[PDF] On the infinite width limit of neural networks with ... Greg Yang: Title: Feature Learning in Infinite-Width ... Feature Learning in Infinite-Width Neural Networks ... One essential assumption is, that at initialization (given infinite width) a neural network is equivalent to a Gaussian Process [].The evolution that occurs when training the network can then be described by a kernel as has been shown by researchers at the Ecole Polytechnique . This correspondence enables exact Bayesian inference for infinite width neural networks on regression tasks by means of evaluating the corresponding GP. However, we show that the standard and NTK parametrizations of a neural network do not admit infinite-width limits that can learn features, which is . Reminder Subject: TALK: Greg Yang: Title: Feature Learning in Infinite-Width Neural Networks Abstract: As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. Our results are directly applicable to infinite-width limit of neural networks that admit a kernel description (including feedforward, convolutional and recurrent neural networks) 13,55,56,57,58 . We find that they outperform both NTK baselines and finite-width networks, with the latter approaching the infinite-width feature learning performance as width increases. On Exact Computation with an Infinitely Wide Neural Net Talk: Jascha Sohl-Dickstein | MIT CSAIL For example, below we compare three different infinite-width neural network architectures on image recognition using the CIFAR-10 dataset. However, we show that the standard and NTK . Hinton et al. further generalized the result to infinite width network of arbitrary depth. The evolution of a deep neural network trained by the gradient descent can be described by its neural tangent kernel (NTK) as introduced in [20], where it was proven that in the infinite width limit the NTK converges to an explicit limiting kernel and it stays constant during training. This is a highly valuable outcome because the kernel ridge regressor (i.e., the predictor from the algorithm . 11/30/2020 ∙ by Greg Yang, et al. There are currently two parameterizations used to derive fixed kernels corresponding to infinite width neural networks, the NTK (Neural Tangent Kernel) parameterization and the naive standard parameterization. Understanding infinite neural networks (e.g. . These networks can then be trained and evaluated either at finite-width as usual or in their infinite-width limit. For example, below we compare three different infinite-width neural network architectures on image recognition using the CIFAR-10 dataset. Feature Learning in Infinite-Width Neural Networks Greg Yang1 Edward J. Hu2 3 Abstract As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. Overparameterized neural networks implement associative memory. Based on the experiments, the authors also propose an improved layer-wise scaling for weight decay and improve the performance . The NTK was also implicit in some other recent papers [6,13,14]. which shows that the infinite-width limit of a neural network of any architecture is well-defined (in the technical sense that the tangent kernel (NTK) of any randomly initialized neural network converges in the large width limit) and can be computed. Feature Learning in Infinite-Width Neural NetworksGreg Yang, Edward Hu. Neural Tangents allows researchers to define, train, and evaluate infinite networks as easily as finite ones. random parameters, in the limit of infinite width, is a function drawn from a Gaussian process (GP) [Neal, 1996]. We show 1) any parametrization in this space either admits feature learning or has an infinite-width training dynamics given by kernel gradient descent, but not both; 2) any such infinite-width limit . Theoretical approaches based on a large width limit. We show 1) any parametrization in this space either admits feature learning or has an infinite-width training dynamics given by kernel gradient descent, but not both; 2) any such infinite-width limit . Phase Diagram for Two-layer ReLU Neural Networks at Infinite-width Limit /LQHDUUHJLPH &RQGHQVHGUHJLPH ([DPSOHV ;DYLHU 0HDQILHOG &ULWLFDOUHJLPH 3KDVH'LDJUDP 17. Speaker: Greg YangAffiliation: MicrosoftAbstract: As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplifi. the NTK parametrization). This model as Invited speaker: The Convexity of Learning Infinite-width Deep Neural Networks, Tong Zhang (Talk) » SlidesLive Video » Deep learning has received considerable empirical successes in recent years. Feature Learning in Infinite-Width Neural Networks. ntk += np.sum(scov, axis= (-1, -2)) return dict(ntk=ntk / seqlen**2, dscov=dscov, scov=scov, hcov=hcov, hhcov=hhcov) The below function computes the NTK even when sequences have different lengths, but is not as computationally efficient as the batched function above. The Problem Many previous works proposed that wide neural networks (NN) are kernel machines [1] [2] [3] , the most well-known theory perhaps being the Neural Tangent Kernel (NTK) [1] . 