Survey on Leveraging Uncertainty Estimation Towards Trustworthy Deep Neural Networks: The Case of Reject Option and Post-training Processing

Although neural networks (especially deep neural networks) have achieved \textit{better-than-human} performance in many fields, their real-world deployment is still questionable due to the lack of awareness about the limitation in their knowledge. To incorporate such awareness in the machine learning model, prediction with reject option (also known as selective classification or classification with abstention) has been proposed in literature. In this paper, we present a systematic review of the prediction with the reject option in the context of various neural networks. To the best of our knowledge, this is the first study focusing on this aspect of neural networks. Moreover, we discuss different novel loss functions related to the reject option and post-training processing (if any) of network output for generating suitable measurements for knowledge awareness of the model. Finally, we address the application of the rejection option in reducing the prediction time for the real-time problems and present a comprehensive summary of the techniques related to the reject option in the context of extensive variety of neural networks. Our code is available on GitHub: \url{https://github.com/MehediHasanTutul/Reject_option}

On the tightness of information-theoretic bounds on generalization error of learning algorithms

A recent line of works, initiated by Russo and Xu, has shown that the generalization error of a learning algorithm can be upper bounded by information measures. In most of the relevant works, the convergence rate of the expected generalization error is in the form of $O(\sqrt{\lambda/n})$ where $\lambda$ is some information-theoretic quantities such as the mutual information or conditional mutual information between the data and the learned hypothesis. However, such a learning rate is typically considered to be ``slow", compared to a ``fast rate" of $O(\lambda/n)$ in many learning scenarios. In this work, we first show that the square root does not necessarily imply a slow rate, and a fast rate result can still be obtained using this bound under appropriate assumptions. Furthermore, we identify the critical conditions needed for the fast rate generalization error, which we call the $(\eta,c)$-central condition. Under this condition, we give information-theoretic bounds on the generalization error and excess risk, with a fast convergence rate for specific learning algorithms such as empirical risk minimization and its regularized version. Finally, several analytical examples are given to show the effectiveness of the bounds.

Multi-fidelity Gaussian Process for Biomanufacturing Process Modeling with Small Data

In biomanufacturing, developing an accurate model to simulate the complex dynamics of bioprocesses is an important yet challenging task. This is partially due to the uncertainty associated with bioprocesses, high data acquisition cost, and lack of data availability to learn complex relations in bioprocesses. To deal with these challenges, we propose to use a statistical machine learning approach, multi-fidelity Gaussian process, for process modelling in biomanufacturing. Gaussian process regression is a well-established technique based on probability theory which can naturally consider uncertainty in a dataset via Gaussian noise, and multi-fidelity techniques can make use of multiple sources of information with different levels of fidelity, thus suitable for bioprocess modeling with small data. We apply the multi-fidelity Gaussian process to solve two significant problems in biomanufacturing, bioreactor scale-up and knowledge transfer across cell lines, and demonstrate its efficacy on real-world datasets.

Enhancing Constraint Programming via Supervised Learning for Job Shop Scheduling

Constraint programming (CP) is an effective technique for solving constraint satisfaction and optimization problems. CP solvers typically use a variable ordering strategy to select which variable to explore first in the solving process, which has a large impact on the efficacy of the solvers. In this paper, we propose a novel variable ordering strategy based on supervised learning to solve job shop scheduling problems. We develop a classification model and a regression model to predict the optimal solution of a problem instance, and use the predicted solution to order variables for CP solvers. We show that training machine learning models is very efficient and can achieve a high accuracy. Our extensive experiments demonstrate that the learned variable ordering methods perform competitively compared to four existing methods. Finally, we show that hybridising the machine learning-based variable ordering methods with traditional domain-based methods is beneficial.

An Information-Theoretic Analysis for Transfer Learning: Error Bounds and Applications

Transfer learning, or domain adaptation, is concerned with machine learning problems in which training and testing data come from possibly different probability distributions. In this work, we give an information-theoretic analysis on the generalization error and excess risk of transfer learning algorithms, following a line of work initiated by Russo and Xu. Our results suggest, perhaps as expected, that the Kullback-Leibler (KL) divergence $D(\mu||\mu')$ plays an important role in the characterizations where $\mu$ and $\mu'$ denote the distribution of the training data and the testing test, respectively. Specifically, we provide generalization error upper bounds for the empirical risk minimization (ERM) algorithm where data from both distributions are available in the training phase. We further apply the analysis to approximated ERM methods such as the Gibbs algorithm and the stochastic gradient descent method. We then generalize the mutual information bound with $\phi$-divergence and Wasserstein distance. These generalizations lead to tighter bounds and can handle the case when $\mu$ is not absolutely continuous with respect to $\mu'$. Furthermore, we apply a new set of techniques to obtain an alternative upper bound which gives a fast (and optimal) learning rate for some learning problems. Finally, inspired by the derived bounds, we propose the InfoBoost algorithm in which the importance weights for source and target data are adjusted adaptively in accordance to information measures. The empirical results show the effectiveness of the proposed algorithm.

