Effective regularisation of neural networks is essential to combat overfitting due to the large number of parameters involved. We present an empirical analogue to the Lipschitz constant of a feed-forward neural network, which we refer to as the maximum gain. We hypothesise that constraining the gain of a network will have a regularising effect, similar to how constraining the Lipschitz constant of a network has been shown to improve generalisation. A simple algorithm is provided that involves rescaling the weight matrix of each layer after each parameter update. We conduct a series of studies on common benchmark datasets, and also a novel dataset that we introduce to enable easier significance testing for experiments using convolutional networks. Performance on these datasets compares favourably with other common regularisation techniques.
The idea of constraining the Lipschitz constant of a network is conceptually related to quantifying the flatness of minima. While there is no single formalisation for what constitutes a flat minimum, the unifying intuition is that a minimum is flat when a small perturbation of the model parameters does not have a large impact on the performance of the model. @cite_18 have shown that Lipschitz continuity is not a reliable tool for quantifying the flatness of minima. However, there is a subtle but very important difference between how they employ Lipschitz continuity, and how it is used by @cite_13 and in this work. Neural networks are functions parameterised by two distinct sets of variables: the model parameters, and the features. @cite_18 consider Lipschitz continuity with respect to the model parameters, whereas we consider Lipschitz continuity with respect the features being supplied to the network. The crux of the argument given by is that the Lipschitz constant of a network with respect to its weights is not invariant to reparameterisation.
@cite_18 @cite_13
2796892552 2709553318 2963664410 1899249567 2547508527 2759973161 2962756315 2946521496 1498932870 1988485873 2885062394 2962720813 2519655882 2042249109 2964292098 2766153148 2580360036 2162151148 2964340589 2799107510
We investigate the effect of explicitly enforcing the Lipschitz continuity of neural networks with respect to their inputs. To this end, we provide a simple technique for computing an upper bound to the Lipschitz constant of a feed forward neural network composed of commonly used layer types and demonstrate inaccuracies in previous work on this topic. Our technique is then used to formulate training a neural network with a bounded Lipschitz constant as a constrained optimisation problem that can be solved using projected stochastic gradient methods. Our evaluation study shows that, in isolation, our method performs comparatively to state-of-the-art regularisation techniques. Moreover, when combined with existing approaches to regularising neural networks the performance gains are cumulative. We also provide evidence that the hyperparameters are intuitive to tune and demonstrate how the choice of norm for computing the Lipschitz constant impacts the resulting model. This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for a standard AlexNet network trained with SGD on the mnist and cifar10 datasets, with both original and random labels; the bound, the Lipschitz constants, and the excess risks are all in direct correlation, suggesting both that SGD selects predictors whose complexity scales with the difficulty of the learning task, and secondly that the presented bound is sensitive to this complexity. This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for a standard AlexNet network trained with SGD on the MNIST and CIFAR10 datasets, with both original and random labels; the bound, the Lipschitz constants, and the excess risks are all in direct correlation, suggesting both that SGD selects predictors whose complexity scales with the difficulty of the learning task, and secondly that the presented bound is sensitive to this complexity. We study the connection between the highly non-convex loss function of a simple model of the fully-connected feed-forward neural network and the Hamiltonian of the spherical spin-glass model under the assumptions of: i) variable independence, ii) redundancy in network parametrization, and iii) uniformity. These assumptions enable us to explain the complexity of the fully decoupled neural network through the prism of the results from random matrix theory. We show that for large-size decoupled networks the lowest critical values of the random loss function form a layered structure and they are located in a well-defined band lower-bounded by the global minimum. The number of local minima outside that band diminishes exponentially with the size of the network. We empirically verify that the mathematical model exhibits similar behavior as the computer simulations, despite the presence of high dependencies in real networks. We conjecture that both simulated annealing and SGD converge to the band of low critical points, and that all critical points found there are local minima of high quality measured by the test error. This emphasizes a major difference between largeand small-size