Neural network ensembles, such as Bayesian neural networks (BNNs), have shown success in the areas of uncertainty estimation and robustness. However, a crucial challenge prohibits their use in practice. BNNs require a large number of predictions to produce reliable results, leading to a significant increase in computational cost. To alleviate this issue, we propose spatial smoothing, a method that ensembles neighboring feature map points of convolutional neural networks. By simply adding a few blur layers to the models, we empirically show that spatial smoothing improves accuracy, uncertainty estimation, and robustness of BNNs across a whole range of ensemble sizes. In particular, BNNs incorporating spatial smoothing achieve high predictive performance merely with a handful of ensembles. Moreover, this method also can be applied to canonical deterministic neural networks to improve the performances. A number of evidences suggest that the improvements can be attributed to the stabilized feature maps and the smoothing of the loss landscape. In addition, we provide a fundamental explanation for prior works — namely, global average pooling, pre-activation, and ReLU6 — by addressing them as special cases of spatial smoothing. These not only enhance accuracy, but also improve uncertainty estimation and robustness by making the loss landscape smoother in the same manner as spatial smoothing. The code is available at https://github.com/xxxnell/spatial-smoothing.

It is an important problem in trustworthy machine learning to recognize out-of-distribution (OOD) inputs which are inputs unrelated to the in-distribution task. Many out-of-distribution detection methods have been suggested in recent years. The goal of this paper is to recognize common objectives as well as to identify the implicit scoring functions of different OOD detection methods. We focus on the sub-class of methods that use surrogate OOD data during training in order to learn an OOD detection score that generalizes to new unseen out-distributions at test time.We show that binary discrimination between in- and (different) out-distributions is equivalent to several distinct formulations of the OOD detection problem. When trained in a shared fashion with a standard classifier, this binary discriminator reaches an OOD detection performance similar to that of Outlier Exposure. Moreover, we show that the confidence loss which is used by Outlier Exposure has an implicit scoring function which differs in a non-trivial fashion from the theoretically optimal scoring function in the case where training and test out-distribution are the same, which again is similar to the one used when training an Energy-Based OOD detector or when adding a background class. In practice, when trained in exactly the same way, all these methods perform similarly.

Negative sampling~(NS) loss plays an important role in learning knowledge graph embedding~(KGE) to handle a huge number of entities. However, the performance of KGE degrades without hyperparameters such as the margin term and number of negative samples in NS loss being appropriately selected. Currently, empirical hyperparameter tuning addresses this problem at the cost of computational time. To solve this problem, we theoretically analyzed NS loss to assist hyperparameter tuning and understand the better use of the NS loss in KGE learning. Our theoretical analysis showed that scoring methods with restricted value ranges, such as TransE and RotatE, require appropriate adjustment of the margin term or the number of negative samples different from those without restricted value ranges, such as RESCAL, ComplEx, and DistMult. We also propose subsampling methods specialized for the NS loss in KGE studied from a theoretical aspect. Our empirical analysis on the FB15k-237, WN18RR, and YAGO3-10 datasets showed that the results of actually trained models agree with our theoretical findings.

Certifiable robustness is a highly desirable property for adopting deep neural networks (DNNs) in safety-critical scenarios, but often demands tedious computations to establish. The main hurdle lies in the massive amount of non-linearity in large DNNs. To trade off the DNN expressiveness (which calls for more non-linearity) and robustness certification scalability (which prefers more linearity), we propose a novel solution to strategically manipulate neurons, by "grafting" appropriate levels of linearity. The core of our proposal is to first linearize insignificant ReLU neurons, to eliminate the non-linear components that are both redundant for DNN performance and harmful to its certification. We then optimize the associated slopes and intercepts of the replaced linear activations for restoring model performance while maintaining certifiability. Hence, typical neuron pruning could be viewed as a special case of grafting a linear function of the fixed zero slopes and intercept, that might overly restrict the network flexibility and sacrifice its performance. Extensive experiments on multiple datasets and network backbones show that our linearity grafting can (1) effectively tighten certified bounds; (2) achieve competitive certifiable robustness without certified robust training (i.e., over 30% improvements on CIFAR-10 models); and (3) scale up complete verification to large adversarially trained models with 17M parameters. Codes are available at https://github.com/VITA-Group/Linearity-Grafting.

