Online Learning Algorithms in Hilbert Spaces with $β-$ and $φ-$Mixing Sequences

In this paper, we study an online algorithm in a reproducing kernel Hilbert spaces (RKHS) based on a class of dependent processes, called the mixing process. For such a process, the degree of dependence is measured by various mixing coefficients. As a representative example, we analyze a strictly stationary Markov chain, where the dependence structure is characterized by the \(\beta-\) and \(\phi-\)mixing coefficients. For these dependent samples, we derive nearly optimal convergence rates. Our findings extend existing error bounds for i.i.d. observations, demonstrating that the i.i.d. case is a special instance of our framework. Moreover, we explicitly account for an additional factor introduced by the dependence structure in the Markov chain.

Upper Bounds for Learning in Reproducing Kernel Hilbert Spaces for Orbits of an Iterated Function System

One of the key problems in learning theory is to compute a function $f$ that closely approximates the relationship between some input $x$ and corresponding output $y$, such that $y\approx f(x)$. This approximation is based on sample points $(x_t,y_t)_{t=1}^{m}$, where the function $f$ can be approximated within reproducing kernel Hilbert spaces using various learning algorithms. In the context of learning theory, it is usually customary to assume that the sample points are drawn independently and identically distributed (i.i.d.) from an unknown underlying distribution. However, we relax this i.i.d. assumption by considering an input sequence $(x_t)_{t\in {\mathbb N}}$ as a trajectory generated by an iterated function system, which forms a particular Markov chain, with $(y_t)_{t\in {\mathbb N}}$ corresponding to an observation sequence when the model is in the corresponding state $x_t$. For such a process, we approximate the function $f$ using the Markov chain stochastic gradient algorithm and estimate the error by deriving upper bounds within reproducing kernel Hilbert spaces.

Learning Bounds for Moment-Based Domain Adaptation

Domain adaptation algorithms are designed to minimize the misclassification risk of a discriminative model for a target domain with little training data by adapting a model from a source domain with a large amount of training data. Standard approaches measure the adaptation discrepancy based on distance measures between the empirical probability distributions in the source and target domain. In this setting, we address the problem of deriving learning bounds under practice-oriented general conditions on the underlying probability distributions. As a result, we obtain learning bounds for domain adaptation based on finitely many moments and smoothness conditions.

Robust Unsupervised Domain Adaptation for Neural Networks via Moment Alignment

A novel approach for unsupervised domain adaptation for neural networks is proposed that relies on metric-based regularization of the learning process. The metric-based regularization aims at domain-invariant latent feature representations by means of maximizing the similarity between domain-specific activation distributions. The proposed metric results from modifying an integral probability metric such that it becomes translation-invariant on a polynomial function space. The metric has an intuitive interpretation in the dual space as the sum of differences of higher order central moments of the corresponding activation distributions. Error minimization guarantees are proven for the continuous case. As demonstrated by an analysis of standard benchmark experiments for sentiment analysis, object recognition and digit recognition, the outlined approach is robust regarding parameter changes and achieves higher classification accuracies than comparable approaches.

Central Moment Discrepancy (CMD) for Domain-Invariant Representation Learning

The learning of domain-invariant representations in the context of domain adaptation with neural networks is considered. We propose a new regularization method that minimizes the discrepancy between domain-specific latent feature representations directly in the hidden activation space. Although some standard distribution matching approaches exist that can be interpreted as the matching of weighted sums of moments, e.g. Maximum Mean Discrepancy (MMD), an explicit order-wise matching of higher order moments has not been considered before. We propose to match the higher order central moments of probability distributions by means of order-wise moment differences. Our model does not require computationally expensive distance and kernel matrix computations. We utilize the equivalent representation of probability distributions by moment sequences to define a new distance function, called Central Moment Discrepancy (CMD). We prove that CMD is a metric on the set of probability distributions on a compact interval. We further prove that convergence of probability distributions on compact intervals w.r.t. the new metric implies convergence in distribution of the respective random variables. We test our approach on two different benchmark data sets for object recognition (Office) and sentiment analysis of product reviews (Amazon reviews). CMD achieves a new state-of-the-art performance on most domain adaptation tasks of Office and outperforms networks trained with MMD, Variational Fair Autoencoders and Domain Adversarial Neural Networks on Amazon reviews. In addition, a post-hoc parameter sensitivity analysis shows that the new approach is stable w.r.t. parameter changes in a certain interval. The source code of the experiments is publicly available.