We study the approximation and learning capacities of convolutional neural networks (CNNs). Our first result proves a new approximation bound for CNNs with certain constraint on the weights. Our second result gives a new analysis on the covering number of feed-forward neural networks, which include CNNs as special cases. The analysis carefully takes into account the size of the weights and hence gives better bounds than existing literature in some situations. Using these two results, we are able to derive rates of convergence for estimators based on CNNs in many learning problems. In particular, we establish minimax optimal convergence rates of the least squares based on CNNs for learning smooth functions in the nonparametric regression setting. For binary classification, we derive convergence rates for CNN classifiers with hinge loss and logistic loss. It is also shown that the obtained rates are minimax optimal in several settings.
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神經網絡（Neural Networks）是世界上三個最古老的神經建模學會的檔案期刊:國際神經網絡學會(INNS)、歐洲神經網絡學會(ENNS)和日本神經網絡學會(JNNS)。神經網絡提供了一個論壇，以發展和培育一個國際社會的學者和實踐者感興趣的所有方面的神經網絡和相關方法的計算智能。神經網絡歡迎高質量論文的提交，有助于全面的神經網絡研究，從行為和大腦建模，學習算法，通過數學和計算分析，系統的工程和技術應用，大量使用神經網絡的概念和技術。這一獨特而廣泛的范圍促進了生物和技術研究之間的思想交流，并有助于促進對生物啟發的計算智能感興趣的跨學科社區的發展。因此，神經網絡編委會代表的專家領域包括心理學，神經生物學，計算機科學，工程，數學，物理。該雜志發表文章、信件和評論以及給編輯的信件、社論、時事、軟件調查和專利信息。文章發表在五個部分之一:認知科學，神經科學，學習系統，數學和計算分析、工程和應用。
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Evolutionary game theory assumes that individuals maximize their benefits when choosing strategies. However, an alternative perspective proposes that individuals seek to maximize the benefits of others. To explore the relationship between these perspectives, we develop a model where self- and other-regarding preferences compete in public goods games. We find that other-regarding preferences are more effective in promoting cooperation, even when self-regarding preferences are more productive. Cooperators with different preferences can coexist in a new phase where two classic solutions invade each other, resulting in a dynamical equilibrium. As a consequence, a lower productivity of self-regarding cooperation can provide a higher cooperation level. Our results, which are also valid in a well-mixed population, may explain why other-regarding preferences could be a viable and frequently observed attitude in human society.
The grading of open-ended questions is a high-effort, high-impact task in education. Automating this task promises a significant reduction in workload for education professionals, as well as more consistent grading outcomes for students, by circumventing human subjectivity and error. While recent breakthroughs in AI technology might facilitate such automation, this has not been demonstrated at scale. It this paper, we introduce a novel automatic short answer grading (ASAG) system. The system is based on a fine-tuned open-source transformer model which we trained on large set of exam data from university courses across a large range of disciplines. We evaluated the trained model's performance against held-out test data in a first experiment and found high accuracy levels across a broad spectrum of unseen questions, even in unseen courses. We further compared the performance of our model with that of certified human domain experts in a second experiment: we first assembled another test dataset from real historical exams - the historic grades contained in that data were awarded to students in a regulated, legally binding examination process; we therefore considered them as ground truth for our experiment. We then asked certified human domain experts and our model to grade the historic student answers again without disclosing the historic grades. Finally, we compared the hence obtained grades with the historic grades (our ground truth). We found that for the courses examined, the model deviated less from the official historic grades than the human re-graders - the model's median absolute error was 44 % smaller than the human re-graders', implying that the model is more consistent than humans in grading. These results suggest that leveraging AI enhanced grading can reduce human subjectivity, improve consistency and thus ultimately increase fairness.
Deep neural networks give us a powerful method to model the training dataset's relationship between input and output. We can regard that as a complex adaptive system consisting of many artificial neurons that work as an adaptive memory as a whole. The network's behavior is training dynamics with a feedback loop from the evaluation of the loss function. We already know the training response can be constant or shows power law-like aging in some ideal situations. However, we still have gaps between those findings and other complex phenomena, like network fragility. To fill the gap, we introduce a very simple network and analyze it. We show the training response consists of some different factors based on training stages, activation functions, or training methods. In addition, we show feature space reduction as an effect of stochastic training dynamics, which can result in network fragility. Finally, we discuss some complex phenomena of deep networks.
