Shapley values are widely recognized as a principled method for attributing importance to input features in machine learning. However, the exact computation of Shapley values scales exponentially with the number of features, severely limiting the practical application of this powerful approach. The challenge is further compounded when the predictive model is probabilistic---as in Gaussian processes (GPs)---where the outputs are random variables rather than point estimates, necessitating additional computational effort in modeling higher-order moments. In this work, we demonstrate that for an important class of GPs known as FANOVA GP, which explicitly models all main effects and interactions, exact Shapley attributions for both local and global explanations can be computed in *quadratic* time. For *local, instance-wise explanations*, we define a stochastic cooperative game over function components and compute the *exact stochastic Shapley value* in quadratic time only, capturing both the expected contribution and uncertainty. For *global explanations*, we introduce a deterministic, variance-based value function and compute exact Shapley values that quantify each feature’s contribution to the model’s overall sensitivity. Our methods leverage a closed-form (stochastic) Möbiusrepresentation of the FANOVA decomposition and introduce recursive algorithms, inspired by Newton's identities, to efficiently compute the mean and variance of Shapley values. Our work enhances the utility of explainable AI, as demonstrated by empirical studies, by providing more scalable, axiomatically sound, and uncertainty-aware explanations for predictions generated by structured probabilistic models.

We introduce credal two-sample testing, a new hypothesis testing framework for comparing credal sets—convex sets of probability measures where each element captures aleatoric uncertainty and the set itself represents epistemic uncertainty that arises from the modeller’s partial ignorance. Compared to classical two-sample tests, which focus on comparing precise distributions, the proposed framework provides a broader and more versatile set of hypotheses. This approach enables the direct integration of epistemic uncertainty, effectively addressing the challenges arising from partial ignorance in hypothesis testing. By generalising two-sample test to compare credal sets, our framework enables reasoning for equality, inclusion, intersection, and mutual exclusivity, each offering unique insights into the modeller’s epistemic beliefs. As the first work on nonparametric hypothesis testing for comparing credal sets, we focus on finitely generated credal sets derived from i.i.d. samples from multiple distributions—referred to as \emph{credal samples}. We formalise these tests as two-sample tests with nuisance parameters and introduce the first permutation-based solution for this class of problems, significantly improving upon existing methods. Our approach properly incorporates the modeller’s epistemic uncertainty into hypothesis testing, leading to more robust and credible conclusions, with kernel-based implementations for real-world applications.

Learning robust models under distribution shifts between training and test datasets is a fundamental challenge in machine learning. While learning invariant features across environments is a popular approach, it often assumes that these features are fully observed in both training and test sets—a condition frequently violated in practice. When models rely on invariant features absent in the test set, their robustness in new environments can deteriorate. To tackle this problem, we introduce a novel learning principle called the Sufficient Invariant Learning (SIL) framework, which focuses on learning a sufficient subset of invariant features rather than relying on a single feature. After demonstrating the limitation of existing invariant learning methods, we propose a new algorithm, Adaptive Sharpness-aware Group Distributionally Robust Optimization (ASGDRO), to learn diverse invariant features by seeking common flat minima across the environments. We theoretically demonstrate that finding a common flat minima enables robust predictions based on diverse invariant features. Empirical evaluations on multiple datasets, including our new benchmark, confirm ASGDRO’s robustness against distribution shifts, highlighting the limitations of existing methods. Code: https://github.com/MLAI-Yonsei/SIL-ASGDRO.

Embedding probability distributions into reproducing kernel Hilbert spaces (RKHS) has enabled powerful nonparametric methods such as the maximum mean discrepancy (MMD), a statistical distance with strong theoretical and computational properties. At its core, the MMD relies on kernel mean embeddings to represent distributions as mean functions in RKHS. However, it remains unclear if the mean function is the only meaningful RKHS representation. Inspired by generalised quantiles, we introduce the notion of kernel quantile embeddings (KQEs). We then use KQEs to construct a family of distances that: (i) are probability metrics under weaker kernel conditions than MMD; (ii) recover a kernelised form of the sliced Wasserstein distance; and (iii) can be efficiently estimated with near-linear cost. Through hypothesis testing, we show that these distances offer a competitive alternative to MMD and its fast approximations.

