Recently, many studies have shed light on the high adaptivity of deep neural network methods in nonparametric regression models, and their superior performance has been established for various function classes. Motivated by this development, we study a deep neural network method to estimate the drift coefficient of a multi-dimensional diffusion process from discrete observations. We derive generalization error bounds for least squares estimates based on deep neural networks and show that they achieve the minimax rate of convergence up to a logarithmic factor when the drift function has a compositional structure.
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This research aims to develop kernel GNG, a kernelized version of the growing neural gas (GNG) algorithm, and to investigate the features of the networks generated by the kernel GNG. The GNG is an unsupervised artificial neural network that can transform a dataset into an undirected graph, thereby extracting the features of the dataset as a graph. The GNG is widely used in vector quantization, clustering, and 3D graphics. Kernel methods are often used to map a dataset to feature space, with support vector machines being the most prominent application. This paper introduces the kernel GNG approach and explores the characteristics of the networks generated by kernel GNG. Five kernels, including Gaussian, Laplacian, Cauchy, inverse multiquadric, and log kernels, are used in this study. The results of this study show that the average degree and the average clustering coefficient decrease as the kernel parameter increases for Gaussian, Laplacian, Cauchy, and IMQ kernels. If we avoid more edges and a higher clustering coefficient (or more triangles), the kernel GNG with a larger value of the parameter will be more appropriate.
Within the framework of computational plasticity, recent advances show that the quasi-static response of an elasto-plastic structure under cyclic loadings may exhibit a time multiscale behaviour. In particular, the system response can be computed in terms of time microscale and macroscale modes using a weakly intrusive multi-time Proper Generalized Decomposition (MT-PGD). In this work, such micro-macro characterization of the time response is exploited to build a data-driven model of the elasto-plastic constitutive relation. This can be viewed as a predictor-corrector scheme where the prediction is driven by the macrotime evolution and the correction is performed via a sparse sampling in space. Once the nonlinear term is forecasted, the multi-time PGD algorithm allows the fast computation of the total strain. The algorithm shows considerable gains in terms of computational time, opening new perspectives in the numerical simulation of history-dependent problems defined in very large time intervals.
Early sensory systems in the brain rapidly adapt to fluctuating input statistics, which requires recurrent communication between neurons. Mechanistically, such recurrent communication is often indirect and mediated by local interneurons. In this work, we explore the computational benefits of mediating recurrent communication via interneurons compared with direct recurrent connections. To this end, we consider two mathematically tractable recurrent linear neural networks that statistically whiten their inputs -- one with direct recurrent connections and the other with interneurons that mediate recurrent communication. By analyzing the corresponding continuous synaptic dynamics and numerically simulating the networks, we show that the network with interneurons is more robust to initialization than the network with direct recurrent connections in the sense that the convergence time for the synaptic dynamics in the network with interneurons (resp. direct recurrent connections) scales logarithmically (resp. linearly) with the spectrum of their initialization. Our results suggest that interneurons are computationally useful for rapid adaptation to changing input statistics. Interestingly, the network with interneurons is an overparameterized solution of the whitening objective for the network with direct recurrent connections, so our results can be viewed as a recurrent linear neural network analogue of the implicit acceleration phenomenon observed in overparameterized feedforward linear neural networks.
Deep neural networks (NNs) are known for their high-prediction performances. However, NNs are prone to yield unreliable predictions when encountering completely new situations without indicating their uncertainty. Bayesian variants of NNs (BNNs), such as Monte Carlo (MC) dropout BNNs, do provide uncertainty measures and simultaneously increase the prediction performance. The only disadvantage of BNNs is their higher computation time during test time because they rely on a sampling approach. Here we present a single-shot MC dropout approximation that preserves the advantages of BNNs while being as fast as NNs. Our approach is based on moment propagation (MP) and allows to analytically approximate the expected value and the variance of the MC dropout signal for commonly used layers in NNs, i.e. convolution, max pooling, dense, softmax, and dropout layers. The MP approach can convert an NN into a BNN without re-training given the NN has been trained with standard dropout. We evaluate our approach on different benchmark datasets and a simulated toy example in a classification and regression setting. We demonstrate that our single-shot MC dropout approximation resembles the point estimate and the uncertainty estimate of the predictive distribution that is achieved with an MC approach, while being fast enough for real-time deployments of BNNs. We show that using part of the saved time to combine our MP approach with deep ensemble techniques does further improve the uncertainty measures.
