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Register a new userNAIS-Net: Stable Deep Networks from Non-Autonomous Differential Equations
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Marco Ciccone
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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This paper introduces Non-Autonomous Input-Output Stable Network(NAIS-Net), a very deep architecture where each stacked processing block is derived from a time-invariant non-autonomous dynamical system. Non-autonomy is implemented by skip connections from the block input to each of the unrolled processing stages and allows stability to be enforced so that blocks can be unrolled adaptively to a pattern-dependent processing depth. NAIS-Net induces non-trivial, Lipschitz input-output maps, even for an infinite unroll length. We prove that the network is globally asymptotically stable so that for every initial condition there is exactly one input-dependent equilibrium assuming $tanh$ units, and incrementally stable for ReL units. An efficient implementation that enforces the stability under derived conditions for both fully-connected and convolutional layers is also presented. Experimental results show how NAIS-Net exhibits stability in practice, yielding a significant reduction in generalization gap compared to ResNets.
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We introduce the Genetic-Gated Networks (G2Ns), simple neural networks that combine a gate vector composed of binary genetic genes in the hidden layer(s) of networks. Our method can take both advantages of gradient-free optimization and gradient-based optimization methods, of which the former is effective for problems with multiple local minima, while the latter can quickly find local minima. In addition, multiple chromosomes can define different models, making it easy to construct multiple models and can be effectively applied to problems that require multiple models. We show that this G2N can be applied to typical reinforcement learning algorithms to achieve a large improvement in sample efficiency and performance.
In this work we systematically analyze general properties of differential equations used as machine learning models. We demonstrate that the gradient of the loss function with respect to to the hidden state can be considered as a generalized momentum conjugate to the hidden state, allowing application of the tools of classical mechanics. In addition, we show that not only residual networks, but also feedforward neural networks with small nonlinearities and the weights matrices deviating only slightly from identity matrices can be related to the differential equations. We propose a differential equation describing such networks and investigate its properties.
Random ordinary differential equations (RODEs), i.e. ODEs with random parameters, are often used to model complex dynamics. Most existing methods to identify unknown governing RODEs from observed data often rely on strong prior knowledge. Extracting the governing equations from data with less prior knowledge remains a great challenge. In this paper, we propose a deep neural network, called RODE-Net, to tackle such challenge by fitting a symbolic expression of the differential equation and the distribution of parameters simultaneously. To train the RODE-Net, we first estimate the parameters of the unknown RODE using the symbolic networks cite{long2019pde} by solving a set of deterministic inverse problems based on the measured data, and use a generative adversarial network (GAN) to estimate the true distribution of the RODEs parameters. Then, we use the trained GAN as a regularization to further improve the estimation of the ODEs parameters. The two steps are operated alternatively. Numerical results show that the proposed RODE-Net can well estimate the distribution of model parameters using simulated data and can make reliable predictions. It is worth noting that, GAN serves as a data driven regularization in RODE-Net and is more effective than the $ell_1$ based regularization that is often used in system identifications.
We perform scalable approximate inference in a continuous-depth Bayesian neural network family. In this model class, uncertainty about separate weights in each layer gives hidden units that follow a stochastic differential equation. We demonstrate gradient-based stochastic variational inference in this infinite-parameter setting, producing arbitrarily-flexible approximate posteriors. We also derive a novel gradient estimator that approaches zero variance as the approximate posterior over weights approaches the true posterior. This approach brings continuous-depth Bayesian neural nets to a competitive comparison against discrete-depth alternatives, while inheriting the memory-efficient training and tunable precision of Neural ODEs.
At present, the vast majority of building blocks, techniques, and architectures for deep learning are based on real-valued operations and representations. However, recent work on recurrent neural networks and older fundamental theoretical analysis suggests that complex numbers could have a richer representational capacity and could also facilitate noise-robust memory retrieval mechanisms. Despite their attractive properties and potential for opening up entirely new neural architectures, complex-valued deep neural networks have been marginalized due to the absence of the building blocks required to design such models. In this work, we provide the key atomic components for complex-valued deep neural networks and apply them to convolutional feed-forward networks and convolutional LSTMs. More precisely, we rely on complex convolutions and present algorithms for complex batch-normalization, complex weight initialization strategies for complex-valued neural nets and we use them in experiments with end-to-end training schemes. We demonstrate that such complex-valued models are competitive with their real-valued counterparts. We test deep complex models on several computer vision tasks, on music transcription using the MusicNet dataset and on Speech Spectrum Prediction using the TIMIT dataset. We achieve state-of-the-art performance on these audio-related tasks.
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