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Register a new userConcentration bounds for linear Monge mapping estimation and optimal transport domain adaptation
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This article investigates the quality of the estimator of the linear Monge mapping between distributions. We provide the first concentration result on the linear mapping operator and prove a sample complexity of $n^{-1/2}$ when using empirical estimates of first and second order moments. This result is then used to derive a generalization bound for domain adaptation with optimal transport. As a consequence, this method approaches the performance of theoretical Bayes predictor under mild conditions on the covariance structure of the problem. We also discuss the computational complexity of the linear mapping estimation and show that when the source and target are stationary the mapping is a convolution that can be estimated very efficiently using fast Fourier transforms. Numerical experiments reproduce the behavior of the proven bounds on simulated and real data for mapping estimation and domain adaptation on images.
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We introduce a formulation of optimal transport problem for distributions on function spaces, where the stochastic map between functional domains can be partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt operator mapping a Hilbert space of functions to another. For numerous machine learning tasks, data can be naturally viewed as samples drawn from spaces of functions, such as curves and surfaces, in high dimensions. Optimal transport for functional data analysis provides a useful framework of treatment for such domains. In this work, we develop an efficient algorithm for finding the stochastic transport map between functional domains and provide theoretical guarantees on the existence, uniqueness, and consistency of our estimate for the Hilbert-Schmidt operator. We validate our method on synthetic datasets and study the geometric properties of the transport map. Experiments on real-world datasets of robot arm trajectories further demonstrate the effectiveness of our method on applications in domain adaptation.
Domain adaptation (DA) arises as an important problem in statistical machine learning when the source data used to train a model is different from the target data used to test the model. Recent advances in DA have mainly been application-driven and have largely relied on the idea of a common subspace for source and target data. To understand the empirical successes and failures of DA methods, we propose a theoretical framework via structural causal models that enables analysis and comparison of the prediction performance of DA methods. This framework also allows us to itemize the assumptions needed for the DA methods to have a low target error. Additionally, with insights from our theory, we propose a new DA method called CIRM that outperforms existing DA methods when both the covariates and label distributions are perturbed in the target data. We complement the theoretical analysis with extensive simulations to show the necessity of the devised assumptions. Reproducible synthetic and real data experiments are also provided to illustrate the strengths and weaknesses of DA methods when parts of the assumptions of our theory are violated.
Machine learning models have traditionally been developed under the assumption that the training and test distributions match exactly. However, recent success in few-shot learning and related problems are encouraging signs that these models can be adapted to more realistic settings where train and test distributions differ. Unfortunately, there is severely limited theoretical support for these algorithms and little is known about the difficulty of these problems. In this work, we provide novel information-theoretic lower-bounds on minimax rates of convergence for algorithms that are trained on data from multiple sources and tested on novel data. Our bounds depend intuitively on the information shared between sources of data, and characterize the difficulty of learning in this setting for arbitrary algorithms. We demonstrate these bounds on a hierarchical Bayesian model of meta-learning, computing both upper and lower bounds on parameter estimation via maximum-a-posteriori inference.
In this paper, we provide two main contributions in PAC-Bayesian theory for domain adaptation where the objective is to learn, from a source distribution, a well-performing majority vote on a different target distribution. On the one hand, we propose an improvement of the previous approach proposed by Germain et al. (2013), that relies on a novel distribution pseudodistance based on a disagreement averaging, allowing us to derive a new tighter PAC-Bayesian domain adaptation bound for the stochastic Gibbs classifier. 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. On the other hand, we generalize these results to multisource domain adaptation allowing us to take into account different source domains. This study opens the door to tackle domain adaptation tasks by making use of all the PAC-Bayesian tools.
In the context of K-armed stochastic bandits with distribution only assumed to be supported by [0,1], we introduce the first algorithm, called KL-UCB-switch, that enjoys simultaneously a distribution-free regret bound of optimal order $sqrt{KT}$ and a distribution-dependent regret bound of optimal order as well, that is, matching the $kappaln T$ lower bound by Lai & Robbins (1985) and Burnetas & Katehakis (1996). This self-contained contribution simultaneously presents state-of-the-art techniques for regret minimization in bandit models, and an elementary construction of non-asymptotic confidence bounds based on the empirical likelihood method for bounded distributions.
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