A Convex Framework for Confounding Robust Inference

We study policy evaluation of offline contextual bandits subject to unobserved confounders. Sensitivity analysis methods are commonly used to estimate the policy value under the worst-case confounding over a given uncertainty set. However, existing work often resorts to some coarse relaxation of the uncertainty set for the sake of tractability, leading to overly conservative estimation of the policy value. In this paper, we propose a general estimator that provides a sharp lower bound of the policy value using convex programming. The generality of our estimator enables various extensions such as sensitivity analysis with f-divergence, model selection with cross validation and information criterion, and robust policy learning with the sharp lower bound. Furthermore, our estimation method can be reformulated as an empirical risk minimization problem thanks to the strong duality, which enables us to provide strong theoretical guarantees of the proposed estimator using techniques of the M-estimation.

Denoising Cosine Similarity: A Theory-Driven Approach for Efficient Representation Learning

Representation learning has been increasing its impact on the research and practice of machine learning, since it enables to learn representations that can apply to various downstream tasks efficiently. However, recent works pay little attention to the fact that real-world datasets used during the stage of representation learning are commonly contaminated by noise, which can degrade the quality of learned representations. This paper tackles the problem to learn robust representations against noise in a raw dataset. To this end, inspired by recent works on denoising and the success of the cosine-similarity-based objective functions in representation learning, we propose the denoising Cosine-Similarity (dCS) loss. The dCS loss is a modified cosine-similarity loss and incorporates a denoising property, which is supported by both our theoretical and empirical findings. To make the dCS loss implementable, we also construct the estimators of the dCS loss with statistical guarantees. Finally, we empirically show the efficiency of the dCS loss over the baseline objective functions in vision and speech domains.

Towards Understanding the Mechanism of Contrastive Learning via Similarity Structure: A Theoretical Analysis

Contrastive learning is an efficient approach to self-supervised representation learning. Although recent studies have made progress in the theoretical understanding of contrastive learning, the investigation of how to characterize the clusters of the learned representations is still limited. In this paper, we aim to elucidate the characterization from theoretical perspectives. To this end, we consider a kernel-based contrastive learning framework termed Kernel Contrastive Learning (KCL), where kernel functions play an important role when applying our theoretical results to other frameworks. We introduce a formulation of the similarity structure of learned representations by utilizing a statistical dependency viewpoint. We investigate the theoretical properties of the kernel-based contrastive loss via this formulation. We first prove that the formulation characterizes the structure of representations learned with the kernel-based contrastive learning framework. We show a new upper bound of the classification error of a downstream task, which explains that our theory is consistent with the empirical success of contrastive learning. We also establish a generalization error bound of KCL. Finally, we show a guarantee for the generalization ability of KCL to the downstream classification task via a surrogate bound.

Deep Clustering with a Constraint for Topological Invariance based on Symmetric InfoNCE

We consider the scenario of deep clustering, in which the available prior knowledge is limited. In this scenario, few existing state-of-the-art deep clustering methods can perform well for both non-complex topology and complex topology datasets. To address the problem, we propose a constraint utilizing symmetric InfoNCE, which helps an objective of deep clustering method in the scenario train the model so as to be efficient for not only non-complex topology but also complex topology datasets. Additionally, we provide several theoretical explanations of the reason why the constraint can enhances performance of deep clustering methods. To confirm the effectiveness of the proposed constraint, we introduce a deep clustering method named MIST, which is a combination of an existing deep clustering method and our constraint. Our numerical experiments via MIST demonstrate that the constraint is effective. In addition, MIST outperforms other state-of-the-art deep clustering methods for most of the commonly used ten benchmark datasets.

Learning Domain Invariant Representations by Joint Wasserstein Distance Minimization

Domain shifts in the training data are common in practical applications of machine learning, they occur for instance when the data is coming from different sources. Ideally, a ML model should work well independently of these shifts, for example, by learning a domain-invariant representation. Moreover, privacy concerns regarding the source also require a domain-invariant representation. In this work, we provide theoretical results that link domain invariant representations -- measured by the Wasserstein distance on the joint distributions -- to a practical semi-supervised learning objective based on a cross-entropy classifier and a novel domain critic. Quantitative experiments demonstrate that the proposed approach is indeed able to practically learn such an invariant representation (between two domains), and the latter also supports models with higher predictive accuracy on both domains, comparing favorably to existing techniques.

