Kernel ridge regression, KRR, is a generalization of linear ridge regression that is non-linear in the data, but linear in the parameters. Here, we introduce an equivalent formulation of the objective function of KRR, opening up both for using penalties other than the ridge penalty and for studying kernel ridge regression from the perspective of gradient descent. Using a continuous-time perspective, we derive a closed-form solution for solving kernel regression with gradient descent, something we refer to as kernel gradient flow, KGF, and theoretically bound the differences between KRR and KGF, where, for the latter, regularization is obtained through early stopping. We also generalize KRR by replacing the ridge penalty with the $\ell_1$ and $\ell_\infty$ penalties, respectively, and use the fact that analogous to the similarities between KGF and KRR, $\ell_1$ regularization and forward stagewise regression (also known as coordinate descent), and $\ell_\infty$ regularization and sign gradient descent, follow similar solution paths. We can thus alleviate the need for computationally heavy algorithms based on proximal gradient descent. We show theoretically and empirically how the $\ell_1$ and $\ell_\infty$ penalties, and the corresponding gradient-based optimization algorithms, produce sparse and robust kernel regression solutions, respectively.
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We consider estimation of a functional parameter of a realistically modeled data distribution based on independent and identically distributed observations. Suppose that the true function is defined as the minimizer of the expectation of a specified loss function over its parameter space. Estimators of the true function are provided, viewed as a data-adaptive coordinate transformation for the true function. For any $J$-dimensional real valued cadlag function with finite sectional variation norm, we define a candidate ensemble estimator as the mapping from the data into the composition of the cadlag function and the $J$ estimated functions. Using $V$-fold cross-validation, we define the cross-validated empirical risk of each cadlag function specific ensemble estimator. We then define the Meta Highly Adaptive Lasso Minimum Loss Estimator (M-HAL-MLE) as the cadlag function that minimizes this cross-validated empirical risk over all cadlag functions with a uniform bound on the sectional variation norm. For each of the $V$ training samples, this yields a composition of the M-HAL-MLE ensemble and the $J$ estimated functions trained on the training sample. We can estimate the true function with the average of these $V$ estimated functions, which we call the M-HAL super-learner. The M-HAL super-learner converges to the oracle estimator at a rate $n^{-2/3}$ (up till $\log n$-factor) w.r.t. excess risk, where the oracle estimator minimizes the excess risk among all considered ensembles. The excess risk of the oracle estimator and true function is generally second order. Under weak conditions on the $J$ candidate estimators, target features of the undersmoothed M-HAL super-learner are asymptotically linear estimators of the corresponding target features of true function, with influence curve either the efficient influence curve, or potentially, a super-efficient influence curve.
We investigate the problem of multiplex graph embedding, that is, graphs in which nodes interact through multiple types of relations (dimensions). In recent years, several methods have been developed to address this problem. However, the need for more effective and specialized approaches grows with the production of graph data with diverse characteristics. In particular, real-world multiplex graphs may exhibit a high number of dimensions, making it difficult to construct a single consensus representation. Furthermore, important information can be hidden in complex latent structures scattered in multiple dimensions. To address these issues, we propose HMGE, a novel embedding method based on hierarchical aggregation for high-dimensional multiplex graphs. Hierarchical aggregation consists of learning a hierarchical combination of the graph dimensions and refining the embeddings at each hierarchy level. Non-linear combinations are computed from previous ones, thus uncovering complex information and latent structures hidden in the multiplex graph dimensions. Moreover, we leverage mutual information maximization between local patches and global summaries to train the model without supervision. This allows to capture of globally relevant information present in diverse locations of the graph. Detailed experiments on synthetic and real-world data illustrate the suitability of our approach to downstream supervised tasks, including link prediction and node classification.
Detecting anomalies has become an increasingly critical function in the financial service industry. Anomaly detection is frequently used in key compliance and risk functions such as financial crime detection fraud and cybersecurity. The dynamic nature of the underlying data patterns especially in adversarial environments like fraud detection poses serious challenges to the machine learning models. Keeping up with the rapid changes by retraining the models with the latest data patterns introduces pressures in balancing the historical and current patterns while managing the training data size. Furthermore the model retraining times raise problems in time-sensitive and high-volume deployment systems where the retraining period directly impacts the models ability to respond to ongoing attacks in a timely manner. In this study we propose a temporal knowledge distillation-based label augmentation approach (TKD) which utilizes the learning from older models to rapidly boost the latest model and effectively reduces the model retraining times to achieve improved agility. Experimental results show that the proposed approach provides advantages in retraining times while improving the model performance.