In this paper, we develop a new graph kernel, namely the Hierarchical Transitive-Aligned Kernel, by transitively aligning the vertices between graphs through a family of hierarchical prototype graphs. Comparing to most existing state-of-the-art graph kernels, the proposed kernel has three theoretical advantages. First, it incorporates the locational correspondence information between graphs into the kernel computation, and thus overcomes the shortcoming of ignoring structural correspondences arising in most R-convolution kernels. Second, it guarantees the transitivity between the correspondence information that is not available for most existing matching kernels. Third, it incorporates the information of all graphs under comparisons into the kernel computation process, and thus encapsulates richer characteristics. Experimental evaluations demonstrate the effectiveness of the new transitive-aligned kernel.

We propose an input sparsity time sampling algorithm that can spectrally approximate the Gram matrix corresponding to the q-fold column-wise tensor product of q matrices using a nearly optimal number of samples, improving upon all previously known methods by poly(q) factors. Furthermore, for the important special case of the q-fold self-tensoring of a dataset, which is the feature matrix of the degree-q polynomial kernel, the leading term of our method’s runtime is proportional to the size of the dataset and has no dependence on q. Previous techniques either incur a poly(q) factor slowdown in their runtime or remove the dependence on q at the expense of having sub-optimal target dimension, and depend quadratically on the number of data-points in their runtime. Our sampling technique relies on a collection of q partially correlated random projections which can be simultaneously applied to a dataset X in total time that only depends on the size of X, and at the same time their q-fold Kronecker product acts as a near-isometry for any fixed vector in the column span of $X^{\otimes q}$. We also show that our sampling methods generalize to other classes of kernels beyond polynomial, such as Gaussian and Neural Tangent kernels.

We propose efficient random features for approximating a new and rich class of kernel functions that we refer to as Generalized Zonal Kernels (GZK). Our proposed GZK family, generalizes the zonal kernels (i.e., dot-product kernels on the unit sphere) by introducing radial factors in the Gegenbauer series expansion of these kernel functions. The GZK class of kernels includes a wide range of ubiquitous kernel functions such as the entirety of dot-product kernels as well as the Gaussian and the recently introduced Neural Tangent kernels. Interestingly, by exploiting the reproducing property of the Gegenbauer (Zonal) Harmonics, we can construct efficient random features for the GZK family based on randomly oriented Gegenbauer harmonics. We prove subspace embedding guarantees for our Gegenbauer features which ensures that our features can be used for approximately solving learning problems such as kernel k-means clustering, kernel ridge regression, etc. Empirical results show that our proposed features outperform recent kernel approximation methods.

Recent years have seen a surge of interest in meta-learning techniques for tackling the few-shot learning (FSL) problem. However, the meta-learner is prone to overfitting since there are only a few available samples, which can be identified as sampling noise on a clean dataset. Besides, when handling the data with noisy labels, the meta-learner could be extremely sensitive to label noise on a corrupted dataset. To address these two challenges, we present Eigen-Reptile (ER) that updates the meta-parameters with the main direction of historical task-specific parameters. Specifically, the main direction is computed in a fast way, where the scale of the calculated matrix is related to the number of gradient steps for the specific task instead of the number of parameters. Furthermore, to obtain a more accurate main direction for Eigen-Reptile in the presence of many noisy labels, we further propose Introspective Self-paced Learning (ISPL). We have theoretically and experimentally demonstrated the soundness and effectiveness of the proposed Eigen-Reptile and ISPL. Particularly, our experiments on different tasks show that the proposed method is able to outperform or achieve highly competitive performance compared with other gradient-based methods with or without noisy labels. The code and data for the proposed method are provided for research purposes https://github.com/Anfeather/Eigen-Reptile.

