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We present a general framework for applying learning algorithms and heuristical guidance to the verification of Markov decision processes (MDPs). The primary goal of our techniques is to improve performance by avoiding an exhaustive exploration of the state space, instead focussing on particularly relevant areas of the system, guided by heuristics. Our work builds on the previous results of Br{\'{a}}zdil et al., significantly extending it as well as refining several details and fixing errors. The presented framework focuses on probabilistic reachability, which is a core problem in verification, and is instantiated in two distinct scenarios. The first assumes that full knowledge of the MDP is available, in particular precise transition probabilities. It performs a heuristic-driven partial exploration of the model, yielding precise lower and upper bounds on the required probability. The second tackles the case where we may only sample the MDP without knowing the exact transition dynamics. Here, we obtain probabilistic guarantees, again in terms of both the lower and upper bounds, which provides efficient stopping criteria for the approximation. In particular, the latter is an extension of statistical model-checking (SMC) for unbounded properties in MDPs. In contrast to other related approaches, we do not restrict our attention to time-bounded (finite-horizon) or discounted properties, nor assume any particular structural properties of the MDP.

We study gradient descent (GD) dynamics on logistic regression problems with large, constant step sizes. For linearly-separable data, it is known that GD converges to the minimizer with arbitrarily large step sizes, a property which no longer holds when the problem is not separable. In fact, the behaviour can be much more complex -- a sequence of period-doubling bifurcations begins at the critical step size $2/\lambda$, where $\lambda$ is the largest eigenvalue of the Hessian at the solution. Using a smaller-than-critical step size guarantees convergence if initialized nearby the solution: but does this suffice globally? In one dimension, we show that a step size less than $1/\lambda$ suffices for global convergence. However, for all step sizes between $1/\lambda$ and the critical step size $2/\lambda$, one can construct a dataset such that GD converges to a stable cycle. In higher dimensions, this is actually possible even for step sizes less than $1/\lambda$. Our results show that although local convergence is guaranteed for all step sizes less than the critical step size, global convergence is not, and GD may instead converge to a cycle depending on the initialization.

The emergence of models like GPTs, Claude, LLaMA, and Qwen has reshaped AI applications, presenting vast new opportunities across industries. Yet, the integration of tabular data remains notably underdeveloped, despite its foundational role in numerous real-world domains. This gap is critical for three main reasons. First, database or data warehouse data integration is essential for advanced applications; second, the vast and largely untapped resource of tabular data offers immense potential for analysis; and third, the business intelligence domain specifically demands adaptable, precise solutions that many current LLMs may struggle to provide. In response, we introduce TableGPT2, a model rigorously pre-trained and fine-tuned with over 593.8K tables and 2.36M high-quality query-table-output tuples, a scale of table-related data unprecedented in prior research. This extensive training enables TableGPT2 to excel in table-centric tasks while maintaining strong general language and coding abilities. One of TableGPT2's key innovations is its novel table encoder, specifically designed to capture schema-level and cell-level information. This encoder strengthens the model's ability to handle ambiguous queries, missing column names, and irregular tables commonly encountered in real-world applications. Similar to visual language models, this pioneering approach integrates with the decoder to form a robust large multimodal model. We believe the results are compelling: over 23 benchmarking metrics, TableGPT2 achieves an average performance improvement of 35.20% in the 7B model and 49.32% in the 72B model over prior benchmark-neutral LLMs, with robust general-purpose capabilities intact.

We propose a low-complexity sign-dependent metric for sequence selection and study the nonlinear shaping gain achievable for a given computational cost, establishing a benchmark for future research. Small gains are obtained with feasible complexity. Higher gains are achievable in principle, but with high complexity or a more sophisticated metric.

Graph Edit Distance (GED) measures the (dis-)similarity between two given graphs, in terms of the minimum-cost edit sequence that transforms one graph to the other. However, the exact computation of GED is NP-Hard, which has recently motivated the design of neural methods for GED estimation. However, they do not explicitly account for edit operations with different costs. In response, we propose GRAPHEDX, a neural GED estimator that can work with general costs specified for the four edit operations, viz., edge deletion, edge addition, node deletion and node addition. We first present GED as a quadratic assignment problem (QAP) that incorporates these four costs. Then, we represent each graph as a set of node and edge embeddings and use them to design a family of neural set divergence surrogates. We replace the QAP terms corresponding to each operation with their surrogates. Computing such neural set divergence require aligning nodes and edges of the two graphs. We learn these alignments using a Gumbel-Sinkhorn permutation generator, additionally ensuring that the node and edge alignments are consistent with each other. Moreover, these alignments are cognizant of both the presence and absence of edges between node-pairs. Experiments on several datasets, under a variety of edit cost settings, show that GRAPHEDX consistently outperforms state-of-the-art methods and heuristics in terms of prediction error.