networks where for the latter poor quality local minima have nonzero probability of being recovered. Finally, we prove that recovering the global minimum becomes harder as the network size increases and that it is in practice irrelevant as global minimum often leads to overfitting. We present an algorithm for the statistical learning setting with a bounded exp-concave loss in @math dimensions that obtains excess risk @math with probability at least @math . The core technique is to boost the confidence of recent in-expectation @math excess risk bounds for empirical risk minimization (ERM), without sacrificing the rate, by leveraging a Bernstein condition which holds due to exp-concavity. We also show that with probability @math the standard ERM method obtains excess risk @math . We further show that a regret bound for any online learner in this setting translates to a high probability excess risk bound for the corresponding online-to-batch conversion of the online learner. Lastly, we present two high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. The first bound is a purely exponential weights type algorithm, obtains a nearly optimal rate, and has no explicit dependence on the Lipschitz continuity of the loss. The second bound requires Lipschitz continuity but obtains the optimal rate. Since their invention, generative adversarial networks (GANs) have become a popular approach for learning to model a distribution of real (unlabeled) data. Convergence problems during training are overcome by Wasserstein GANs which minimize the distance between the model and the empirical distribution in terms of a different metric, but thereby introduce a Lipschitz constraint into the optimization problem. A simple way to enforce the Lipschitz constraint on the class of functions, which can be modeled by the neural network, is weight clipping. It was proposed that training can be improved by instead augmenting the loss by a regularization term that penalizes the deviation of the gradient of the critic (as a function of the network's input) from one. We present theoretical arguments why using a weaker regularization term enforcing the Lipschitz constraint is preferable. These arguments are supported by experimental results on toy data sets. In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in O(1 √ t ), the structure of the communication network only impacts a second-order term in O(1 t), where t is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a d1 4 multiplicative factor of the optimal convergence rate, where d is the underlying dimension. We present a theoretical and experimental investigation of the quantization problem for artificial neural networks. We provide a mathematical definition of quantized neural networks and analyze their approximation capabilities, showing in particular that any Lipschitz-continuous map defined on a hypercube can be uniformly approximated by a quantized neural network. We then focus on the regularization effect of additive noise on the arguments of multi-step functions inherent to the quantization of continuous variables. In particular, when the expectation operator is applied to a non-differentiable multi-step random function, and if the underlying probability density is differentiable (in either classical or weak sense), then a differentiable function is retrieved, with explicit bounds on its Lipschitz constant. Based on these results, we propose a novel gradient-based training algorithm for quantized neural networks that generalizes the straight-through estimator, acting on noise applied to the network's parameters. We evaluate our algorithm on the CIFAR-10 and ImageNet image classification benchmarks, showing state-of-the-art performance on AlexNet and MobileNetV2 for ternary networks. Ability of deep networks to extract high level features and of recurrent networks to perform time-series inference have been studied. In view of universality of one hidden layer network at approximating functions under weak constraints, the benefit of multiple layers is to enlarge the space of dynamical systems approximated or, given the space, reduce the number of units required for a certain error. Traditionally shallow networks with manually engineered features are used, back-propagation extent is limited to one and attempt to choose a large number of hidden units to satisfy the Markov condition is made. In case of Markov models, it has been shown that many systems need to be modeled as higher order. In the present work, we present deep recurrent networks with longer backpropagation through time extent as a solution to modeling systems that are high order and to predicting ahead. We study epileptic seizure suppression electro-stimulator. Extraction of manually engineered complex features and prediction employing them has not allowed small low-power implementations as, to avoid possibility of surgery, extraction of any features that may be required has to be included. In this solution, a recurrent neural network performs both feature extraction and prediction. We prove analytically that adding hidden layers or increasing backpropagation extent increases the rate of decrease of approximation