Residual connections significantly boost the performance of deep neural networks. However, there are few theoretical results that address the influence of residuals on the hypothesis complexity and the generalization ability of deep neural networks. This paper studies the influence of residual connections on the hypothesis complexity of the neural network in terms of the covering number of its hypothesis space. We prove that the upper bound of the covering number is the same as chain-like neural networks, if the total numbers of the weight matrices and nonlinearities are fixed, no matter whether they are in the residuals or not. This result demonstrates that residual connections may not increase the hypothesis complexity of the neural network compared with the chain-like counterpart. Based on the upper bound of the covering number, we then obtain an @math margin-based multi-class generalization bound for ResNet, as an exemplary case of any deep neural network with residual connections. Generalization guarantees for similar state-of-the-art neural network architectures, such as DenseNet and ResNeXt, are straight-forward. From our generalization bound, a practical implementation is summarized: to approach a good generalization ability, we need to use regularization terms to control the magnitude of the norms of weight matrices not to increase too much, which justifies the standard technique of weight decay.
discuss many open problems regarding the excellent generalization ability of deep neural networks despite the large capacity, complexity, possible algorithmic instability, nonrobustness, and sharp minima @cite_2 . They also provide some insights to solve the problems.
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This paper provides non-vacuous and numerically-tight generalization guarantees for deep learning, as well as theoretical insights into why and how deep learning can generalize well, despite its large capacity, complexity, possible algorithmic instability, nonrobustness, and sharp minima, responding to an open question in the literature. We also propose new open problems and discuss the limitations of our results. Using deep neural nets as function approximator for reinforcement learning tasks have recently been shown to be very powerful for solving problems approaching real-world complexity. Using these results as a benchmark, we discuss the role that the discount factor may play in the quality of the learning process of a deep Q-network (DQN). When the discount factor progressively increases up to its final value, we empirically show that it is possible to significantly reduce the number of learning steps. When used in conjunction with a varying learning rate, we empirically show that it outperforms original DQN on several experiments. We relate this phenomenon with the instabilities of neural networks when they are used in an approximate Dynamic Programming setting. We also describe the possibility to fall within a local optimum during the learning process, thus connecting our discussion with the exploration exploitation dilemma. In this chapter, we focus on two important areas in neural computation, i. e., deep and modular neural networks, given the fact that both deep and modular neural networks are among the most powerful machine learning and pattern recognition techniques for complex AI problem solving. We begin by providing a general overview of deep and modular neural networks to describe the general motivation behind such neural architectures and fundamental requirements imposed by complex AI problems. Next, we describe background and motivation, methodologies, major building blocks, and the state-of-the-art hybrid learning strategy in context of deep neural architectures. Then, we describe background and motivation, taxonomy, and learning algorithms pertaining to various typical modular neural networks in a wide context. Furthermore, we also examine relevant issues and discuss open problems in deep and modular neural network research areas. The stochastic gradient descent (SGD) method and its variants are algorithms of choice for many Deep Learning tasks. These methods operate in a small-batch regime wherein a fraction of the training data, say @math - @math data points, is sampled to compute an approximation to the gradient. It has been observed in practice that when using a larger batch there is a degradation in the quality of the model, as measured by its ability to generalize. We investigate the cause for this generalization drop in the large-batch regime and present numerical evidence that supports the view that large-batch methods tend to converge to sharp minimizers of the training and testing functions - and as is well known, sharp minima lead to poorer generalization. In contrast, small-batch methods consistently converge to flat minimizers, and our experiments support a commonly held view that this is due to the inherent noise in the gradient estimation. We discuss several strategies to attempt to help large-batch methods eliminate this generalization gap. It is widely observed that deep learning models with learned parameters generalize well, even with much more model parameters than the number of training samples. We systematically investigate the underlying reasons why deep neural networks often generalize well, and reveal the difference between the minima (with the same training error) that generalize well and those they don't. We show that it is the characteristics the landscape of the loss function that explains the good generalization capability. For the landscape of loss function for deep networks, the volume of basin of attraction of good minima dominates over that of