01/21/2020 ∙ by Jascha Sohl-Dickstein, et al. It provides a high-level API for specifying complex and hierarchical neural network architectures. These networks can then be trained and evaluated either at finite-width as usual or in their infinite-width limit. As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. And Turing machines have an infinite tape. For example, we calculated the \(μP\) limit of Word2Vec and found it outperformed both the NTK and NNGP limits as well as finite-width networks. On the infinite width limit of neural networks with a standard parameterization. These networks can then be trained and evaluated either at finite-width as usual or in their infinite-width limit. As neural networks become wider their accuracy improves, and their behavior becomes easier to analyze theoretically. During training, the evolution of the function represented by the infinite-width neural network matches the evolution of the function represented by the kernel machine. Reminder Subject: TALK: Greg Yang: Title: Feature Learning in Infinite-Width Neural Networks Abstract: As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. In short, the code here will allow you to train feature learning infinite-width neural networks on Word2Vec and on Omniglot (via MAML). Two essential kernels — our gates to infinity. However, networks built using Neural Tangents can be applied to any problem on which you could apply a regular neural network. We consider neural networks where the number of neurons in all hidden layers are increased to infinity. the NTK parametrization). It provides a high-level API for specifying complex and hierarchical neural network architectures. Feature Learning in Infinite-Width Neural Networks. The marked ex-amples are studied in existing literature (see Table 1 for details.) We empirically show that a single bottleneck in infinite networks dramatically accelerates training when compared to purely infinite networks, with an improved overall performance. Using Monte Carlo approximations, we derive a novel data- and task-dependent weight initialisation scheme for finite-width networks that incorporates the structure of the data and information about the task at hand into the network. ∙ 6 ∙ share . I'm excited to share with you my new paper [2011.14522] Feature Learning in Infinite-Width Neural Networks (arxiv.org). Neural Architecture Search on ImageNet in Four GPU Hours: 2021.10.08: Adit Radha: Simple, Fast, and Flexible Framework for Matrix Completion with Infinite Width Neural Networks: 2021.10.01: Ilan Price: Dense for the Price of Sparse: Improved Performance of Sparsely Initialized Networks via a Subspace Offset: 2021.09.24: Preetum Nakkiran Neural Tangents is a high-level neural network API for specifying complex, hierarchical, neural networks of both finite and infinite width. However, networks built using Neural Tangents can be applied to any problem on which you could apply a regular neural network. I will give an introduction to a rapidly growing body of work which examines the learning dynamics and prior over functions induced by infinitely wide, randomly initialized, neural networks. By doing so, a lot of interesting observations are made. Co-authors Radhakrishnan A, Stefanakis G, Belkin M, Uhler C. JOURNAL ARTICLE. Abstract: Neural Tangents is a library for working with infinite-width neural networks. In this article, analytic forms are derived for the covariance function of the gaussian processes corresponding to networks with sigmoidal and gaussian hidden units. I will argue that this growing understanding of neural networks in the limit of infinite width is foundational for future theoretical and practical understanding of deep learning. Add to Calendar 2019-11-18 16:30:00 2019-11-18 17:30:00 America/New_York Talk: Jascha Sohl-Dickstein Title: Understanding infinite width neural networksAbstract: As neural networks become wider their accuracy improves, and their behavior becomes easier to analyze theoretically. neural networks whose number of neurons is infinite in the hidden layers) is much easier than finite ones. given by the Ne. When seen in function space, the neural network and its equivalent kernel machine both roll down a simple, bowl-shaped landscape in some hyper-dimensional space. Understanding the Neural Tangent Kernel. A standard deep neural network (DNN) is, technically speaking, parametric since it has a fixed number of parameters. Feature Learning in Infinite-Width Neural Networks. As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. Feature