Robust modal regression with direct log-density derivative estimation

Modal regression is aimed at estimating the global mode (i.e., global maximum) of the conditional density function of the output variable given input variables, and has led to regression methods robust against heavy-tailed or skewed noises. The conditional mode is often estimated through maximization of the modal regression risk (MRR). In order to apply a gradient method for the maximization, the fundamental challenge is accurate approximation of the gradient of MRR, not MRR itself. To overcome this challenge, in this paper, we take a novel approach of directly approximating the gradient of MRR. To approximate the gradient, we develop kernelized and neural-network-based versions of the least-squares log-density derivative estimator, which directly approximates the derivative of the log-density without density estimation. With direct approximation of the MRR gradient, we first propose a modal regression method with kernels, and derive a new parameter update rule based on a fixed-point method. Then, the derived update rule is theoretically proved to have a monotonic hill-climbing property towards the conditional mode. Furthermore, we indicate that our approach of directly approximating the gradient is compatible with recent sophisticated stochastic gradient methods (e.g., Adam), and then propose another modal regression method based on neural networks. Finally, the superior performance of the proposed methods is demonstrated on various artificial and benchmark datasets.

Estimating Density Models with Complex Truncation Boundaries

Truncated densities are probability density functions defined on truncated input domains. These densities share the same parametric form with their non-truncated counterparts up to a normalization term. However, normalization terms usually cannot be obtained in closed form for these distributions, due to complicated truncation domains. Score Matching is a powerful tool for fitting parameters in unnormalized models. However, it cannot be straightforwardly applied here as boundary conditions used to derive a tractable objective are usually not satisfied by truncated distributions. In this paper, we propose a maximally weighted Score Matching objective function which takes the geometry of the truncation boundary into account when fitting unnormalized density models. We show the weighting function that maximizes the objective function can be constructed easily and the boundary conditions for deriving a tradable objective are satisfied. Experiments on toy datasets and Chicago crime dataset show promising results.

Unified estimation framework for unnormalized models with statistical efficiency

Parameter estimation of unnormalized models is a challenging problem because normalizing constants are not calculated explicitly and maximum likelihood estimation is computationally infeasible. Although some consistent estimators have been proposed earlier, the problem of statistical efficiency does remain. In this study, we propose a unified, statistically efficient estimation framework for unnormalized models and several novel efficient estimators with reasonable computational time regardless of whether the sample space is discrete or continuous. The loss functions of the proposed estimators are derived by combining the following two methods: (1) density-ratio matching using Bregman divergence, and (2) plugging-in nonparametric estimators. We also analyze the properties of the proposed estimators when the unnormalized model is misspecified. Finally, the experimental results demonstrate the advantages of our method over existing approaches.

Variable Selection for Nonparametric Learning with Power Series Kernels

In this paper, we propose a variable selection method for general nonparametric kernel-based estimation. The proposed method consists of two-stage estimation: (1) construct a consistent estimator of the target function, (2) approximate the estimator using a few variables by l1-type penalized estimation. We see that the proposed method can be applied to various kernel nonparametric estimation such as kernel ridge regression, kernel-based density and density-ratio estimation. We prove that the proposed method has the property of the variable selection consistency when the power series kernel is used. This result is regarded as an extension of the variable selection consistency for the non-negative garrote to the kernel-based estimators. Several experiments including simulation studies and real data applications show the effectiveness of the proposed method.

Mode-Seeking Clustering and Density Ridge Estimation via Direct Estimation of Density-Derivative-Ratios

Modes and ridges of the probability density function behind observed data are useful geometric features. Mode-seeking clustering assigns cluster labels by associating data samples with the nearest modes, and estimation of density ridges enables us to find lower-dimensional structures hidden in data. A key technical challenge both in mode-seeking clustering and density ridge estimation is accurate estimation of the ratios of the first- and second-order density derivatives to the density. A naive approach takes a three-step approach of first estimating the data density, then computing its derivatives, and finally taking their ratios. However, this three-step approach can be unreliable because a good density estimator does not necessarily mean a good density derivative estimator, and division by the estimated density could significantly magnify the estimation error. To cope with these problems, we propose a novel estimator for the \emph{density-derivative-ratios}. The proposed estimator does not involve density estimation, but rather \emph{directly} approximates the ratios of density derivatives of any order. Moreover, we establish a convergence rate of the proposed estimator. Based on the proposed estimator, novel methods both for mode-seeking clustering and density ridge estimation are developed, and the respective convergence rates to the mode and ridge of the underlying density are also established. Finally, we experimentally demonstrate that the developed methods significantly outperform existing methods, particularly for relatively high-dimensional data.