We propose statistically robust and computationally efficient linear learning methods in the high-dimensional batch setting, where the number of features $d$ may exceed the sample size $n$. We employ, in a generic learning setting, two algorithms depending on whether the considered loss function is gradient-Lipschitz or not. Then, we instantiate our framework on several applications including vanilla sparse, group-sparse and low-rank matrix recovery. This leads, for each application, to efficient and robust learning algorithms, that reach near-optimal estimation rates under heavy-tailed distributions and the presence of outliers. For vanilla $s$-sparsity, we are able to reach the $s\log (d)/n$ rate under heavy-tails and $\eta$-corruption, at a computational cost comparable to that of non-robust analogs. We provide an efficient implementation of our algorithms in an open-source $\mathtt{Python}$ library called $\mathtt{linlearn}$, by means of which we carry out numerical experiments which confirm our theoretical findings together with a comparison to other recent approaches proposed in the literature.
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Maximizing long-term rewards is the primary goal in sequential decision-making problems. The majority of existing methods assume that side information is freely available, enabling the learning agent to observe all features' states before making a decision. In real-world problems, however, collecting beneficial information is often costly. That implies that, besides individual arms' reward, learning the observations of the features' states is essential to improve the decision-making strategy. The problem is aggravated in a non-stationary environment where reward and cost distributions undergo abrupt changes over time. To address the aforementioned dual learning problem, we extend the contextual bandit setting and allow the agent to observe subsets of features' states. The objective is to maximize the long-term average gain, which is the difference between the accumulated rewards and the paid costs on average. Therefore, the agent faces a trade-off between minimizing the cost of information acquisition and possibly improving the decision-making process using the obtained information. To this end, we develop an algorithm that guarantees a sublinear regret in time. Numerical results demonstrate the superiority of our proposed policy in a real-world scenario.
In many industrial applications, obtaining labeled observations is not straightforward as it often requires the intervention of human experts or the use of expensive testing equipment. In these circumstances, active learning can be highly beneficial in suggesting the most informative data points to be used when fitting a model. Reducing the number of observations needed for model development alleviates both the computational burden required for training and the operational expenses related to labeling. Online active learning, in particular, is useful in high-volume production processes where the decision about the acquisition of the label for a data point needs to be taken within an extremely short time frame. However, despite the recent efforts to develop online active learning strategies, the behavior of these methods in the presence of outliers has not been thoroughly examined. In this work, we investigate the performance of online active linear regression in contaminated data streams. Our study shows that the currently available query strategies are prone to sample outliers, whose inclusion in the training set eventually degrades the predictive performance of the models. To address this issue, we propose a solution that bounds the search area of a conditional D-optimal algorithm and uses a robust estimator. Our approach strikes a balance between exploring unseen regions of the input space and protecting against outliers. Through numerical simulations, we show that the proposed method is effective in improving the performance of online active learning in the presence of outliers, thus expanding the potential applications of this powerful tool.
In this paper, we focus on the high-dimensional double sparse structure, where the parameter of interest simultaneously encourages group-wise sparsity and element-wise sparsity in each group. By combining the Gilbert-Varshamov bound and its variants, we develop a novel lower bound technique for the metric entropy of the parameter space, specifically tailored for the double sparse structure over $\ell_u(\ell_q)$-balls with $u,q \in [0,1]$. We prove lower bounds on the estimation error using an information-theoretic approach, leveraging our proposed lower bound technique and Fano's inequality. To complement the lower bounds, we establish matching upper bounds through a direct analysis of constrained least-squares estimators and utilize results from empirical processes. A significant finding of our study is the discovery of a phase transition phenomenon in the minimax rates for $u,q \in (0, 1]$. Furthermore, we extend the theoretical results to the double sparse regression model and determine its minimax rate for estimation error. To tackle double sparse linear regression, we develop the DSIHT (Double Sparse Iterative Hard Thresholding) algorithm, demonstrating its optimality in the minimax sense. Finally, we demonstrate the superiority of our method through numerical experiments.
We adopt a maximum-likelihood framework to estimate parameters of a stochastic susceptible-infected-recovered (SIR) model with contact tracing on a rooted random tree. Given the number of detectees per index case, our estimator allows to determine the degree distribution of the random tree as well as the tracing probability. Since we do not discover all infectees via contact tracing, this estimation is non-trivial. To keep things simple and stable, we develop an approximation suited for realistic situations (contract tracing probability small, or the probability for the detection of index cases small). In this approximation, the only epidemiological parameter entering the estimator is $R_0$. The estimator is tested in a simulation study and is furthermore applied to covid-19 contact tracing data from India. The simulation study underlines the efficiency of the method. For the empirical covid-19 data, we compare different degree distributions and perform a sensitivity analysis. We find that particularly a power-law and a negative binomial degree distribution fit the data well and that the tracing probability is rather large. The sensitivity analysis shows no strong dependency of the estimates on the reproduction number. Finally, we discuss the relevance of our findings.