Neural dynamical systems with stable attractor structures, such as point attractors and continuous attractors, are hypothesized to underlie meaningful temporal behavior that requires working memory. However, working memory may not support useful learning signals necessary to adapt to changes in the temporal structure of the environment. We show that in addition to the continuous attractors that are widely implicated, periodic and quasi-periodic attractors can also support learning arbitrarily long temporal relationships. Unlike the continuous attractors that suffer from the fine-tuning problem, the less explored quasi-periodic attractors are uniquely qualified for learning to produce temporally structured behavior. Our theory has broad implications for the design of artificial learning systems and makes predictions about observable signatures of biological neural dynamics that can support temporal dependence learning and working memory. Based on our theory, we developed a new initialization scheme for artificial recurrent neural networks that outperforms standard methods for tasks that require learning temporal dynamics. Moreover, we propose a robust recurrent memory mechanism for integrating and maintaining head direction without a ring attractor.
We explore the data-parallel acceleration of physics-informed machine learning (PIML) schemes, with a focus on physics-informed neural networks (PINNs) for multiple graphics processing units (GPUs) architectures. In order to develop scale-robust and high-throughput PIML models for sophisticated applications which may require a large number of training points (e.g., involving complex and high-dimensional domains, non-linear operators or multi-physics), we detail a novel protocol based on $h$-analysis and data-parallel acceleration through the Horovod training framework. The protocol is backed by new convergence bounds for the generalization error and the train-test gap. We show that the acceleration is straightforward to implement, does not compromise training, and proves to be highly efficient and controllable, paving the way towards generic scale-robust PIML. Extensive numerical experiments with increasing complexity illustrate its robustness and consistency, offering a wide range of possibilities for real-world simulations.
Sparsity is a highly desired feature in deep neural networks (DNNs) since it ensures numerical efficiency, improves the interpretability of models (due to the smaller number of relevant features), and robustness. In machine learning approaches based on linear models, it is well known that there exists a connecting path between the sparsest solution in terms of the $\ell^1$ norm (i.e., zero weights) and the non-regularized solution, which is called the regularization path. Very recently, there was a first attempt to extend the concept of regularization paths to DNNs by means of treating the empirical loss and sparsity ($\ell^1$ norm) as two conflicting criteria and solving the resulting multiobjective optimization problem. However, due to the non-smoothness of the $\ell^1$ norm and the high number of parameters, this approach is not very efficient from a computational perspective. To overcome this limitation, we present an algorithm that allows for the approximation of the entire Pareto front for the above-mentioned objectives in a very efficient manner. We present numerical examples using both deterministic and stochastic gradients. We furthermore demonstrate that knowledge of the regularization path allows for a well-generalizing network parametrization.
In sampling-based Bayesian models of brain function, neural activities are assumed to be samples from probability distributions that the brain uses for probabilistic computation. However, a comprehensive understanding of how mechanistic models of neural dynamics can sample from arbitrary distributions is still lacking. We use tools from functional analysis and stochastic differential equations to explore the minimum architectural requirements for $\textit{recurrent}$ neural circuits to sample from complex distributions. We first consider the traditional sampling model consisting of a network of neurons whose outputs directly represent the samples (sampler-only network). We argue that synaptic current and firing-rate dynamics in the traditional model have limited capacity to sample from a complex probability distribution. We show that the firing rate dynamics of a recurrent neural circuit with a separate set of output units can sample from an arbitrary probability distribution. We call such circuits reservoir-sampler networks (RSNs). We propose an efficient training procedure based on denoising score matching that finds recurrent and output weights such that the RSN implements Langevin sampling. We empirically demonstrate our model's ability to sample from several complex data distributions using the proposed neural dynamics and discuss its applicability to developing the next generation of sampling-based brain models.
Temporal networks are essential for modeling and understanding systems whose behavior varies in time, from social interactions to biological systems. Often, however, real-world data are prohibitively expensive to collect in a large scale or unshareable due to privacy concerns. A promising way to bypass the problem consists in generating arbitrarily large and anonymized synthetic graphs with the properties of real-world networks, namely `surrogate networks'. Until now, the generation of realistic surrogate temporal networks has remained an open problem, due to the difficulty of capturing both the temporal and topological properties of the input network, as well as their correlations, in a scalable model. Here, we propose a novel and simple method for generating surrogate temporal networks. Our method decomposes the input network into star-like structures evolving in time. Then those structures are used as building blocks to generate a surrogate temporal network. Our model vastly outperforms current methods across multiple examples of temporal networks in terms of both topological and dynamical similarity. We further show that beyond generating realistic interaction patterns, our method is able to capture intrinsic temporal periodicity of temporal networks, all with an execution time lower than competing methods by multiple orders of magnitude. The simplicity of our algorithm makes it easily interpretable, extendable and algorithmically scalable.
We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.
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