Daily Papers

We study the implicit bias of momentum-based optimizers on smooth homogeneous models. We show that momentum steepest descent algorithms like Muon (spectral norm), MomentumGD (ell_2 norm), and Signum (ell_infty norm) are approximate steepest descent trajectories under a decaying learning rate schedule, proving that these algorithms have a bias towards KKT points of the corresponding margin maximization problem. We extend the analysis to Adam (without the stability constant), which maximizes the ell_infty margin, and to Muon-Signum and Muon-Adam, which maximize a hybrid norm. Our experiments corroborate the theory and show that the identity of the margin maximized depends on the choice of optimizer. Overall, our results extend earlier lines of work on steepest descent in homogeneous models and momentum-based optimizers in linear models.

Gradient methods have become mainstream techniques for Bi-Level Optimization (BLO) in learning fields. The validity of existing works heavily rely on either a restrictive Lower- Level Strong Convexity (LLSC) condition or on solving a series of approximation subproblems with high accuracy or both. In this work, by averaging the upper and lower level objectives, we propose a single loop Bi-level Averaged Method of Multipliers (sl-BAMM) for BLO that is simple yet efficient for large-scale BLO and gets rid of the limited LLSC restriction. We further provide non-asymptotic convergence analysis of sl-BAMM towards KKT stationary points, and the comparative advantage of our analysis lies in the absence of strong gradient boundedness assumption, which is always required by others. Thus our theory safely captures a wider variety of applications in deep learning, especially where the upper-level objective is quadratic w.r.t. the lower-level variable. Experimental results demonstrate the superiority of our method.

Nonnegative matrix factorization (NMF) seeks a low-rank approximation X approx UV^T with nonnegative factors and is commonly solved using interior methods that enforce feasibility throughout optimization. We show that such constraint-driven approaches can impede progress in the nonconvex landscape, leading to slow convergence or convergence to suboptimal stationary points. We propose an exterior framework for NMF (eNMF) that separates low-rank approximation from nonnegativity enforcement. Our method initializes from the optimal unconstrained factorization and introduces a rotation procedure that maps unconstrained factors to an exterior point closest to the nonnegative orthant. This viewpoint yields an algorithmic framework in which simple iterative updates converge to KKT-satisfying stationary points on the boundary of the positive orthant. The exterior formulation also enables a geometric interpretation of NMF solutions, clarifying equivalence classes of factorizations under permutation and orthogonal transformations. An intriguing numerical result, involving 400 NMF experiments across both real and synthetic datasets, show that in 99% of the cases, different algorithms tend to converge towards equivalent factor matrices. We benchmark eNMF against 9 state-of-the-art NMF algorithms with 9 initialization schemes across 3 real-world and 2 synthetic datasets. eNMF consistently outperforms all 81 competitors, achieving up to 30% lower reconstruction error under equal-time settings and up to 150% speedup under equal-error settings. The downstream experiments further demonstrate substantial performance gains in audio processing and recommendation tasks, corroborating the practical benefits of the proposed exterior optimization framework. Code is available at https://github.com/roychowdhuryresearch/eNMF

Determinantal point processes (DPPs) are a useful probabilistic model for selecting a small diverse subset out of a large collection of items, with applications in summarization, stochastic optimization, active learning and more. Given a kernel function and a subset size k, our goal is to sample k out of n items with probability proportional to the determinant of the kernel matrix induced by the subset (a.k.a. k-DPP). Existing k-DPP sampling algorithms require an expensive preprocessing step which involves multiple passes over all n items, making it infeasible for large datasets. A naïve heuristic addressing this problem is to uniformly subsample a fraction of the data and perform k-DPP sampling only on those items, however this method offers no guarantee that the produced sample will even approximately resemble the target distribution over the original dataset. In this paper, we develop an algorithm which adaptively builds a sufficiently large uniform sample of data that is then used to efficiently generate a smaller set of k items, while ensuring that this set is drawn exactly from the target distribution defined on all n items. We show empirically that our algorithm produces a k-DPP sample after observing only a small fraction of all elements, leading to several orders of magnitude faster performance compared to the state-of-the-art.

Knowledge Tracing (KT) is a task of tracing evolving knowledge state of students with respect to one or more concepts as they engage in a sequence of learning activities. One important purpose of KT is to personalize the practice sequence to help students learn knowledge concepts efficiently. However, existing methods such as Bayesian Knowledge Tracing and Deep Knowledge Tracing either model knowledge state for each predefined concept separately or fail to pinpoint exactly which concepts a student is good at or unfamiliar with. To solve these problems, this work introduces a new model called Dynamic Key-Value Memory Networks (DKVMN) that can exploit the relationships between underlying concepts and directly output a student's mastery level of each concept. Unlike standard memory-augmented neural networks that facilitate a single memory matrix or two static memory matrices, our model has one static matrix called key, which stores the knowledge concepts and the other dynamic matrix called value, which stores and updates the mastery levels of corresponding concepts. Experiments show that our model consistently outperforms the state-of-the-art model in a range of KT datasets. Moreover, the DKVMN model can automatically discover underlying concepts of exercises typically performed by human annotations and depict the changing knowledge state of a student.