In this paper, we propose an online convex optimization method with two different levels of adaptivity. On a higher level, our method is agnostic to the specific type and curvature of the loss functions, while at a lower level, it can exploit the niceness of the environments and attain problem-dependent guarantees. To be specific, we obtain $\mathcal{O}(\ln V_T)$, $\mathcal{O}(d \ln V_T)$ and $\hat{\mathcal{O}}(\sqrt{V_T})$ regret bounds for strongly convex, exp-concave and convex loss functions, respectively, where $d$ is the dimension, $V_T$ denotes problem-dependent gradient variations and $\hat{\mathcal{O}}(\cdot)$-notation omits logarithmic factors on $V_T$. Our result finds broad implications and applications. It not only safeguards the worst-case guarantees, but also implies the small-loss bounds in analysis directly. Besides, it draws deep connections with adversarial/stochastic convex optimization and game theory, further validating its practical potential. Our method is based on a multi-layer online ensemble incorporating novel ingredients, including carefully-designed optimism for unifying diverse function types and cascaded corrections for algorithmic stability. Remarkably, despite its multi-layer structure, our algorithm necessitates only one gradient query per round, making it favorable when the gradient evaluation is time-consuming. This is facilitated by a novel regret decomposition equipped with customized surrogate losses.
We consider gradient-related methods for low-rank matrix optimization with a smooth cost function. The methods operate on single factors of the low-rank factorization and share aspects of both alternating and Riemannian optimization. Two possible choices for the search directions based on Gauss-Southwell type selection rules are compared: one using the gradient of a factorized non-convex formulation, the other using the Riemannian gradient. While both methods provide gradient convergence guarantees that are similar to the unconstrained case, numerical experiments on a quadratic cost function indicate that the version based on the Riemannian gradient is significantly more robust with respect to small singular values and the condition number of the cost function. As a side result of our approach, we also obtain new convergence results for the alternating least squares method.
In this article, we propose two kinds of neural networks inspired by power method and inverse power method to solve linear eigenvalue problems. These neural networks share similar ideas with traditional methods, in which the differential operator is realized by automatic differentiation. The eigenfunction of the eigenvalue problem is learned by the neural network and the iterative algorithms are implemented by optimizing the specially defined loss function. The largest positive eigenvalue, smallest eigenvalue and interior eigenvalues with the given prior knowledge can be solved efficiently. We examine the applicability and accuracy of our methods in the numerical experiments in one dimension, two dimensions and higher dimensions. Numerical results show that accurate eigenvalue and eigenfunction approximations can be obtained by our methods.
We design new algorithms for $k$-clustering in high-dimensional Euclidean spaces. These algorithms run in the Massively Parallel Computation (MPC) model, and are fully scalable, meaning that the local memory in each machine is $n^{\sigma}$ for arbitrarily small fixed $\sigma>0$. Importantly, the local memory may be substantially smaller than $k$. Our algorithms take $O(1)$ rounds and achieve $O(1)$-bicriteria approximation for $k$-Median and for $k$-Means, namely, they compute $(1+\varepsilon)k$ clusters of cost within $O(1/\varepsilon^2)$-factor of the optimum. Previous work achieves only $\mathrm{poly}(\log n)$-bicriteria approximation [Bhaskara et al., ICML'18], or handles a special case [Cohen-Addad et al., ICML'22]. Our results rely on an MPC algorithm for $O(1)$-approximation of facility location in $O(1)$ rounds. A primary technical tool that we develop, and may be of independent interest, is a new MPC primitive for geometric aggregation, namely, computing certain statistics on an approximate neighborhood of every data point, which includes range counting and nearest-neighbor search. Our implementation of this primitive works in high dimension, and is based on consistent hashing (aka sparse partition), a technique that was recently used for streaming algorithms [Czumaj et al., FOCS'22].
Deep neural networks (DNNs) have achieved unprecedented success in the field of artificial intelligence (AI), including computer vision, natural language processing and speech recognition. However, their superior performance comes at the considerable cost of computational complexity, which greatly hinders their applications in many resource-constrained devices, such as mobile phones and Internet of Things (IoT) devices. Therefore, methods and techniques that are able to lift the efficiency bottleneck while preserving the high accuracy of DNNs are in great demand in order to enable numerous edge AI applications. This paper provides an overview of efficient deep learning methods, systems and applications. We start from introducing popular model compression methods, including pruning, factorization, quantization as well as compact model design. To reduce the large design cost of these manual solutions, we discuss the AutoML framework for each of them, such as neural architecture search (NAS) and automated pruning and quantization. We then cover efficient on-device training to enable user customization based on the local data on mobile devices. Apart from general acceleration techniques, we also showcase several task-specific accelerations for point cloud, video and natural language processing by exploiting their spatial sparsity and temporal/token redundancy. Finally, to support all these algorithmic advancements, we introduce the efficient deep learning system design from both software and hardware perspectives.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
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