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In transfer learning, we wish to make inference about a target population when we have access to data both from the distribution itself, and from a different but related source distribution. We introduce a flexible framework for transfer learning in the context of binary classification, allowing for covariate-dependent relationships between the source and target distributions that are not required to preserve the Bayes decision boundary. Our main contributions are to derive the minimax optimal rates of convergence (up to poly-logarithmic factors) in this problem, and show that the optimal rate can be achieved by an algorithm that adapts to key aspects of the unknown transfer relationship, as well as the smoothness and tail parameters of our distributional classes. This optimal rate turns out to have several regimes, depending on the interplay between the relative sample sizes and the strength of the transfer relationship, and our algorithm achieves optimality by careful, decision tree-based calibration of local nearest-neighbour procedures.
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Consider the problem: given the data pair $(mathbf{x}, mathbf{y})$ drawn from a population with $f_*(x) = mathbf{E}[mathbf{y} | mathbf{x} = x]$, specify a neural network model and run gradient flow on the weights over time until reaching any stationarity. How does $f_t$, the function computed by the neural network at time $t$, relate to $f_*$, in terms of approximation and representation? What are the provable benefits of the adaptive representation by neural networks compared to the pre-specified fixed basis representation in the classical nonparametric literature? We answer the above questions via a dynamic reproducing kernel Hilbert space (RKHS) approach indexed by the training process of neural networks. Firstly, we show that when reaching any local stationarity, gradient flow learns an adaptive RKHS representation and performs the global least-squares projection onto the adaptive RKHS, simultaneously. Secondly, we prove that as the RKHS is data-adaptive and task-specific, the residual for $f_*$ lies in a subspace that is potentially much smaller than the orthogonal complement of the RKHS. The result formalizes the representation and approximation benefits of neural networks. Lastly, we show that the neural network function computed by gradient flow converges to the kernel ridgeless regression with an adaptive kernel, in the limit of vanishing regularization. The adaptive kernel viewpoint provides new angles of studying the approximation, representation, generalization, and optimization advantages of neural networks.
Distance metric learning (DML) approaches learn a transformation to a representation space where distance is in correspondence with a predefined notion of similarity. While such models offer a number of compelling benefits, it has been difficult for these to compete with modern classification algorithms in performance and even in feature extraction. In this work, we propose a novel approach explicitly designed to address a number of subtle yet important issues which have stymied earlier DML algorithms. It maintains an explicit model of the distributions of the different classes in representation space. It then employs this knowledge to adaptively assess similarity, and achieve local discrimination by penalizing class distribution overlap. We demonstrate the effectiveness of this idea on several tasks. Our approach achieves state-of-the-art classification results on a number of fine-grained visual recognition datasets, surpassing the standard softmax classifier and outperforming triplet loss by a relative margin of 30-40%. In terms of computational performance, it alleviates training inefficiencies in the traditional triplet loss, reaching the same error in 5-30 times fewer iterations. Beyond classification, we further validate the saliency of the learnt representations via their attribute concentration and hierarchy recovery properties, achieving 10-25% relative gains on the softmax classifier and 25-50% on triplet loss in these tasks.
The count-min sketch (CMS) is a time and memory efficient randomized data structure that provides estimates of tokens frequencies in a data stream, i.e. point queries, based on random hashed data. Learning-augmented CMSs improve the CMS by learning models that allow to better exploit data properties. In this paper, we focus on the learning-augmented CMS of Cai, Mitzenmacher and Adams (textit{NeurIPS} 2018), which relies on Bayesian nonparametric (BNP) modeling of a data stream via Dirichlet process (DP) priors. This is referred to as the CMS-DP, and it leads to BNP estimates of a point query as posterior means of the point query given the hashed data. While BNPs is proved to be a powerful tool for developing robust learning-augmented CMSs, ideas and methods behind the CMS-DP are tailored to point queries under DP priors, and they can not be used for other priors or more general queries. In this paper, we present an alternative, and more flexible, derivation of the CMS-DP such that: i) it allows to make use of the Pitman-Yor process (PYP) prior, which is arguably the most popular generalization of the DP prior; ii) it can be readily applied to the more general problem of estimating range queries. This leads to develop a novel learning-augmented CMS under power-law data streams, referred to as the CMS-PYP, which relies on BNP modeling of the stream via PYP priors. Applications to synthetic and real data show that the CMS-PYP outperforms the CMS and the CMS-DP in the estimation of low-frequency tokens; this known to be a critical feature in natural language processing, where it is indeed common to encounter power-law data streams.
In the context of supervised learning of a function by a Neural Network (NN), we claim and empirically justify that a NN yields better results when the distribution of the data set focuses on regions where the function to learn is steeper. We first traduce this assumption in a mathematically workable way using Taylor expansion. Then, theoretical derivations allow to construct a methodology that we call Variance Based Samples Weighting (VBSW). VBSW uses local variance of the labels to weight the training points. This methodology is general, scalable, cost effective, and significantly increases the performances of a large class of NNs for various classification and regression tasks on image, text and multivariate data. We highlight its benefits with experiments involving NNs from shallow linear NN to Resnet or Bert.
We consider ill-posed inverse problems where the forward operator $T$ is unknown, and instead we have access to training data consisting of functions $f_i$ and their noisy images $Tf_i$. This is a practically relevant and challenging problem which current methods are able to solve only under strong assumptions on the training set. Here we propose a new method that requires minimal assumptions on the data, and prove reconstruction rates that depend on the number of training points and the noise level. We show that, in the regime of many training data, the method is minimax optimal. The proposed method employs a type of convolutional neural networks (U-nets) and empirical risk minimization in order to fit the unknown operator. In a nutshell, our approach is based on two ideas: the first is to relate U-nets to multiscale decompositions such as wavelets, thereby linking them to the existing theory, and the second is to use the hierarchical structure of U-nets and the low number of parameters of convolutional neural nets to prove entropy bounds that are practically useful. A significant difference with the existing works on neural networks in nonparametric statistics is that we use them to approximate operators and not functions, which we argue is mathematically more natural and technically more convenient.
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