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Miscellaneous Aspects of Machine Learning/Sequential, Network, and Time Series Modeling

YIXUAN ZHANG, Quyu Kong, Feng Zhou

In this study, we propose a novel deep spatio-temporal point process model, Deep Kernel Mixture Point Processes (DKMPP), that incorporates multimodal covariate information. DKMPP is an enhanced version of Deep Mixture Point Processes (DMPP), which uses a more flexible deep kernel to model complex relationships between events and covariate data, improving the model's expressiveness. To address the intractable training procedure of DKMPP due to the non-integrable deep kernel, we utilize an integration-free method based on score matching, and further improve efficiency by adopting a scalable denoising score matching method. Our experiments demonstrate that DKMPP and its corresponding score-based estimators outperform baseline models, showcasing the advantages of incorporating covariate information, utilizing a deep kernel, and employing score-based estimators.

Theory/Learning Theory

Moritz Haas, David Holzmüller, Ulrike Luxburg, Ingo Steinwart

The success of over-parameterized neural networks trained to near-zero training error has caused great interest in the phenomenon of benign overfitting, where estimators are statistically consistent even though they interpolate noisy training data. While benign overfitting in fixed dimension has been established for some learning methods, current literature suggests that for regression with typical kernel methods and wide neural networks, benign overfitting requires a high-dimensional setting, where the dimension grows with the sample size. In this paper, we show that the smoothness of the estimators, and not the dimension, is the key: benign overfitting is possible if and only if the estimator's derivatives are large enough. We generalize existing inconsistency results to non-interpolating models and more kernels to show that benign overfitting with moderate derivatives is impossible in fixed dimension. Conversely, we show that benign overfitting is possible for regression with a sequence of spiky-smooth kernels with large derivatives. Using neural tangent kernels, we translate our results to wide neural networks. We prove that while infinite-width networks do not overfit benignly with the ReLU activation, this can be fixed by adding small high-frequency fluctuations to the activation function. Our experiments verify that such neural networks, while overfitting, can indeed generalize well even on low-dimensional data sets.

Optimization/Zero-order and Black-box Optimization

Zhongxiang Dai, Gregory Kang Ruey Lau, Arun Verma, YAO SHU, Bryan Kian Hsiang Low, Patrick Jaillet

Kernelized bandits, also known as Bayesian optimization (BO), has been a prevalent method for optimizing complicated black-box reward functions. Various BO algorithms have been theoretically shown to enjoy upper bounds on their cumulative regret which are sub-linear in the number $T$ of iterations, and a regret lower bound of $\Omega(\sqrt{T})$ has been derived which represents the unavoidable regrets for any classical BO algorithm. Recent works on quantum bandits have shown that with the aid of quantum computing, it is possible to achieve tighter regret upper bounds better than their corresponding classical lower bounds. However, these works are restricted to either multi-armed or linear bandits, and are hence not able to solve sophisticated real-world problems with non-linear reward functions. To this end, we introduce the quantum-Gaussian process-upper confidence bound (Q-GP-UCB) algorithm. To the best of our knowledge, our Q-GP-UCB is the first BO algorithm able to achieve a regret upper bound of $\mathcal{O}(\text{poly}\log T)$, which is significantly smaller than its regret lower bound of $\Omega(\sqrt{T})$ in the classical setting. Moreover, thanks to our novel analysis of the confidence ellipsoid, our Q-GP-UCB with the linear kernel achieves a smaller regret than the quantum linear UCB algorithm from the previous work. We use simulations, as well as an experiment using a real quantum computer, to verify that the theoretical quantum speedup achieved by our Q-GP-UCB is also potentially relevant in practice.