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 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 minn2018deep,yeung2018addressing. Therefore, in this paper, we provide a strong but simple baseline method to deal with the KT task named 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 https://github.com/pykt-team/pykt-toolkitWe merged our model to the \textsc{pyKT benchmark at https://pykt.org/.}.

Recently, an intriguing class of non-convex optimization problems has emerged in the context of learning directed acyclic graphs (DAGs). These problems involve minimizing a given loss or score function, subject to a non-convex continuous constraint that penalizes the presence of cycles in a graph. In this work, we delve into the optimization challenges associated with this class of non-convex programs. To address these challenges, we propose a bi-level algorithm that leverages the non-convex constraint in a novel way. The outer level of the algorithm optimizes over topological orders by iteratively swapping pairs of nodes within the topological order of a DAG. A key innovation of our approach is the development of an effective method for generating a set of candidate swapping pairs for each iteration. At the inner level, given a topological order, we utilize off-the-shelf solvers that can handle linear constraints. The key advantage of our proposed algorithm is that it is guaranteed to find a local minimum or a KKT point under weaker conditions compared to previous work and finds solutions with lower scores. Extensive experiments demonstrate that our method outperforms state-of-the-art approaches in terms of achieving a better score. Additionally, our method can also be used as a post-processing algorithm to significantly improve the score of other algorithms. Code implementing the proposed method is available at https://github.com/duntrain/topo.

Knowledge tracing (KT) refers to the problem of predicting future learner performance given their past performance in educational applications. Recent developments in KT using flexible deep neural network-based models excel at this task. However, these models often offer limited interpretability, thus making them insufficient for personalized learning, which requires using interpretable feedback and actionable recommendations to help learners achieve better learning outcomes. In this paper, we propose attentive knowledge tracing (AKT), which couples flexible attention-based neural network models with a series of novel, interpretable model components inspired by cognitive and psychometric models. AKT uses a novel monotonic attention mechanism that relates a learner's future responses to assessment questions to their past responses; attention weights are computed using exponential decay and a context-aware relative distance measure, in addition to the similarity between questions. Moreover, we use the Rasch model to regularize the concept and question embeddings; these embeddings are able to capture individual differences among questions on the same concept without using an excessive number of parameters. We conduct experiments on several real-world benchmark datasets and show that AKT outperforms existing KT methods (by up to 6% in AUC in some cases) on predicting future learner responses. We also conduct several case studies and show that AKT exhibits excellent interpretability and thus has potential for automated feedback and personalization in real-world educational settings.

Knowledge tracing (KT) models, e.g., the deep knowledge tracing (DKT) model, track an individual learner's acquisition of skills over time by examining the learner's performance on questions related to those skills. A practical limitation in most existing KT models is that all questions nested under a particular skill are treated as equivalent observations of a learner's ability, which is an inaccurate assumption in real-world educational scenarios. To overcome this limitation we introduce qDKT, a variant of DKT that models every learner's success probability on individual questions over time. First, qDKT incorporates graph Laplacian regularization to smooth predictions under each skill, which is particularly useful when the number of questions in the dataset is big. Second, qDKT uses an initialization scheme inspired by the fastText algorithm, which has found success in a variety of language modeling tasks. Our experiments on several real-world datasets show that qDKT achieves state-of-art performance on predicting learner outcomes. Because of this, qDKT can serve as a simple, yet tough-to-beat, baseline for new question-centric KT models.

Identifying keyphrases (KPs) from text documents is a fundamental task in natural language processing and information retrieval. Vast majority of the benchmark datasets for this task are from the scientific domain containing only the document title and abstract information. This limits keyphrase extraction (KPE) and keyphrase generation (KPG) algorithms to identify keyphrases from human-written summaries that are often very short (approx 8 sentences). This presents three challenges for real-world applications: human-written summaries are unavailable for most documents, the documents are almost always long, and a high percentage of KPs are directly found beyond the limited context of title and abstract. Therefore, we release two extensive corpora mapping KPs of ~1.3M and ~100K scientific articles with their fully extracted text and additional metadata including publication venue, year, author, field of study, and citations for facilitating research on this real-world problem.