The focus of the paper is functional output regression (FOR) with convoluted losses. While most existing work consider the square loss setting, we leverage extensions of the Huber and the $\epsilon$-insensitive loss (induced by infimal convolution) and propose a flexible framework capable of handling various forms of outliers and sparsity in the FOR family. We derive computationally tractable algorithms relying on duality to tackle the resulting tasks in the context of vector-valued reproducing kernel Hilbert spaces. The efficiency of the approach is demonstrated and contrasted with the classical squared loss setting on both synthetic and real-world benchmarks.

Measures of similarity (or dissimilarity) are a key ingredient to many machine learning algorithms. We introduce DID, a pairwise dissimilarity measure applicable to a wide range of data spaces, which leverages the data's internal structure to be invariant to diffeomorphisms. We prove that DID enjoys properties which make it relevant for theoretical study and practical use. By representing each datum as a function, DID is defined as the solution to an optimization problem in a Reproducing Kernel Hilbert Space and can be expressed in closed-form. In practice, it can be efficiently approximated via Nyström sampling. Empirical experiments support the merits of DID.

We study a nonparametric approach to Bayesian computation via feature means, where the expectation of prior features is updated to yield expected posterior features, based on regression from kernel or neural net features of the observations. All quantities involved in the Bayesian update are learned from observed data, making the method entirely model-free. The resulting algorithm is a novel instance of a kernel Bayes' rule (KBR). Our approach is based on importance weighting, which results in superior numerical stability to the existing approach to KBR, which requires operator inversion. We show the convergence of the estimator using a novel consistency analysis on the importance weighting estimator in the infinity norm. We evaluate our KBR on challenging synthetic benchmarks, including a filtering problem with a state-space model involving high dimensional image observations. The proposed method yields uniformly better empirical performance than the existing KBR, and competitive performance with other competing methods.We evaluate our KBR on challenging synthetic benchmarks, including a filtering problem with a state-space model involving high dimensional image observations. The proposed method yields uniformly better empirical performance than the existing KBR, and competitive performance with other competing methods.

We propose a new conditional dependence measure and a statistical test for conditional independence. The measure is based on the difference between analytic kernel embeddings of two well-suited distributions evaluated at a finite set of locations. We obtain its asymptotic distribution under the null hypothesis of conditional independence and design a consistent statistical test from it. We conduct a series of experiments showing that our new test outperforms state-of-the-art methods both in terms of type-I and type-II errors even in the high dimensional setting.

Kernel mean embeddings are a powerful tool to represent probability distributions over arbitrary spaces as single points in a Hilbert space. Yet, the cost of computing and storing such embeddings prohibits their direct use in large-scale settings. We propose an efficient approximation procedure based on the Nyström method, which exploits a small random subset of the dataset. Our main result is an upper bound on the approximation error of this procedure. It yields sufficient conditions on the subsample size to obtain the standard (1/sqrt(n)) rate while reducing computational costs. We discuss applications of this result for the approximation of the maximum mean discrepancy and quadrature rules, and we illustrate our theoretical findings with numerical experiments.

The problem of learning functions over spaces of probabilities - or distribution regression - is gaining significant interest in the machine learning community. The main challenge in these settings is to identify a suitable representation capturing all relevant properties of a distribution. The well-established approach in this sense is to use kernel mean embeddings, which lift kernel-induced similarity on the input domain at the probability level. This strategy effectively tackles the two-stage sampling nature of the problem, enabling one to derive estimators with strong statistical guarantees, such as universal consistency and excess risk bounds. However, kernel mean embeddings implicitly hinge on the maximum mean discrepancy (MMD), a metric on probabilities, which is not the most suited to capture geometrical relations between distributions. In contrast, optimal transport (OT) metrics, are potentially more appealing. In this work, we propose an OT-based estimator for distribution regression. We build on the Sliced Wasserstein distance to obtain an OT-based representation. We study the theoretical properties of a kernel ridge regression estimator based on such representation, for which we prove universal consistency and excess risk bounds. Preliminary experiments complement our theoretical findings by showing the effectiveness of the proposed approach and compare it with MMD-based estimators.