138 research outputs found
Online Infinite-Dimensional Regression: Learning Linear Operators
We consider the problem of learning linear operators under squared loss
between two infinite-dimensional Hilbert spaces in the online setting. We show
that the class of linear operators with uniformly bounded -Schatten norm is
online learnable for any . On the other hand, we prove an
impossibility result by showing that the class of uniformly bounded linear
operators with respect to the operator norm is \textit{not} online learnable.
Moreover, we show a separation between online uniform convergence and online
learnability by identifying a class of bounded linear operators that is online
learnable but uniform convergence does not hold. Finally, we prove that the
impossibility result and the separation between uniform convergence and
learnability also hold in the agnostic PAC setting.Comment: 17 page
The learnability of unknown quantum measurements
© Rinton Press. In this work, we provide an elegant framework to analyze learning matrices in the Schatten class by taking advantage of a recently developed methodology—matrix concentration inequalities. We establish the fat-shattering dimension, Rademacher/Gaussian complexity, and the entropy number of learning bounded operators and trace class operators. By characterising the tasks of learning quantum states and two-outcome quantum measurements into learning matrices in the Schatten-1 and ∞ classes, our proposed approach directly solves the sample complexity problems of learning quantum states and quantum measurements. Our main result in the paper is that, for learning an unknown quantum measurement, the upper bound, given by the fat-shattering dimension, is linearly proportional to the dimension of the underlying Hilbert space. Learning an unknown quantum state becomes a dual problem to ours, and as a byproduct, we can recover Aaronson’s famous result [Proc. R. Soc. A 463, 3089–3144 (2007)] solely using a classical machine learning technique. In addition, other famous complexity measures like covering numbers and Rademacher/Gaussian complexities are derived explicitly under the same framework. We are able to connect measures of sample complexity with various areas in quantum information science, e.g. quantum state/measurement tomography, quantum state discrimination and quantum random access codes, which may be of independent interest. Lastly, with the assistance of general Bloch-sphere representation, we show that learning quantum measurements/states can be mathematically formulated as a neural network. Consequently, classical ML algorithms can be applied to efficiently accomplish the two quantum learning tasks
Generalization Bounds: Perspectives from Information Theory and PAC-Bayes
A fundamental question in theoretical machine learning is generalization.
Over the past decades, the PAC-Bayesian approach has been established as a
flexible framework to address the generalization capabilities of machine
learning algorithms, and design new ones. Recently, it has garnered increased
interest due to its potential applicability for a variety of learning
algorithms, including deep neural networks. In parallel, an
information-theoretic view of generalization has developed, wherein the
relation between generalization and various information measures has been
established. This framework is intimately connected to the PAC-Bayesian
approach, and a number of results have been independently discovered in both
strands. In this monograph, we highlight this strong connection and present a
unified treatment of generalization. We present techniques and results that the
two perspectives have in common, and discuss the approaches and interpretations
that differ. In particular, we demonstrate how many proofs in the area share a
modular structure, through which the underlying ideas can be intuited. We pay
special attention to the conditional mutual information (CMI) framework;
analytical studies of the information complexity of learning algorithms; and
the application of the proposed methods to deep learning. This monograph is
intended to provide a comprehensive introduction to information-theoretic
generalization bounds and their connection to PAC-Bayes, serving as a
foundation from which the most recent developments are accessible. It is aimed
broadly towards researchers with an interest in generalization and theoretical
machine learning.Comment: 222 page
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