error. A Dynamic Programming (DP) training procedure employing matrix operations is derived. DP and use of matrix operations makes the procedure efficient particularly when using data-parallel computing. The simulation studies show the geometry of the parameter space, that the network learns the temporal structure, that parameters converge while model output displays same dynamic behavior as the system and greater than .99 Average Detection Rate on all real seizure data tried. Local minima and plateaus pose a serious problem in learning of neural networks. We investigate the hierarchical geometric structure of the parameter space of three-layer perceptrons in order to show the existence of local minima and plateaus. It is proved that a critical point of the model with H−1 hidden units always gives many critical points of the model with H hidden units. These critical points consist of many lines in the parameter space, which can cause plateaus in learning of neural networks. Based on this result, we prove that a point in the critical lines corresponding to the global minimum of the smaller model can be a local minimum or a saddle point of the larger model. We give a necessary and sufficient condition for this, and show that this kind of local minima exist as a line segment if any. The results are universal in the sense that they do not require special properties of the target, loss functions and activation functions, but only use the hierarchical structure of the model. Multi-layer neural networks have lead to remarkable performance on many kinds of benchmark tasks in text, speech and image processing. Nonlinear parameter estimation in hierarchical models is known to be subject to overfitting and misspecification. One approach to these estimation and related problems (local minima, colinearity, feature discovery etc.) is called Dropout (Hinton, et al 2012, 2016). The Dropout algorithm removes hidden units according to a Bernoulli random variable with probability @math prior to each update, creating random "shocks" to the network that are averaged over updates. In this paper we will show that Dropout is a special case of a more general model published originally in 1990 called the Stochastic Delta Rule, or SDR (Hanson, 1990). SDR redefines each weight in the network as a random variable with mean @math and standard deviation @math . Each weight random variable is sampled on each forward activation, consequently creating an exponential number of potential networks with shared weights. Both parameters are updated according to prediction error, thus resulting in weight noise injections that reflect a local history of prediction error and local model averaging. SDR therefore implements a more sensitive local gradient-dependent simulated annealing per weight converging in the limit to a Bayes optimal network. Tests on standard benchmarks (CIFAR) using a modified version of DenseNet shows the SDR outperforms standard Dropout in test error by approx. @math with DenseNet-BC 250 on CIFAR-100 and approx. @math in smaller networks. We also show that SDR reaches the same accuracy that Dropout attains in 100 epochs in as few as 35 epochs. Given a weighted graph with N vertices, consider a real-valued regression problem in a semisupervised setting, where one observes n labeled vertices, and the task is to label the remaining ones. We present a theoretical study of p-based Laplacian regularization under a d-dimensional geometric random graph model. We provide a variational characterization of the performance of this regularized learner as N grows to infinity while n stays constant; the associated optimality conditions lead to a partial differential equation that must be satisfied by the associated function estimate f . From this formulation we derive several predictions on the limiting behavior the function f , including (a) a phase transition in its smoothness at the threshold p = d + 1; and (b) a tradeoff between smoothness and sensitivity to the underlying unlabeled data distribution P . Thus, over the range p ≤ d, the function estimate f is degenerate and “spiky,” whereas for p ≥ d + 1, the function estimate f is smooth. We show that the effect of the underlying density vanishes monotonically with p, such that in the limit p = ∞, corresponding to the so-called Absolutely Minimal Lipschitz Extension, the estimate f is independent of the distribution P . Under the assumption of semi-supervised smoothness, ignoring P can lead to poor statistical performance; in particular, we construct a specific example for d = 1 to demonstrate that p = 2 has lower risk than p =∞ due to the former penalty adapting to P and the latter ignoring it. We also provide simulations that verify the accuracy of our predictions for finite sample sizes. Together, these properties show that p = d + 1 is an optimal choice, yielding a function estimate f that is both smooth and non-degenerate, while remaining maximally sensitive to P . We present a theoretically grounded approach to train deep neural networks, including recurrent networks, subject to class-dependent label noise. We propose two procedures for loss correction that are agnostic to both application domain