The technique of data augmentation (DA) is often used in machine learning for regularization purposes to better generalize under i.i.d. settings. In this work, we present a unifying framework with topics in causal inference to make a case for the use of DA beyond just the i.i.d. setting, but for generalization across interventions as well. Specifically, we argue that when the outcome generating mechanism is invariant to our choice of DA, then such augmentations can effectively be thought of as interventions on the treatment generating mechanism itself. This can potentially help to reduce bias in causal effect estimation arising from hidden confounders. In the presence of such unobserved confounding we typically make use of instrumental variables (IVs)—sources of treatment randomization that are conditionally independent of the outcome. However, IVs may not be as readily available as DA for many applications, which is the main motivation behind this work. By appropriately regularizing IV based estimators, we introduce the concept of *IV-like (IVL)* regression for mitigating confounding bias and improving predictive performance across interventions even when certain IV properties are relaxed. Finally, we cast parameterized DA as an IVL regression problem and show that when used in composition can simulate a worst-case application of such DA, further improving performance on causal estimation and generalization tasks beyond what simple DA may offer. This is shown both theoretically for the population case and via simulation experiments for the finite sample case using a simple linear example. We also present real data experiments to support our case.

Quantifying differences between probability distributions is fundamental to statistics and machine learning, primarily for comparing statistical uncertainty. In contrast, epistemic uncertainty---due to incomplete knowledge---requires richer representations than those offered by classical probability. Imprecise probability (IP) theory offers such models, capturing ambiguity and partial belief. This has driven growing interest in imprecise probabilistic machine learning (IPML), where inference and decision-making rely on broader uncertainty models---highlighting the need for metrics beyond classical probability. This work introduces the Integral Imprecise Probability Metric (IIPM) framework, a Choquet integral-based generalisation of classical Integral Probability Metric to the setting of capacities---a broad class of IP models encompassing many existing ones, including lower probabilities, probability intervals, belief functions, and more. Theoretically, we establish conditions under which IIPM serves as a valid metric and metrises a form of weak convergence of capacities. Practically, IIPM not only enables comparison across different IP models but also supports the quantification of epistemic uncertainty~(EU) within a single IP model. In particular, by comparing an IP model with its conjugate, IIPM gives rise to a new class of epistemic uncertainty measures---Maximum Mean Imprecision (MMI) ---which satisfy key axiomatic properties proposed in the Uncertainty Quantification literature. We validate MMI through selective classification experiments, demonstrating strong empirical performance against established EU measures, and outperforming them when classical methods struggle to scale to a large number of classes. Our work advances both theory and practice in Imprecise Probabilistic Machine Learning, offering a principled framework for comparing and quantifying epistemic uncertainty under imprecision.

The quality of probabilistic forecasts is crucial for decision-making under uncertainty. While proper scoring rules incentivize truthful reporting of precise forecasts, they fall short when forecasters face epistemic uncertainty about their beliefs, limiting their use in safety-critical domains where decision-makers (DMs) prioritize proper uncertainty management. To address this, we propose a framework for scoring \emph{imprecise forecasts}—forecasts given as a set of beliefs. Despite existing impossibility results for deterministic scoring rules, we enable truthful elicitation by drawing connection to social choice theory and introducing a two-way communication framework where DMs first share their aggregation rules (e.g., averaging or min-max) used in downstream decisions for resolving forecast ambiguity. This, in turn, helps forecasters resolve indecision during elicitation. We further show that truthful elicitation of imprecise forecasts is achievable using proper scoring rules randomized over the aggregation procedure. Our approach allows DM to elicit and integrate the forecaster’s epistemic uncertainty into their decision-making process, improving the credibility of downstream decisions.

