Ridgeless regression has garnered attention among researchers, particularly in light of the ``Benign Overfitting'' phenomenon, where models interpolating noisy samples demonstrate robust generalization. However, kernel ridgeless regression does not always perform well due to the lack of flexibility. This paper enhances kernel ridgeless regression with Locally-Adaptive-Bandwidths (LAB) RBF kernels, incorporating kernel learning techniques to improve performance in both experiments and theory. For the first time, we demonstrate that functions learned from LAB RBF kernels belong to an integral space of Reproducible Kernel Hilbert Spaces (RKHSs). Despite the absence of explicit regularization in the proposed model, its optimization is equivalent to solving an $\ell_0$-regularized problem in the integral space of RKHSs, elucidating the origin of its generalization ability. Taking an approximation analysis viewpoint, we introduce an $l_q$-norm analysis technique (with $0

Equivariant machine learning is an approach for designing deep learning models that respect the symmetries of the problem, with the aim of reducing model complexity and improving generalization. In this paper, we focus on an extension of shift equivariance, which is the basis of convolution networks on images, to general graphs. Unlike images, graphs do not have a natural notion of domain translation. Therefore, we consider the graph functional shifts as the symmetry group: the unitary operators that commute with the graph shift operator. Notably, such symmetries operate in the signal space rather than directly in the spatial space. We remark that each linear filter layer of a standard spectral graph neural network (GNN) commutes with graph functional shifts, but the activation function breaks this symmetry. Instead, we propose nonlinear spectral filters (NLSFs) that are fully equivariant to graph functional shifts and show that they have universal approximation properties. The proposed NLSFs are based on a new form of spectral domain that is transferable between graphs. We demonstrate the superior performance of NLSFs over existing spectral GNNs in node and graph classification benchmarks.

Many questions in science center around the fundamental problem of understanding causal relationships. However, most constraint-based causal discovery algorithms, including the well-celebrated PC algorithm, often incur an exponential number of conditional independence (CI) tests, posing limitations in various applications. Addressing this, our work focuses on characterizing what can be learned about the underlying causal graph with a reduced number of CI tests. We show that it is possible to a learn a coarser representation of the hidden causal graph with a polynomial number of tests. This coarser representation, named Causal Consistent Partition Graph (CCPG), comprises of a partition of the vertices and a directed graph defined over its components. CCPG satisfies consistency of orientations and additional constraints which favor finer partitions. Furthermore, it reduces to the underlying causal graph when the causal graph is identifiable. As a consequence, our results offer the first efficient algorithm for recovering the true causal graph with a polynomial number of tests, in special cases where the causal graph is fully identifiable through observational data and potentially additional interventions.

We give the first polynomial-time, differentially node-private, and robust algorithm for estimating the edge density of Erd\H{o}s-R\'enyi random graphs and their generalization, inhomogeneous random graphs. We further prove information-theoretical lower bounds, showing that the error rate of our algorithm is optimal up to logarithmic factors. Previous algorithms incur either exponential running time or suboptimal error rates. Two key ingredients of our algorithm are (1) a new sum-of-squares algorithm for robust edge density estimation, and (2) the reduction from privacy to robustness based on sum-of-squares exponential mechanisms due to Hopkins et al. (STOC 2023).

Current language models decode text token by token according to probabilistic distribution, and determining the appropriate candidates for the next token is crucial to ensure generation quality. This study introduces adaptive decoding, a mechanism that dynamically empowers language models to ascertain a sensible candidate set during generation. Specifically, we introduce an entropy-based metric called confidence and conceptualize determining the optimal candidate set as a confidence-increasing process. The rationality of including a token in the candidate set is assessed by leveraging the increment of confidence. Experimental results reveal that our method balances diversity and coherence well. The human evaluation shows that our method can generate human-preferred text. Additionally, our method can potentially improve the reasoning ability of language models.

SARS-CoV-2, the highly contagious pathogen responsible for the COVID-19 pandemic, has persistent effects that begin four weeks after initial infection and last for an undetermined duration. These chronic effects are more harmful than acute ones. This review explores the long-term impact of the virus on various human organs, including the pulmonary, cardiovascular, neurological, reproductive, gastrointestinal, musculoskeletal, endocrine, and lymphoid systems, particularly in older adults. Regarding diagnosis, RT-PCR is the gold standard for detecting COVID-19, though it requires specialized equipment, skilled personnel, and considerable time to produce results. To address these limitations, artificial intelligence in imaging and microfluidics technologies offers promising alternatives for diagnosing COVID-19 efficiently. Pharmacological and non-pharmacological strategies are effective in mitigating the persistent impacts of COVID-19. These strategies enhance immunity in post-COVID-19 patients by reducing cytokine release syndrome, improving T cell response, and increasing the circulation of activated natural killer and CD8 T cells in blood and tissues. This, in turn, alleviates symptoms such as fever, nausea, fatigue, muscle weakness, and pain. Vaccines, including inactivated viral, live attenuated viral, protein subunit, viral vectored, mRNA, DNA, and nanoparticle vaccines, significantly reduce the adverse long-term effects of the virus. However, no vaccine has been reported to provide lifetime protection against COVID-19. Consequently, protective measures such as physical distancing, mask usage, and hand hygiene remain essential strategies. This review offers a comprehensive understanding of the persistent effects of COVID-19 on individuals of varying ages, along with insights into diagnosis, treatment, vaccination, and future preventative measures against the spread of SARS-CoV-2.

