A review of single-source unsupervised domain adaptation <s> Generalization error <s> Empirical risk minimization offers well-known learning guarantees when training and test data come from the same domain. In the real world, though, we often wish to adapt a classifier from a source domain with a large amount of training data to different target domain with very little training data. In this work we give uniform convergence bounds for algorithms that minimize a convex combination of source and target empirical risk. The bounds explicitly model the inherent trade-off between training on a large but inaccurate source data set and a small but accurate target training set. Our theory also gives results when we have multiple source domains, each of which may have a different number of instances, and we exhibit cases in which minimizing a non-uniform combination of source risks can achieve much lower target error than standard empirical risk minimization. <s> BIB001 </s> A review of single-source unsupervised domain adaptation <s> Generalization error <s> Discriminative learning methods for classification perform well when training and test data are drawn from the same distribution. Often, however, we have plentiful labeled training data from a source domain but wish to learn a classifier which performs well on a target domain with a different distribution and little or no labeled training data. In this work we investigate two questions. First, under what conditions can a classifier trained from source data be expected to perform well on target data? Second, given a small amount of labeled target data, how should we combine it during training with the large amount of labeled source data to achieve the lowest target error at test time?We address the first question by bounding a classifier’s target error in terms of its source error and the divergence between the two domains. We give a classifier-induced divergence measure that can be estimated from finite, unlabeled samples from the domains. Under the assumption that there exists some hypothesis that performs well in both domains, we show that this quantity together with the empirical source error characterize the target error of a source-trained classifier.We answer the second question by bounding the target error of a model which minimizes a convex combination of the empirical source and target errors. Previous theoretical work has considered minimizing just the source error, just the target error, or weighting instances from the two domains equally. We show how to choose the optimal combination of source and target error as a function of the divergence, the sample sizes of both domains, and the complexity of the hypothesis class. The resulting bound generalizes the previously studied cases and is always at least as tight as a bound which considers minimizing only the target error or an equal weighting of source and target errors. <s> BIB002 </s> A review of single-source unsupervised domain adaptation <s> Generalization error <s> This paper addresses the general problem of domain adaptation which arises in a variety of applications where the distribution of the labeled sample available somewhat differs from that of the test data. Building on previous work by Ben-David et al. (2007), we introduce a novel distance between distributions, discrepancy distance, that is tailored to adaptation problems with arbitrary loss functions. We give Rademacher complexity bounds for estimating the discrepancy distance from finite samples for different loss functions. Using this distance, we derive new generalization bounds for domain adaptation for a wide family of loss functions. We also present a series of novel adaptation bounds for large classes of regularization-based algorithms, including support vector machines and kernel ridge regression based on the empirical discrepancy. This motivates our analysis of the problem of minimizing the empirical discrepancy for various loss functions for which we also give several algorithms. We report the results of preliminary experiments that demonstrate the benefits of our discrepancy minimization algorithms for domain adaptation. <s> BIB003 </s> A review of single-source unsupervised domain adaptation <s> Generalization error <s> We provide a first PAC-Bayesian analysis for domain adaptation (DA) which arises when the learning and test distributions differ. It relies on a novel distribution pseudodistance based on a disagreement averaging. Using this measure, we derive a PAC-Bayesian DA bound for the stochastic Gibbs classifier. This bound has the advantage of being directly optimizable for any hypothesis space. We specialize it to linear classifiers, and design a learning algorithm which shows interesting results on a synthetic problem and on a popular sentiment annotation task. This opens the door to tackling DA tasks by making use of all the PAC-Bayesian tools. <s> BIB004
If the difference between the source risk and the target risk can be bounded, then new insights can be obtained into when a classifier will generalize from a source to a target domain. Components of bounds can often be exploited to design new adaptive classifiers. We will first discuss a generalization error bound that looks at the target error of an unadaptive source classifier. In later sections, we discuss bounds for adaptive classifiers. One of the first cross-domain generalization error bounds is based on the error of the ideal joint hypothesis: e * S,T = min h∈H [e S (h) + e T (h)] BIB002 ,  , BIB001 . The difference between the true target error of a source classifier e T (ĥ S ) and the true target error of the optimal target classifier e T (h * T ) depends on this value e * S,T . If e * S,T is too large, then the source classifier is not guaranteed be approximately correct in the target domain. Given e * S,T and the H∆Hdivergence, one can state that with probability 1 − δ, for δ ∈ (0, 1), the following holds: where ν is the VC-dimension of the hypothesis space H (adapted from Theorem 3 in BIB002 by setting β = 0 and α = 0). This bound is specific to error functions, but can be generalized to bounded real-valued loss functions BIB003 . A similar form has been found using the PAC-Bayesian framework BIB004 . Overall, the larger e * S,T and D H∆H are for a given domain adaptation problem, the less a source classifier will generalize to the target domain. As said, this generalization error bound describes how well an unadaptive source classifier generalizes to the target domain. Now, the challenge is to find relationships between domains that can be exploited to form tighter bounds and construct new adaptive classifiers. The following three sections describe approaches based on different types of relationships between domains.