We study the problem of agent selection in causal strategic learning under multiple decision makers and address two key challenges that come with it. Firstly, while much of prior work focuses on studying a fixed pool of agents that remains static regardless of their evaluations, we consider the impact of selection procedure by which agents are not only evaluated, but also selected. When each decision maker unilaterally selects agents by maximising their own utility, we show that the optimal selection rule is a trade-off between selecting the best agents and providing incentives to maximise the agents' improvement. Furthermore, this optimal selection rule relies on incorrect predictions of agents' outcomes. Hence, we study the conditions under which a decision maker's optimal selection rule will not lead to deterioration of agents' outcome nor cause unjust reduction in agents' selection chance. To that end, we provide an analytical form of the optimal selection rule and a mechanism to retrieve the causal parameters from observational data, under certain assumptions on agents' behaviour. Secondly, when there are multiple decision makers, the interference between selection rules introduces another source of biases in estimating the underlying causal parameters. To address this problem, we provide a cooperative protocol which all decision makers must collectively adopt to recover the true causal parameters. Lastly, we complement our theoretical results with simulation studies. Our results highlight not only the importance of causal modeling as a strategy to mitigate the effect of gaming, as suggested by previous work, but also the need of a benevolent regulator to enable it.

Like many optimizers, Bayesian optimization often falls short of gaining user trust due to opacity. While attempts have been made to develop human-centric optimizers, they typically assume user knowledge is well-specified and error-free, employing users mainly as supervisors of the optimization process. We relax these assumptions and propose a more balanced human-AI partnership with our Collaborative and Explainable Bayesian Optimization (CoExBO) framework. Instead of explicitly requiring a user to provide a knowledge model, CoExBO employs preference learning to seamlessly integrate human insights into the optimization, resulting in algorithmic suggestions that resonate with user preference. CoExBO explains its candidate selection every iteration to foster trust, empowering users with a clearer grasp of the optimization. Furthermore, CoExBO offers a no-harm guarantee, allowing users to make mistakes; even with extreme adversarial interventions, the algorithm converges asymptotically to a vanilla Bayesian optimization. We validate CoExBO’s efficacy through human-AI teaming experiments in lithium-ion battery design, highlighting substantial improvements over conventional methods. Code is available https://github.com/ma921/CoExBO.

Out-of-distribution (OOD) generalisation is challenging because it involves not only learning from empirical data, but also deciding among various notions of generalisation, e.g. optimise based on the average-case risk, worst-case risk, or interpolations thereof. While this decision should in principle be decided by the model operator like medical doctors in practice, this information might not always be available at training time. This situation leads to arbitrary commitments to specific generalisation strategies by machine learners due to these deployment uncertainties. We introduce the Imprecise Domain Generalisation framework to mitigate this, featuring an imprecise risk optimisation that allows learners to stay imprecise by optimising against a continuous spectrum of generalisation strategies during training, and a model framework that allows operators to specify their generalisation preference at deployment. Our work, supported by theoretical and empirical evidence, showcases the benefits of integrating imprecision into domain generalisation.

This paper provides some first steps in developing empirical process theory for functions taking values in a vector space. Our main results provide bounds on the entropy of classes of smooth functions taking values in a Hilbert space, by leveraging theory from differential calculus of vector-valued functions and fractal dimension theory of metric spaces. We demonstrate how these entropy bounds can be used to show the uniform law of large numbers and asymptotic equicontinuity of the function classes, and also apply it to statistical learning theory in which the output space is a Hilbert space. We conclude with a discussion on the extension of Rademacher complexities to vector-valued function classes.

Being able to provide explanations for a model’s decision has become a central requirement for the development, deployment, and adoption of machine learning models. However, we are yet to understand what explanation methods can and cannot do. How do upstream factors such as data, model prediction, hyperparameters, and random initialization influence downstream explanations? While previous work raised concerns that explanations (E) may have little relationship with the prediction (Y), there is a lack of conclusive study to quantify this relationship. Our work borrows tools from causal inference to systematically assay this relationship. More specifically, we study the relationship between E and Y by measuring the treatment effect when intervening on their causal ancestors, i.e., on hyperparameters and inputs used to generate saliency-based Es or Ys. Our results suggest that the relationships between E and Y is far from ideal. In fact, the gap between ’ideal’ case only increase in higher-performing models — models that are likely to be deployed. Our work is a promising first step towards providing a quantitative measure of the relationship between E and Y, which could also inform the future development of methods for E with a quantitative metric.

