Bhattacharya et al. (2015) introduce a novel prior, the Dirichlet-Laplace (DL) prior, and propose a Markov chain Monte Carlo (MCMC) method to simulate posterior draws under this prior in a conditionally Gaussian setting. The original algorithm samples from conditional distributions in the wrong order, i.e., it does not correctly sample from the joint posterior distribution of all latent variables. This note details the issue and provides two simple solutions: A correction to the original algorithm and a new algorithm based on an alternative, yet equivalent, formulation of the prior. This corrigendum does not affect the theoretical results in Bhattacharya et al. (2015). A slightly modified version of this article is included in Luis Gruber's master thesis (Gruber, 2025).

Although nearly 20 years have passed since its conception, the feasibility pump algorithm remains a widely used heuristic to find feasible primal solutions to mixed-integer linear problems. Many extensions of the initial algorithm have been proposed. Yet, its core algorithm remains centered around two key steps: solving the linear relaxation of the original problem to obtain a solution that respects the constraints, and rounding it to obtain an integer solution. This paper shows that the traditional feasibility pump and many of its follow-ups can be seen as gradient-descent algorithms with specific parameters. A central aspect of this reinterpretation is observing that the traditional algorithm differentiates the solution of the linear relaxation with respect to its cost. This reinterpretation opens many opportunities for improving the performance of the original algorithm. We study how to modify the gradient-update step as well as extending its loss function. We perform extensive experiments on MIPLIB instances and show that these modifications can substantially reduce the number of iterations needed to find a solution.

Summary: This paper proposes a modification of Mondorian Forest which is a variant of Random Forest, a majority vote of decision trees. The authors show that the modified algorithm has the consistency property while the original algorithm does not have one. In particular, when the conditional probability function is Lipschitz, the proposed algorithm achieves the minimax error rate, where the lower bound is previously known. Comments: The technical contribution is to refine the original version of the Mondorian Forest and prove its consistency. The theoretical results are nice and solid. The main idea comes from the original algorithm, thus the originality of the paper is a bit incremental.

Trustworthiness reasoning is crucial in multiplayer games with incomplete information, enabling agents to identify potential allies and adversaries, thereby enhancing reasoning and decision-making processes. Traditional approaches relying on pre-trained models necessitate extensive domain-specific data and considerable reward feedback, with their lack of real-time adaptability hindering their effectiveness in dynamic environments. In this paper, we introduce the Graph Retrieval Augmented Reasoning (GRATR) framework, leveraging the Retrieval-Augmented Generation (RAG) technique to bolster trustworthiness reasoning in agents. GRATR constructs a dynamic trustworthiness graph, updating it in real-time with evidential information, and retrieves relevant trust data to augment the reasoning capabilities of Large Language Models (LLMs). We validate our approach through experiments on the multiplayer game "Werewolf," comparing GRATR against baseline LLM and LLM enhanced with Native RAG and Rerank RAG. Our results demonstrate that GRATR surpasses the baseline methods by over 30\% in winning rate, with superior reasoning performance. Moreover, GRATR effectively mitigates LLM hallucinations, such as identity and objective amnesia, and crucially, it renders the reasoning process more transparent and traceable through the use of the trustworthiness graph.

An improved version of the sparse multiway kernel spectral clustering (KSC) is presented in this brief. The original algorithm is derived from weighted kernel principal component (KPCA) analysis formulated within the primal-dual least-squares support vector machine (LS-SVM) framework. Sparsity is achieved then by the combination of the incomplete Cholesky decomposition (ICD) based low rank approximation of the kernel matrix with the so called reduced set method. The original ICD based sparse KSC algorithm was reported to be computationally far too demanding, especially when applied on large scale data clustering problems that actually it was designed for, which has prevented to gain more than simply theoretical relevance so far. This is altered by the modifications reported in this brief that drastically improve the computational characteristics. Solving the alternative, symmetrized version of the computationally most demanding core eigenvalue problem eliminates the necessity of forming and SVD of large matrices during the model construction. This results in solving clustering problems now within seconds that were reported to require hours without altering the results. Furthermore, sparsity is also improved significantly, leading to more compact model representation, increasing further not only the computational efficiency but also the descriptive power. These transform the original, only theoretically relevant ICD based sparse KSC algorithm applicable for large scale practical clustering problems. Theoretical results and improvements are demonstrated by computational experiments on carefully selected synthetic data as well as on real life problems such as image segmentation.

