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Recently, there has been a growing interest in leveraging Large Language Models (LLMs) for recommendation systems, which usually adapt a pre-trained LLM to the recommendation scenario through supervised fine-tuning (SFT). However, both the pre-training and SFT stages fail to explicitly model the comparative relationships of a user's preferences on different items. To construct a "helpful and harmless" LLM-based recommender, we propose a general framework -- Recommendation with smoothing personalized Preference Optimization (RosePO), which better aligns with customized human values during the post-training stage. Specifically, in addition to the input and chosen response that naturally align with SFT data, we design a rejected sampling strategy tailored for enhancing helpfulness, along with two strategies aimed at mitigating biases to promote harmlessness. To ensure robustness against uncertain labels present in automatically constructed preference data, we introduce a personalized smoothing factor predicted by a preference oracle into the optimization objective. Evaluation on three real-world datasets demonstrates the effectiveness of our method, showcasing not only improved recommendation performance but also mitigation of semantic hallucination and popularity bias.

In this paper, we introduce an HPR-LP solver, an implementation of a Halpern Peaceman-Rachford (HPR) method with semi-proximal terms for solving linear programming (LP). The HPR method enjoys the iteration complexity of $O(1/k)$ in terms of the Karush-Kuhn-Tucker residual and the objective error. Based on the complexity results, we design an adaptive strategy of restart and penalty parameter update to improve the efficiency and robustness of the HPR method. We conduct extensive numerical experiments on different LP benchmark datasets using NVIDIA A100-SXM4-80GB GPU in different stopping tolerances. Our solver's Julia version achieves a $\textbf{2.39x}$ to $\textbf{5.70x}$ speedup measured by SGM10 on benchmark datasets with presolve ($\textbf{2.03x}$ to $\textbf{4.06x}$ without presolve) over the award-winning solver PDLP with the tolerance of $10^{-8}$.

This paper proposes an efficient HOT algorithm for solving the optimal transport (OT) problems with finite supports. We particularly focus on an efficient implementation of the HOT algorithm for the case where the supports are in $\mathbb{R}^2$ with ground distances calculated by $L_2^2$-norm. Specifically, we design a Halpern accelerating algorithm to solve the equivalent reduced model of the discrete OT problem. Moreover, we derive a novel procedure to solve the involved linear systems in the HOT algorithm in linear time complexity. Consequently, we can obtain an $\varepsilon$-approximate solution to the optimal transport problem with $M$ supports in $O(M^{1.5}/\varepsilon)$ flops, which significantly improves the best-known computational complexity. We further propose an efficient procedure to recover an optimal transport plan for the original OT problem based on a solution to the reduced model, thereby overcoming the limitations of the reduced OT model in applications that require the transport map. We implement the HOT algorithm in PyTorch and extensive numerical results show the superior performance of the HOT algorithm compared to existing state-of-the-art algorithms for solving the OT problems.

We introduce a novel self-supervised deep clustering approach tailored for unstructured data without requiring prior knowledge of the number of clusters, termed Adaptive Self-supervised Robust Clustering (ASRC). In particular, ASRC adaptively learns the graph structure and edge weights to capture both local and global structural information. The obtained graph enables us to learn clustering-friendly feature representations by an enhanced graph auto-encoder with contrastive learning technique. It further leverages the clustering results adaptively obtained by robust continuous clustering (RCC) to generate prototypes for negative sampling, which can further contribute to promoting consistency among positive pairs and enlarging the gap between positive and negative samples. ASRC obtains the final clustering results by applying RCC to the learned feature representations with their consistent graph structure and edge weights. Extensive experiments conducted on seven benchmark datasets demonstrate the efficacy of ASRC, demonstrating its superior performance over other popular clustering models. Notably, ASRC even outperforms methods that rely on prior knowledge of the number of clusters, highlighting its effectiveness in addressing the challenges of clustering unstructured data.