We present a novel approach for explaining Gaussian processes (GPs) that can utilize the full analytical covariance structure present in GPs. Our method is based on the popular solution concept of Shapley values extended to stochastic cooperative games, resulting in explanations that are random variables. The GP explanations generated using our approach satisfy similar favorable axioms to standard Shapley values and possess a tractable covariance function across features and data observations. This covariance allows for quantifying explanation uncertainties and studying the statistical dependencies between explanations. We further extend our framework to the problem of predictive explanation, and propose a Shapley prior over the explanation function to predict Shapley values for new data based on previously computed ones. Our extensive illustrations demonstrate the effectiveness of the proposed approach.

Causality is a central concept in a wide range of research areas, yet there is still no universally agreed axiomatisation of causality. We view causality both as an extension of probability theory and as a study of what happens when one intervenes on a system, and argue in favour of taking Kolmogorov's measure-theoretic axiomatisation of probability as the starting point towards an axiomatisation of causality. To that end, we propose the notion of a causal space, consisting of a probability space along with a collection of transition probability kernels, called causal kernels, that encode the causal information of the space. Our proposed framework is not only rigorously grounded in measure theory, but it also sheds light on long-standing limitations of existing frameworks including, for example, cycles, latent variables and stochastic processes.

The Maximum Mean Discrepancy (MMD) has been the state-of-the-art nonparametric test for tackling the two-sample problem. Its statistic is given by the difference in expectations of the witness function, a real-valued function defined as a weighted sum of kernel evaluations on a set of basis points. Typically the kernel is optimized on a training set, and hypothesis testing is performed on a separate test set to avoid overfitting (i.e., control type-I error). That is, the test set is used to simultaneously estimate the expectations and define the basis points, while the training set only serves to select the kernel and is discarded. In this work, we propose to use the training data to also define the weights and the basis points for better data efficiency. We show that 1) the new test is consistent and has a well-controlled type-I error; 2) the optimal witness function is given by a precision-weighted mean in the reproducing kernel Hilbert space associated with the kernel; and 3) the test power of the proposed test is comparable or exceeds that of the MMD and other modern tests, as verified empirically on challenging synthetic and real problems (e.g., Higgs data).

Important problems in causal inference, economics, and, more generally, robust machine learning can be expressed as conditional moment restrictions, but estimation becomes challenging as it requires solving a continuum of unconditional moment restrictions. Previous works addressed this problem by extending the generalized method of moments (GMM) to continuum moment restrictions. In contrast, generalized empirical likelihood (GEL) provides a more general framework and has been shown to enjoy favorable small-sample properties compared to GMM-based estimators. To benefit from recent developments in machine learning, we provide a functional reformulation of GEL in which arbitrary models can be leveraged. Motivated by a dual formulation of the resulting infinite dimensional optimization problem, we devise a practical method and explore its asymptotic properties. Finally, we provide kernel- and neural network-based implementations of the estimator, which achieve state-of-the-art empirical performance on two conditional moment restriction problems.

Two-sample tests are important in statistics and machine learning, both as tools for scientific discovery as well as to detect distribution shifts.This led to the development of many sophisticated test procedures going beyond the standard supervised learning frameworks, whose usage can require specialized knowledge about two-sample testing. We use a simple test that takes the mean discrepancy of a witness function as the test statistic and prove that minimizing a squared loss leads to a witness with optimal testing power. This allows us to leverage recent advancements in AutoML. Without any user input about the problems at hand, and using the same method for all our experiments, our AutoML two-sample test achieves competitive performance on a diverse distribution shift benchmark as well as on challenging two-sample testing problems.

