We consider a contextual online learning (multi-armed bandit) problem with high-dimensional covariate $\mathbf{x}$ and decision $\mathbf{y}$. The reward function to learn, $f(\mathbf{x},\mathbf{y})$, does not have a particular parametric form. The literature has shown that the optimal regret is $\tilde{O}(T^{(d_x+d_y+1)/(d_x+d_y+2)})$, where $d_x$ and $d_y$ are the dimensions of $\mathbf x$ and $\mathbf y$, and thus it suffers from the curse of dimensionality. In many applications, only a small subset of variables in the covariate affect the value of $f$, which is referred to as \textit{sparsity} in statistics. To take advantage of the sparsity structure of the covariate, we propose a variable selection algorithm called \textit{BV-LASSO}, which incorporates novel ideas such as binning and voting to apply LASSO to nonparametric settings. Our algorithm achieves the regret $\tilde{O}(T^{(d_x^*+d_y+1)/(d_x^*+d_y+2)})$, where $d_x^*$ is the effective covariate dimension. The regret matches the optimal regret when the covariate is $d^*_x$-dimensional and thus cannot be improved. Our algorithm may serve as a general recipe to achieve dimension reduction via variable selection in nonparametric settings.
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Fast transforms correspond to factorizations of the form $\mathbf{Z} = \mathbf{X}^{(1)} \ldots \mathbf{X}^{(J)}$, where each factor $ \mathbf{X}^{(\ell)}$ is sparse and possibly structured. This paper investigates essential uniqueness of such factorizations, i.e., uniqueness up to unavoidable scaling ambiguities. Our main contribution is to prove that any $N \times N$ matrix having the so-called butterfly structure admits an essentially unique factorization into $J$ butterfly factors (where $N = 2^{J}$), and that the factors can be recovered by a hierarchical factorization method, which consists in recursively factorizing the considered matrix into two factors. This hierarchical identifiability property relies on a simple identifiability condition in the two-layer and fixed-support setting. This approach contrasts with existing ones that fit the product of butterfly factors to a given matrix via gradient descent. The proposed method can be applied in particular to retrieve the factorization of the Hadamard or the discrete Fourier transform matrices of size $N=2^J$. Computing such factorizations costs $\mathcal{O}(N^{2})$, which is of the order of dense matrix-vector multiplication, while the obtained factorizations enable fast $\mathcal{O}(N \log N)$ matrix-vector multiplications and have the potential to be applied to compress deep neural networks.
Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method to approximate the action of the Koopman operator on a linear function space spanned by a dictionary of functions. The accuracy of EDMD model critically depends on the quality of the particular dictionary's span, specifically on how close it is to being invariant under the Koopman operator. Motivated by the observation that the residual error of EDMD, typically used for dictionary learning, does not encode the quality of the function space and is sensitive to the choice of basis, we introduce the novel concept of consistency index. We show that this measure, based on using EDMD forward and backward in time, enjoys a number of desirable qualities that make it suitable for data-driven modeling of dynamical systems: it measures the quality of the function space, it is invariant under the choice of basis, can be computed in closed form from the data, and provides a tight upper-bound for the relative root mean square error of all function predictions on the entire span of the dictionary.
In the past decade, the technology industry has adopted online randomized controlled experiments (a.k.a. A/B testing) to guide product development and make business decisions. In practice, A/B tests are often implemented with increasing treatment allocation: the new treatment is gradually released to an increasing number of units through a sequence of randomized experiments. In scenarios such as experimenting in a social network setting or in a bipartite online marketplace, interference among units may exist, which can harm the validity of simple inference procedures. In this work, we introduce a widely applicable procedure to test for interference in A/B testing with increasing allocation. Our procedure can be implemented on top of an existing A/B testing platform with a separate flow and does not require a priori a specific interference mechanism. In particular, we introduce two permutation tests that are valid under different assumptions. Firstly, we introduce a general statistical test for interference requiring no additional assumption. Secondly, we introduce a testing procedure that is valid under a time fixed effect assumption. The testing procedure is of very low computational complexity, it is powerful, and it formalizes a heuristic algorithm implemented already in industry. We demonstrate the performance of the proposed testing procedure through simulations on synthetic data. Finally, we discuss one application at LinkedIn, where a screening step is implemented to detect potential interference in all their marketplace experiments with the proposed methods in the paper.
We propose an iterative variable selection scheme for high-dimensional data with binary outcomes. The scheme adopts a structured screen-and-select framework and uses non-local prior-based Bayesian model selection within the same. The structured screening is based on the association of the independent variables with the outcome which is measured in terms of the maximum marginal likelihood estimator. Performance comparison with several well-known methods in terms of true positive rate and false discovery rate shows that our proposed method stands to be a competitive alternative for sparse high-dimensional variable selection with binary outcomes. The method has been implemented within the R package GWASinlps.
Feature selection is of great importance in Machine Learning, where it can be used to reduce the dimensionality of classification, ranking and prediction problems. The removal of redundant and noisy features can improve both the accuracy and scalability of the trained models. However, feature selection is a computationally expensive task with a solution space that grows combinatorically. In this work, we consider in particular a quadratic feature selection problem that can be tackled with the Quantum Approximate Optimization Algorithm (QAOA), already employed in combinatorial optimization. First we represent the feature selection problem with the QUBO formulation, which is then mapped to an Ising spin Hamiltonian. Then we apply QAOA with the goal of finding the ground state of this Hamiltonian, which corresponds to the optimal selection of features. In our experiments, we consider seven different real-world datasets with dimensionality up to 21 and run QAOA on both a quantum simulator and, for small datasets, the 7-qubit IBM (ibm-perth) quantum computer. We use the set of selected features to train a classification model and evaluate its accuracy. Our analysis shows that it is possible to tackle the feature selection problem with QAOA and that currently available quantum devices can be used effectively. Future studies could test a wider range of classification models as well as improve the effectiveness of QAOA by exploring better performing optimizers for its classical step.
