It is common to split a dataset into training and testing sets before fitting a statistical or machine learning model. However, there is no clear guidance on how much data should be used for training and testing. In this article we show that the optimal splitting ratio is $\sqrt{p}:1$, where $p$ is the number of parameters in a linear regression model that explains the data well.
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Due to the high human cost of annotation, it is non-trivial to curate a large-scale medical dataset that is fully labeled for all classes of interest. Instead, it would be convenient to collect multiple small partially labeled datasets from different matching sources, where the medical images may have only been annotated for a subset of classes of interest. This paper offers an empirical understanding of an under-explored problem, namely partially supervised multi-label classification (PSMLC), where a multi-label classifier is trained with only partially labeled medical images. In contrast to the fully supervised counterpart, the partial supervision caused by medical data scarcity has non-trivial negative impacts on the model performance. A potential remedy could be augmenting the partial labels. Though vicinal risk minimization (VRM) has been a promising solution to improve the generalization ability of the model, its application to PSMLC remains an open question. To bridge the methodological gap, we provide the first VRM-based solution to PSMLC. The empirical results also provide insights into future research directions on partially supervised learning under data scarcity.
Covariance estimation for matrix-valued data has received an increasing interest in applications. Unlike previous works that rely heavily on matrix normal distribution assumption and the requirement of fixed matrix size, we propose a class of distribution-free regularized covariance estimation methods for high-dimensional matrix data under a separability condition and a bandable covariance structure. Under these conditions, the original covariance matrix is decomposed into a Kronecker product of two bandable small covariance matrices representing the variability over row and column directions. We formulate a unified framework for estimating bandable covariance, and introduce an efficient algorithm based on rank one unconstrained Kronecker product approximation. The convergence rates of the proposed estimators are established, and the derived minimax lower bound shows our proposed estimator is rate-optimal under certain divergence regimes of matrix size. We further introduce a class of robust covariance estimators and provide theoretical guarantees to deal with heavy-tailed data. We demonstrate the superior finite-sample performance of our methods using simulations and real applications from a gridded temperature anomalies dataset and a S&P 500 stock data analysis.
We investigate the feature compression of high-dimensional ridge regression using the optimal subsampling technique. Specifically, based on the basic framework of random sampling algorithm on feature for ridge regression and the A-optimal design criterion, we first obtain a set of optimal subsampling probabilities. Considering that the obtained probabilities are uneconomical, we then propose the nearly optimal ones. With these probabilities, a two step iterative algorithm is established which has lower computational cost and higher accuracy. We provide theoretical analysis and numerical experiments to support the proposed methods. Numerical results demonstrate the decent performance of our methods.
SVD (singular value decomposition) is one of the basic tools of machine learning, allowing to optimize basis for a given matrix. However, sometimes we have a set of matrices $\{A_k\}_k$ instead, and would like to optimize a single common basis for them: find orthogonal matrices $U$, $V$, such that $\{U^T A_k V\}$ set of matrices is somehow simpler. For example DCT-II is orthonormal basis of functions commonly used in image/video compression - as discussed here, this kind of basis can be quickly automatically optimized for a given dataset. While also discussed gradient descent optimization might be computationally costly, there is proposed CSVD (common SVD): fast general approach based on SVD. Specifically, we choose $U$ as built of eigenvectors of $\sum_i (w_k)^q (A_k A_k^T)^p$ and $V$ of $\sum_k (w_k)^q (A_k^T A_k)^p$, where $w_k$ are their weights, $p,q>0$ are some chosen powers e.g. 1/2, optionally with normalization e.g. $A \to A - rc^T$ where $r_i=\sum_j A_{ij}, c_j =\sum_i A_{ij}$.
We propose a multiple-splitting projection test (MPT) for one-sample mean vectors in high-dimensional settings. The idea of projection test is to project high-dimensional samples to a 1-dimensional space using an optimal projection direction such that traditional tests can be carried out with projected samples. However, estimation of the optimal projection direction has not been systematically studied in the literature. In this work, we bridge the gap by proposing a consistent estimation via regularized quadratic optimization. To retain type I error rate, we adopt a data-splitting strategy when constructing test statistics. To mitigate the power loss due to data-splitting, we further propose a test via multiple splits to enhance the testing power. We show that the $p$-values resulted from multiple splits are exchangeable. Unlike existing methods which tend to conservatively combine dependent $p$-values, we develop an exact level $\alpha$ test that explicitly utilizes the exchangeability structure to achieve better power. Numerical studies show that the proposed test well retains the type I error rate and is more powerful than state-of-the-art tests.
This paper studies how well generative adversarial networks (GANs) learn probability distributions from finite samples. Our main results establish the convergence rates of GANs under a collection of integral probability metrics defined through H\"older classes, including the Wasserstein distance as a special case. We also show that GANs are able to adaptively learn data distributions with low-dimensional structures or have H\"older densities, when the network architectures are chosen properly. In particular, for distributions concentrated around a low-dimensional set, we show that the learning rates of GANs do not depend on the high ambient dimension, but on the lower intrinsic dimension. Our analysis is based on a new oracle inequality decomposing the estimation error into the generator and discriminator approximation error and the statistical error, which may be of independent interest.
Policy gradient (PG) estimation becomes a challenge when we are not allowed to sample with the target policy but only have access to a dataset generated by some unknown behavior policy. Conventional methods for off-policy PG estimation often suffer from either significant bias or exponentially large variance. In this paper, we propose the double Fitted PG estimation (FPG) algorithm. FPG can work with an arbitrary policy parameterization, assuming access to a Bellman-complete value function class. In the case of linear value function approximation, we provide a tight finite-sample upper bound on policy gradient estimation error, that is governed by the amount of distribution mismatch measured in feature space. We also establish the asymptotic normality of FPG estimation error with a precise covariance characterization, which is further shown to be statistically optimal with a matching Cramer-Rao lower bound. Empirically, we evaluate the performance of FPG on both policy gradient estimation and policy optimization, using either softmax tabular or ReLU policy networks. Under various metrics, our results show that FPG significantly outperforms existing off-policy PG estimation methods based on importance sampling and variance reduction techniques.
Obtaining a dynamic population distribution is key to many decision-making processes such as urban planning, disaster management and most importantly helping the government to better allocate socio-technical supply. For the aspiration of these objectives, good population data is essential. The traditional method of collecting population data through the census is expensive and tedious. In recent years, machine learning methods have been developed to estimate the population distribution. Most of the methods use data sets that are either developed on a small scale or not publicly available yet. Thus, the development and evaluation of the new methods become challenging. We fill this gap by providing a comprehensive data set for population estimation in 98 European cities. The data set comprises digital elevation model, local climate zone, land use classifications, nighttime lights in combination with multi-spectral Sentinel-2 imagery, and data from the Open Street Map initiative. We anticipate that it would be a valuable addition to the research community for the development of sophisticated machine learning-based approaches in the field of population estimation.
Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.
Modern neural network training relies heavily on data augmentation for improved generalization. After the initial success of label-preserving augmentations, there has been a recent surge of interest in label-perturbing approaches, which combine features and labels across training samples to smooth the learned decision surface. In this paper, we propose a new augmentation method that leverages the first and second moments extracted and re-injected by feature normalization. We replace the moments of the learned features of one training image by those of another, and also interpolate the target labels. As our approach is fast, operates entirely in feature space, and mixes different signals than prior methods, one can effectively combine it with existing augmentation methods. We demonstrate its efficacy across benchmark data sets in computer vision, speech, and natural language processing, where it consistently improves the generalization performance of highly competitive baseline networks.
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