We introduce LArge Model Based Data Agent (LAMBDA), a novel open-source, code-free multi-agent data analysis system that leverages the power of large models. LAMBDA is designed to address data analysis challenges in complex data-driven applications through innovatively designed data agents that operate iteratively and generatively using natural language. At the core of LAMBDA are two key agent roles: the programmer and the inspector, which are engineered to work together seamlessly. Specifically, the programmer generates code based on the user's instructions and domain-specific knowledge, enhanced by advanced models. Meanwhile, the inspector debugs the code when necessary. To ensure robustness and handle adverse scenarios, LAMBDA features a user interface that allows direct user intervention in the operational loop. Additionally, LAMBDA can flexibly integrate external models and algorithms through our proposed Knowledge Integration Mechanism, catering to the needs of customized data analysis. LAMBDA has demonstrated strong performance on various data analysis tasks. It has the potential to enhance data analysis paradigms by seamlessly integrating human and artificial intelligence, making it more accessible, effective, and efficient for users from diverse backgrounds. The strong performance of LAMBDA in solving data analysis problems is demonstrated using real-world data examples. Videos of several case studies are available at https://xxxlambda.github.io/lambda_webpage., Comment: 51 pages, 23 figures and 6 tables

In this paper, we aim to accelerate a preconditioned alternating direction method of multipliers (pADMM), whose proximal terms are convex quadratic functions, for solving linearly constrained convex optimization problems. To achieve this, we first reformulate the pADMM into a form of proximal point method (PPM) with a positive semidefinite preconditioner which can be degenerate due to the lack of strong convexity of the proximal terms in the pADMM. Then we accelerate the pADMM by accelerating the reformulated degenerate PPM (dPPM). Specifically, we first propose an accelerated dPPM by integrating the Halpern iteration and the fast Krasnosel'ski\u{i}-Mann iteration into it, achieving asymptotic $o(1/k)$ and non-asymptotic $O(1/k)$ convergence rates. Subsequently, building upon the accelerated dPPM, we develop an accelerated pADMM algorithm that exhibits both asymptotic $o(1/k)$ and non-asymptotic $O(1/k)$ nonergodic convergence rates concerning the Karush-Kuhn-Tucker residual and the primal objective function value gap. Preliminary numerical experiments validate the theoretical findings, demonstrating that the accelerated pADMM outperforms the pADMM in solving convex quadratic programming problems.

We investigate certified robustness for GNNs under graph injection attacks. Existing research only provides sample-wise certificates by verifying each node independently, leading to very limited certifying performance. In this paper, we present the first collective certificate, which certifies a set of target nodes simultaneously. To achieve it, we formulate the problem as a binary integer quadratic constrained linear programming (BQCLP). We further develop a customized linearization technique that allows us to relax the BQCLP into linear programming (LP) that can be efficiently solved. Through comprehensive experiments, we demonstrate that our collective certification scheme significantly improves certification performance with minimal computational overhead. For instance, by solving the LP within 1 minute on the Citeseer dataset, we achieve a significant increase in the certified ratio from 0.0% to 81.2% when the injected node number is 5% of the graph size. Our step marks a crucial step towards making provable defense more practical.

Large language models (LLMs) have shown an impressive ability to perform a wide range of tasks using in-context learning (ICL), where a few examples are used to describe a task to the model. However, the performance of ICL varies significantly with the choice of demonstrations, and it is still unclear why this happens or what factors will influence its choice. In this work, we first revisit the factors contributing to this variance from both data and model aspects, and find that the choice of demonstration is both data- and model-dependent. We further proposed a data- and model-dependent demonstration selection method, \textbf{TopK + ConE}, based on the assumption that \textit{the performance of a demonstration positively correlates with its contribution to the model's understanding of the test samples}, resulting in a simple and effective recipe for ICL. Empirically, our method yields consistent improvements in both language understanding and generation tasks with different model scales. Further analyses confirm that, besides the generality and stability under different circumstances, our method provides a unified explanation for the effectiveness of previous methods. Code will be released., Comment: ACL 2024

