Mathematical reasoning remains a challenging area for large language models (LLMs), prompting the development of math-specific LLMs such as LLEMMA, DeepSeekMath, and Qwen2-Math, among others. These models typically follow a two-stage training paradigm: pre-training with math-related corpora and posttraining with problem datasets for supervised fine-tuning (SFT). Despite these efforts, the improvements in mathematical reasoning achieved through continued pre-training (CPT) are often less significant compared to those obtained via SFT. We investigate three primary research questions: (1) Can problem-solving data enhance the model's mathematical reasoning capabilities more effectively than general mathematical corpora during CPT? (2) Are synthetic data from the same source equally effective, and which synthesis methods are most efficient? Our findings indicate that problem-solving data significantly enhances the model's mathematical capabilities compared to general mathematical corpora. We also identify effective data synthesis methods, demonstrating that the tutorship amplification synthesis method achieves the best performance. Furthermore, while SFT facilitates instruction-following abilities, it underperforms compared to CPT with the same data, which can be partially attributed to its poor learning capacity for more challenging problem-solving data. To address the challenge of insufficient mathematical reasoning capabilities in large language models (LLMs), various math-specific LLMs are developed. These models generally follow a common training paradigm. During the pre-training stage, math-related corpora are filtered from extensive internet data to augment the model's mathematical knowledge. Work done during Zui Chen's internship at Guangdong Institute of Smart Education, Jinan University, Guangzhou, China. Model is available at https://huggingface.co/ai4ed/MathGPT-8B This enables the models to follow instructions and produce outputs in the desired format. Recently, there is a growing focus on constructing preference datasets for the solution process to perform Step-DPO (Lai et al., 2024) or online-RLHF (Dong et al., 2024). These approaches aim to obtain more accurate reasoning pathways, thereby significantly enhancing the mathematical reasoning capabilities of the models.

Step-level reward models (SRMs) can significantly enhance mathematical reasoning performance through process supervision or step-level preference alignment based on reinforcement learning. The performance of SRMs is pivotal, as they serve as critical guidelines, ensuring that each step in the reasoning process is aligned with desired outcomes. Recently, AlphaZero-like methods, where Monte Carlo Tree Search (MCTS) is employed for automatic step-level preference annotation, have proven particularly effective. However, the precise mechanisms behind the success of SRMs remain largely unexplored. To address this gap, this study delves into the counterintuitive aspects of SRMs, particularly focusing on MCTS-based approaches. Our findings reveal that the removal of natural language descriptions of thought processes has minimal impact on the efficacy of SRMs. Furthermore, we demonstrate that SRMs are adept at assessing the complex logical coherence present in mathematical language while having difficulty in natural language. These insights provide a nuanced understanding of the core elements that drive effective step-level reward modeling in mathematical reasoning. By shedding light on these mechanisms, this study offers valuable guidance for developing more efficient and streamlined SRMs, which can be achieved by focusing on the crucial parts of mathematical reasoning.

Belief revision and update, two significant types of belief change, both focus on how an agent modify her beliefs in presence of new information. The most striking difference between them is that the former studies the change of beliefs in a static world while the latter concentrates on a dynamically-changing world. The famous AGM and KM postulates were proposed to capture rational belief revision and update, respectively. However, both of them are too permissive to exclude some unreasonable changes in the iteration. In response to this weakness, the DP postulates and its extensions for iterated belief revision were presented. Furthermore, Rodrigues integrated these postulates in belief update. Unfortunately, his approach does not meet the basic requirement of iterated belief update. This paper is intended to solve this problem of Rodrigues's approach. Firstly, we present a modification of the original KM postulates based on belief states. Subsequently, we migrate several well-known postulates for iterated belief revision to iterated belief update. Moreover, we provide the exact semantic characterizations based on partial preorders for each of the proposed postulates. Finally, we analyze the compatibility between the above iterated postulates and the KM postulates for belief update.

Sentence completion (SC) questions present a sentence with one or more blanks that need to be filled in, three to five possible words or phrases as options. SC questions are widely used for students learning English as a Second Language (ESL). In this paper, we present a large-scale SC dataset, \textsc{SC-Ques}, which is made up of 289,148 ESL SC questions from real-world standardized English examinations. Furthermore, we build a comprehensive benchmark of automatically solving the SC questions by training the large-scale pre-trained language models on the proposed \textsc{SC-Ques} dataset. We conduct detailed analysis of the baseline models performance, limitations and trade-offs. The data and our code are available for research purposes from: \url{https://github.com/ai4ed/SC-Ques}.

Knowledge tracing (KT) is a crucial technique to predict students' future performance by observing their historical learning processes. Due to the powerful representation ability of deep neural networks, remarkable progress has been made by using deep learning techniques to solve the KT problem. The majority of existing approaches rely on the \emph{homogeneous question} assumption that questions have equivalent contributions if they share the same set of knowledge components. Unfortunately, this assumption is inaccurate in real-world educational scenarios. Furthermore, it is very challenging to interpret the prediction results from the existing deep learning based KT models. Therefore, in this paper, we present QIKT, a question-centric interpretable KT model to address the above challenges. The proposed QIKT approach explicitly models students' knowledge state variations at a fine-grained level with question-sensitive cognitive representations that are jointly learned from a question-centric knowledge acquisition module and a question-centric problem solving module. Meanwhile, the QIKT utilizes an item response theory based prediction layer to generate interpretable prediction results. The proposed QIKT model is evaluated on three public real-world educational datasets. The results demonstrate that our approach is superior on the KT prediction task, and it outperforms a wide range of deep learning based KT models in terms of prediction accuracy with better model interpretability. To encourage reproducible results, we have provided all the datasets and code at \url{https://pykt.org/}.