Miscellaneous Aspects of Machine Learning/Transfer, Multitask and Meta-learning

Tianjun Ke, Haoqun Cao, Zenan Ling, Feng Zhou

Meta-learning has demonstrated promising results in few-shot classification (FSC) by learning to solve new problems using prior knowledge. Bayesian methods are effective at characterizing uncertainty in FSC, which is crucial in high-risk fields. In this context, the logistic-softmax likelihood is often employed as an alternative to the softmax likelihood in multi-class Gaussian process classification due to its conditional conjugacy property. However, the theoretical property of logistic-softmax is not clear and previous research indicated that the inherent uncertainty of logistic-softmax leads to suboptimal performance. To mitigate these issues, we revisit and redesign the logistic-softmax likelihood, which enables control of the \textit{a priori} confidence level through a temperature parameter. Furthermore, we theoretically and empirically show that softmax can be viewed as a special case of logistic-softmax and logistic-softmax induces a larger family of data distribution than softmax. Utilizing modified logistic-softmax, we integrate the data augmentation technique into the deep kernel based Gaussian process meta-learning framework, and derive an analytical mean-field approximation for task-specific updates. Our approach yields well-calibrated uncertainty estimates and achieves comparable or superior results on standard benchmark datasets. Code is publicly available at \url{https://github.com/keanson/revisit-logistic-softmax}.

Miscellaneous Aspects of Machine Learning/Kernel methods

Mingzhen He, FAN He, Ruikai Yang, Xiaolin Huang

As a nonlinear dimension reduction technique, the diffusion map (DM) has been widely used. In DM, kernels play an important role for capturing the nonlinear relationship of data. However, only symmetric kernels can be used now, which prevents the use of DM in directed graphs, trophic networks, and other real-world scenarios where the intrinsic and extrinsic geometries in data are asymmetric. A promising technique is the magnetic transform which converts an asymmetric matrix to a Hermitian one. However, we are facing essential problems, including how diffusion distance could be preserved and how divergence could be avoided during diffusion process. Via theoretical proof, we successfully establish a diffusion representation framework with the magnetic transform, named MagDM. The effectiveness and robustness for dealing data endowed with asymmetric proximity are demonstrated on three synthetic datasets and two trophic networks.

Miscellaneous Aspects of Machine Learning/Kernel methods

Sangwoong Yoon, Frank Park, Gunsu YUN, Iljung Kim, Yung-Kyun Noh

Kernel density estimation (KDE) is integral to a range of generative and discriminative tasks in machine learning. Drawing upon tools from the multidimensional calculus of variations, we derive an optimal weight function that reduces bias in standard kernel density estimates for density ratios, leading to improved estimates of prediction posteriors and information-theoretic measures. In the process, we shed light on some fundamental aspects of density estimation, particularly from the perspective of algorithms that employ KDEs as their main building blocks.

Miscellaneous Aspects of Machine Learning/Kernel methods

Christian Fiedler, Michael Herty, Sebastian Trimpe

In many applications of machine learning, a large number of variables are considered. Motivated by machine learning of interacting particle systems, we consider the situation when the number of input variables goes to infinity. First, we continue the recent investigation of the mean field limit of kernels and their reproducing kernel Hilbert spaces, completing the existing theory. Next, we provide results relevant for approximation with such kernels in the mean field limit, including a representer theorem. Finally, we use these kernels in the context of statistical learning in the mean field limit, focusing on Support Vector Machines. In particular, we show mean field convergence of empirical and infinite-sample solutions as well as the convergence of the corresponding risks. On the one hand, our results establish rigorous mean field limits in the context of kernel methods, providing new theoretical tools and insights for large-scale problems. On the other hand, our setting corresponds to a new form of limit of learning problems, which seems to have not been investigated yet in the statistical learning theory literature.

Theory/Reinforcement Learning and Planning

Gellert Weisz, András György, Csaba Szepesvari

We consider online reinforcement learning (RL) in episodic Markov decision processes (MDPs) under the linear $q^\pi$-realizability assumption, where it is assumed that the action-values of all policies can be expressed as linear functions of state-action features. This class is known to be more general than linear MDPs, where the transition kernel and the reward function are assumed to be linear functions of the feature vectors. As our first contribution, we show that the difference between the two classes is the presence of states in linearly $q^\pi$-realizable MDPs where for any policy, all the actions have approximately equal values, and skipping over these states by following an arbitrarily fixed policy in those states transforms the problem to a linear MDP. Based on this observation, we derive a novel (computationally inefficient) learning algorithm for linearly $q^\pi$-realizable MDPs that simultaneously learns what states should be skipped over and runs another learning algorithm on the linear MDP hidden in the problem. The method returns an $\epsilon$-optimal policy after $\text{polylog}(H, d)/\epsilon^2$ interactions with the MDP, where $H$ is the time horizon and $d$ is the dimension of the feature vectors, giving the first polynomial-sample-complexity online RL algorithm for this setting. The results are proved for the misspecified case, where the sample complexity is shown to degrade gracefully with the misspecification error.