Predicting in-hospital mortality for intensive care unit (ICU) patients is key to final clinical outcomes. AI has shown advantaged accuracy but suffers from the lack of explainability. To address this issue, this paper proposes an eXplainable Multimodal Mortality Predictor (X-MMP) approaching an efficient, explainable AI solution for predicting in-hospital mortality via multimodal ICU data. We employ multimodal learning in our framework, which can receive heterogeneous inputs from clinical data and make decisions. Furthermore, we introduce an explainable method, namely Layer-Wise Propagation to Transformer, as a proper extension of the LRP method to Transformers, producing explanations over multimodal inputs and revealing the salient features attributed to prediction. Moreover, the contribution of each modality to clinical outcomes can be visualized, assisting clinicians in understanding the reasoning behind decision-making. We construct a multimodal dataset based on MIMIC-III and MIMIC-III Waveform Database Matched Subset. Comprehensive experiments on benchmark datasets demonstrate that our proposed framework can achieve reasonable interpretation with competitive prediction accuracy. In particular, our framework can be easily transferred to other clinical tasks, which facilitates the discovery of crucial factors in healthcare research.

Sequential recommendation aims to predict users' next interaction with items based on their past engagement sequence. Recently, the advent of Large Language Models (LLMs) has sparked interest in leveraging them for sequential recommendation, viewing it as language modeling. Previous studies represent items within LLMs' input prompts as either ID indices or textual metadata. However, these approaches often fail to either encapsulate comprehensive world knowledge or exhibit sufficient behavioral understanding. To combine the complementary strengths of conventional recommenders in capturing behavioral patterns of users and LLMs in encoding world knowledge about items, we introduce Large Language-Recommendation Assistant (LLaRA). Specifically, it uses a novel hybrid prompting method that integrates ID-based item embeddings learned by traditional recommendation models with textual item features. Treating the "sequential behaviors of users" as a distinct modality beyond texts, we employ a projector to align the traditional recommender's ID embeddings with the LLM's input space. Moreover, rather than directly exposing the hybrid prompt to LLMs, a curriculum learning strategy is adopted to gradually ramp up training complexity. Initially, we warm up the LLM using text-only prompts, which better suit its inherent language modeling ability. Subsequently, we progressively transition to the hybrid prompts, training the model to seamlessly incorporate the behavioral knowledge from the traditional sequential recommender into the LLM. Empirical results validate the effectiveness of our proposed framework. Codes are available at https://github.com/ljy0ustc/LLaRA., Comment: 11 pages, 5 figures

Sequential recommendation is to predict the next item of interest for a user, based on her/his interaction history with previous items. In conventional sequential recommenders, a common approach is to model item sequences using discrete IDs, learning representations that encode sequential behaviors and reflect user preferences. Inspired by recent success in empowering large language models (LLMs) to understand and reason over diverse modality data (e.g., image, audio, 3D points), a compelling research question arises: ``Can LLMs understand and work with hidden representations from ID-based sequential recommenders?''.To answer this, we propose a simple framework, RecInterpreter, which examines the capacity of open-source LLMs to decipher the representation space of sequential recommenders. Specifically, with the multimodal pairs (\ie representations of interaction sequence and text narrations), RecInterpreter first uses a lightweight adapter to map the representations into the token embedding space of the LLM. Subsequently, it constructs a sequence-recovery prompt that encourages the LLM to generate textual descriptions for items within the interaction sequence. Taking a step further, we propose a sequence-residual prompt instead, which guides the LLM in identifying the residual item by contrasting the representations before and after integrating this residual into the existing sequence. Empirical results showcase that our RecInterpreter enhances the exemplar LLM, LLaMA, to understand hidden representations from ID-based sequential recommenders, especially when guided by our sequence-residual prompts. Furthermore, RecInterpreter enables LLaMA to instantiate the oracle items generated by generative recommenders like DreamRec, concreting the item a user would ideally like to interact with next. Codes are available at https://github.com/YangZhengyi98/RecInterpreter.