Learning in Infinite-Width Neural Networks Greg Yang, Edward Hu. However, we show that the standard and NTK parametrizations of a neural network do not admit infinite-width limits that can learn features, which is . Core results that I will discuss include: that the distribution over functions computed . Abstract: As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. The infinite-width limit replaces the inner loop of training a finite-width neural network with a simple kernel regression. Simplicity and speed come from the connection between the infinite width limit of neural networks and kernels known as neural tangent kernels (NTK). We explicitly compute several such infinite-width networks in this repo. Feb 24, 2021, 04:00 PM - 05:00 PM | Zoom id: 97648161149. With the addition of a regularizing term, the kernel regression becomes a kernel ridge-regression (KRR) problem. Abstract: It has long been known that a single-layer fully-connected neural network with an i.i.d. given by the Neural Tangent Kernel (NTK)), if it is parametrized appropriately (e.g. The Neural Network Gaussian Process (NNGP) corresponds to the infinite width limit of Bayesian neural networks, and to the distribution over functions realized by non-Bayesian neural networks after random initialization. More generally, we classify a natural space of neural network parametrizations that generalizes standard, NTK, and Mean Field parametrizations. Let's suppose e.g. Allowing width to go to infinity also connects deep learning in an interesting way with other areas of machine learning. At first, this limit may seem impractical and even pointless . I'm excited to share with you my new paper [2011.14522] Feature Learning in Infinite-Width Neural Networks (arxiv.org). For neural networks with a wide class of weight priors, it can be shown that in the limit of an infinite number of hidden units, the prior over functions tends to a gaussian process. : The parameters of the GP are: (Note that outputs are independent because have Normal joint and zero covariance.) 30 Jul 2021 arXiv. However, the extrapolation of both of these . Feature Learning is Crucial in Deep Learning Imagenetand Resnet BERT and GPT3 In short, the code here will allow you to train feature learning infinite-width neural networks on Word2Vec and on Omniglot (via MAML). 2. A different thing I noticed from the article was the researchers refer to "reduction": During training, the evolution of the function represented by the infinite-width neural network matches the evolution of the function represented by the kernel machine But this is reducing a kernel to a neural net. the NTK parametrization). Typically we consider networks with a Gaussian-initialized weights, and scale the variance at initialization as 1 √H, where H is . the NTK parametrization). On the infinite width limit of neural networks with a standard parameterization. Edward Hu As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. There are currently two parameterizations used to derive fixed kernels corresponding to infinite width neural networks, the NTK (Neural Tangent Kernel) parameterization and the naive standard parameterization. ; The same underlying computations that are used to derive the NNGP kernel are also used in deep information propagation to . Simple, fast, and flexible framework for matrix completion with infinite width neural networks. Our results on Word2Vec: Our Results on MAML: Please see the README in individual folders for more details. Shallow Neural Networks and GP Priors Follows from the Central Limit Theorem. For example, below we compare three different infinite-width neural network architectures on image recognition using the CIFAR-10 dataset. Allowing width to go to infinity also connects deep learning in an interesting way with other areas of machine learning. Distilling the Knowledge in a Neural Network, 2014. The Problem Many previous works proposed that wide neural networks (NN) are kernel machines [1] [2] [3] , the most well-known theory perhaps being the Neural Tangent Kernel (NTK) [1] . And since the tangent kernel stays constant during training, the training dynamics is now reduced to a simple linear ordinary differential equation. Infinite neural networks have a Gaussian distribution that can be described by a kernel (as it is the case in Support Vector Machines or Bayesian inference) determined by the network architecture. The theoretical analysis of infinite-width neural networks has led to many interesting practical results (choice of initialization schemes, choice of Bayesian priors etc.). width neural networks in terms of weight initialisation. Feature Learning in Infinite-Width Neural NetworksGreg Yang, Edward Hu. given by the Neural