and network architecture. They simply amount to at most a matrix inversion and multiplication, provided that we know the probability of each class being corrupted into another. We further show how one can estimate these probabilities, adapting a recent technique for noise estimation to the multi-class setting, and thus providing an end-to-end framework. Extensive experiments on MNIST, IMDB, CIFAR-10, CIFAR-100 and a large scale dataset of clothing images employing a diversity of architectures --- stacking dense, convolutional, pooling, dropout, batch normalization, word embedding, LSTM and residual layers --- demonstrate the noise robustness of our proposals. Incidentally, we also prove that, when ReLU is the only non-linearity, the loss curvature is immune to class-dependent label noise. There has been a long history of using neural networks for combinatorial optimization and constraint satisfaction problems. Symmetric Hopfield networks and similar approaches use steepest descent dynamics, and they always converge to the closest local minimum of the energy landscape. For finding global minima additional parameter-sensitive techniques are used, such as classical simulated annealing or the so-called chaotic simulated annealing, which induces chaotic dynamics by addition of extra terms to the energy landscape. Here we show that asymmetric continuous-time neural networks can solve constraint satisfaction problems without getting trapped in non-solution attractors. We concentrate on a model solving Boolean satisfiability (k-SAT), which is a quintessential NP-complete problem. There is a one-to-one correspondence between the stable fixed points of the neural network and the k-SAT solutions and we present numerical evidence that limit cycles may also be avoided by appropriately choosing the parameters of the model. This optimal parameter region is fairly independent of the size and hardness of instances, this way parameters can be chosen independently of the properties of problems and no tuning is required during the dynamical process. The model is similar to cellular neural networks already used in CNN computers. On an analog device solving a SAT problem would take a single operation: the connection weights are determined by the k-SAT instance and starting from any initial condition the system searches until finding a solution. In this new approach transient chaotic behavior appears as a natural consequence of optimization hardness and not as an externally induced effect. We present a theoretically grounded approach to train deep neural networks, including recurrent networks, subject to class-dependent label noise. We propose two procedures for loss correction that are agnostic to both application domain and network architecture. They simply amount to at most a matrix inversion and multiplication, provided that we know the probability of each class being corrupted into another. We further show how one can estimate these probabilities, adapting a recent technique for noise estimation to the multi-class setting, and thus providing an end-to-end framework. Extensive experiments on MNIST, IMDB, CIFAR-10, CIFAR-100 and a large scale dataset of clothing images employing a diversity of architectures &#x2014; stacking dense, convolutional, pooling, dropout, batch normalization, word embedding, LSTM and residual layers &#x2014; demonstrate the noise robustness of our proposals. Incidentally, we also prove that, when ReLU is the only non-linearity, the loss curvature is immune to class-dependent label noise. The usual approach to developing and analyzing first-order methods for non-smooth (stochastic or deterministic) convex optimization assumes that the objective function is uniformly Lipschitz continuous with parameter @math . However, in many settings the non-differentiable convex function @math is not uniformly Lipschitz continuous -- for example (i) the classical support vector machine (SVM) problem, (ii) the problem of minimizing the maximum of convex quadratic functions, and even (iii) the univariate setting with @math . Herein we develop a notion of "relative continuity" that is determined relative to a user-specified "reference function" @math (that should be computationally tractable for algorithms), and we show that many non-differentiable convex functions are relatively continuous with respect to a correspondingly fairly-simple reference function @math . We also similarly develop a notion of "relative stochastic continuity" for the stochastic setting. We analysis two standard algorithms -- the (deterministic) mirror descent algorithm and the stochastic mirror descent algorithm -- for solving optimization problems in these two new settings, and we develop for the first time computational guarantees for instances where the objective function is not uniformly Lipschitz continuous. This paper is a companion paper for non-differentiable convex optimization to the recent paper by Lu, Freund, and Nesterov, which developed similar sorts of results for differentiable convex optimization. In this paper, we present the Lipschitz regularization theory and algorithms for a novel Loss-Sensitive Generative Adversarial Network (LS-GAN). Specifically, it trains a loss function to distinguish between real and fake samples by designated margins, while learning a generator alternately to produce realistic samples by minimizing their losses. The LS-GAN further regularizes its loss function with a Lipschitz regularity condition on the density of real data, yielding a regularized model that can better generalize to produce new data from a reasonable number of training examples than the classic GAN. We will further present a Generalized LS-GAN (GLS-GAN) and show it contains a large family of regularized GAN models, including both LS-GAN and Wasserstein GAN, as its special cases. Compared with the other GAN models, we will conduct experiments to show both LS-GAN and GLS-GAN exhibit competitive ability in generating new images in terms of the Minimum Reconstruction Error (MRE) assessed on a separate test set. We further extend the LS-GAN to a conditional form for supervised and semi-supervised learning problems, and demonstrate its outstanding performance on image classification tasks. We introduce a new class of multifunctions whose graphs under certain "kernel inverting" matrices, are locally equal to the graphs of Lipschitzian (single-valued) mappings. We characterize the existence of Lipschitzian localizations of these multifunctions in terms of a natural condition on a generalized Jacobian mapping. One corollary to our main result is a Lipschitzian inverse mapping theorem for the broad class of "max hypomonotone" multifunctions. We apply our theoretical results to the sensitivity analysis of solution mappings associated with parameterized optimization problems. In particular, we obtain new characterizations of the Lipschitzian stability of stationary points and Karush-Kuhn-Tucker pairs associated with parameterized nonlinear programs. Conventional convolutional neural networks use either a linear or a nonlinear filter to extract features from an image patch (region) of spatial size @math (typically, @math is small and is equal to @math , e.g., @math is 5 or 7). Generally, the size of the filter is equal to the size @math of the input patch. We argue that the representational ability of equal-size strategy is not strong enough. To overcome the drawback, we propose to use subpatch filter whose spatial size @math is smaller than @math . The proposed subpatch filter consists of two subsequent filters. The first one is a linear filter of spatial size @math and is aimed at extracting features from spatial domain. The second one is of spatial size @math and is used for strengthening the connection between different input feature channels and for reducing the number of parameters. The subpatch filter convolves with the input patch and the resulting network is called a subpatch network. Taking the output of one subpatch network as input, we further repeat constructing subpatch networks until the output contains only one neuron in spatial domain. These subpatch networks form a new network called the cascaded subpatch network (CSNet). The feature layer generated by CSNet is called the csconv layer. For the whole input image, we construct a deep neural network by stacking a sequence of csconv layers. Experimental results on five benchmark data sets demonstrate the effectiveness and compactness of the proposed CSNet. For example, our CSNet reaches a test error of 5.68 on the CIFAR10 data set without model averaging. To the best of our knowledge, this is the best result ever obtained on the CIFAR10 data set. Verifying the robustness property of a general Rectified Linear Unit (ReLU) network is an NP-complete problem [Katz, Barrett, Dill, Julian and Kochenderfer CAV17]. Although finding the exact minimum adversarial distortion is hard, giving a certified lower bound of the minimum distortion is possible. Current available methods of computing such a bound are either time-consuming or delivering low quality bounds that are too loose to be useful. In this paper, we exploit the special structure of ReLU networks and provide two computationally efficient algorithms Fast-Lin and Fast-Lip that are able to certify non-trivial lower bounds of minimum distortions, by bounding the ReLU units with appropriate linear functions Fast-Lin, or by bounding the local Lipschitz constant Fast-Lip. Experiments show that (1) our proposed methods deliver bounds close to (the gap is 2-3X) exact minimum distortion found by Reluplex in small MNIST networks while our algorithms are more than 10,000 times faster; (2) our methods deliver similar quality of bounds (the gap is within 35 and usually around 10 ; sometimes our bounds are even better) for larger networks compared to the methods based on solving linear programming problems but our algorithms are 33-14,000 times faster; (3) our method is capable of solving large MNIST and CIFAR networks up to 7 layers with more than 10,000 neurons within tens of seconds on a single CPU core. In addition, we show that, in fact, there is no polynomial time algorithm that can approximately find the minimum @math adversarial distortion of a ReLU network with a @math approximation ratio unless @math = @math , where @math is the number of neurons in the network.