Sequential recommendation aims to recommend the next item that matches a user's interest, based on the sequence of items he/she interacted with before. Scrutinizing previous studies, we can summarize a common learning-to-classify paradigm -- given a positive item, a recommender model performs negative sampling to add negative items and learns to classify whether the user prefers them or not, based on his/her historical interaction sequence. Although effective, we reveal two inherent limitations:(1) it may differ from human behavior in that a user could imagine an oracle item in mind and select potential items matching the oracle; and (2) the classification is limited in the candidate pool with noisy or easy supervision from negative samples, which dilutes the preference signals towards the oracle item. Yet, generating the oracle item from the historical interaction sequence is mostly unexplored. To bridge the gap, we reshape sequential recommendation as a learning-to-generate paradigm, which is achieved via a guided diffusion model, termed DreamRec.Specifically, for a sequence of historical items, it applies a Transformer encoder to create guidance representations. Noising target items explores the underlying distribution of item space; then, with the guidance of historical interactions, the denoising process generates an oracle item to recover the positive item, so as to cast off negative sampling and depict the true preference of the user directly. We evaluate the effectiveness of DreamRec through extensive experiments and comparisons with existing methods. Codes and data are open-sourced at https://github.com/YangZhengyi98/DreamRec.

In this paper, we propose an efficient sieving based secant method to address the computational challenges of solving sparse optimization problems with least-squares constraints. A level-set method has been introduced in [X. Li, D.F. Sun, and K.-C. Toh, SIAM J. Optim., 28 (2018), pp. 1842--1866] that solves these problems by using the bisection method to find a root of a univariate nonsmooth equation $\varphi(\lambda) = \varrho$ for some $\varrho > 0$, where $\varphi(\cdot)$ is the value function computed by a solution of the corresponding regularized least-squares optimization problem. When the objective function in the constrained problem is a polyhedral gauge function, we prove that (i) for any positive integer $k$, $\varphi(\cdot)$ is piecewise $C^k$ in an open interval containing the solution $\lambda^*$ to the equation $\varphi(\lambda) = \varrho$; (ii) the Clarke Jacobian of $\varphi(\cdot)$ is always positive. These results allow us to establish the essential ingredients of the fast convergence rates of the secant method. Moreover, an adaptive sieving technique is incorporated into the secant method to effectively reduce the dimension of the level-set subproblems for computing the value of $\varphi(\cdot)$. The high efficiency of the proposed algorithm is demonstrated by extensive numerical results.

In this paper, we propose an adaptive sieving (AS) strategy for solving general sparse machine learning models by effectively exploring the intrinsic sparsity of the solutions, wherein only a sequence of reduced problems with much smaller sizes need to be solved. We further apply the proposed AS strategy to generate solution paths for large-scale sparse optimization problems efficiently. We establish the theoretical guarantees for the proposed AS strategy including its finite termination property. Extensive numerical experiments are presented in this paper to demonstrate the effectiveness and flexibility of the AS strategy to solve large-scale machine learning models.

The exclusive lasso (also known as elitist lasso) regularizer has become popular recently due to its superior performance on intra-group feature selection. Its complex nature poses difficulties for the computation of high-dimensional machine learning models involving such a regularizer. In this paper, we propose a highly efficient dual Newton method based proximal point algorithm (PPDNA) for solving large-scale exclusive lasso models. As important ingredients, we systematically study the proximal mapping of the weighted exclusive lasso regularizer and the corresponding generalized Jacobian. These results also make popular first-order algorithms for solving exclusive lasso models more practical. Extensive numerical results are presented to demonstrate the superior performance of the PPDNA against other popular numerical algorithms for solving the exclusive lasso problems., Comment: arXiv admin note: substantial text overlap with arXiv:2009.08719