We address the problem of causal effect estima-tion in the presence of unobserved confounding,but where proxies for the latent confounder(s) areobserved. We propose two kernel-based meth-ods for nonlinear causal effect estimation in thissetting: (a) a two-stage regression approach, and(b) a maximum moment restriction approach. Wefocus on the proximal causal learning setting, butour methods can be used to solve a wider classof inverse problems characterised by a Fredholmintegral equation. In particular, we provide a uni-fying view of two-stage and moment restrictionapproaches for solving this problem in a nonlin-ear setting. We provide consistency guaranteesfor each algorithm, and demonstrate that these ap-proaches achieve competitive results on syntheticdata and data simulating a real-world task. In par-ticular, our approach outperforms earlier methodsthat are not suited to leveraging proxy variables.

We propose to analyse the conditional distributional treatment effect (CoDiTE), which, in contrast to the more common conditional average treatment effect (CATE), is designed to encode a treatment’s distributional aspects beyond the mean. We first introduce a formal definition of the CoDiTE associated with a distance function between probability measures. Then we discuss the CoDiTE associated with the maximum mean discrepancy via kernel conditional mean embeddings, which, coupled with a hypothesis test, tells us whether there is any conditional distributional effect of the treatment. Finally, we investigate what kind of conditional distributional effect the treatment has, both in an exploratory manner via the conditional witness function, and in a quantitative manner via U-statistic regression, generalising the CATE to higher-order moments. Experiments on synthetic, semi-synthetic and real datasets demonstrate the merits of our approach.

Counterfactual inference has become a ubiquitous tool in online advertisement, recommendation systems, medical diagnosis, and econometrics. Accurate modelling of outcome distributions associated with different interventions---known as counterfactual distributions---is crucial for the success of these applications. In this work, we propose to model counterfactual distributions using a novel Hilbert space representation called counterfactual mean embedding (CME). The CME embeds the associated counterfactual distribution into a reproducing kernel Hilbert space (RKHS) endowed with a positive definite kernel, which allows us to perform causal inference over the entire landscape of the counterfactual distribution. Based on this representation, we propose a distributional treatment effect (DTE) which can quantify the causal effect over entire outcome distributions. Our approach is nonparametric as the CME can be estimated under the unconfoundedness assumption from observational data without requiring any parametric assumption about the underlying distributions. We also establish a rate of convergence of the proposed estimator which depends on the smoothness of the conditional mean and the Radon-Nikodym derivative of the underlying marginal distributions. Furthermore, our framework allows for more complex outcomes such as images, sequences, and graphs. Our experimental results on synthetic data and off-policy evaluation tasks demonstrate the advantages of the proposed estimator.

Robotic grasping of house-hold objects has made remarkable progress in recent years. Yet, human grasps are still difficult to synthesize realistically. There are several key reasons: (1) the human hand has many degrees of freedom (more than robotic manipulators); (2) the synthesized hand should conform to the surface of the object; and (3) it should interact with the object in a semantically and physically plausible manner. To make progress in this direction, we draw inspiration from the recent progress on learning-based implicit representations for 3D object reconstruction. Specifically, we propose an expressive representation for human grasp modelling that is efficient and easy to integrate with deep neural networks. Our insight is that every point in a three-dimensional space can be characterized by the signed distances to the surface of the hand and the object, respectively. Consequently, the hand, the object, and the contact area can be represented by implicit surfaces in a common space, in which the proximity between the hand and the object can be modelled explicitly. We name this 3D to 2D mapping as Grasping Field, parameterize it with a deep neural network, and learn it from data. We demonstrate that the proposed grasping field is an effective and expressive representation for human grasp generation. Specifically, our generative model is able to synthesize high-quality human grasps, given only on a 3D object point cloud. The extensive experiments demonstrate that our generative model compares favorably with a strong baseline and approaches the level of natural human grasps. Furthermore, based on the grasping field representation, we propose a deep network for the challenging task of 3D hand-object interaction reconstruction from a single RGB image. Our method improves the physical plausibility of the hand-object contact reconstruction and achieves comparable performance for 3D hand reconstruction compared to state-of-the-art methods. Our model and code are available for research purpose at https://github.com/korrawe/grasping_field.