Tangent Kernel (NTK)), if it is parametrized appropriately (e.g. the NTK parametrization). Add to Calendar 2021-03-15 17:00:00 2021-03-15 18:30:00 America/New_York Greg Yang: Title: Feature Learning in Infinite-Width Neural Networks Abstract: As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. More generally, we classify a natural space of neural network parametrizations that generalizes standard, NTK, and Mean Field parametrizations. This is the 4th paper in the . the NTK parametrization). given by the Neural Tangent Kernel (NTK)), if it is parametrized appropriately (e.g. An improved extrapolation of the standard parameterization that preserves all of these properties as width is taken to infinity and yields a well-defined neural tangent kernel is proposed. Back to 199 5, Radford M. Neal showed that a single layer neural network with random parameters would converge to a Gaussian process as the width goes to infinity.In 2018, Lee et al. We will not use it here but we present it for future reference. We consider neural networks where the number of neurons in all hidden layers are increased to infinity. Maximal Update Parametrization \((μP)\), which follows the principles we discussed and learns features maximally in the infinite-width limit, has the potential to change the way we train neural networks. Evans Hall | Happening As Scheduled. Radhakrishnan, A., Stefanakis, G., Belkin, M. and Uhler, C. %0 Conference Paper %T Tensor Programs IV: Feature Learning in Infinite-Width Neural Networks %A Greg Yang %A Edward J. Hu %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-yang21c %I PMLR %P 11727--11737 %U https://proceedings . Infinitely wide neural networks are written using the neural tangents library developed by Google Research. It is based on JAX, and provides a neural network library that lets us analytically obtain the infinite-width kernel corresponding to the particular neural network architecture specified. This is a highly valuable outcome because the kernel ridge regressor (i.e., the predictor from the algorithm . More generally, we classify a natural space of neural network parametrizations that generalizes standard, NTK, and Mean Field parametrizations. the NTK parametrization). Our results on Word2Vec: Our Results on MAML: Please see the README in individual folders for more details. The most exciting recent developments in the theory of neural networks have focused the infinite-width limit. No passcode. When seen in function space . During training, the evolution of the function represented by the infinite-width neural network matches the evolution of the function represented by the kernel machine. As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. the NTK parametrization). The most exciting recent developments in the theory of neural networks have focused the infinite-width limit. As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. Simple, Fast, and Flexible Framework for Matrix Completion with Infinite Width Neural Networks. Greg Yang, Microsoft Research. A flurry of recent papers in theoretical deep learning tackles the common theme of analyzing neural networks in the infinite-width limit. given by the Neural Tangent Kernel (NTK)), if it is parametrized appropriately (e.g. Neural Tangents is a high-level neural network API for specifying complex, hierarchical, neural networks of both finite and infinite width. However, the traditional infinite-width framework focuses on fixed depth networks and omits the large depth behavior of these models. However, networks built using Neural Tangents can be applied to any problem on which you could apply a regular neural network. In the infinite width limit, every finite collection of will have a joint multivariate Normal distribution. Related Works Speaker Bio. Infinite Width Nets: Initialization . random parameters, in the limit of infinite width, is a function drawn from a Gaussian Process (GP) (Neal, 1996).This model as well as analogous ones with multiple layers (Lee et al., 2018; Matthews et al., 2018) and . An attraction of such ideas is that a pure kernel-based method is used to capture the power of a . I will give an introduction to a rapidly growing body of work which examines the learning dynamics and prior over . Infinitely Wide Neural Networks In the limit of infinite width, neural networks become tractable: NN with MSE loss kernel ridge-regression with . Photo by Benton Sherman on Unsplash. A recent paper [Jacot et al., 2018] introduced the Neural Tangent Kernel (NTK) which captures the behavior of fully-connected deep nets in the infinite width limit trained by gradient descent; this object was implicit in some other recent papers. the NTK parametrization). 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