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In supervised learning, the regularization path is sometimes used as a convenient theoretical proxy for the optimization path of gradient descent initialized from zero. In this paper, we study a modification of the regularization path for infinite-width 2-layer ReLU neural networks with nonzero initial distribution of the weights at different scales. By exploiting a link with unbalanced optimal-transport theory, we show that, despite the non-convexity of the 2-layer network training, this problem admits an infinite-dimensional convex counterpart. We formulate the corresponding functional-optimization problem and investigate its main properties. In particular, we show that, as the scale of the initialization ranges between $0$ and $+\infty$, the associated path interpolates continuously between the so-called kernel and rich regimes. Numerical experiments confirm that, in our setting, the scaling path and the final states of the optimization path behave similarly, even beyond these extreme points.

Search query variation poses a challenge in e-commerce search, as equivalent search intents can be expressed through different queries with surface-level differences. This paper introduces a framework to recognize and leverage query equivalence to enhance searcher and business outcomes. The proposed approach addresses three key problems: mapping queries to vector representations of search intent, identifying nearest neighbor queries expressing equivalent or similar intent, and optimizing for user or business objectives. The framework utilizes both surface similarity and behavioral similarity to determine query equivalence. Surface similarity involves canonicalizing queries based on word inflection, word order, compounding, and noise words. Behavioral similarity leverages historical search behavior to generate vector representations of query intent. An offline process is used to train a sentence similarity model, while an online nearest neighbor approach supports processing of unseen queries. Experimental evaluations demonstrate the effectiveness of the proposed approach, outperforming popular sentence transformer models and achieving a Pearson correlation of 0.85 for query similarity. The results highlight the potential of leveraging historical behavior data and training models to recognize and utilize query equivalence in e-commerce search, leading to improved user experiences and business outcomes. Further advancements and benchmark datasets are encouraged to facilitate the development of solutions for this critical problem in the e-commerce domain.

An experimental comparison of two or more optimization algorithms requires the same computational resources to be assigned to each algorithm. When a maximum runtime is set as the stopping criterion, all algorithms need to be executed in the same machine if they are to use the same resources. Unfortunately, the implementation code of the algorithms is not always available, which means that running the algorithms to be compared in the same machine is not always possible. And even if they are available, some optimization algorithms might be costly to run, such as training large neural-networks in the cloud. In this paper, we consider the following problem: how do we compare the performance of a new optimization algorithm B with a known algorithm A in the literature if we only have the results (the objective values) and the runtime in each instance of algorithm A? Particularly, we present a methodology that enables a statistical analysis of the performance of algorithms executed in different machines. The proposed methodology has two parts. First, we propose a model that, given the runtime of an algorithm in a machine, estimates the runtime of the same algorithm in another machine. This model can be adjusted so that the probability of estimating a runtime longer than what it should be is arbitrarily low. Second, we introduce an adaptation of the one-sided sign test that uses a modified p-value and takes into account that probability. Such adaptation avoids increasing the probability of type I error associated with executing algorithms A and B in different machines.

The curse-of-dimensionality (CoD) taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional PDEs as Richard Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerically partial differential equations (PDEs) in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. In this paper, we develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs into pieces corresponding to different dimensions and samples randomly a subset of these dimensional pieces in each iteration of training PINNs. We theoretically prove the convergence guarantee and other desired properties of the proposed method. We experimentally demonstrate that the proposed method allows us to solve many notoriously hard high-dimensional PDEs, including the Hamilton-Jacobi-Bellman (HJB) and the Schr\"{o}dinger equations in thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. For instance, we solve nontrivial nonlinear PDEs (one HJB equation and one Black-Scholes equation) in 100,000 dimensions in 6 hours on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, SDGD can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