Chain of thought (CoT) is a reasoning framework that can enhance the performance of Large Language Models (LLMs) on complex inference tasks. In particular, among various studies related to CoT, multi-path inference stands out as a simple yet effective improvement. However, there is no optimal setting for the number of inference paths. Therefore, we have to increase the number of inference paths to obtain better results, which in turn increases the inference cost. To address this limitation, we can utilize question-related role templates to guide LLMs into relevant roles, thereby increasing the possibility of correct inferences for each path and further reducing dependence on the number of inference paths while improving reasoning accuracy. However, placing LLMs into specific roles may reduce their reasoning diversity and performance on a few tasks where role dependence is low. To alleviate the excessive immersion of the LLM into a specific role, we propose Nash CoT by constructing a competitive system on each path that balances the generation from role-specific LLMs' and the general LLMs' generation, thereby ensuring both effective role adoption and diversity in LLM generation further maintaining the performance of multi-path inference while reducing the requirement of the number of inference paths. We evaluate Nash CoT across various inference tasks, including Arabic Reasoning, Commonsense Question Answering, and Symbolic Inference, achieving results that are comparable to or better than those of multi-path CoT with the equal number of inference paths.

Adaptive Mixed-Criticality (AMC) is a fixed-priority preemptive scheduling algorithm for mixed-criticality hard real-time systems. It dominates many other scheduling algorithms for mixed-criticality systems, but does so at the cost of occasionally dropping jobs of less important/critical tasks, when low-priority jobs overrun their time budgets. In this paper we enhance AMC with a deep reinforcement learning (DRL) approach based on a Deep-Q Network. The DRL agent is trained off-line, and at run-time adjusts the low-criticality budgets of tasks to avoid budget overruns, while ensuring that no job misses its deadline if it does not overrun its budget. We have implemented and evaluated this approach by simulating realistic workloads from the automotive domain. The results show that the agent is able to reduce budget overruns by at least up to 50%, even when the budget of each task is chosen based on sampling the distribution of its execution time. To the best of our knowledge, this is the first use of DRL in AMC reported in the literature.

This paper employs a localized orthogonal decomposition (LOD) method with $H^1$ interpolation for solving the multiscale elliptic problem. This method does not need any assumptions on scale separation. We give a priori error estimate for the proposed method. The theoretical results are conformed by various numerical experiments.

Many real-world applications of tabular data involve using historic events to predict properties of new ones, for example whether a credit card transaction is fraudulent or what rating a customer will assign a product on a retail platform. Existing approaches to event prediction include costly, brittle, and application-dependent techniques such as time-aware positional embeddings, learned row and field encodings, and oversampling methods for addressing class imbalance. Moreover, these approaches often assume specific use-cases, for example that we know the labels of all historic events or that we only predict a pre-specified label and not the data's features themselves. In this work, we propose a simple but flexible baseline using standard autoregressive LLM-style transformers with elementary positional embeddings and a causal language modeling objective. Our baseline outperforms existing approaches across popular datasets and can be employed for various use-cases. We demonstrate that the same model can predict labels, impute missing values, or model event sequences.

Retrieval-Augmented Generation (RAG) merges retrieval methods with deep learning advancements to address the static limitations of large language models (LLMs) by enabling the dynamic integration of up-to-date external information. This methodology, focusing primarily on the text domain, provides a cost-effective solution to the generation of plausible but incorrect responses by LLMs, thereby enhancing the accuracy and reliability of their outputs through the use of real-world data. As RAG grows in complexity and incorporates multiple concepts that can influence its performance, this paper organizes the RAG paradigm into four categories: pre-retrieval, retrieval, post-retrieval, and generation, offering a detailed perspective from the retrieval viewpoint. It outlines RAG's evolution and discusses the field's progression through the analysis of significant studies. Additionally, the paper introduces evaluation methods for RAG, addressing the challenges faced and proposing future research directions. By offering an organized framework and categorization, the study aims to consolidate existing research on RAG, clarify its technological underpinnings, and highlight its potential to broaden the adaptability and applications of LLMs.