Highly over-parameterized classifiers generalize since bad solutions are rare

We study the generalization of over-parameterized classifiers where Empirical Risk Minimization (ERM) for learning leads to zero training error. In these over-parameterized settings there are many global minima with zero training error, some of which generalize better than others. We show that under certain conditions the fraction of "bad" global minima with a true error larger than {\epsilon} decays to zero exponentially fast with the number of training data n. The bound depends on the distribution of the true error over the set of classifier functions used for the given classification problem, and does not necessarily depend on the size or complexity (e.g. the number of parameters) of the classifier function set. This might explain the unexpectedly good generalization even of highly over-parameterized Neural Networks. We support our mathematical framework with experiments on a synthetic data set and a subset of MNIST.

Large Neural Networks Learning from Scratch with Very Few Data and without Regularization

Recent findings have shown that Neural Networks generalize also in over-parametrized regimes with zero training error. This is surprising, since it is completely against traditional machine learning wisdom. In our empirical study we fortify these findings in the domain of fine-grained image classification. We show that very large Convolutional Neural Networks with millions of weights do learn with only a handful of training samples and without image augmentation, explicit regularization or pretraining. We train the architectures ResNet018, ResNet101 and VGG19 on subsets of the difficult benchmark datasets Caltech101, CUB_200_2011, FGVCAircraft, Flowers102 and StanfordCars with 100 classes and more, perform a comprehensive comparative study and draw implications for the practical application of CNNs. Finally, we show that VGG19 with 140 million weights learns to distinguish airplanes and motorbikes up to 95% accuracy with only 20 samples per class.

Explainable COVID-19 Detection Using Chest CT Scans and Deep Learning

This paper explores how well deep learning models trained on chest CT images can diagnose COVID-19 infected people in a fast and automated process. To this end, we adopt advanced deep network architectures and propose a transfer learning strategy using custom-sized input tailored for each deep architecture to achieve the best performance. We conduct extensive sets of experiments on two CT image datasets, namely the SARS-CoV-2 CT-scan and the COVID19-CT. The obtained results show superior performances for our models compared with previous studies, where our best models achieve average accuracy, precision, sensitivity, specificity and F1 score of 99.4%, 99.6%, 99.8%, 99.6% and 99.4% on the SARS-CoV-2 dataset; and 92.9%, 91.3%, 93.7%, 92.2% and 92.5% on the COVID19-CT dataset, respectively. Furthermore, we apply two visualization techniques to provide visual explanations for the models' predictions. The visualizations show well-separated clusters for CT images of COVID-19 from other lung diseases, and accurate localizations of the COVID-19 associated regions.

Feature Products Yield Efficient Networks

We introduce Feature-Product networks (FP-nets) as a novel deep-network architecture based on a new building block inspired by principles of biological vision. For each input feature map, a so-called FP-block learns two different filters, the outputs of which are then multiplied. Such FP-blocks are inspired by models of end-stopped neurons, which are common in cortical areas V1 and especially in V2. Convolutional neural networks can be transformed into parameter-efficient FP-nets by substituting conventional blocks of regular convolutions with FP-blocks. In this way, we create several novel FP-nets based on state-of-the-art networks and evaluate them on the Cifar-10 and ImageNet challenges. We show that the use of FP-blocks reduces the number of parameters significantly without decreasing generalization capability. Since so far heuristics and search algorithms have been used to find more efficient networks, it seems remarkable that we can obtain even more efficient networks based on a novel bio-inspired design principle.

Solving Raven's Progressive Matrices with Multi-Layer Relation Networks

Raven's Progressive Matrices are a benchmark originally designed to test the cognitive abilities of humans. It has recently been adapted to test relational reasoning in machine learning systems. For this purpose the so-called Procedurally Generated Matrices dataset was set up, which is so far one of the most difficult relational reasoning benchmarks. Here we show that deep neural networks are capable of solving this benchmark, reaching an accuracy of 98.0 percent over the previous state-of-the-art of 62.6 percent by combining Wild Relation Networks with Multi-Layer Relation Networks and introducing Magnitude Encoding, an encoding scheme designed for late fusion architectures.

Multi-layer Relation Networks

Relational Networks (RN) as introduced by Santoro et al. (2017) have demonstrated strong relational reasoning capabilities with a rather shallow architecture. Its single-layer design, however, only considers pairs of information objects, making it unsuitable for problems requiring reasoning across a higher number of facts. To overcome this limitation, we propose a multi-layer relation network architecture which enables successive refinements of relational information through multiple layers. We show that the increased depth allows for more complex relational reasoning by applying it to the bAbI 20 QA dataset, solving all 20 tasks with joint training and surpassing the state-of-the-art results.

Deep Convolutional Neural Networks as Generic Feature Extractors

Recognizing objects in natural images is an intricate problem involving multiple conflicting objectives. Deep convolutional neural networks, trained on large datasets, achieve convincing results and are currently the state-of-the-art approach for this task. However, the long time needed to train such deep networks is a major drawback. We tackled this problem by reusing a previously trained network. For this purpose, we first trained a deep convolutional network on the ILSVRC2012 dataset. We then maintained the learned convolution kernels and only retrained the classification part on different datasets. Using this approach, we achieved an accuracy of 67.68 % on CIFAR-100, compared to the previous state-of-the-art result of 65.43 %. Furthermore, our findings indicate that convolutional networks are able to learn generic feature extractors that can be used for different tasks.

Recursive Autoconvolution for Unsupervised Learning of Convolutional Neural Networks

In visual recognition tasks, such as image classification, unsupervised learning exploits cheap unlabeled data and can help to solve these tasks more efficiently. We show that the recursive autoconvolution operator, adopted from physics, boosts existing unsupervised methods by learning more discriminative filters. We take well established convolutional neural networks and train their filters layer-wise. In addition, based on previous works we design a network which extracts more than 600k features per sample, but with the total number of trainable parameters greatly reduced by introducing shared filters in higher layers. We evaluate our networks on the MNIST, CIFAR-10, CIFAR-100 and STL-10 image classification benchmarks and report several state of the art results among other unsupervised methods.

Committees of deep feedforward networks trained with few data

Deep convolutional neural networks are known to give good results on image classification tasks. In this paper we present a method to improve the classification result by combining multiple such networks in a committee. We adopt the STL-10 dataset which has very few training examples and show that our method can achieve results that are better than the state of the art. The networks are trained layer-wise and no backpropagation is used. We also explore the effects of dataset augmentation by mirroring, rotation, and scaling.

Genetic Algorithms in Time-Dependent Environments

The influence of time-dependent fitnesses on the infinite population dynamics of simple genetic algorithms (without crossover) is analyzed. Based on general arguments, a schematic phase diagram is constructed that allows one to characterize the asymptotic states in dependence on the mutation rate and the time scale of changes. Furthermore, the notion of regular changes is raised for which the population can be shown to converge towards a generalized quasispecies. Based on this, error thresholds and an optimal mutation rate are approximately calculated for a generational genetic algorithm with a moving needle-in-the-haystack landscape. The so found phase diagram is fully consistent with our general considerations.