The Weisfeiler-Lehman (WL) test is a classical procedure for graph isomorphism testing. The WL test has also been widely used both for designing graph kernels and for analyzing graph neural networks. In this paper, we propose the Weisfeiler-Lehman (WL) distance, a notion of distance between labeled measure Markov chains (LMMCs), of which labeled graphs are special cases. The WL distance is polynomial time computable and is also compatible with the WL test in the sense that the former is positive if and only if the WL test can distinguish the two involved graphs. The WL distance captures and compares subtle structures of the underlying LMMCs and, as a consequence of this, it is more discriminating than the distance between graphs used for defining the state-of-the-art Wasserstein Weisfeiler-Lehman graph kernel. Inspired by the structure of the WL distance we identify a neural network architecture on LMMCs which turns out to be universal w.r.t. continuous functions defined on the space of all LMMCs (which includes all graphs) endowed with the WL distance. Finally, the WL distance turns out to be stable w.r.t. a natural variant of the Gromov-Wasserstein (GW) distance for comparing metric Markov chains that we identify. Hence, the WL distance can also be construed as a polynomial time lower bound for the GW distance which is in general NP-hard to compute.

Mixup is a data augmentation method that generates new data points by mixing a pair of input data. While mixup generally improves the prediction performance, it sometimes degrades the performance. In this paper, we first identify the main causes of this phenomenon by theoretically and empirically analyzing the mixup algorithm. To resolve this, we propose GenLabel, a simple yet effective relabeling algorithm designed for mixup. In particular, GenLabel helps the mixup algorithm correctly label mixup samples by learning the class-conditional data distribution using generative models. Via theoretical and empirical analysis, we show that mixup, when used together with GenLabel, can effectively resolve the aforementioned phenomenon, improving the accuracy of mixup-trained model.

In many machine learning applications, it is important for the model to provide confidence scores that accurately capture its prediction uncertainty. Although modern learning methods have achieved great success in predictive accuracy, generating calibrated confidence scores remains a major challenge. Mixup, a popular yet simple data augmentation technique based on taking convex combinations of pairs of training examples, has been empirically found to significantly improve confidence calibration across diverse applications. However, when and how Mixup helps calibration is still a mystery. In this paper, we theoretically prove that Mixup improves calibration in \textit{high-dimensional} settings by investigating natural statistical models. Interestingly, the calibration benefit of Mixup increases as the model capacity increases. We support our theories with experiments on common architectures and datasets. In addition, we study how Mixup improves calibration in semi-supervised learning. While incorporating unlabeled data can sometimes make the model less calibrated, adding Mixup training mitigates this issue and provably improves calibration. Our analysis provides new insights and a framework to understand Mixup and calibration.

Mini-batch optimal transport (m-OT) has been successfully used in practical applications that involve probability measures with a very high number of supports. The m-OT solves several smaller optimal transport problems and then returns the average of their costs and transportation plans. Despite its scalability advantage, the m-OT does not consider the relationship between mini-batches which leads to undesirable estimation. Moreover, the m-OT does not approximate a proper metric between probability measures since the identity property is not satisfied. To address these problems, we propose a novel mini-batch scheme for optimal transport, named Batch of Mini-batches Optimal Transport (BoMb-OT), that finds the optimal coupling between mini-batches and it can be seen as an approximation to a well-defined distance on the space of probability measures. Furthermore, we show that the m-OT is a limit of the entropic regularized version of the BoMb-OT when the regularized parameter goes to infinity. Finally, we carry out experiments on various applications including deep generative models, deep domain adaptation, approximate Bayesian computation, color transfer, and gradient flow to show that the BoMb-OT can be widely applied and performs well in various applications.

Continual Learning (CL) is the problem of sequentially learning a set of tasks and preserving all the knowledge acquired. Many existing methods assume that the data stream is explicitly divided into a sequence of known contexts (tasks), and use this information to know when to transfer knowledge from one context to another. Unfortunately, many real-world CL scenarios have no clear task nor context boundaries, motivating the study of task-agnostic CL, where neither the specific tasks nor their switches are known both in training and testing. This paper proposes a variational architecture growing framework dubbed VariGrow. By interpreting dynamically growing neural networks as a Bayesian approximation, and defining flexible implicit variational distributions, VariGrow detects if a new task is arriving through an energy-based novelty score. If the novelty score is high and the sample is detected" as a new task, VariGrow will grow a new expert module to be responsible for it. Otherwise, the sample will be assigned to one of the existing experts who is mostfamiliar" with it (i.e., one with the lowest novelty score). We have tested VariGrow on several CIFAR and ImageNet-based benchmarks for the strict task-agnostic CL setting and demonstrate its consistent superior performance. Perhaps surprisingly, its performance can even be competitive compared to task-aware methods.

