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In the last decade, it has been shown that many hard AI tasks, especially in NLP, can be naturally modeled as extreme classification problems leading to improved precision. However, such models are prohibitively expensive to train due to the memory bottleneck in the last layer. For example, a reasonable softmax layer for the dataset of interest in this paper can easily reach well beyond 100 billion parameters (> 400 GB memory). To alleviate this problem, we present Merged-Average Classifiers via Hashing (MACH), a generic $K$-classification algorithm where memory provably scales at $O(\log K)$ without any assumption on the relation between classes. MACH is subtly a count-min sketch structure in disguise, which uses universal hashing to reduce classification with a large number of classes to few embarrassingly parallel and independent classification tasks with a small (constant) number of classes.

Robust Principal Component Analysis (RPCA) aims to recover a low-rank structure from noisy, partially observed data that is also corrupted by sparse, potentially large-magnitude outliers. Traditional RPCA models rely on convex relaxations, such as nuclear norm and $\ell_1$ norm, to approximate the rank of a matrix and the $\ell_0$ functional (the number of non-zero elements) of another. In this work, we advocate a nonconvex regularization method, referred to as transformed $\ell_1$ (TL1), to improve both approximations. The rationale is that by varying the internal parameter of TL1, its behavior asymptotically approaches either $\ell_0$ or $\ell_1$. Since the rank is equal to the number of non-zero singular values and the nuclear norm is defined as their sum, applying TL1 to the singular values can approximate either the rank or the nuclear norm, depending on its internal parameter. We conduct a fine-grained theoretical analysis of statistical convergence rates, measured in the Frobenius norm, for both the low-rank and sparse components under general sampling schemes. These rates are comparable to those of the classical RPCA model based on the nuclear norm and $\ell_1$ norm. Moreover, we establish constant-order upper bounds on the estimated rank of the low-rank component and the cardinality of the sparse component in the regime where TL1 behaves like $\ell_0$, assuming that the respective matrices are exactly low-rank and exactly sparse. Extensive numerical experiments on synthetic data and real-world applications demonstrate that the proposed approach achieves higher accuracy than the classic convex model, especially under non-uniform sampling schemes.

In the last decade, it has been shown that many hard AI tasks, especially in NLP, can be naturally modeled as extreme classification problems leading to improved precision. However, such models are prohibitively expensive to train due to the memory bottleneck in the last layer. For example, a reasonable softmax layer for the dataset of interest in this paper can easily reach well beyond 100 billion parameters ( 400 GB memory). To alleviate this problem, we present Merged-Average Classifiers via Hashing (MACH), a generic K -classification algorithm where memory provably scales at O(\log K) without any assumption on the relation between classes. MACH is subtly a count-min sketch structure in disguise, which uses universal hashing to reduce classification with a large number of classes to few embarrassingly parallel and independent classification tasks with a small (constant) number of classes.

Aiming to generalize the label knowledge from a source domain with continuous outputs to an unlabeled target domain, Domain Adaptation Regression (DAR) is developed for complex practical learning problems. However, due to the continuity problem in regression, existing conditional distribution alignment theory and methods with discrete prior, which are proven to be effective in classification settings, are no longer applicable. In this work, focusing on the feasibility problems in DAR, we establish the sufficiency theory for the regression model, which shows the generalization error can be sufficiently dominated by the cross-domain conditional discrepancy. Further, to characterize conditional discrepancy with continuous conditioning variable, a novel Conditional Operator Discrepancy (COD) is proposed, which admits the metric property on conditional distributions via the kernel embedding theory. Finally, to minimize the discrepancy, a COD-based conditional invariant representation learning model is proposed, and the reformulation is derived to show that reasonable modifications on moment statistics can further improve the discriminability of the adaptation model. Extensive experiments on standard DAR datasets verify the validity of theoretical results and the superiority over SOTA DAR methods.

This paper presents a novel non-rigid point set registration method that is inspired by unsupervised clustering analysis. Unlike previous approaches that treat the source and target point sets as separate entities, we develop a holistic framework where they are formulated as clustering centroids and clustering members, separately. We then adopt Tikhonov regularization with an $\ell_1$-induced Laplacian kernel instead of the commonly used Gaussian kernel to ensure smooth and more robust displacement fields. Our formulation delivers closed-form solutions, theoretical guarantees, independence from dimensions, and the ability to handle large deformations. Subsequently, we introduce a clustering-improved Nystr\"om method to effectively reduce the computational complexity and storage of the Gram matrix to linear, while providing a rigorous bound for the low-rank approximation. Our method achieves high accuracy results across various scenarios and surpasses competitors by a significant margin, particularly on shapes with substantial deformations. Additionally, we demonstrate the versatility of our method in challenging tasks such as shape transfer and medical registration.

We consider the problem of deep fair clustering, which partitions data into clusters via the representations extracted by deep neural networks while hiding sensitive data attributes. To achieve fairness, existing methods present a variety of fairness-related objective functions based on the group fairness criterion. However, these works typically assume that the sensitive attributes are discrete and do not work for continuous sensitive variables, such as the proportion of the female population in an area. Besides, the potential of the representations learned from clustering tasks to improve performance on other tasks is ignored by existing works. In light of these limitations, we propose a flexible deep fair clustering method that can handle discrete and continuous sensitive attributes simultaneously. Specifically, we design an information bottleneck style objective function to learn fair and clustering-friendly representations. Furthermore, we explore for the first time the transferability of the extracted representations to other downstream tasks. Unlike existing works, we impose fairness at the representation level, which could guarantee fairness for the transferred task regardless of clustering results. To verify the effectiveness of the proposed method, we perform extensive experiments on datasets with discrete and continuous sensitive attributes, demonstrating the advantage of our method in comparison with state-of-the-art methods.

We study the performance of standard online learning algorithms when the feedback is delayed by an adversary. We show that online-gradient-descent [1] and follow-the-perturbed-leader [2] achieve regret O( D) in the delayed setting, where D is the sum of delays of each round's feedback. This bound collapses to an optimal O( T) bound in the usual setting of no delays (where D = T). Our main contribution is to show that standard algorithms for online learning already have simple regret bounds in the most general setting of delayed feedback, making adjustments to the analysis and not to the algorithms themselves. Our results help affirm and clarify the success of recent algorithms in optimization and machine learning that operate in a delayed feedback model.