poor minima, which guarantees optimization methods with random initialization to converge to good minima. We theoretically justify our findings through analyzing 2-layer neural networks; and show that the low-complexity solutions have a small norm of Hessian matrix with respect to model parameters. For deeper networks, extensive numerical evidence helps to support our arguments. In deep learning, , as well as , create non-convex loss surfaces. Then, does depth alone create bad local minima? In this paper, we prove that without nonlinearity, depth alone does not create bad local minima, although it induces non-convex loss surface. Using this insight, we greatly simplify a recently proposed proof to show that all of the local minima of feedforward deep linear neural networks are global minima. Our theoretical results generalize previous results with fewer assumptions, and this analysis provides a method to show similar results beyond square loss in deep linear models. Whereas optimizing deep neural networks using stochastic gradient descent has been showed great performances in practice, the rule for setting step size (i.e. learning rate) of gradient descent is not well studied. Although it appears that some intriguing learning rate rules such as ADAM (Kingma and Ba, 2014) have since been developed, they concentrated on improving convergence, not on improving generalization capabilities. Recently, the improved generalization property of the flat minima was revisited, and this research guides us towards promising solutions to many current optimization problems. In this paper, we analyze the flatness of loss surfaces in the lens of robustness to input perturbations and advocate that gradient descent should be guided to reach flatter region of loss surfaces to achieve generalization. Finally, we suggest a learning rate rule for escaping sharp regions of loss surfaces, and we demonstrate the capacity of our approach by performing numerous experiments. This paper introduces a novel measure-theoretic learning theory to analyze generalization behaviors of practical interest. The proposed learning theory has the following abilities: 1) to utilize the qualities of each learned representation on the path from raw inputs to outputs in representation learning, 2) to guarantee good generalization errors possibly with arbitrarily rich hypothesis spaces (e.g., arbitrarily large capacity and Rademacher complexity) and non-stable non-robust learning algorithms, and 3) to clearly distinguish each individual problem instance from each other. Our generalization bounds are relative to a representation of the data, and hold true even if the representation is learned. We discuss several consequences of our results on deep learning, one-shot learning and curriculum learning. Unlike statistical learning theory, the proposed learning theory analyzes each problem instance individually via measure theory, rather than a set of problem instances via statistics. Because of the differences in the assumptions and the objectives, the proposed learning theory is meant to be complementary to previous learning theory and is not designed to compete with it. We propose a very simple modification of gradient descent and stochastic gradient descent. We show that when applied to a variety of machine learning models including softmax regression, convolutional neural nets, generative adversarial nets, and deep reinforcement learning, this very simple surrogate can dramatically reduce the variance and improve the accuracy of the generalization. The new algorithm, (which depends on one nonnegative parameter) when applied to non-convex minimization, tends to avoid sharp local minima. Instead it seeks somewhat flatter local (and often global) minima. The method only involves preconditioning the gradient by the inverse of a tri-diagonal matrix that is positive definite. The motivation comes from the theory of Hamilton-Jacobi partial differential equations. This theory demonstrates that the new algorithm is almost the same as doing gradient descent on a new function which (a) has the same global minima as the original function and (b) is "more convex". Again, the programming effort in doing this is minimal, in cost, complexity and effort. We implement our algorithm into both PyTorch and Tensorflow platforms, which will be made publicly available. Deep convolutional networks have become a popular tool for image generation and restoration. Generally, their excellent performance is imputed to their ability to learn realistic image priors from a large number of example images. In this paper, we show that, on the contrary, the structure of a generator network is sufficient to capture a great deal of low-level image statistics prior to any learning. In order to do so, we show that a randomly-initialized neural network can be used as a handcrafted prior with excellent results in standard inverse problems such as denoising, superresolution, and inpainting. Furthermore, the same prior can be used to invert deep neural representations to diagnose them, and to restore images based on flash-no flash input pairs. Apart from its diverse applications, our approach highlights the inductive bias captured by standard generator network architectures. It also bridges the gap between two very popular families of image restoration methods: learning-based methods using deep convolutional networks and learning-free methods based on handcrafted image priors such as self-similarity. Deep convolutional neural networks (DCNNs) have shown remarkable performance in image classification tasks in recent years. Generally, deep neural network architectures are stacks consisting of a large number of