We present an effective framework for improving the breakdown point of robust regression algorithms. Robust regression has attracted widespread attention due to the ubiquity of outliers, which significantly affect the estimation results. However, many existing robust least-squares regression algorithms suffer from a low breakdown point, as they become stuck around local optima when facing severe attacks. By expanding on the previous work, we propose a novel framework that enhances the breakdown point of these algorithms by inserting a prior distribution in each iteration step, and adjusting the prior distribution according to historical information. We apply this framework to a specific algorithm and derive the consistent robust regression algorithm with iterative local search (CORALS). The relationship between CORALS and momentum gradient descent is described, and a detailed proof of the theoretical convergence of CORALS is presented. Finally, we demonstrate that the breakdown point of CORALS is indeed higher than that of the algorithm from which it is derived. We apply the proposed framework to other robust algorithms, and show that the improved algorithms achieve better results than the original algorithms, indicating the effectiveness of the proposed framework.

Online weighted matching problem is a fundamental problem in machine learning due to its numerous applications. Despite many efforts in this area, existing algorithms are either too slow or don't take $\mathrm{deadline}$ (the longest time a node can be matched) into account. In this paper, we introduce a market model with $\mathrm{deadline}$ first. Next, we present our two optimized algorithms (\textsc{FastGreedy} and \textsc{FastPostponedGreedy}) and offer theoretical proof of the time complexity and correctness of our algorithms. In \textsc{FastGreedy} algorithm, we have already known if a node is a buyer or a seller. But in \textsc{FastPostponedGreedy} algorithm, the status of each node is unknown at first. Then, we generalize a sketching matrix to run the original and our algorithms on both real data sets and synthetic data sets. Let $\epsilon \in (0,0.1)$ denote the relative error of the real weight of each edge. The competitive ratio of original \textsc{Greedy} and \textsc{PostponedGreedy} is $\frac{1}{2}$ and $\frac{1}{4}$ respectively. Based on these two original algorithms, we proposed \textsc{FastGreedy} and \textsc{FastPostponedGreedy} algorithms and the competitive ratio of them is $\frac{1 - \epsilon}{2}$ and $\frac{1 - \epsilon}{4}$ respectively. At the same time, our algorithms run faster than the original two algorithms. Given $n$ nodes in $\mathbb{R} ^ d$, we decrease the time complexity from $O(nd)$ to $\widetilde{O}(\epsilon^{-2} \cdot (n + d))$.

In recent work Long and Servedio LS05short presented a martingale boosting'' algorithm that works by constructing a branching program over weak classifiers and has a simple analysis based on elementary properties of random walks. LS05short showed that this martingale booster can tolerate random classification noise when it is run with a noise-tolerant weak learner; however, a drawback of the algorithm is that it is not adaptive, i.e. it cannot effectively take advantage of variation in the quality of the weak classifiers it receives. In this paper we present a variant of the original martingale boosting algorithm and prove that it is adaptive. This adaptiveness is achieved by modifying the original algorithm so that the random walks that arise in its analysis have different step size depending on the quality of the weak learner at each stage. The new algorithm inherits the desirable properties of the original LS05short algorithm, such as random classification noise tolerance, and has several other advantages besides adaptiveness: it requires polynomially fewer calls to the weak learner than the original algorithm, and it can be used with confidence-rated weak hypotheses that output real values rather than Boolean predictions.

We demonstrate that locally linear embedding (LLE) inherently admits some unwanted results when no regularization is used, even for cases in which regularization is not supposed to be needed in the original algorithm. The existence of one special type of result, which we call ``projection pattern'', is mathematically proved in the situation that an exact local linear relation is achieved in each neighborhood of the data. These special patterns as well as some other bizarre results that may occur in more general situations are shown by numerical examples on the Swiss roll with a hole embedded in a high dimensional space. It is observed that all these bad results can be effectively prevented by using regularization.

With the continuous development of machine learning technology, major e-commerce platforms have launched recommendation systems based on it to serve a large number of customers with different needs more efficiently. Compared with traditional supervised learning, reinforcement learning can better capture the user's state transition in the decision-making process, and consider a series of user actions, not just the static characteristics of the user at a certain moment. In theory, it will have a long-term perspective, producing a more effective recommendation. The special requirements of reinforcement learning for data make it need to rely on an offline virtual system for training. Our project mainly establishes a virtual user environment for offline training. At the same time, we tried to improve a reinforcement learning algorithm based on bi-clustering to expand the action space and recommended path space of the recommendation agent.