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Quantum Relative Entropy (QRE) programming is a recently popular and challenging class of convex optimization problems with significant applications in quantum computing and quantum information theory. We are interested in modern interior point (IP) methods based on optimal self-concordant barriers for the QRE cone. A range of theoretical and numerical challenges associated with such barrier functions and the QRE cones have hindered the scalability of IP methods. To address these challenges, we propose a series of numerical and linear algebraic techniques and heuristics aimed at enhancing the efficiency of gradient and Hessian computations for the self-concordant barrier function, solving linear systems, and performing matrix-vector products. We also introduce and deliberate about some interesting concepts related to QRE such as symmetric quantum relative entropy (SQRE). We also introduce a two-phase method for performing facial reduction that can significantly improve the performance of QRE programming. Our new techniques have been implemented in the latest version (DDS 2.2) of the software package DDS. In addition to handling QRE constraints, DDS accepts any combination of several other conic and non-conic convex constraints. Our comprehensive numerical experiments encompass several parts including 1) a comparison of DDS 2.2 with Hypatia for the nearest correlation matrix problem, 2) using DDS for combining QRE constraints with various other constraint types, and 3) calculating the key rate for quantum key distribution (QKD) channels and presenting results for several QKD protocols.

Despite the frequent use of agent-based models (ABMs) for studying social phenomena, parameter estimation remains a challenge, often relying on costly simulation-based heuristics. This work uses variational inference to estimate the parameters of an opinion dynamics ABM, by transforming the estimation problem into an optimization task that can be solved directly. Our proposal relies on probabilistic generative ABMs (PGABMs): we start by synthesizing a probabilistic generative model from the ABM rules. Then, we transform the inference process into an optimization problem suitable for automatic differentiation. In particular, we use the Gumbel-Softmax reparameterization for categorical agent attributes and stochastic variational inference for parameter estimation. Furthermore, we explore the trade-offs of using variational distributions with different complexity: normal distributions and normalizing flows. We validate our method on a bounded confidence model with agent roles (leaders and followers). Our approach estimates both macroscopic (bounded confidence intervals and backfire thresholds) and microscopic ($200$ categorical, agent-level roles) more accurately than simulation-based and MCMC methods. Consequently, our technique enables experts to tune and validate their ABMs against real-world observations, thus providing insights into human behavior in social systems via data-driven analysis.

In the era of Large Language Models (LLMs), Knowledge Distillation (KD) emerges as a pivotal methodology for transferring advanced capabilities from leading proprietary LLMs, such as GPT-4, to their open-source counterparts like LLaMA and Mistral. Additionally, as open-source LLMs flourish, KD plays a crucial role in both compressing these models, and facilitating their self-improvement by employing themselves as teachers. This paper presents a comprehensive survey of KD's role within the realm of LLM, highlighting its critical function in imparting advanced knowledge to smaller models and its utility in model compression and self-improvement. Our survey is meticulously structured around three foundational pillars: \textit{algorithm}, \textit{skill}, and \textit{verticalization} -- providing a comprehensive examination of KD mechanisms, the enhancement of specific cognitive abilities, and their practical implications across diverse fields. Crucially, the survey navigates the intricate interplay between data augmentation (DA) and KD, illustrating how DA emerges as a powerful paradigm within the KD framework to bolster LLMs' performance. By leveraging DA to generate context-rich, skill-specific training data, KD transcends traditional boundaries, enabling open-source models to approximate the contextual adeptness, ethical alignment, and deep semantic insights characteristic of their proprietary counterparts. This work aims to provide an insightful guide for researchers and practitioners, offering a detailed overview of current methodologies in KD and proposing future research directions. Importantly, we firmly advocate for compliance with the legal terms that regulate the use of LLMs, ensuring ethical and lawful application of KD of LLMs. An associated Github repository is available at //github.com/Tebmer/Awesome-Knowledge-Distillation-of-LLMs.

This study addresses the challenges in parameter estimation of stochastic differential equations driven by non-Gaussian noises, which are critical in understanding dynamic phenomena such as price fluctuations and the spread of infectious diseases. Previous research highlighted the potential of LSTM networks in estimating parameters of alpha stable Levy driven SDEs but faced limitations including high time complexity and constraints of the LSTM chaining property. To mitigate these issues, we introduce the PEnet, a novel CNN-LSTM-based three-stage model that offers an end to end approach with superior accuracy and adaptability to varying data structures, enhanced inference speed for long sequence observations through initial data feature condensation by CNN, and high generalization capability, allowing its application to various complex SDE scenarios. Experiments on synthetic datasets confirm PEnet significant advantage in estimating SDE parameters associated with noise characteristics, establishing it as a competitive method for SDE parameter estimation in the presence of Levy noise.