This paper introduces Fortune Analytics Language Model (FALM). FALM empowers users with direct access to comprehensive business analysis, including market trends, company performance metrics, and expert insights. Unlike generic LLMs, FALM leverages a curated knowledge base built from professional journalism, enabling it to deliver precise and in-depth answers to intricate business questions. Users can further leverage natural language queries to directly visualize financial data, generating insightful charts and graphs to understand trends across diverse business sectors clearly. FALM fosters user trust and ensures output accuracy through three novel methods: 1) Time-aware reasoning guarantees accurate event registration and prioritizes recent updates. 2) Thematic trend analysis explicitly examines topic evolution over time, providing insights into emerging business landscapes. 3) Content referencing and task decomposition enhance answer fidelity and data visualization accuracy. We conduct both automated and human evaluations, demonstrating FALM's significant performance improvements over baseline methods while prioritizing responsible AI practices. These benchmarks establish FALM as a cutting-edge LLM in the business and media domains, with exceptional accuracy and trustworthiness.

While there are a plethora of methods for link prediction in knowledge graphs, state-of-the-art approaches are often black box, obfuscating model reasoning and thereby limiting the ability of users to make informed decisions about model predictions. Recently, methods have emerged to generate prediction explanations for Knowledge Graph Embedding models, a widely-used class of methods for link prediction. The question then becomes, how well do these explanation systems work? To date this has generally been addressed anecdotally, or through time-consuming user research. In this work, we present an in-depth exploration of a simple link prediction explanation method we call LinkLogic, that surfaces and ranks explanatory information used for the prediction. Importantly, we construct the first-ever link prediction explanation benchmark, based on family structures present in the FB13 dataset. We demonstrate the use of this benchmark as a rich evaluation sandbox, probing LinkLogic quantitatively and qualitatively to assess the fidelity, selectivity and relevance of the generated explanations. We hope our work paves the way for more holistic and empirical assessment of knowledge graph prediction explanation methods in the future.

We consider the problem of recovering the ground truth ordering (ranking, top-$k$, or others) over a large number of alternatives. The wisdom of crowd is a heuristic approach based on Condorcet's Jury theorem to address this problem through collective opinions. This approach fails to recover the ground truth when the majority of the crowd is misinformed. The surprisingly popular (SP) algorithm cite{prelec2017solution} is an alternative approach that is able to recover the ground truth even when experts are in minority. The SP algorithm requires the voters to predict other voters' report in the form of a full probability distribution over all rankings of alternatives. However, when the number of alternatives, $m$, is large, eliciting the prediction report or even the vote over $m$ alternatives might be too costly. In this paper, we design a scalable alternative of the SP algorithm which only requires eliciting partial preferences from the voters, and propose new variants of the SP algorithm. In particular, we propose two versions -- Aggregated-SP and Partial-SP -- that ask voters to report vote and prediction on a subset of size $k$ ($\ll m$) in terms of top alternative, partial rank, or an approval set. Through a large-scale crowdsourcing experiment on MTurk, we show that both of our approaches outperform conventional preference aggregation algorithms for the recovery of ground truth rankings, when measured in terms of Kendall-Tau distance and Spearman's $\rho$. We further analyze the collected data and demonstrate that voters' behavior in the experiment, including the minority of the experts, and the SP phenomenon, can be correctly simulated by a concentric mixtures of Mallows model. Finally, we provide theoretical bounds on the sample complexity of SP algorithms with partial rankings to demonstrate the theoretical guarantees of the proposed methods.

Mathematical equations have been unreasonably effective in describing complex natural phenomena across various scientific disciplines. However, discovering such insightful equations from data presents significant challenges due to the necessity of navigating extremely high-dimensional combinatorial and nonlinear hypothesis spaces. Traditional methods of equation discovery, commonly known as symbolic regression, largely focus on extracting equations from data alone, often neglecting the rich domain-specific prior knowledge that scientists typically depend on. To bridge this gap, we introduce LLM-SR, a novel approach that leverages the extensive scientific knowledge and robust code generation capabilities of Large Language Models (LLMs) to discover scientific equations from data in an efficient manner. Specifically, LLM-SR treats equations as programs with mathematical operators and combines LLMs' scientific priors with evolutionary search over equation programs. The LLM iteratively proposes new equation skeleton hypotheses, drawing from its physical understanding, which are then optimized against data to estimate skeleton parameters. We demonstrate LLM-SR's effectiveness across three diverse scientific domains, where it discovers physically accurate equations that provide significantly better fits to in-domain and out-of-domain data compared to the well-established symbolic regression baselines. Incorporating scientific prior knowledge also enables LLM-SR to search the equation space more efficiently than baselines. Code is available at: https://github.com/deep-symbolic-mathematics/LLM-SR