Information-Theoretic Characterizations of Generalization Error for the Gibbs Algorithm

Various approaches have been developed to upper bound the generalization error of a supervised learning algorithm. However, existing bounds are often loose and even vacuous when evaluated in practice. As a result, they may fail to characterize the exact generalization ability of a learning algorithm. Our main contributions are exact characterizations of the expected generalization error of the well-known Gibbs algorithm (a.k.a. Gibbs posterior) using different information measures, in particular, the symmetrized KL information between the input training samples and the output hypothesis. Our result can be applied to tighten existing expected generalization errors and PAC-Bayesian bounds. Our information-theoretic approach is versatile, as it also characterizes the generalization error of the Gibbs algorithm with a data-dependent regularizer and that of the Gibbs algorithm in the asymptotic regime, where it converges to the standard empirical risk minimization algorithm. Of particular relevance, our results highlight the role the symmetrized KL information plays in controlling the generalization error of the Gibbs algorithm

To Each Optimizer a Norm, To Each Norm its Generalization

We study the implicit regularization of optimization methods for linear models interpolating the training data in the under-parametrized and over-parametrized regimes. Since it is difficult to determine whether an optimizer converges to solutions that minimize a known norm, we flip the problem and investigate what is the corresponding norm minimized by an interpolating solution. Using this reasoning, we prove that for over-parameterized linear regression, projections onto linear spans can be used to move between different interpolating solutions. For under-parameterized linear classification, we prove that for any linear classifier separating the data, there exists a family of quadratic norms ||.||_P such that the classifier's direction is the same as that of the maximum P-margin solution. For linear classification, we argue that analyzing convergence to the standard maximum l2-margin is arbitrary and show that minimizing the norm induced by the data results in better generalization. Furthermore, for over-parameterized linear classification, projections onto the data-span enable us to use techniques from the under-parameterized setting. On the empirical side, we propose techniques to bias optimizers towards better generalizing solutions, improving their test performance. We validate our theoretical results via synthetic experiments, and use the neural tangent kernel to handle non-linear models

Хемометричні методи в розв'язанні задач якісного хімічного аналізу та класифікації фізико-хімічних даних

В монографії обговорено зміст та актуальні завдання сучасного якісного хімічного аналізу та хемометричні методи, що найчастіше використовують для обробки хіміко-аналітичних і фізико-хімічних даних. Особливу увагу приділено засобам контролю автентичності продуктів харчування і напоїв, сільсько- господарської сировини, лікарських засобів, ідентифікації об’єктів довкілля. Розглянуто застосування апарату штучних нейронних мереж та нечітких множин для розв’язання задач якісного хімічного аналізу (ідентифікації аналітів та кластеризації багатопараметричних масивів даних). Для фахівців у царинах хемометрії, якісного хімічного аналізу, фізичної хімії

Subspace Representations and Learning for Visual Recognition

Pervasive and affordable sensor and storage technology enables the acquisition of an ever-rising amount of visual data. The ability to extract semantic information by interpreting, indexing and searching visual data is impacting domains such as surveillance, robotics, intelligence, human- computer interaction, navigation, healthcare, and several others. This further stimulates the investigation of automated extraction techniques that are more efficient, and robust against the many sources of noise affecting the already complex visual data, which is carrying the semantic information of interest. We address the problem by designing novel visual data representations, based on learning data subspace decompositions that are invariant against noise, while being informative for the task at hand. We use this guiding principle to tackle several visual recognition problems, including detection and recognition of human interactions from surveillance video, face recognition in unconstrained environments, and domain generalization for object recognition.;By interpreting visual data with a simple additive noise model, we consider the subspaces spanned by the model portion (model subspace) and the noise portion (variation subspace). We observe that decomposing the variation subspace against the model subspace gives rise to the so-called parity subspace. Decomposing the model subspace against the variation subspace instead gives rise to what we name invariant subspace. We extend the use of kernel techniques for the parity subspace. This enables modeling the highly non-linear temporal trajectories describing human behavior, and performing detection and recognition of human interactions. In addition, we introduce supervised low-rank matrix decomposition techniques for learning the invariant subspace for two other tasks. We learn invariant representations for face recognition from grossly corrupted images, and we learn object recognition classifiers that are invariant to the so-called domain bias.;Extensive experiments using the benchmark datasets publicly available for each of the three tasks, show that learning representations based on subspace decompositions invariant to the sources of noise lead to results comparable or better than the state-of-the-art

