm of the Hankel matrix, the tightest convex relaxation of its rank, as the spectral regularizer. To cope with the fact that the Hankel matrix is bi-infinite, we propose an unbiased stochastic estimator for…
Convolutional neural networks are among the most successful architectures in deep learning with this success at least partially attributable to the efficacy of spatial invariance as an inductive bias. Locally connected layers, which differ from convolutional layers only in their lack of spatial invariance, usually perform poorly in practice. However, these observations still leave open the possibility that some degree of relaxation of spatial invariance may yield a better inductive bias than either convolution or local connectivity. To test this hypothesis, we design a method to relax the spat…
Scaling adaptive traffic-signal control involves dealing with combinatorial state and action spaces. Multi-agent reinforcement learning attempts to address this challenge by distributing control to specialized agents. However, specialization hinders generalization and transferability, and the computational graphs underlying neural-networks architectures -- dominating in the multi-agent setting -- do not offer the flexibility to handle an arbitrary number of entities which changes both between road networks, and over time as vehicles traverse the network. We introduce Inductive Graph Reinforcem…
Characterising the Inductive Biases of Neural Networks on Boolean Data
Department of Theoretical Physics, University of Oxford, UK（理论物理系，牛津大学）；Department of Theoretical Chemistry, University of Oxford, UK（理论化学系，牛津大学）；Green Templeton College, University of Oxford, UK（格林·泰尔顿学院，牛津大学）；Department of Computer Science, University of Oxford, UK（计算机科学系，牛津大学）
Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系）；Institute for Computational & Mathematical Engineering, Stanford University（斯坦福大学计算与数学工程研究所）；DAMO Academy, Alibaba Group（阿里巴巴达摩院）
Graph Convolution Networks (GCNs) are becoming more and more popular for learning node representations on graphs. Though there exist various developments on sampling and aggregation to accelerate the training process and improve the performances, limited works focus on dealing with the dimensional information imbalance of node representations. To bridge the gap, we propose a method named Dimensional reweighting Graph Convolution Network (DrGCN). We theoretically prove that our DrGCN can guarantee to improve the stability of GCNs via mean field theory. Our dimensional reweighting method is very…
University of Toronto（多伦多大学）；Vector Institute（矢量研究所）
Distances are pervasive in machine learning. They serve as similarity measures, loss functions, and learning targets; it is said that a good distance measure solves a task. When defining distances, the triangle inequality has proven to be a useful constraint, both theoretically--to prove convergence and optimality guarantees--and empirically--as an inductive bias. Deep metric learning architectures that respect the triangle inequality rely, almost exclusively, on Euclidean distance in the latent space. Though effective, this fails to model two broad classes of subadditive distances, common in …
Deep convolutional networks have witnessed unprecedented success in various machine learning applications. Formal understanding on what makes these networks so successful is gradually unfolding, but for the most part there are still significant mysteries to unravel. The inductive bias, which reflects prior knowledge embedded in the network architecture, is one of them. In this work, we establish a fundamental connection between the fields of quantum physics and deep learning. We use this connection for asserting novel theoretical observations regarding the role that the number of channels in e…
University of Science and Technology of China（中国科学技术大学）
There still remains an extreme performance gap between Vision Transformers (ViTs) and Convolutional Neural Networks (CNNs) when training from scratch on small datasets, which is concluded to the lack of inductive bias. In this paper, we further consider this problem and point out two weaknesses of ViTs in inductive biases, that is, the spatial relevance and diverse channel representation. First, on spatial aspect, objects are locally compact and relevant, thus fine-grained feature needs to be extracted from a token and its neighbors. While the lack of data hinders ViTs to attend the spatial re…
Elliptic；The University of Sheffield（谢菲尔德大学）；Telecom SudParis（南巴黎电信学院）；Polytechnic Institute of Paris（巴黎综合理工学院）
We show that a modification of the first layer of a Graph Convolutional Network (GCN) can be used to effectively propagate label information across neighbor nodes, for binary and multi-class classification problems. This is done by selectively eliminating self-loops for the label features during the training phase of a GCN. The GCN architecture is otherwise unchanged, without any extra hyper-parameters, and can be used in both a transductive and inductive setting. We show through several experiments that, depending on how many labels are available during the inference phase, this strategy can …
