Quantum learning: optimal classification of qubit states

Pattern recognition is a central topic in Learning Theory with numerous applications such as voice and text recognition, image analysis, computer diagnosis. The statistical set-up in classification is the following: we are given an i.i.d. training set $(X_{1},Y_{1}),... (X_{n},Y_{n})$ where $X_{i}$ represents a feature and $Y_{i}\in \{0,1\}$ is a label attached to that feature. The underlying joint distribution of $(X,Y)$ is unknown, but we can learn about it from the training set and we aim at devising low error classifiers $f:X\to Y$ used to predict the label of new incoming features. Here we solve a quantum analogue of this problem, namely the classification of two arbitrary unknown qubit states. Given a number of `training' copies from each of the states, we would like to `learn' about them by performing a measurement on the training set. The outcome is then used to design mesurements for the classification of future systems with unknown labels. We find the asymptotically optimal classification strategy and show that typically, it performs strictly better than a plug-in strategy based on state estimation. The figure of merit is the excess risk which is the difference between the probability of error and the probability of error of the optimal measurement when the states are known, that is the Helstrom measurement. We show that the excess risk has rate $n^{-1}$ and compute the exact constant of the rate.Comment: 24 pages, 4 figure

Generative Adversarial Networks for Mitigating Biases in Machine Learning Systems

In this paper, we propose a new framework for mitigating biases in machine learning systems. The problem of the existing mitigation approaches is that they are model-oriented in the sense that they focus on tuning the training algorithms to produce fair results, while overlooking the fact that the training data can itself be the main reason for biased outcomes. Technically speaking, two essential limitations can be found in such model-based approaches: 1) the mitigation cannot be achieved without degrading the accuracy of the machine learning models, and 2) when the data used for training are largely biased, the training time automatically increases so as to find suitable learning parameters that help produce fair results. To address these shortcomings, we propose in this work a new framework that can largely mitigate the biases and discriminations in machine learning systems while at the same time enhancing the prediction accuracy of these systems. The proposed framework is based on conditional Generative Adversarial Networks (cGANs), which are used to generate new synthetic fair data with selective properties from the original data. We also propose a framework for analyzing data biases, which is important for understanding the amount and type of data that need to be synthetically sampled and labeled for each population group. Experimental results show that the proposed solution can efficiently mitigate different types of biases, while at the same time enhancing the prediction accuracy of the underlying machine learning model