Deep Learning/Graph Neural Networks

Maksim Zhdanov, Nico Hoffmann, Gabriele Cesa

Steerable convolutional neural networks (CNNs) provide a general framework for building neural networks equivariant to translations and transformations of an origin-preserving group $G$, such as reflections and rotations. They rely on standard convolutions with $G$-steerable kernels obtained by analytically solving the group-specific equivariance constraint imposed onto the kernel space. As the solution is tailored to a particular group $G$, implementing a kernel basis does not generalize to other symmetry transformations, complicating the development of general group equivariant models. We propose using implicit neural representation via multi-layer perceptrons (MLPs) to parameterize $G$-steerable kernels. The resulting framework offers a simple and flexible way to implement Steerable CNNs and generalizes to any group $G$ for which a $G$-equivariant MLP can be built. We prove the effectiveness of our method on multiple tasks, including N-body simulations, point cloud classification and molecular property prediction.

Theory/Learning Theory

Yatin Dandi, Ludovic Stephan, Florent Krzakala, Bruno Loureiro, Lenka Zdeborová

A recent line of work in high-dimensional statistics working under the Gaussian mixture hypothesis has led to a number of results in the context of empirical risk minimization, Bayesian uncertainty quantification, separation of kernel methods and neural networks, ensembling and fluctuation of random features. We provide rigorous proofs for the applicability of these results to a general class of datasets $(\mathbf{x_i},y_i, {i=1,\dots,n})$ containing independent samples from a mixture distribution $\sum_{c\in\mathcal{C}} \rho_{c}P_{c}^{\mathbf{x}}$. Specifically, we consider the hypothesis class of generalized linear models $\hat{y} = F(\mathbf{\Theta}^{\top}\mathbf{x})$ and investigate the asymptotic joint statistics of a family of generalized linear estimators $(\mathbf{\Theta}^{(1)}, \dots, \mathbf{\Theta}^{(M)})$, obtained either from (a) minimizing an empirical risk $\hat{R_n}^{(m)}(\mathbf{\Theta}^{(m)};\mathbf{X},\mathbf{y})$ or (b) sampling from the associated Gibbs measure $\exp(-\beta n \hat{R_n}^{(m)}(\mathbf{\Theta}^{(m)};\mathbf{X},\mathbf{y}))$. Our main contribution is to characterize under which conditions the asymptotic joint statistics of this family depends (on a weak sense) only on the means and covariances of the class conditional features distribution $P_{c}^{\mathbf{x}}$. This allows us to prove the universality of different quantities of interest, including training, generalization errors, as well as the geometrical properties and correlations of the estimators.

Applications/Chemistry and Drug Discovery

Siddhartha Laghuvarapu, Zhen Lin, Jimeng Sun

In drug discovery, it is vital to confirm the predictions of pharmaceutical properties from computational models using costly wet-lab experiments. Hence, obtaining reliable uncertainty estimates is crucial for prioritizing drug molecules for subsequent experimental validation. Conformal Prediction (CP) is a promising tool for creating such prediction sets for molecular properties with a coverage guarantee. However, the exchangeability assumption of CP is often challenged with covariate shift in drug discovery tasks: Most datasets contain limited labeled data, which may not be representative of the vast chemical space from which molecules are drawn. To address this limitation, we propose a method called CoDrug that employs an energy-based model leveraging both training data and unlabelled data, and Kernel Density Estimation (KDE) to assess the densities of a molecule set. The estimated densities are then used to weigh the molecule samples while building prediction sets and rectifying for distribution shift. In extensive experiments involving realistic distribution drifts in various small-molecule drug discovery tasks, we demonstrate the ability of CoDrug to provide valid prediction sets and its utility in addressing the distribution shift arising from de novo drug design models. On average, using CoDrug can reduce the coverage gap by over 35% when compared to conformal prediction sets not adjusted for covariate shift.