Randomized Controlled Trials (RCTs) may suffer from limited scope. In particular, samples may be unrepresentative: some RCTs over- or under- sample individuals with certain characteristics compared to the target population, for which one wants conclusions on treatment effectiveness. Re-weighting trial individuals to match the target population can improve the treatment effect estimation. In this work, we establish the exact expressions of the bias and variance of such reweighting procedures -- also called Inverse Propensity of Sampling Weighting (IPSW) -- in presence of categorical covariates for any sample size. Such results allow us to compare the theoretical performance of different versions of IPSW estimates. Besides, our results show how the performance (bias, variance, and quadratic risk) of IPSW estimates depends on the two sample sizes (RCT and target population). A by-product of our work is the proof of consistency of IPSW estimates. Results also reveal that IPSW performances are improved when the trial probability to be treated is estimated (rather than using its oracle counterpart). In addition, we study choice of variables: how including covariates that are not necessary for identifiability of the causal effect may impact the asymptotic variance. Including covariates that are shifted between the two samples but not treatment effect modifiers increases the variance while non-shifted but treatment effect modifiers do not. We illustrate all the takeaways in a didactic example, and on a semi-synthetic simulation inspired from critical care medicine.
Analyzing large samples of high-dimensional data under dependence is a challenging statistical problem as long time series may have change points, most importantly in the mean and the marginal covariances, for which one needs valid tests. Inference for large covariance matrices is especially difficult due to noise accumulation, resulting in singular estimates and poor power of related tests. The singularity of the sample covariance matrix in high dimensions can be overcome by considering a linear combination with a regular, more structured target matrix. This approach is known as shrinkage, and the target matrix is typically of diagonal form. In this paper, we consider covariance shrinkage towards structured nonparametric estimators of the bandable or Toeplitz type, respectively, aiming at improved estimation accuracy and statistical power of tests even under nonstationarity. We derive feasible Gaussian approximation results for bilinear projections of the shrinkage estimators which are valid under nonstationarity and dependence. These approximations especially enable us to formulate a statistical test for structural breaks in the marginal covariance structure of high-dimensional time series without restrictions on the dimension, and which is robust against nonstationarity of nuisance parameters. We show via simulations that shrinkage helps to increase the power of the proposed tests. Moreover, we suggest a data-driven choice of the shrinkage weights, and assess its performance by means of a Monte Carlo study. The results indicate that the proposed shrinkage estimator is superior for non-Toeplitz covariance structures close to fractional Gaussian noise.
In offline reinforcement learning (RL), a learner leverages prior logged data to learn a good policy without interacting with the environment. A major challenge in applying such methods in practice is the lack of both theoretically principled and practical tools for model selection and evaluation. To address this, we study the problem of model selection in offline RL with value function approximation. The learner is given a nested sequence of model classes to minimize squared Bellman error and must select among these to achieve a balance between approximation and estimation error of the classes. We propose the first model selection algorithm for offline RL that achieves minimax rate-optimal oracle inequalities up to logarithmic factors. The algorithm, ModBE, takes as input a collection of candidate model classes and a generic base offline RL algorithm. By successively eliminating model classes using a novel one-sided generalization test, ModBE returns a policy with regret scaling with the complexity of the minimally complete model class. In addition to its theoretical guarantees, it is conceptually simple and computationally efficient, amounting to solving a series of square loss regression problems and then comparing relative square loss between classes. We conclude with several numerical simulations showing it is capable of reliably selecting a good model class.
We study best-arm identification with a fixed budget and contextual (covariate) information in stochastic multi-armed bandit problems. In each round, after observing contextual information, we choose a treatment arm using past observations and current context. Our goal is to identify the best treatment arm, a treatment arm with the maximal expected reward marginalized over the contextual distribution, with a minimal probability of misidentification. First, we derive semiparametric lower bounds of the misidentification probability for this problem, where we regard the gaps between the expected rewards of the best and suboptimal treatment arms as parameters of interest, and all other parameters, such as the expected rewards conditioned on contexts, as the nuisance parameters. We then develop the ``Contextual RS-AIPW strategy,'' which consists of the random sampling (RS) rule tracking a target allocation ratio and the recommendation rule using the augmented inverse probability weighting (AIPW) estimator. Our proposed Contextual RS-AIPW strategy is optimal because the upper bound for the probability of misidentification by the strategy matches the semiparametric lower bound, when the budget goes to infinity and the gaps converge to zero.
In this paper, we propose the joint learning attention and recurrent neural network (RNN) models for multi-label classification. While approaches based on the use of either model exist (e.g., for the task of image captioning), training such existing network architectures typically require pre-defined label sequences. For multi-label classification, it would be desirable to have a robust inference process, so that the prediction error would not propagate and thus affect the performance. Our proposed model uniquely integrates attention and Long Short Term Memory (LSTM) models, which not only addresses the above problem but also allows one to identify visual objects of interests with varying sizes without the prior knowledge of particular label ordering. More importantly, label co-occurrence information can be jointly exploited by our LSTM model. Finally, by advancing the technique of beam search, prediction of multiple labels can be efficiently achieved by our proposed network model.
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