Consequential decisions are increasingly informed by sophisticated data-driven predictive models. However, consistently learning accurate predictive models requires access to ground truth labels. Unfortunately, in practice, labels may only exist conditional on certain decisions—if a loan is denied, there is not even an option for the individual to pay back the loan. In this paper, we show that, in this selective labels setting, learning to predict is suboptimal in terms of both fairness and utility. To avoid this undesirable behavior, we propose to directly learn stochastic decision policies that maximize utility under fairness constraints. In the context of fair machine learning, our results suggest the need for a paradigm shift from "learning to predict" to "learning to decide". Experiments on synthetic and real-world data illustrate the favorable properties of learning to decide, in terms of both utility and fairness.

We introduce a novel conditional density estimationmodel termed the conditional densityoperator (CDO). It naturally captures multivariate,multimodal output densities andshows performance that is competitive withrecent neural conditional density models andGaussian processes. The proposed model isbased on a novel approach to the reconstructionof probability densities from their kernelmean embeddings by drawing connections toestimation of Radon-Nikodym derivatives inthe reproducing kernel Hilbert space (RKHS).We prove finite sample bounds for the estimationerror in a standard density reconstructionscenario, independent of problem dimensionality.Interestingly, when a kernel is used thatis also a probability density, the CDO allowsus to both evaluate and sample the outputdensity efficiently. We demonstrate the versatilityand performance of the proposed modelon both synthetic and real-world data.

Meta-learning improves generalization of machine learning models when faced with previously unseen tasks by leveraging experiences from different, yet related prior tasks. To allow for better generalization, we propose a novel task representation called model-aware task embedding (MATE) that incorporates not only the data distributions of different tasks, but also the complexity of the tasks through the models used. The task complexity is taken into account by a novel variant of kernel mean embedding, combined with an instance-adaptive attention mechanism inspired by an SVM-based feature selection algorithm. Together with conditioning layers in deep neural networks, MATE can be easily incorporated into existing meta learners as a plug-and-play module. While MATE is widely applicable to general tasks where the concept of task/environment is involved, we demonstrate its effectiveness in few-shot learning by improving a state-of-the-art model consistently on two benchmarks. Source codes for this paper are available at https://github.com/VITA-Group/MATE.

Modern large-scale kernel-based tests such as maximum mean discrepancy (MMD) and kernelized Stein discrepancy (KSD) optimize kernel hyperparameters on a held-out sample via data splitting to obtain the most powerful test statistics. While data splitting results in a tractable null distribution, it suffers from a reduction in test power due to smaller test sample size. Inspired by the selective inference framework, we propose an approach that enables learning the hyperparameters and testing on the full sample without data splitting. Our approach can correctly calibrate the test in the presence of such dependency, and yield a test threshold in closed form. At the same significance level, our approach’s test power is empirically larger than that of the data-splitting approach, regardless of its split proportion.

We present a novel algorithm for non-linear instrumental variable (IV) regression, DualIV, which simplifies traditional two-stage methods via a dual formulation. Inspired by problems in stochastic programming, we show that two-stage procedures for non-linear IV regression can be reformulated as a convex-concave saddle-point problem. Our formulation enables us to circumvent the first-stage regression which is a potential bottleneck in real-world applications. We develop a simple kernel-based algorithm with an analytic solution based on this formulation. Empirical results show that we are competitive to existing, more complicated algorithms for non-linear instrumental variable regression.

We present a new operator-free, measure-theoretic approach to the conditional mean embedding as a random variable taking values in a reproducing kernel Hilbert space. While the kernel mean embedding of marginal distributions has been defined rigorously, the existing operator-based approach of the conditional version lacks a rigorous treatment, and depends on strong assumptions that hinder its analysis. Our approach does not impose any of the assumptions that the operator-based counterpart requires. We derive a natural regression interpretation to obtain empirical estimates, and provide a thorough analysis of its properties, including universal consistency with improved convergence rates. As natural by-products, we obtain the conditional analogues of the Maximum Mean Discrepancy and Hilbert-Schmidt Independence Criterion, and demonstrate their behaviour via simulations.