In this paper, we propose a randomly projected convex clustering model for clustering a collection of $n$ high dimensional data points in $\mathbb{R}^d$ with $K$ hidden clusters. Compared to the convex clustering model for clustering original data with dimension $d$, we prove that, under some mild conditions, the perfect recovery of the cluster membership assignments of the convex clustering model, if exists, can be preserved by the randomly projected convex clustering model with embedding dimension $m = O(\epsilon^{-2}\log(n))$, where $0 < \epsilon < 1$ is some given parameter. We further prove that the embedding dimension can be improved to be $O(\epsilon^{-2}\log(K))$, which is independent of the number of data points. Extensive numerical experiment results will be presented in this paper to demonstrate the robustness and superior performance of the randomly projected convex clustering model. The numerical results presented in this paper also demonstrate that the randomly projected convex clustering model can outperform the randomly projected K-means model in practice.

Collaborative Filtering (CF) models, despite their great success, suffer from severe performance drops due to popularity distribution shifts, where these changes are ubiquitous and inevitable in real-world scenarios. Unfortunately, most leading popularity debiasing strategies, rather than tackling the vulnerability of CF models to varying popularity distributions, require prior knowledge of the test distribution to identify the degree of bias and further learn the popularity-entangled representations to mitigate the bias. Consequently, these models result in significant performance benefits in the target test set, while dramatically deviating the recommendation from users' true interests without knowing the popularity distribution in advance. In this work, we propose a novel learning framework, Invariant Collaborative Filtering (InvCF), to discover disentangled representations that faithfully reveal the latent preference and popularity semantics without making any assumption about the popularity distribution. At its core is the distillation of unbiased preference representations (i.e., user preference on item property), which are invariant to the change of popularity semantics, while filtering out the popularity feature that is unstable or outdated. Extensive experiments on five benchmark datasets and four evaluation settings (i.e., synthetic long-tail, unbiased, temporal split, and out-of-distribution evaluations) demonstrate that InvCF outperforms the state-of-the-art baselines in terms of popularity generalization ability on real recommendations. Visualization studies shed light on the advantages of InvCF for disentangled representation learning. Our codes are available at https://github.com/anzhang314/InvCF.

In this paper, we propose and analyze an efficient Halpern-Peaceman-Rachford (HPR) algorithm for solving the Wasserstein barycenter problem (WBP) with fixed supports. While the Peaceman-Rachford (PR) splitting method itself may not be convergent for solving the WBP, the HPR algorithm can achieve an $O(1/\varepsilon)$ non-ergodic iteration complexity with respect to the Karush-Kuhn-Tucker (KKT) residual. More interestingly, we propose an efficient procedure with linear time computational complexity to solve the linear systems involved in the subproblems of the HPR algorithm. As a consequence, the HPR algorithm enjoys an $O({\rm Dim(P)}/\varepsilon)$ non-ergodic computational complexity in terms of flops for obtaining an $\varepsilon$-optimal solution measured by the KKT residual for the WBP, where ${\rm Dim(P)}$ is the dimension of the variable of the WBP. This is better than the best-known complexity bound for the WBP. Moreover, the extensive numerical results on both the synthetic and real data sets demonstrate the superior performance of the HPR algorithm for solving the large-scale WBP.

This paper discovers that the neural network with lower decision boundary (DB) variability has better generalizability. Two new notions, algorithm DB variability and $(\epsilon, \eta)$-data DB variability, are proposed to measure the decision boundary variability from the algorithm and data perspectives. Extensive experiments show significant negative correlations between the decision boundary variability and the generalizability. From the theoretical view, two lower bounds based on algorithm DB variability are proposed and do not explicitly depend on the sample size. We also prove an upper bound of order $\mathcal{O}\left(\frac{1}{\sqrt{m}}+\epsilon+\eta\log\frac{1}{\eta}\right)$ based on data DB variability. The bound is convenient to estimate without the requirement of labels, and does not explicitly depend on the network size which is usually prohibitively large in deep learning., Comment: Accepted by IEEE TNNLS