Fast Rate Generalization Error Bounds: Variations on a Theme

A recent line of works, initiated by Russo and Xu, has shown that the generalization error of a learning algorithm can be upper bounded by information measures. In most of the relevant works, the convergence rate of the expected generalization error is in the form of O(sqrt{lambda/n}) where lambda is some information-theoretic quantities such as the mutual information between the data sample and the learned hypothesis. However, such a learning rate is typically considered to be "slow", compared to a "fast rate" of O(1/n) in many learning scenarios. In this work, we first show that the square root does not necessarily imply a slow rate, and a fast rate (O(1/n)) result can still be obtained using this bound under appropriate assumptions. Furthermore, we identify the key conditions needed for the fast rate generalization error, which we call the (eta,c)-central condition. Under this condition, we give information-theoretic bounds on the generalization error and excess risk, with a convergence rate of O(\lambda/{n}) for specific learning algorithms such as empirical risk minimization. Finally, analytical examples are given to show the effectiveness of the bounds.

On Causality in Domain Adaptation and Semi-Supervised Learning: an Information-Theoretic Analysis

The establishment of the link between causality and unsupervised domain adaptation (UDA)/semi-supervised learning (SSL) has led to methodological advances in these learning problems in recent years. However, a formal theory that explains the role of causality in the generalization performance of UDA/SSL is still lacking. In this paper, we consider the UDA/SSL setting where we access m labeled source data and n unlabeled target data as training instances under a parametric probabilistic model. We study the learning performance (e.g., excess risk) of prediction in the target domain. Specifically, we distinguish two scenarios: the learning problem is called causal learning if the feature is the cause and the label is the effect, and is called anti-causal learning otherwise. We show that in causal learning, the excess risk depends on the size of the source sample at a rate of O(1/m) only if the labelling distribution between the source and target domains remains unchanged. In anti-causal learning, we show that the unlabeled data dominate the performance at a rate of typically O(1/n). Our analysis is based on the notion of potential outcome random variables and information theory. These results bring out the relationship between the data sample size and the hardness of the learning problem with different causal mechanisms.

Multi-objective Semi-supervised Clustering for Finding Predictive Clusters

This study concentrates on clustering problems and aims to find compact clusters that are informative regarding the outcome variable. The main goal is partitioning data points so that observations in each cluster are similar and the outcome variable can be predicated using these clusters simultaneously. We model this semi-supervised clustering problem as a multi-objective optimization problem with considering deviation of data points in clusters and prediction error of the outcome variable as two objective functions to be minimized. For finding optimal clustering solutions, we employ a non-dominated sorting genetic algorithm II approach and local regression is applied as prediction method for the output variable. For comparing the performance of the proposed model, we compute seven models using five real-world data sets. Furthermore, we investigate the impact of using local regression for predicting the outcome variable in all models, and examine the performance of the multi-objective models compared to single-objective models.

A Bayesian Approach to (Online) Transfer Learning: Theory and Algorithms

Transfer learning is a machine learning paradigm where knowledge from one problem is utilized to solve a new but related problem. While conceivable that knowledge from one task could be useful for solving a related task, if not executed properly, transfer learning algorithms can impair the learning performance instead of improving it -- commonly known as negative transfer. In this paper, we study transfer learning from a Bayesian perspective, where a parametric statistical model is used. Specifically, we study three variants of transfer learning problems, instantaneous, online, and time-variant transfer learning. For each problem, we define an appropriate objective function, and provide either exact expressions or upper bounds on the learning performance using information-theoretic quantities, which allow simple and explicit characterizations when the sample size becomes large. Furthermore, examples show that the derived bounds are accurate even for small sample sizes. The obtained bounds give valuable insights into the effect of prior knowledge for transfer learning, at least with respect to our Bayesian formulation of the transfer learning problem. In particular, we formally characterize the conditions under which negative transfer occurs. Lastly, we devise two (online) transfer learning algorithms that are amenable to practical implementations, one of which does not require the parametric assumption. We demonstrate the effectiveness of our algorithms with real data sets, focusing primarily on when the source and target data have strong similarities.