In this article, we consider designs of simple analog artificial neural networks based on adiabatic Josephson cells with a sigmoid activation function. A new approach based on the gradient descent method is developed to adjust the circuit parameters, allowing efficient signal transmission between the network layers. The proposed solution is demonstrated on the example of the system implementing XOR and OR logical operations.
We study how to construct a stochastic process on a finite interval with given `roughness' and finite joint moments of marginal distributions. We first extend Ciesielski's isomorphism along a general sequence of partitions, and provide a characterization of H\"older regularity of a function in terms of its Schauder coefficients. Using this characterization we provide a better (pathwise) estimator of H\"older exponent. As an additional application, we construct fake (fractional) Brownian motions with some path properties and finite moments of marginal distributions same as (fractional) Brownian motions. These belong to non-Gaussian families of stochastic processes which are statistically difficult to distinguish from real (fractional) Brownian motions.
Generative adversarial learning with optimal input dimension and its adaptive generator architecture
We investigate the impact of the input dimension on the generalization error in generative adversarial networks (GANs). In particular, we first provide both theoretical and practical evidence to validate the existence of an optimal input dimension (OID) that minimizes the generalization error. Then, to identify the OID, we introduce a novel framework called generalized GANs (G-GANs), which includes existing GANs as a special case. By incorporating the group penalty and the architecture penalty developed in the paper, G-GANs have several intriguing features. First, our framework offers adaptive dimensionality reduction from the initial dimension to a dimension necessary for generating the target distribution. Second, this reduction in dimensionality also shrinks the required size of the generator network architecture, which is automatically identified by the proposed architecture penalty. Both reductions in dimensionality and the generator network significantly improve the stability and the accuracy of the estimation and prediction. Theoretical support for the consistent selection of the input dimension and the generator network is provided. Third, the proposed algorithm involves an end-to-end training process, and the algorithm allows for dynamic adjustments between the input dimension and the generator network during training, further enhancing the overall performance of G-GANs. Extensive experiments conducted with simulated and benchmark data demonstrate the superior performance of G-GANs. In particular, compared to that of off-the-shelf methods, G-GANs achieves an average improvement of 45.68% in the CT slice dataset, 43.22% in the MNIST dataset and 46.94% in the FashionMNIST dataset in terms of the maximum mean discrepancy or Frechet inception distance. Moreover, the features generated based on the input dimensions identified by G-GANs align with visually significant features.
Neural network (NN) designed for challenging machine learning tasks is in general a highly nonlinear mapping that contains massive variational parameters. High complexity of NN, if unbounded or unconstrained, might unpredictably cause severe issues including over-fitting, loss of generalization power, and unbearable cost of hardware. In this work, we propose a general compression scheme that significantly reduces the variational parameters of NN by encoding them to deep automatically-differentiable tensor network (ADTN) that contains exponentially-fewer free parameters. Superior compression performance of our scheme is demonstrated on several widely-recognized NN's (FC-2, LeNet-5, AlextNet, ZFNet and VGG-16) and datasets (MNIST, CIFAR-10 and CIFAR-100). For instance, we compress two linear layers in VGG-16 with approximately $10^{7}$ parameters to two ADTN's with just 424 parameters, where the testing accuracy on CIFAR-10 is improved from $90.17 \%$ to $91.74\%$. Our work suggests TN as an exceptionally efficient mathematical structure for representing the variational parameters of NN's, which exhibits superior compressibility over the commonly-used matrices and multi-way arrays.
Rectified Linear Units (ReLU) have become the main model for the neural units in current deep learning systems. This choice has been originally suggested as a way to compensate for the so called vanishing gradient problem which can undercut stochastic gradient descent (SGD) learning in networks composed of multiple layers. Here we provide analytical results on the effects of ReLUs on the capacity and on the geometrical landscape of the solution space in two-layer neural networks with either binary or real-valued weights. We study the problem of storing an extensive number of random patterns and find that, quite unexpectedly, the capacity of the network remains finite as the number of neurons in the hidden layer increases, at odds with the case of threshold units in which the capacity diverges. Possibly more important, a large deviation approach allows us to find that the geometrical landscape of the solution space has a peculiar structure: while the majority of solutions are close in distance but still isolated, there exist rare regions of solutions which are much more dense than the similar ones in the case of threshold units. These solutions are robust to perturbations of the weights and can tolerate large perturbations of the inputs. The analytical results are corroborated by numerical findings.
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.
When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.
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