Tennenbaum's theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit, analyze and generalize this result. The chosen framework allows for a synthetic approach to computability theory, exploiting that, externally, all functions definable in constructive type theory can be shown computable. We then build on this viewpoint, and furthermore internalize it by assuming a version of Church's thesis, which expresses that any function on natural numbers is representable by a formula in PA. This assumption provides for a conveniently abstract setup to carry out rigorous computability arguments, even in the theorem's mechanization. Concretely, we constructivize several classical proofs and present one inherently constructive rendering of Tennenbaum's theorem, all following arguments from the literature. Concerning the classical proofs in particular, the constructive setting allows us to highlight differences in their assumptions and conclusions which are not visible classically. All versions are accompanied by a unified mechanization in the Coq proof assistant.

Knowledge distillation, the technique of transferring knowledge from large, complex models to smaller ones, marks a pivotal step towards efficient AI deployment. Distilling Step-by-Step (DSS), a novel method utilizing chain-of-thought (CoT) distillation, has demonstrated promise by imbuing smaller models with the superior reasoning capabilities of their larger counterparts. In DSS, the distilled model acquires the ability to generate rationales and predict labels concurrently through a multi-task learning framework. However, DSS overlooks the intrinsic relationship between the two training tasks, leading to ineffective integration of CoT knowledge with the task of label prediction. To this end, we investigate the mutual relationship of the two tasks from Information Bottleneck perspective and formulate it as maximizing the mutual information of the representation features of the two tasks. We propose a variational approach to solve this optimization problem using a learning-based method. Our experimental results across four datasets demonstrate that our method outperforms the state-of-the-art DSS. Our findings offer insightful guidance for future research on language model distillation as well as applications involving CoT. Code and models will be released soon.

Mathematical reasoning is a fundamental aspect of human intelligence and is applicable in various fields, including science, engineering, finance, and everyday life. The development of artificial intelligence (AI) systems capable of solving math problems and proving theorems has garnered significant interest in the fields of machine learning and natural language processing. For example, mathematics serves as a testbed for aspects of reasoning that are challenging for powerful deep learning models, driving new algorithmic and modeling advances. On the other hand, recent advances in large-scale neural language models have opened up new benchmarks and opportunities to use deep learning for mathematical reasoning. In this survey paper, we review the key tasks, datasets, and methods at the intersection of mathematical reasoning and deep learning over the past decade. We also evaluate existing benchmarks and methods, and discuss future research directions in this domain.

In pace with developments in the research field of artificial intelligence, knowledge graphs (KGs) have attracted a surge of interest from both academia and industry. As a representation of semantic relations between entities, KGs have proven to be particularly relevant for natural language processing (NLP), experiencing a rapid spread and wide adoption within recent years. Given the increasing amount of research work in this area, several KG-related approaches have been surveyed in the NLP research community. However, a comprehensive study that categorizes established topics and reviews the maturity of individual research streams remains absent to this day. Contributing to closing this gap, we systematically analyzed 507 papers from the literature on KGs in NLP. Our survey encompasses a multifaceted review of tasks, research types, and contributions. As a result, we present a structured overview of the research landscape, provide a taxonomy of tasks, summarize our findings, and highlight directions for future work.

Learning on big data brings success for artificial intelligence (AI), but the annotation and training costs are expensive. In future, learning on small data is one of the ultimate purposes of AI, which requires machines to recognize objectives and scenarios relying on small data as humans. A series of machine learning models is going on this way such as active learning, few-shot learning, deep clustering. However, there are few theoretical guarantees for their generalization performance. Moreover, most of their settings are passive, that is, the label distribution is explicitly controlled by one specified sampling scenario. This survey follows the agnostic active sampling under a PAC (Probably Approximately Correct) framework to analyze the generalization error and label complexity of learning on small data using a supervised and unsupervised fashion. With these theoretical analyses, we categorize the small data learning models from two geometric perspectives: the Euclidean and non-Euclidean (hyperbolic) mean representation, where their optimization solutions are also presented and discussed. Later, some potential learning scenarios that may benefit from small data learning are then summarized, and their potential learning scenarios are also analyzed. Finally, some challenging applications such as computer vision, natural language processing that may benefit from learning on small data are also surveyed.

Game theory has by now found numerous applications in various fields, including economics, industry, jurisprudence, and artificial intelligence, where each player only cares about its own interest in a noncooperative or cooperative manner, but without obvious malice to other players. However, in many practical applications, such as poker, chess, evader pursuing, drug interdiction, coast guard, cyber-security, and national defense, players often have apparently adversarial stances, that is, selfish actions of each player inevitably or intentionally inflict loss or wreak havoc on other players. Along this line, this paper provides a systematic survey on three main game models widely employed in adversarial games, i.e., zero-sum normal-form and extensive-form games, Stackelberg (security) games, zero-sum differential games, from an array of perspectives, including basic knowledge of game models, (approximate) equilibrium concepts, problem classifications, research frontiers, (approximate) optimal strategy seeking techniques, prevailing algorithms, and practical applications. Finally, promising future research directions are also discussed for relevant adversarial games.