Expected Generalization Error Research Articles

A data-driven data-augmentation method based on Slim-generative Adversarial Imputation Networks for short-term ship-motion attitude prediction

The performance of data-driven complex ship-motion attitude prediction models not only depends on the structure and algorithm design of the model but is also affected by the quantity and quality of training samples. Insufficient available data, resulting from factors such as low sampling rates and high costs, could inhibit the achievement of optimal prediction accuracy. This limitation in turn increases the model’s expected generalization error of the model and causes overfitting in regression models. Nevertheless, by artificially expanding the training dataset, data-augmentation methods enhance the resistance of the model to overfitting, thereby improving the prediction performance. This study introduces a novel data-driven model for ship-motion prediction based on the principles of Slim Generative Adversarial Imputation Networks (SGAINs). The model not only increases the amount of data but also considers temporal correlation features in the enhancement process, thus uniquely addressing the issue of poor accuracy of prediction models caused by insufficient training datasets. The simulation results demonstrate the effectiveness and superiority of the proposed model for short-term ship-motion attitude data augmentation.

Going Deeper, Generalizing Better: An Information-Theoretic View for Deep Learning.

Deep learning has transformed computer vision, natural language processing, and speech recognition. However, two critical questions remain obscure: 1) why do deep neural networks (DNNs) generalize better than shallow networks and 2) does it always hold that a deeper network leads to better performance? In this article, we first show that the expected generalization error of neural networks (NNs) can be upper bounded by the mutual information between the learned features in the last hidden layer and the parameters of the output layer. This bound further implies that as the number of layers increases in the network, the expected generalization error will decrease under mild conditions. Layers with strict information loss, such as the convolutional or pooling layers, reduce the generalization error for the whole network; this answers the first question. However, algorithms with zero expected generalization error do not imply a small test error. This is because the expected training error is large when the information for fitting the data is lost as the number of layers increases. This suggests that the claim "the deeper the better" is conditioned on a small training error. Finally, we show that deep learning satisfies a weak notion of stability and provides some generalization error bounds for noisy stochastic gradient decent (SGD) and binary classification in DNNs.

Learning Maximum Margin Channel Decoders

The problem of learning a channel decoder is considered for two channel models. The first model is an additive noise channel whose noise distribution is unknown and nonparametric. The learner is provided with a fixed codebook and a dataset comprised of independent samples of the noise, and is required to select a precision matrix for a nearest neighbor decoder in terms of the Mahalanobis distance. The second model is a non-linear channel with additive white Gaussian noise and unknown channel transformation. The learner is provided with a fixed codebook and a dataset comprised of independent input-output samples of the channel, and is required to select a matrix for a nearest neighbor decoder with a linear kernel. For both models, the objective of maximizing the margin of the decoder is addressed. Accordingly, for each channel model, a regularized loss minimization problem with a codebook-related regularization term and hinge-like loss function is developed, which is inspired by the support vector machine paradigm for classification problems. Expected generalization error bounds for the error probability loss function are provided for both models, under optimal choice of the regularization parameter. For the additive noise channel, a theoretical guidance for choosing the training signal-to-noise ratio is proposed based on this bound. In addition, for the non-linear channel, a high probability uniform generalization error bound is provided for the hypothesis class. For each channel, a stochastic sub-gradient descent algorithm for solving the regularized loss minimization problem is proposed, and an optimization error bound is stated. The performance of the proposed algorithms is demonstrated through several examples.

Deep Learning approximation of diffeomorphisms via linear-control systems

<p style='text-indent:20px;'>In this paper we propose a Deep Learning architecture to approximate diffeomorphisms diffeotopic to the identity. We consider a control system of the form <inline-formula><tex-math id="M3">\begin{document}$ \dot x = \sum_{i = 1}^lF_i(x)u_i $\end{document}</tex-math></inline-formula>, with linear dependence in the controls, and we use the corresponding flow to approximate the action of a diffeomorphism on a compact ensemble of points. Despite the simplicity of the control system, it has been recently shown that a Universal Approximation Property holds. The problem of minimizing the sum of the training error and of a regularizing term induces a gradient flow in the space of admissible controls. A possible training procedure for the discrete-time neural network consists in projecting the gradient flow onto a finite-dimensional subspace of the admissible controls. An alternative approach relies on an iterative method based on Pontryagin Maximum Principle for the numerical resolution of Optimal Control problems. Here the maximization of the Hamiltonian can be carried out with an extremely low computational effort, owing to the linear dependence of the system in the control variables. Finally, we use tools from <inline-formula><tex-math id="M4">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence to provide an estimate of the expected generalization error.</p>

Improved Information-Theoretic Generalization Bounds for Distributed, Federated, and Iterative Learning.

We consider information-theoretic bounds on the expected generalization error for statistical learning problems in a network setting. In this setting, there are K nodes, each with its own independent dataset, and the models from the K nodes have to be aggregated into a final centralized model. We consider both simple averaging of the models as well as more complicated multi-round algorithms. We give upper bounds on the expected generalization error for a variety of problems, such as those with Bregman divergence or Lipschitz continuous losses, that demonstrate an improved dependence of on the number of nodes. These “per node” bounds are in terms of the mutual information between the training dataset and the trained weights at each node and are therefore useful in describing the generalization properties inherent to having communication or privacy constraints at each node.