We propose a new family of specification tests called kernel conditional moment (KCM) tests. Our tests are built on a novel representation of conditional moment restrictions in a reproducing kernel Hilbert space (RKHS) called conditional moment embedding (CMME). After transforming the conditional moment restrictions into a continuum of unconditional counterparts, the test statistic is defined as the maximum moment restriction (MMR) within the unit ball of the RKHS. We show that the MMR not only fully characterizes the original conditional moment restrictions, leading to consistency in both hypothesis testing and parameter estimation, but also has an analytic expression that is easy to compute as well as closed-form asymptotic distributions. Our empirical studies show that the KCM test has a promising finite-sample performance compared to existing tests.

Fine-grained temporal action parsing is important in many applications, such as daily activity understanding, human motion analysis, surgical robotics and others requiring subtle and precise operations over a long-term period. In this paper we propose a novel bilinear pooling operation, which is used in intermediate layers of a temporal convolutional encoder-decoder net. In contrast to previous work, our proposed bilinear pooling is learnable and hence can capture more complex local statistics than the conventional counterpart. In addition, we introduce exact lower-dimension representations of our bilinear forms, so that the dimensionality is reduced without suffering from information loss nor requiring extra computation. We perform extensive experiments to quantitatively analyze our model and show the superior performances to other state-of-the-art pooling work on various datasets.

Neural Information Processing Systems (NIPS) is a top-tier annual conference in machine learning. The 2016 edition of the conference comprised more than 2,400 paper submissions, 3,000 reviewers, and 8,000 attendees. This represents a growth of nearly 40% in terms of submissions, 96% in terms of reviewers, and over 100% in terms of attendees as compared to the previous year. The massive scale as well as rapid growth of the conference calls for a thorough quality assessment of the peer-review process and novel means of improvement. In this paper, we analyze several aspects of the data collected during the review process, including an experiment investigating the efficacy of collecting ordinal rankings from reviewers. We make a number of key observations, provide suggestions that may be useful for subsequent conferences, and discuss open problems towards the goal of improving peer review.

In this paper, we study the minimax estimation of the Bochner integral \[ \mu_k(P) := \int_\mathcal{X} k(\cdot,x)\, dP(x), \] also called the kernel mean embedding, based on random samples drawn i.i.d. from $P$, where $k:\mathcal{X}\times\mathcal{X}\rightarrow \mathbb{R}$ is a positive definite kernel. Various estimators (including the empirical estimator), $\hat{\theta}_n$ of $\mu_k(P)$ are studied in the literature wherein all of them satisfy $\|\hat{\theta}_n-\mu_k(P)\|_{\mathcal{H}_k}=O_P(n^{-1/2})$ with $\mathcal{H}_k$ being the reproducing kernel Hilbert space induced by $k$. The main contribution of the paper is in showing that the above mentioned rate of $n^{-1/2}$ is minimax in $\|\cdot\|_{\mathcal{H}_k}$ and $\|\cdot\|_{L^2(\mathbb{R}^d)}$-norms over the class of discrete measures and the class of measures that has an infinitely differentiable density, with $k$ being a continuous translation- invariant kernel on $\mathbb{R}^d$. The interesting aspect of this result is that the minimax rate is independent of the smoothness of the kernel and the density of $P$ (if it exists).

For large-scale multi-class classification problems, consisting of tens of thousand target categories, recent works have emphasized the need to store billions of parameters. For instance, the classical l2-norm regularization employed by a state-of-the-art method results in the model size of 17GB for a training set whose size is only 129MB. To the contrary, by using a mixed-norm regularization approach, we show that around 99.5% of the stored parameters is dispensable noise. Using this strategy, we can extract the information relevant for classification, which is constituted in remaining 0.5% of the parameters, and hence demonstrate drastic reduction in model sizes. Furthermore, the proposed method leads to improvement in generalization performance compared to state-of-the-art methods, especially for under-represented categories. Lastly, our method enjoys easy parallelization, and scales well to tens of thousand target categories.