Abstractive dialogue summarization is to generate a concise and fluent summary covering the salient information in a dialogue among two or more interlocutors. It has attracted great attention in recent years based on the massive emergence of social communication platforms and an urgent requirement for efficient dialogue information understanding and digestion. Different from news or articles in traditional document summarization, dialogues bring unique characteristics and additional challenges, including different language styles and formats, scattered information, flexible discourse structures and unclear topic boundaries. This survey provides a comprehensive investigation on existing work for abstractive dialogue summarization from scenarios, approaches to evaluations. It categorizes the task into two broad categories according to the type of input dialogues, i.e., open-domain and task-oriented, and presents a taxonomy of existing techniques in three directions, namely, injecting dialogue features, designing auxiliary training tasks and using additional data.A list of datasets under different scenarios and widely-accepted evaluation metrics are summarized for completeness. After that, the trends of scenarios and techniques are summarized, together with deep insights on correlations between extensively exploited features and different scenarios. Based on these analyses, we recommend future directions including more controlled and complicated scenarios, technical innovations and comparisons, publicly available datasets in special domains, etc.

We consider the problem of computing bounds for causal queries on causal graphs with unobserved confounders and discrete valued observed variables, where identifiability does not hold. Existing non-parametric approaches for computing such bounds use linear programming (LP) formulations that quickly become intractable for existing solvers because the size of the LP grows exponentially in the number of edges in the causal graph. We show that this LP can be significantly pruned, allowing us to compute bounds for significantly larger causal inference problems compared to existing techniques. This pruning procedure allows us to compute bounds in closed form for a special class of problems, including a well-studied family of problems where multiple confounded treatments influence an outcome. We extend our pruning methodology to fractional LPs which compute bounds for causal queries which incorporate additional observations about the unit. We show that our methods provide significant runtime improvement compared to benchmarks in experiments and extend our results to the finite data setting. For causal inference without additional observations, we propose an efficient greedy heuristic that produces high quality bounds, and scales to problems that are several orders of magnitude larger than those for which the pruned LP can be solved.

We consider the problem of estimating the false-/ true-positive-rate (FPR/TPR) for a binary classification model when there are incorrect labels (label noise) in the validation set. Our motivating application is fraud prevention where accurate estimates of FPR are critical to preserving the experience for good customers, and where label noise is highly asymmetric. Existing methods seek to minimize the total error in the cleaning process - to avoid cleaning examples that are not noise, and to ensure cleaning of examples that are. This is an important measure of accuracy but insufficient to guarantee good estimates of the true FPR or TPR for a model, and we show that using the model to directly clean its own validation data leads to underestimates even if total error is low. This indicates a need for researchers to pursue methods that not only reduce total error but also seek to de-correlate cleaning error with model scores.

Fractional (hyper-)graph theory is concerned with the specific problems that arise when fractional analogues of otherwise integer-valued (hyper-)graph invariants are considered. The focus of this paper is on fractional edge covers of hypergraphs. Our main technical result generalizes and unifies previous conditions under which the size of the support of fractional edge covers is bounded independently of the size of the hypergraph itself. This allows us to extend previous tractability results for checking if the fractional hypertree width of a given hypergraph is $\leq k$ for some constant $k$. We also show how our results translate to fractional vertex covers.

We analyze the behavior of stochastic approximation algorithms where iterates, in expectation, make progress towards an objective at each step. When progress is proportional to the step size of the algorithm, we prove exponential concentration bounds. These tail-bounds contrast asymptotic normality results which are more frequently associated with stochastic approximation. The methods that we develop rely on a geometric ergodicity proof. This extends a result on Markov chains due to Hajek (1982) to the area of stochastic approximation algorithms. For Projected Stochastic Gradient Descent with a non-vanishing gradient, our results can be used to prove $O(1/t)$ and linear convergence rates.

This work considers the question of how convenient access to copious data impacts our ability to learn causal effects and relations. In what ways is learning causality in the era of big data different from -- or the same as -- the traditional one? To answer this question, this survey provides a comprehensive and structured review of both traditional and frontier methods in learning causality and relations along with the connections between causality and machine learning. This work points out on a case-by-case basis how big data facilitates, complicates, or motivates each approach.