Knowledge tracing (KT) is the problem of predicting students' future performance based on their historical interactions with intelligent tutoring systems. Recently, many works present lots of special methods for applying deep neural networks to KT from different perspectives like model architecture, adversarial augmentation and etc., which make the overall algorithm and system become more and more complex. Furthermore, due to the lack of standardized evaluation protocol \citep{liu2022pykt}, there is no widely agreed KT baselines and published experimental comparisons become inconsistent and self-contradictory, i.e., the reported AUC scores of DKT on ASSISTments2009 range from 0.721 to 0.821 \citep{minn2018deep,yeung2018addressing}. Therefore, in this paper, we provide a strong but simple baseline method to deal with the KT task named \textsc{simpleKT}. Inspired by the Rasch model in psychometrics, we explicitly model question-specific variations to capture the individual differences among questions covering the same set of knowledge components that are a generalization of terms of concepts or skills needed for learners to accomplish steps in a task or a problem. Furthermore, instead of using sophisticated representations to capture student forgetting behaviors, we use the ordinary dot-product attention function to extract the time-aware information embedded in the student learning interactions. Extensive experiments show that such a simple baseline is able to always rank top 3 in terms of AUC scores and achieve 57 wins, 3 ties and 16 loss against 12 DLKT baseline methods on 7 public datasets of different domains. We believe this work serves as a strong baseline for future KT research. Code is available at \url{https://github.com/pykt-team/pykt-toolkit}\footnote{We merged our model to the \textsc{pyKT} benchmark at \url{https://pykt.org/}.}.

Knowledge tracing (KT) is the problem of predicting students' future performance based on their historical interactions with intelligent tutoring systems. Recent studies have applied multiple types of deep neural networks to solve the KT problem. However, there are two important factors in real-world educational data that are not well represented. First, most existing works augment input representations with the co-occurrence matrix of questions and knowledge components\footnote{\label{ft:kc}A KC is a generalization of everyday terms like concept, principle, fact, or skill.} (KCs) but fail to explicitly integrate such intrinsic relations into the final response prediction task. Second, the individualized historical performance of students has not been well captured. In this paper, we proposed \emph{AT-DKT} to improve the prediction performance of the original deep knowledge tracing model with two auxiliary learning tasks, i.e., \emph{question tagging (QT) prediction task} and \emph{individualized prior knowledge (IK) prediction task}. Specifically, the QT task helps learn better question representations by predicting whether questions contain specific KCs. The IK task captures students' global historical performance by progressively predicting student-level prior knowledge that is hidden in students' historical learning interactions. We conduct comprehensive experiments on three real-world educational datasets and compare the proposed approach to both deep sequential KT models and non-sequential models. Experimental results show that \emph{AT-DKT} outperforms all sequential models with more than 0.9\% improvements of AUC for all datasets, and is almost the second best compared to non-sequential models. Furthermore, we conduct both ablation studies and quantitative analysis to show the effectiveness of auxiliary tasks and the superior prediction outcomes of \emph{AT-DKT}.

Knowledge tracing (KT) is the task of using students' historical learning interaction data to model their knowledge mastery over time so as to make predictions on their future interaction performance. Recently, remarkable progress has been made of using various deep learning techniques to solve the KT problem. However, the success behind deep learning based knowledge tracing (DLKT) approaches is still left somewhat unknown and proper measurement and analysis of these DLKT approaches remain a challenge. First, data preprocessing procedures in existing works are often private and custom, which limits experimental standardization. Furthermore, existing DLKT studies often differ in terms of the evaluation protocol and are far away real-world educational contexts. To address these problems, we introduce a comprehensive python based benchmark platform, \textsc{pyKT}, to guarantee valid comparisons across DLKT methods via thorough evaluations. The \textsc{pyKT} library consists of a standardized set of integrated data preprocessing procedures on 7 popular datasets across different domains, and 10 frequently compared DLKT model implementations for transparent experiments. Results from our fine-grained and rigorous empirical KT studies yield a set of observations and suggestions for effective DLKT, e.g., wrong evaluation setting may cause label leakage that generally leads to performance inflation; and the improvement of many DLKT approaches is minimal compared to the very first DLKT model proposed by Piech et al. \cite{piech2015deep}. We have open sourced \textsc{pyKT} and our experimental results at https://pykt.org/. We welcome contributions from other research groups and practitioners.

In this paper, we investigate the opportunities of automating the judgment process in online one-on-one math classes. We build a Wide & Deep framework to learn fine-grained predictive representations from a limited amount of noisy classroom conversation data that perform better student judgments. We conducted experiments on the task of predicting students' levels of mastery of example questions and the results demonstrate the superiority and availability of our model in terms of various evaluation metrics.

In recent years, quantitative investment methods combined with artificial intelligence have attracted more and more attention from investors and researchers. Existing related methods based on the supervised learning are not very suitable for learning problems with long-term goals and delayed rewards in real futures trading. In this paper, therefore, we model the price prediction problem as a Markov decision process (MDP), and optimize it by reinforcement learning with expert trajectory. In the proposed method, we employ more than 100 short-term alpha factors instead of price, volume and several technical factors in used existing methods to describe the states of MDP. Furthermore, unlike DQN (deep Q-learning) and BC (behavior cloning) in related methods, we introduce expert experience in training stage, and consider both the expert-environment interaction and the agent-environment interaction to design the temporal difference error so that the agents are more adaptable for inevitable noise in financial data. Experimental results evaluated on share price index futures in China, including IF (CSI 300) and IC (CSI 500), show that the advantages of the proposed method compared with three typical technical analysis and two deep leaning based methods.