convolutional layers, and they perform downsampling along the spatial dimension via pooling to reduce memory usage. Concurrently, the feature map dimension (i.e., the number of channels) is sharply increased at downsampling locations, which is essential to ensure effective performance because it increases the diversity of high-level attributes. This also applies to residual networks and is very closely related to their performance. In this research, instead of sharply increasing the feature map dimension at units that perform downsampling, we gradually increase the feature map dimension at all units to involve as many locations as possible. This design, which is discussed in depth together with our new insights, has proven to be an effective means of improving generalization ability. Furthermore, we propose a novel residual unit capable of further improving the classification accuracy with our new network architecture. Experiments on benchmark CIFAR-10, CIFAR-100, and ImageNet datasets have shown that our network architecture has superior generalization ability compared to the original residual networks. Code is available at this https URL In this research, we propose a deep learning based approach for speeding up the topology optimization methods. The problem we seek to solve is the layout problem. The main novelty of this work is to state the problem as an image segmentation task. We leverage the power of deep learning methods as the efficient pixel-wise image labeling technique to perform the topology optimization. We introduce convolutional encoder-decoder architecture and the overall approach of solving the above-described problem with high performance. The conducted experiments demonstrate the significant acceleration of the optimization process. The proposed approach has excellent generalization properties. We demonstrate the ability of the application of the proposed model to other problems. The successful results, as well as the drawbacks of the current method, are discussed. We treat the use of more complex higher degree polynomial curves and surfaces of degree higher than 2, which have many desirable properties for object recognition and position estimation, and attack the instability problem arising in their use with partial and noisy data. The scenario discussed in this paper is one where we have a set of objects that are modeled as implicit polynomial functions, or a set of representations of classes of objects with each object in a class modeled as an implicit polynomial function, stored in the database. Then, given partial data from one of the objects, we want to recognize the object (or the object class) or collect more data in order to get better parameter estimates for more reliable recognition. Two problems arising in this scenario are discussed: 1) the problem of recognizing these polynomials by comparing them in terms of their coefficients; and 2) the problem of where to collect data so as to improve the parameter estimates as quickly as possible. We use an asymptotic Bayesian approximation for solving the two problems. The intrinsic dimensionality of polynomials and the use of the Mahalanobis distance are discussed. While the optimization problem behind deep neural networks is highly non-convex, it is frequently observed in practice that training deep networks seems possible without getting stuck in suboptimal points. It has been argued that this is the case as all local minima are close to being globally optimal. We show that this is (almost) true, in fact almost all local minima are globally optimal, for a fully connected network with squared loss and analytic activation function given that the number of hidden units of one layer of the network is larger than the number of training points and the network structure from this layer on is pyramidal. Deep convolutional neural networks (DCNNs) have shown remarkable performance in image classification tasks in recent years. Generally, deep neural network architectures are stacks consisting of a large number of convolutional layers, and they perform downsampling along the spatial dimension via pooling to reduce memory usage. Concurrently, the feature map dimension (i.e., the number of channels) is sharply increased at downsampling locations, which is essential to ensure effective performance because it increases the diversity of high-level attributes. This also applies to residual networks and is very closely related to their performance. In this research, instead of sharply increasing the feature map dimension at units that perform downsampling, we gradually increase the feature map dimension at all units to involve as many locations as possible. This design, which is discussed in depth together with our new insights, has proven to be an effective means of improving generalization ability. Furthermore, we propose a novel residual unit capable of further improving the classification accuracy with our new network architecture. Experiments on benchmark CIFAR-10, CIFAR-100, and ImageNet datasets have shown that our network architecture has superior generalization ability compared to the original residual networks. Code is available at https: github.com jhkim89 PyramidNet. Deep convolutional networks have become a popular tool for image generation and restoration. Generally, their excellent performance is imputed to their ability to learn realistic image priors from a large number of example images. In this paper, we show that, on the contrary, the structure of a generator network is sufficient to capture a great deal of low-level image statistics prior to any learning. In order to do so, we show that a randomly-initialized neural network can be used