Semi-supervised machine learning techniques for classification of evolving data in pattern recognition

The amount of data recorded and processed over recent years has increased exponentially. To create intelligent systems that can learn from this data, we need to be able to identify patterns hidden in the data itself, learn these pattern and predict future results based on our current observations. If we think about this system in the context of time, the data itself evolves and so does the nature of the classification problem. As more data become available, different classification algorithms are suitable for a particular setting. At the beginning of the learning cycle when we have a limited amount of data, online learning algorithms are more suitable. When truly large amounts of data become available, we need algorithms that can handle large amounts of data that might be only partially labeled as a result of the bottleneck in the learning pipeline from human labeling of the data. An excellent example of evolving data is gesture recognition, and it is present throughout our work. We need a gesture recognition system to work fast and with very few examples at the beginning. Over time, we are able to collect more data and the system can improve. As the system evolves, the user expects it to work better and not to have to become involved when the classifier is unsure about decisions. This latter situation produces additional unlabeled data. Another example of an application is medical classification, where experts’ time is a rare resource and the amount of received and labeled data disproportionately increases over time. Although the process of data evolution is continuous, we identify three main discrete areas of contribution in different scenarios. When the system is very new and not enough data are available, online learning is used to learn after every single example and to capture the knowledge very fast. With increasing amounts of data, offline learning techniques are applicable. Once the amount of data is overwhelming and the teacher cannot provide labels for all the data, we have another setup that combines labeled and unlabeled data. These three setups define our areas of contribution; and our techniques contribute in each of them with applications to pattern recognition scenarios, such as gesture recognition and sketch recognition. An online learning setup significantly restricts the range of techniques that can be used. In our case, the selected baseline technique is the Evolving TS-Fuzzy Model. The semi-supervised aspect we use is a relation between rules created by this model. Specifically, we propose a transductive similarity model that utilizes the relationship between generated rules based on their decisions about a query sample during the inference time. The activation of each of these rules is adjusted according to the transductive similarity, and the new decision is obtained using the adjusted activation. We also propose several new variations to the transductive similarity itself. Once the amount of data increases, we are not limited to the online learning setup, and we can take advantage of the offline learning scenario, which normally performs better than the online one because of the independence of sample ordering and global optimization with respect to all samples. We use generative methods to obtain data outside of the training set. Specifically, we aim to improve the previously mentioned TS Fuzzy Model by incorporating semi-supervised learning in the offline learning setup without unlabeled data. We use the Universum learning approach and have developed a method called UFuzzy. This method relies on artificially generated examples with high uncertainty (Universum set), and it adjusts the cost function of the algorithm to force the decision boundary to be close to the Universum data. We were able to prove the hypothesis behind the design of the UFuzzy classifier that Universum learning can improve the TS Fuzzy Model and have achieved improved performance on more than two dozen datasets and applications. With increasing amounts of data, we use the last scenario, in which the data comprises both labeled data and additional non-labeled data. This setting is one of the most common ones for semi-supervised learning problems. In this part of our work, we aim to improve the widely popular tecjniques of self-training (and its successor help-training) that are both meta-frameworks over regular classifier methods but require probabilistic representation of output, which can be hard to obtain in the case of discriminative classifiers. Therefore, we develop a new algorithm that uses the modified active learning technique Query-by-Committee (QbC) to sample data with high certainty from the unlabeled set and subsequently embed them into the original training set. Our new method allows us to achieve increased performance over both a range of datasets and a range of classifiers. These three works are connected by gradually relaxing the constraints on the learning setting in which we operate. Although our main motivation behind the development was to increase performance in various real-world tasks (gesture recognition, sketch recognition), we formulated our work as general methods in such a way that they can be used outside a specific application setup, the only restriction being that the underlying data evolve over time. Each of these methods can successfully exist on its own. The best setting in which they can be used is a learning problem where the data evolve over time and it is possible to discretize the evolutionary process. Overall, this work represents a significant contribution to the area of both semi-supervised learning and pattern recognition. It presents new state-of-the-art techniques that overperform baseline solutions, and it opens up new possibilities for future research