Fraunhofer Institute for Integrated Circuits IIS（弗劳恩霍夫集成电路研究所IIS）
This report to our stage 2 submission to the NeurIPS 2019 disentanglement challenge presents a simple image preprocessing method for learning disentangled latent factors. We propose to train a variational autoencoder on regionally aggregated feature maps obtained from networks pretrained on the ImageNet database, utilizing the implicit inductive bias contained in those features for disentanglement. This bias can be further enhanced by explicitly fine-tuning the feature maps on auxiliary tasks useful for the challenge, such as angle, position estimation, or color classification. Our approach ac…
Stratifying cancer patients based on their gene expression levels allows improving diagnosis, survival analysis and treatment planning. However, such data is extremely highly dimensional as it contains expression values for over 20000 genes per patient, and the number of samples in the datasets is low. To deal with such settings, we propose to incorporate prior biological knowledge about genes from ontologies into the machine learning system for the task of patient classification given their gene expression data. We use ontology embeddings that capture the semantic similarities between the gen…
University of Amsterdam（阿姆斯特丹大学）；Owlin；VU Amsterdam（阿姆斯特丹自由大学）；ING（荷兰国际集团）；WBAA；Heineken（喜力）
We study the problem of end-to-end learning from complex multigraphs with potentially very large numbers of edges between two vertices, each edge labeled with rich information. Examples range from communication networks to flights between airports or financial transaction graphs. We propose Latent-Graph Convolutional Networks (L-GCNs), which propagate information from these complex edges to a latent adjacency tensor, after which further downstream tasks can be performed, such as node classification. We evaluate the performance of several variations of the model on two synthetic datasets simula…
Renault Automotive（雷诺汽车公司）；Axionable Labs（Axionable实验室）
We propose a methodology to extend the concept of Two-Stream Convolutional Networks to perform end-to-end learning for self-driving cars with temporal cues. The system has the ability to learn spatiotemporal features by simultaneously mapping raw images and pre-calculated optical flows directly to steering commands. Although optical flows encode temporal-rich information, we found that 2D-CNNs are prone to capturing features only as spatial representations. We show how the use of Multitask Learning favors the learning of temporal features via inductive transfer from a shared spatiotemporal rep…
In this work, we analyze the role of the network architecture in shaping the inductive bias of deep classifiers. To that end, we start by focusing on a very simple problem, i.e., classifying a class of linearly separable distributions, and show that, depending on the direction of the discriminative feature of the distribution, many state-of-the-art deep convolutional neural networks (CNNs) have a surprisingly hard time solving this simple task. We then define as neural anisotropy directions (NADs) the vectors that encapsulate the directional inductive bias of an architecture. These vectors, wh…
We define the bicategory of Graph Convolutional Neural Networks $\mathbf{GCNN}_n$ for an arbitrary graph with $n$ nodes. We show it can be factored through the already existing categorical constructions for deep learning called $\mathbf{Para}$ and $\mathbf{Lens}$ with the base category set to the CoKleisli category of the product comonad. We prove that there exists an injective-on-objects, faithful 2-functor $\mathbf{GCNN}_n \to \mathbf{Para}(\mathsf{CoKl}(\mathbb{R}^{n \times n} \times -))$. We show that this construction allows us to treat the adjacency matrix of a GCNN as a global parameter…
F. Hoffmann-La Roche AG（罗氏制药）；Genentech, Inc.（基因泰克公司）
Recently, vision transformers were shown to be capable of outperforming convolutional neural networks when pretrained on sufficient amounts of data. In comparison to convolutional neural networks, vision transformers have a weaker inductive bias and therefore allow a more flexible feature detection. Due to their promising feature detection, this work explores vision transformers for tumor detection in digital pathology whole slide images in four tissue types, and for tissue type identification. We compared the patch-wise classification performance of the vision transformer DeiT-Tiny to the sta…
MIT（麻省理工学院）；Stanford University（斯坦福大学）；National University of Singapore（新加坡国立大学）
Progress in machine learning (ML) stems from a combination of data availability, computational resources, and an appropriate encoding of inductive biases. Useful biases often exploit symmetries in the prediction problem, such as convolutional networks relying on translation equivariance. Automatically discovering these useful symmetries holds the potential to greatly improve the performance of ML systems, but still remains a challenge. In this work, we focus on sequential prediction problems and take inspiration from Noether's theorem to reduce the problem of finding inductive biases to meta-l…