A mean function in a reproducing kernel Hilbert space (RKHS), or a kernel mean, is central to kernel methods in that it is used by many classical algorithms such as kernel principal component analysis, and it also forms the core inference step of modern kernel methods that rely on embedding probability distributions in RKHSs. Given a finite sample, an empirical average has been used commonly as a standard estimator of the true kernel mean. Despite a widespread use of this estimator, we show that it can be improved thanks to the well-known Stein phenomenon. We propose a new family of estimators called kernel mean shrinkage estimators (KMSEs), which benefit from both theoretical justifications and good empirical performance. The results demonstrate that the proposed estimators outperform the standard one, especially in a "large $d$, small $n$" paradigm.

We pose causal inference as the problem of learning to classify probability distributions. In particular, we assume access to a collection {(S_i,l_i)}_i=1^n, where each S_i is a sample drawn from the probability distribution of X_i \times Y_i, and l_i is a binary label indicating whether “X_i \to Y_i” or “X_i ←Y_i”. Given these data, we build a causal inference rule in two steps. First, we featurize each S_i using the kernel mean embedding associated with some characteristic kernel. Second, we train a binary classifier on such embeddings to distinguish between causal directions. We present generalization bounds showing the statistical consistency and learning rates of the proposed approach, and provide a simple implementation that achieves state-of-the-art cause-effect inference. Furthermore, we extend our ideas to infer causal relationships between more than two variables.

A mean function in reproducing kernel Hilbert space (RKHS), or a kernel mean, is an important part of many algorithms ranging from kernel principal component analysis to Hilbert-space embedding of distributions. Given a finite sample, an empirical average is the standard estimate for the true kernel mean. We show that this estimator can be improved due to a well-known phenomenon in statistics called Stein phenomenon. After consideration, our theoretical analysis reveals the existence of a wide class of estimators that are better than the standard one. Focusing on a subset of this class, we propose efficient shrinkage estimators for the kernel mean. Empirical evaluations on several applications clearly demonstrate that the proposed estimators outperform the standard kernel mean estimator.

This paper investigates domain generalization: How to take knowledge acquired from an arbitrary number of related domains and apply it to previously unseen domains? We propose Domain-Invariant Component Analysis (DICA), a kernel-based optimization algorithm that learns an invariant transformation by minimizing the dissimilarity across domains, whilst preserving the functional relationship between input and output variables. A learning-theoretic analysis shows that reducing dissimilarity improves the expected generalization ability of classifiers on new domains, motivating the proposed algorithm. Experimental results on synthetic and real-world datasets demonstrate that DICA successfully learns invariant features and improves classifier performance in practice.

Let X denote the feature and Y the target. We consider domain adaptation under three possible scenarios: (1) the marginal P_Y changes, while the conditional P_X|Y stays the same (\it target shift), (2) the marginal P_Y is fixed, while the conditional P_X|Y changes with certain constraints (\it conditional shift), and (3) the marginal P_Y changes, and the conditional P_X|Y changes with constraints (\it generalized target shift). Using background knowledge, causal interpretations allow us to determine the correct situation for a problem at hand. We exploit importance reweighting or sample transformation to find the learning machine that works well on test data, and propose to estimate the weights or transformations by \it reweighting or transforming training data to reproduce the covariate distribution on the test domain. Thanks to kernel embedding of conditional as well as marginal distributions, the proposed approaches avoid distribution estimation, and are applicable for high-dimensional problems. Numerical evaluations on synthetic and real-world datasets demonstrate the effectiveness of the proposed framework.

This paper presents a kernel-based discriminative learning framework on probability measures. Rather than relying on large collections of vectorial training examples, our framework learns using a collection of probability distributions that have been constructed to meaningfully represent training data. By representing these probability distributions as mean embeddings in the reproducing kernel Hilbert space (RKHS), we are able to apply many standard kernel-based learning techniques in straightforward fashion. To accomplish this, we construct a generalization of the support vector machine (SVM) called a support measure machine (SMM). Our analyses of SMMs provides several insights into their relationship to traditional SVMs. Based on such insights, we propose a flexible SVM (Flex-SVM) that places different kernel functions on each training example. Experimental results on both synthetic and real-world data demonstrate the effectiveness of our proposed framework.