The label noise transition matrix, denoting the transition probabilities from clean labels to noisy labels, is crucial for designing statistically robust solutions. Existing estimators for noise transition matrices, e.g., using either anchor points or clusterability, focus on computer vision tasks that are relatively easier to obtain high-quality representations. We observe that tasks with lower-quality features fail to meet the anchor-point or clusterability condition, due to the coexistence of both uninformative and informative representations. To handle this issue, we propose a generic and practical information-theoretic approach to down-weight the less informative parts of the lower-quality features. This improvement is crucial to identifying and estimating the label noise transition matrix. The salient technical challenge is to compute the relevant information-theoretical metrics using only noisy labels instead of clean ones. We prove that the celebrated $f$-mutual information measure can often preserve the order when calculated using noisy labels. We then build our transition matrix estimator using this distilled version of features. The necessity and effectiveness of the proposed method are also demonstrated by evaluating the estimation error on a varied set of tabular data and text classification tasks with lower-quality features. Code is available at github.com/UCSC-REAL/BeyondImages.

We propose a model-agnostic randomized learning framework based on Random Hypothesis Subspace Sampling (RHSS). Given any hypothesis class, it randomly samples $k$ hypotheses and learns a near-optimal model from their span by simply solving a linear least square problem in $O(n k^2)$ time, where $n$ is the number of training instances. On the theory side, we derive the performance guarantee of RHSS from a generic subspace approximation perspective, leveraging properties of metric entropy and random matrices. On the practical side, we apply the RHSS framework to learn kernel, network and tree based models. Experimental results show they converge efficiently as $k$ increases and outperform their model-specific counterparts including random Fourier feature, random vector functional link and extra tree on real-world data sets.

When one observes a sequence of variables $(x_1, y_1), \ldots, (x_n, y_n)$, Conformal Prediction (CP) is a methodology that allows to estimate a confidence set for $y_{n+1}$ given $x_{n+1}$ by merely assuming that the distribution of the data is exchangeable. CP sets have guaranteed coverage for any finite population size $n$. While appealing, the computation of such a set turns out to be infeasible in general, \eg when the unknown variable $y_{n+1}$ is continuous. The bottleneck is that it is based on a procedure that readjusts a prediction model on data where we replace the unknown target by all its possible values in order to select the most probable one. This requires computing an infinite number of models, which often makes it intractable. In this paper, we combine CP techniques with classical algorithmic stability bounds to derive a prediction set computable with a single model fit. We demonstrate that our proposed confidence set does not lose any coverage guarantees while avoiding the need for data splitting as currently done in the literature. We provide some numerical experiments to illustrate the tightness of our estimation when the sample size is sufficiently large, on both synthetic and real datasets.

We propose a fundamental theory on ensemble learning that evaluates a given ensemble system by a well-grounded set of metrics.Previous studies used a variant of Fano's inequality of information theory and derived a lower bound of the classification error rate on the basis of the accuracy and diversity of models.We revisit the original Fano's inequality and argue that the studies did not take into account the information lost when multiple model predictions are combined into a final prediction.To address this issue, we generalize the previous theory to incorporate the information loss.Further, we empirically validate and demonstrate the proposed theory through extensive experiments on actual systems.The theory reveals the strengths and weaknesses of systems on each metric, which will push the theoretical understanding of ensemble learning and give us insights into designing systems.

Technology improvements have made it easier than ever to collect diverse telemetry at high resolution from any cyber or physical system, for both monitoring and control. In the domain of monitoring, anomaly detection has become an important problem in many research areas ranging from IoT and sensor networks to devOps. These systems operate in real, noisy and non-stationary environments. A fundamental question is then, ‘\emph{How to quickly spot anomalies in a data-stream, and differentiate them from either sudden or gradual drifts in the normal behaviour?}’ Although several heuristics have been proposed for detecting anomalies on streams, no known method has formalized the desiderata and rigorously proven that they can be achieved. We begin by formalizing the problem as a sequential estimation task. We propose \name, (\textbf{Fi}ne \textbf{T}une on \textbf{Ne}w and \textbf{S}imilar \textbf{S}amples), a flexible framework for detecting anomalies on data streams. We show that in the case when the data stream has a gaussian distribution, FITNESS is provably both robust and adaptive. The core of our method is to fine-tune the anomaly detection system only on recent, similar examples, before predicting an anomaly score. We prove that this is sufficient for robustness and adaptivity. We further experimentally demonstrate that \name\; is \emph{flexible} in practice, i.e., it can convert existing offline AD algorithms in to robust and adaptive online ones.