Upper Bounds on the Generalization Error of Private Algorithms for Discrete Data

In this work, we study the generalization capability of algorithms from an information-theoretic perspective. It has been shown that the expected generalization error of an algorithm is bounded from above by a function of the relative entropy between the conditional probability distribution of the algorithm's output hypothesis, given the dataset with which it was trained, and its marginal probability distribution. We build upon this fact and introduce a mathematical formulation to obtain upper bounds on this relative entropy. Assuming that the data is discrete, we then develop a strategy using this formulation, based on the method of types and typicality, to find explicit upper bounds on the generalization error of stable algorithms, i.e., algorithms that produce similar output hypotheses given similar input datasets. In particular, we show the bounds obtained with this strategy for the case of $\epsilon$-DP and $\mu$-GDP algorithms.

Competitive Caching with Machine Learned Advice

Traditional online algorithms encapsulate decision making under uncertainty, and give ways to hedge against all possible future events, while guaranteeing a nearly optimal solution, as compared to an offline optimum. On the other hand, machine learning algorithms are in the business of extrapolating patterns found in the data to predict the future, and usually come with strong guarantees on the expected generalization error. In this work, we develop a framework for augmenting online algorithms with a machine learned predictor to achieve competitive ratios that provably improve upon unconditional worst-case lower bounds when the predictor has low error. Our approach treats the predictor as a complete black box and is not dependent on its inner workings or the exact distribution of its errors. We apply this framework to the traditional caching problem—creating an eviction strategy for a cache of size k . We demonstrate that naively following the oracle’s recommendations may lead to very poor performance, even when the average error is quite low. Instead, we show how to modify the Marker algorithm to take into account the predictions and prove that this combined approach achieves a competitive ratio that both (i) decreases as the predictor’s error decreases and (ii) is always capped by O (log k ), which can be achieved without any assistance from the predictor. We complement our results with an empirical evaluation of our algorithm on real-world datasets and show that it performs well empirically even when using simple off-the-shelf predictions.

A simulation-approximation approach to sample size planning for high-dimensional classification studies

Classification studies with high-dimensional measurements and relatively small sample sizes are increasingly common. Prospective analysis of the role of sample sizes in the performance of such studies is important for study design and interpretation of results, but the complexity of typical pattern discovery methods makes this problem challenging. The approach developed here combines Monte Carlo methods and new approximations for linear discriminant analysis, assuming multivariate normal distributions. Monte Carlo methods are used to sample the distribution of which features are selected for a classifier and the mean and variance of features given that they are selected. Given selected features, the linear discriminant problem involves different distributions of training data and generalization data, for which 2 approximations are compared: one based on Taylor series approximation of the generalization error and the other on approximating the discriminant scores as normally distributed. Combining the Monte Carlo and approximation approaches to different aspects of the problem allows efficient estimation of expected generalization error without full simulations of the entire sampling and analysis process. To evaluate the method and investigate realistic study design questions, full simulations are used to ask how validation error rate depends on the strength and number of informative features, the number of noninformative features, the sample size, and the number of features allowed into the pattern. Both approximation methods perform well for most cases but only the normal discriminant score approximation performs well for cases of very many weakly informative or uninformative dimensions. The simulated cases show that many realistic study designs will typically estimate substantially suboptimal patterns and may have low probability of statistically significant validation results.

Generalization of finite size Boolean perceptrons with genetic algorithms

We present an investigation of the generalization ability of finite size perceptrons with binary couplings. The results for the expected generalization error provide a guide for practical applications by establishing limits for the learning capacity of finite systems. The method applied to find solutions was the genetic algorithm, which showed to be efficient, even for values of α larger then the Gardner–Derrida storage capacity α GD = 1.245 , for which the number of solutions is largely reduced. We show that the generalization error of finite size networks for α up to α GD coincides with the value calculated through the statistical mechanical analysis in the thermodynamic limit.

On the problem in model selection of neural network regression in overrealizable scenario.

In considering a statistical model selection of neural networks and radial basis functions under an overrealizable case, the problem of unidentifiability emerges. Because the model selection criterion is an unbiased estimator of the generalization error based on the training error, this article analyzes the expected training error and the expected generalization error of neural networks and radial basis functions in overrealizable cases and clarifies the difference from regular models, for which identifiability holds. As a special case of an overrealizable scenario, we assumed a gaussian noise sequence as training data. In the least-squares estimation under this assumption, we first formulated the problem, in which the calculation of the expected errors of unidentifiable networks is reduced to the calculation of the expectation of the supremum of the chi2 process. Under this formulation, we gave an upper bound of the expected training error and a lower bound of the expected generalization error, where the generalization is measured at a set of training inputs. Furthermore, we gave stochastic bounds on the training error and the generalization error. The obtained upper bound of the expected training error is smaller than in regular models, and the lower bound of the expected generalization error is larger than in regular models. The result tells us that the degree of overfitting in neural networks and radial basis functions is higher than in regular models. Correspondingly, it also tells us that the generalization capability is worse than in the case of regular models. The article may be enough to show a difference between neural networks and regular models in the context of the least-squares estimation in a simple situation. This is a first step in constructing a model selection criterion in an overrealizable case. Further important problems in this direction are also included in this article.

Model Selection for Small Sample Regression

Model selection is an important ingredient of many machine learning algorithms, in particular when the sample size in small, in order to strike the right trade-off between overfitting and underfitting. Previous classical results for linear regression are based on an asymptotic analysis. We present a new penalization method for performing model selection for regression that is appropriate even for small samples. Our penalization is based on an accurate estimator of the ratio of the expected training error and the expected generalization error, in terms of the expected eigenvalues of the input covariance matrix.

No free lunch for early stopping.

We show that with a uniform prior on models having the same training error, early stopping at some fixed training error above the training error minimum results in an increase in the expected generalization error.