Learning causal structure from observational data is a fundamental challenge in machine learning. However, the majority of commonly used differentiable causal discovery methods are non-identifiable, turning this problem into a continuous optimization task prone to data biases. In many real-life situations, data is collected from different environments, in which the functional relations remain consistent across environments, while the distribution of additive noises may vary. This paper proposes Differentiable Invariant Causal Discovery (DICD), utilizing the multi-environment information based on a differentiable framework to avoid learning spurious edges and wrong causal directions. Specifically, DICD aims to discover the environment-invariant causation while removing the environment-dependent correlation. We further formulate the constraint that enforces the target structure equation model to maintain optimal across the environments. Theoretical guarantees for the identifiability of proposed DICD are provided under mild conditions with enough environments. Extensive experiments on synthetic and real-world datasets verify that DICD outperforms state-of-the-art causal discovery methods up to 36% in SHD. Our code will be open-sourced., Comment: 22 pages, 11 figures

In this paper, we propose a novel adaptive sieving (AS) technique and an enhanced AS (EAS) technique, which are solver independent and could accelerate optimization algorithms for solving large scale convex optimization problems with intrinsic structured sparsity. We establish the finite convergence property of the AS technique and the EAS technique with inexact solutions of the reduced subproblems. As an important application, we apply the AS technique and the EAS technique on the convex clustering model, which could accelerate the state-of-the-art algorithm SSNAL by more than 7 times and the algorithm ADMM by more than 14 times.

Knowledge graph (KG) plays an increasingly important role in recommender systems. A recent technical trend is to develop end-to-end models founded on graph neural networks (GNNs). However, existing GNN-based models are coarse-grained in relational modeling, failing to (1) identify user-item relation at a fine-grained level of intents, and (2) exploit relation dependencies to preserve the semantics of long-range connectivity. In this study, we explore intents behind a user-item interaction by using auxiliary item knowledge, and propose a new model, Knowledge Graph-based Intent Network (KGIN). Technically, we model each intent as an attentive combination of KG relations, encouraging the independence of different intents for better model capability and interpretability. Furthermore, we devise a new information aggregation scheme for GNN, which recursively integrates the relation sequences of long-range connectivity (i.e., relational paths). This scheme allows us to distill useful information about user intents and encode them into the representations of users and items. Experimental results on three benchmark datasets show that, KGIN achieves significant improvements over the state-of-the-art methods like KGAT, KGNN-LS, and CKAN. Further analyses show that KGIN offers interpretable explanations for predictions by identifying influential intents and relational paths. The implementations are available at https://github.com/huangtinglin/Knowledge_Graph_based_Intent_Network., Comment: WWW 2021 oral presentation

The exclusive lasso (also known as elitist lasso) regularization has become popular recently due to its superior performance on structured sparsity. Its complex nature poses difficulties for the computation of high-dimensional machine learning models involving such a regularizer. In this paper, we propose an adaptive sieving (AS) strategy for generating solution paths of machine learning models with the exclusive lasso regularizer, wherein a sequence of reduced problems with much smaller sizes need to be solved. In order to solve these reduced problems, we propose a highly efficient dual Newton method based proximal point algorithm (PPDNA). As important ingredients, we systematically study the proximal mapping of the weighted exclusive lasso regularizer and the corresponding generalized Jacobian. These results also make popular first-order algorithms for solving exclusive lasso models practical. Various numerical experiments for the exclusive lasso models have demonstrated the effectiveness of the AS strategy for generating solution paths and the superior performance of the PPDNA.