Mini-batch optimal transport (m-OT) has been widely used recently to deal with the memory issue of OT in large-scale applications. Despite their practicality, m-OT suffers from misspecified mappings, namely, mappings that are optimal on the mini-batch level but are partially wrong in the comparison with the optimal transportation plan between the original measures. Motivated by the misspecified mappings issue, we propose a novel mini-batch method by using partial optimal transport (POT) between mini-batch empirical measures, which we refer to as mini-batch partial optimal transport (m-POT). Leveraging the insight from the partial transportation, we explain the source of misspecified mappings from the m-OT and motivate why limiting the amount of transported masses among mini-batches via POT can alleviate the incorrect mappings. Finally, we carry out extensive experiments on various applications such as deep domain adaptation, partial domain adaptation, deep generative model, color transfer, and gradient flow to demonstrate the favorable performance of m-POT compared to current mini-batch methods.

Missing values arise in most real-world data sets due to the aggregation of multiple sources and intrinsically missing information (sensor failure, unanswered questions in surveys...). In fact, the very nature of missing values usually prevents us from running standard learning algorithms. In this paper, we focus on the extensively-studied linear models, but in presence of missing values, which turns out to be quite a challenging task. Indeed, the Bayes predictor can be decomposed as a sum of predictors corresponding to each missing pattern. This eventually requires to solve a number of learning tasks, exponential in the number of input features, which makes predictions impossible for current real-world datasets. First, we propose a rigorous setting to analyze a least-square type estimator and establish a bound on the excess risk which increases exponentially in the dimension. Consequently, we leverage the missing data distribution to propose a new algorithm, and derive associated adaptive risk bounds that turn out to be minimax optimal. Numerical experiments highlight the benefits of our method compared to state-of-the-art algorithms used for predictions with missing values.

Recent works put much effort into \emph{tensor network structure search} (TN-SS), aiming to select suitable tensor network (TN) structures, involving the TN-ranks, formats, and so on, for the decomposition or learning tasks. In this paper, we consider a practical variant of TN-SS, dubbed \emph{TN permutation search}~(TN-PS), in which we search for good mappings from tensor modes onto TN vertices (core tensors) for compact TN representations. We conduct a theoretical investigation of TN-PS and propose a practically-efficient algorithm to resolve the problem. Theoretically, we prove the counting and metric properties of search spaces of TN-PS, analyzing for the first time the impact of TN structures on these unique properties. Numerically, we propose a novel \emph{meta-heuristic} algorithm, in which the searching is done by randomly sampling in a neighborhood established in our theory, and then recurrently updating the neighborhood until convergence. Numerical results demonstrate that the new algorithm can reduce the required model size of TNs in extensive benchmarks, implying the improvement in the expressive power of TNs. Furthermore, the computational cost for the new algorithm is significantly less than that in (Li and Sun, 2020).

This work investigates the compatibility between label smoothing (LS) and knowledge distillation (KD). Contemporary findings addressing this thesis statement take dichotomous standpoints: Muller et al. (2019) and Shen et al. (2021b). Critically, there is no effort to understand and resolve these contradictory findings, leaving the primal question − to smooth or not to smooth a teacher network? − unanswered. The main contributions of our work are the discovery, analysis and validation of systematic diffusion as the missing concept which is instrumental in understanding and resolving these contradictory findings. This systematic diffusion essentially curtails the benefits of distilling from an LS-trained teacher, thereby rendering KD at increased temperatures ineffective. Our discovery is comprehensively supported by large-scale experiments, analyses and case studies including image classification, neural machine translation and compact student distillation tasks spanning across multiple datasets and teacher-student architectures. Based on our analysis, we suggest practitioners to use an LS-trained teacher with a low-temperature transfer to achieve high performance students. Code and models are available at https://keshik6.github.io/revisiting-ls-kd-compatibility/

K-nearest neighbors (KNN) is one of the earliest and most established algorithms in machine learning. For regression tasks, KNN averages the targets within a neighborhood which poses a number of challenges: the neighborhood definition is crucial for the predictive performance as neighbors might be selected based on uninformative features, and averaging does not account for how the function changes locally. We propose a novel method called Differential Nearest Neighbors Regression (DNNR) that addresses both issues simultaneously: during training, DNNR estimates local gradients to scale the features; during inference, it performs an n-th order Taylor approximation using estimated gradients. In a large-scale evaluation on over 250 datasets, we find that DNNR performs comparably to state-of-the-art gradient boosting methods and MLPs while maintaining the simplicity and transparency of KNN. This allows us to derive theoretical error bounds and inspect failures. In times that call for transparency of ML models, DNNR provides a good balance between performance and interpretability.