Miscellaneous Aspects of Machine Learning/Kernel methods

Yanfang Xue, Pengfei Fang, Jinyue Tian, Shipeng Zhu, hui xue

Complex-valued representation exists inherently in the time-sequential data that can be derived from the integration of harmonic waves. The non-stationary spectral kernel, realizing a complex-valued feature mapping, has shown its potential to analyze the time-varying statistical characteristics of the time-sequential data, as a result of the modeling frequency parameters. However, most existing spectral kernel-based methods eliminate the imaginary part, thereby limiting the representation power of the spectral kernel. To tackle this issue, we propose a generalized spectral kernel network, namely, \underline{Co}mplex-valued \underline{s}pectral kernel \underline{Net}work (CosNet), which includes spectral kernel mapping generalization (SKMG) module and complex-valued spectral kernel embedding (CSKE) module. Concretely, the SKMG module is devised to generalize the spectral kernel mapping in the real number domain to the complex number domain, recovering the inherent complex-valued representation for the real-valued data. Then a following CSKE module is further developed to combine the complex-valued spectral kernels and neural networks to effectively capture long-range or periodic relations of the data. Along with the CosNet, we study the effect of the complex-valued spectral kernel mapping via theoretically analyzing the bound of covering number and generalization error. Extensive experiments demonstrate that CosNet performs better than the mainstream kernel methods and complex-valued neural networks.

Deep Learning/Generative Models and Autoencoders

Mohammad Jalali, Cheuk Ting Li, Farzan Farnia

The evaluation of generative models has received significant attention in the machine learning community. When applied to a multi-modal distribution which is common among image datasets, an intuitive evaluation criterion is the number of modes captured by the generative model. While several scores have been proposed to evaluate the quality and diversity of a model's generated data, the correspondence between existing scores and the number of modes in the distribution is unclear. In this work, we propose an information-theoretic diversity evaluation method for multi-modal underlying distributions. We utilize the R\'enyi Kernel Entropy (RKE) as an evaluation score based on quantum information theory to measure the number of modes in generated samples. To interpret the proposed evaluation method, we show that the RKE score can output the number of modes of a mixture of sub-Gaussian components. We also prove estimation error bounds for estimating the RKE score from limited data, suggesting a fast convergence of the empirical RKE score to the score for the underlying data distribution. Utilizing the RKE score, we conduct an extensive evaluation of state-of-the-art generative models over standard image datasets. The numerical results indicate that while the recent algorithms for training generative models manage to improve the mode-based diversity over the earlier architectures, they remain incapable of capturing the full diversity of real data. Our empirical results provide a ranking of widely-used generative models based on the RKE score of their generated samples.

Deep Learning/Theory

Arvind Mahankali, Haochen Zhang, Kefan Dong, Margalit Glasgow, Tengyu Ma

Despite recent theoretical progress on the non-convex optimization of two-layer neural networks, it is still an open question whether gradient descent on neural networks without unnatural modifications can achieve better sample complexity than kernel methods. This paper provides a clean mean-field analysis of projected gradient flow on polynomial-width two-layer neural networks. Different from prior works, our analysis does not require unnatural modifications of the optimization algorithm. We prove that with sample size $n = O(d^{3.1})$ where $d$ is the dimension of the inputs, the network trained with projected gradient flow converges in polynomial time to a non-trivial error that is not achievable by kernel methods using $n \ll d^4$ samples, hence demonstrating a clear separation between unmodified gradient descent and NTK. As a corollary, we show that projected gradient descent with a positive learning rate and a polynomial number of iterations converges to low error with the same sample complexity.

Social Aspects/Accountability, Transparency and Interpretability

Hao Sun, Boris van Breugel, Jonathan Crabbé, Nabeel Seedat, Mihaela van der Schaar

Uncertainty quantification (UQ) is essential for creating trustworthy machine learning models. Recent years have seen a steep rise in UQ methods that can flag suspicious examples, however, it is often unclear what exactly these methods identify. In this work, we propose a framework for categorizing uncertain examples flagged by UQ methods. We introduce the confusion density matrix---a kernel-based approximation of the misclassification density---and use this to categorize suspicious examples identified by a given uncertainty method into three classes: out-of-distribution (OOD) examples, boundary (Bnd) examples, and examples in regions of high in-distribution misclassification (IDM). Through extensive experiments, we show that our framework provides a new and distinct perspective for assessing differences between uncertainty quantification methods, thereby forming a valuable assessment benchmark.