The exclusive lasso (also known as elitist lasso) regularization has become popular recently due to its superior performance on group sparsity. Compared to the group lasso regularization which enforces the competition on variables among different groups, the exclusive lasso regularization also enforces the competition within each group. In this paper, we propose a highly efficient dual Newton based preconditioned proximal point algorithm (PPDNA) to solve machine learning models involving the exclusive lasso regularizer. As an important ingredient, we provide a rigorous proof for deriving the closed-form solution to the proximal mapping of the weighted exclusive lasso regularizer. In addition, we derive the corresponding HS-Jacobian to the proximal mapping and analyze its structure --- which plays an essential role in the efficient computation of the PPA subproblem via applying a semismooth Newton method on its dual. Various numerical experiments in this paper demonstrate the superior performance of the proposed PPDNA against other state-of-the-art numerical algorithms.

Clustering is a fundamental problem in unsupervised learning. Popular methods like K-means, may suffer from poor performance as they are prone to get stuck in its local minima. Recently, the sum-of-norms (SON) model (also known as the clustering path) has been proposed in Pelckmans et al. (2005), Lindsten et al. (2011) and Hocking et al. (2011). The perfect recovery properties of the convex clustering model with uniformly weighted all pairwise-differences regularization have been proved by Zhu et al. (2014) and Panahi et al. (2017). However, no theoretical guarantee has been established for the general weighted convex clustering model, where better empirical results have been observed. In the numerical optimization aspect, although algorithms like the alternating direction method of multipliers (ADMM) and the alternating minimization algorithm (AMA) have been proposed to solve the convex clustering model (Chi and Lange, 2015), it still remains very challenging to solve large-scale problems. In this paper, we establish sufficient conditions for the perfect recovery guarantee of the general weighted convex clustering model, which include and improve existing theoretical results as special cases. In addition, we develop a semismooth Newton based augmented Lagrangian method for solving large-scale convex clustering problems. Extensive numerical experiments on both simulated and real data demonstrate that our algorithm is highly efficient and robust for solving large-scale problems. Moreover, the numerical results also show the superior performance and scalability of our algorithm comparing to the existing first-order methods. In particular, our algorithm is able to solve a convex clustering problem with 200,000 points in $\mathbb{R}^3$ in about 6 minutes., Comment: arXiv admin note: substantial text overlap with arXiv:1802.07091

Clustering may be the most fundamental problem in unsupervised learning which is still active in machine learning research because its importance in many applications. Popular methods like K-means, may suffer from instability as they are prone to get stuck in its local minima. Recently, the sum-of-norms (SON) model (also known as clustering path), which is a convex relaxation of hierarchical clustering model, has been proposed in [7] and [5] Although numerical algorithms like ADMM and AMA are proposed to solve convex clustering model [2], it is known to be very challenging to solve large-scale problems. In this paper, we propose a semi-smooth Newton based augmented Lagrangian method for large-scale convex clustering problems. Extensive numerical experiments on both simulated and real data demonstrate that our algorithm is highly efficient and robust for solving large-scale problems. Moreover, the numerical results also show the superior performance and scalability of our algorithm compared to existing first-order methods.

SDPNAL+ is a {\sc Matlab} software package that implements an augmented Lagrangian based method to solve large scale semidefinite programming problems with bound constraints. The implementation was initially based on a majorized semismooth Newton-CG augmented Lagrangian method, here we designed it within an inexact symmetric Gauss-Seidel based semi-proximal ADMM/ALM (alternating direction method of multipliers/augmented Lagrangian method) framework for the purpose of deriving simpler stopping conditions and closing the gap between the practical implementation of the algorithm and the theoretical algorithm. The basic code is written in {\sc Matlab}, but some subroutines in C language are incorporated via Mex files. We also design a convenient interface for users to input their SDP models into the solver. Numerous problems arising from combinatorial optimization and binary integer quadratic programming problems have been tested to evaluate the performance of the solver. Extensive numerical experiments conducted in [Yang, Sun, and Toh, Mathematical Programming Computation, 7 (2015), pp. 331--366] show that the proposed method is quite efficient and robust, in that it is able to solve 98.9\% of the 745 test instances of SDP problems arising from various applications to the accuracy of $ 10^{-6}$ in the relative KKT residual.