as a handcrafted prior with excellent results in standard inverse problems such as denoising, super-resolution, and inpainting. Furthermore, the same prior can be used to invert deep neural representations to diagnose them, and to restore images based on flash-no flash input pairs. Apart from its diverse applications, our approach highlights the inductive bias captured by standard generator network architectures. It also bridges the gap between two very popular families of image restoration methods: learning-based methods using deep convolutional networks and learning-free methods based on handcrafted image priors such as self-similarity. Code and supplementary material are available at this https URL . Neural networks with their inherent learning ability have been widely applied to solve the robot manipulator inverse kinematics problems. However, there are still two open problems: (1) without knowing inverse kinematic expressions, these solutions have the difficulty of how to collect training sets, and (2) the gradient-based learning algorithms can cause a very slow training process, especially for a complex configuration, or a large set of training data. Unlike these traditional implementations, the proposed metho trains neural network in joint subspace which can be easily calculated with electromagnetism-like method. The kinematics equation and its inverse are one-to-one mapping within the subspace. Thus the constrained training sets can be easily collected by forward kinematics relations. For issue 2, this paper uses a novel learning algorithm called extreme learning machine (ELM) which randomly choose the input weights and analytically determines the output weights of the single hidden layer feedforward neural networks (SLFNs). In theory, this algorithm tends to provide the best generalization performance at extremely fast learning speed. The results show that the proposed approach has not only greatly reduced the computation time but also improved the precision. Despite their overwhelming capacity to overfit, deep learning architectures tend to generalize relatively well to unseen data, allowing them to be deployed in practice. However, explaining why this is the case is still an open area of research. One standing hypothesis that is gaining popularity, e.g. Hochreiter & Schmidhuber (1997); (2017), is that the flatness of minima of the loss function found by stochastic gradient based methods results in good generalization. This paper argues that most notions of flatness are problematic for deep models and can not be directly applied to explain generalization. Specifically, when focusing on deep networks with rectifier units, we can exploit the particular geometry of parameter space induced by the inherent symmetries that these architectures exhibit to build equivalent models corresponding to arbitrarily sharper minima. Furthermore, if we allow to reparametrize a function, the geometry of its parameters can change drastically without affecting its generalization properties. Model-free deep reinforcement learning algorithms have been shown to be capable of learning a wide range of robotic skills, but typically require a very large number of samples to achieve good performance. Model-based algorithms, in principle, can provide for much more efficient learning, but have proven difficult to extend to expressive, high-capacity models such as deep neural networks. In this work, we demonstrate that medium-sized neural network models can in fact be combined with model predictive control (MPC) to achieve excellent sample complexity in a model-based reinforcement learning algorithm, producing stable and plausible gaits to accomplish various complex locomotion tasks. We also propose using deep neural network dynamics models to initialize a model-free learner, in order to combine the sample efficiency of model-based approaches with the high task-specific performance of model-free methods. We empirically demonstrate on MuJoCo locomotion tasks that our pure model-based approach trained on just random action data can follow arbitrary trajectories with excellent sample efficiency, and that our hybrid algorithm can accelerate model-free learning on high-speed benchmark tasks, achieving sample efficiency gains of 3-5x on swimmer, cheetah, hopper, and ant agents. Videos can be found at this https URL Model-free deep reinforcement learning algorithms have been shown to be capable of learning a wide range of robotic skills, but typically require a very large number of samples to achieve good performance. Model-based algorithms, in principle, can provide for much more efficient learning, but have proven difficult to extend to expressive, high-capacity models such as deep neural networks. In this work, we demonstrate that neural network dynamics models can in fact be combined with model predictive control (MPC) to achieve excellent sample complexity in a model-based reinforcement learning algorithm, producing stable and plausible gaits that accomplish various complex locomotion tasks. We further propose using deep neural network dynamics models to initialize a model-free learner, in order to combine the sample efficiency of model-based approaches with the high task-specific performance of model-free methods. We empirically demonstrate on MuJoCo locomotion tasks that our pure model-based approach trained on just random action data can follow arbitrary trajectories with excellent sample efficiency, and that our hybrid algorithm can accelerate model-free learning on high-speed benchmark tasks, achieving sample efficiency gains of @math on swimmer, cheetah, hopper, and ant agents. Videos can be found at https: sites.google.com view mbmf