Optimization/Convex

Jungbin Kim, Insoon Yang

We propose a novel methodology that systematically analyzes ordinary differential equation (ODE) models for first-order optimization methods by converting the task of proving convergence rates into verifying the positive semidefiniteness of specific Hilbert-Schmidt integral operators. Our approach is based on the performance estimation problems (PEP) introduced by Drori and Teboulle. Unlike previous works on PEP, which rely on finite-dimensional linear algebra, we use tools from functional analysis. Using the proposed method, we establish convergence rates of various accelerated gradient flow models, some of which are new. As an immediate consequence of our framework, we show a correspondence between minimizing function values and minimizing gradient norms.

Miscellaneous Aspects of Machine Learning/Kernel methods

Xingdong Feng, Xin HE, Caixing Wang, Chao Wang, Jingnan Zhang

Covariate shift occurs prevalently in practice, where the input distributions of the source and target data are substantially different. Despite its practical importance in various learning problems, most of the existing methods only focus on some specific learning tasks and are not well validated theoretically and numerically. To tackle this problem, we propose a unified analysis of general nonparametric methods in a reproducing kernel Hilbert space (RKHS) under covariate shift. Our theoretical results are established for a general loss belonging to a rich loss function family, which includes many commonly used methods as special cases, such as mean regression, quantile regression, likelihood-based classification, and margin-based classification. Two types of covariate shift problems are the focus of this paper and the sharp convergence rates are established for a general loss function to provide a unified theoretical analysis, which concurs with the optimal results in literature where the squared loss is used. Extensive numerical studies on synthetic and real examples confirm our theoretical findings and further illustrate the effectiveness of our proposed method.

Theory/Reinforcement Learning and Planning

Navdeep Kumar, Esther Derman, Matthieu Geist, Kfir Y. Levy, Shie Mannor

Policy gradient methods have become a standard for training reinforcement learning agents in a scalable and efficient manner. However, they do not account for transition uncertainty, whereas learning robust policies can be computationally expensive. In this paper, we introduce robust policy gradient (RPG), a policy-based method that efficiently solves rectangular robust Markov decision processes (MDPs). We provide a closed-form expression for the worst occupation measure. Incidentally, we find that the worst kernel is a rank-one perturbation of the nominal. Combining the worst occupation measure with a robust Q-value estimation yields an explicit form of the robust gradient. Our resulting RPG can be estimated from data with the same time complexity as its non-robust equivalent. Hence, it relieves the computational burden of convex optimization problems required for training robust policies by current policy gradient approaches.

Miscellaneous Aspects of Machine Learning/Kernel methods

Peter Macgregor, He Sun

Constructing a similarity graph from a set $X$ of data points in $ \mathbb{R}^d$ is the first step of many modern clustering algorithms. However, typical constructions of a similarity graph have high time complexity, and a quadratic space dependency with respect to $|X|$. We address this limitation and present a new algorithmic framework that constructs a sparse approximation of the fully connected similarity graph while preserving its cluster structure. Our presented algorithm is based on the kernel density estimation problem, and is applicable for arbitrary kernel functions. We compare our designed algorithm with the well-known implementations from the scikit-learn library and the FAISS library, and find that our method significantly outperforms the implementation from both libraries on a variety of datasets.

Miscellaneous Aspects of Machine Learning/Kernel methods

Felix Biggs, Antonin Schrab, Arthur Gretton

We propose novel statistics which maximise the power of a two-sample test based on the Maximum Mean Discrepancy (MMD), byadapting over the set of kernels used in defining it.For finite sets, this reduces to combining (normalised) MMD values under each of these kernels via a weighted soft maximum.Exponential concentration bounds are proved for our proposed statistics under the null and alternative.We further show how these kernels can be chosen in a data-dependent but permutation-independent way, in a well-calibrated test, avoiding data splitting.This technique applies more broadly to general permutation-based MMD testing, and includes the use of deep kernels with features learnt using unsupervised models such as auto-encoders.We highlight the applicability of our MMD-Fuse tests on both synthetic low-dimensional and real-world high-dimensional data, and compare its performance in terms of power against current state-of-the-art kernel tests.