Optimal transport is a fundamental topic that has attracted a great amount of attention from the optimization community in the past decades. In this paper, we consider an interesting discrete dynamic optimal transport problem: can we efficiently update the optimal transport plan when the weights or the locations of the data points change? This problem is naturally motivated by several applications in machine learning. For example, we often need to compute the optimal transport cost between two different data sets; if some changes happen to a few data points, should we re-compute the high complexity cost function or update the cost by some efficient dynamic data structure? We are aware that several dynamic maximum flow algorithms have been proposed before, however, the research on dynamic minimum cost flow problem is still quite limited, to the best of our knowledge. We propose a novel 2D Skip Orthogonal List together with some dynamic tree techniques. Although our algorithm is based on the conventional simplex method, it can efficiently find the variable to pivot within expected $O(1)$ time, and complete each pivoting operation within expected $O(|V|)$ time where $V$ is the set of all supply and demand nodes. Since dynamic modifications typically do not introduce significant changes, our algorithm requires only a few simplex iterations in practice. So our algorithm is more efficient than re-computing the optimal transport cost that needs at least one traversal over all $|E| = O(|V|^2)$ variables, where $|E|$ denotes the number of edges in the network. Our experiments demonstrate that our algorithm significantly outperforms existing algorithms in the dynamic scenarios.
In recent years, considerable attention has been devoted to the regularization models due to the presence of high-dimensional data in scientific research. Sparse support vector machine (SVM) are useful tools in high-dimensional data analysis, and they have been widely used in the area of econometrics. Nevertheless, the non-smoothness of objective functions and constraints present computational challenges for many existing solvers in the presence of ultra-high dimensional covariates. In this paper, we design efficient and parallelizable algorithms for solving sparse SVM problems with high dimensional data through feature space split. The proposed algorithm is based on the alternating direction method of multiplier (ADMM). We establish the rate of convergence of the proposed ADMM method and compare it with existing solvers in various high and ultra-high dimensional settings. The compatibility of the proposed algorithm with parallel computing can further alleviate the storage and scalability limitations of a single machine in large-scale data processing.
Stochastic optimization is a widely used approach for optimization under uncertainty, where uncertain input parameters are modeled by random variables. Exact or approximation algorithms have been obtained for several fundamental problems in this area. However, a significant limitation of this approach is that it requires full knowledge of the underlying probability distributions. Can we still get good (approximation) algorithms if these distributions are unknown, and the algorithm needs to learn them through repeated interactions? In this paper, we resolve this question for a large class of "monotone" stochastic problems, by providing a generic online learning algorithm with $\sqrt{T \log T}$ regret relative to the best approximation algorithm (under known distributions). Importantly, our online algorithm works in a semi-bandit setting, where in each period, the algorithm only observes samples from the r.v.s that were actually probed. Our framework applies to several fundamental problems in stochastic optimization such as prophet inequality, Pandora's box, stochastic knapsack, stochastic matchings and stochastic submodular optimization.
Transformers with linear attention allow for efficient parallel training but can simultaneously be formulated as an RNN with 2D (matrix-valued) hidden states, thus enjoying linear (with respect to output length) inference complexity. Recent works such as RetNet (Sun et al., 2023) and TransNormerLLM (Qin et al., 2023a) observe that adding a global decay term to the additive RNN update rule greatly improves performance, sometimes outperforming standard Transformers with softmax attention when trained at scale. In this work we show that adding a data-dependent gating mechanism further improves performance. We derive a parallel form of this gated linear attention layer that enables efficient training. However, a straightforward, numerically stable implementation of this parallel form requires generalized matrix multiplications in log-space for numerical stability, and thus cannot take advantage of tensor cores on modern GPUs which are optimized for standard matrix multiplications. We develop a hardware-efficient version of the parallel form that can still make use of tensor cores through block-parallel computations over sequence chunks. Experiments on moderate-scale language modeling (340M-parameter models trained on 15B tokens, 1.3B-parameter models trained on 100B tokens) show that gated linear attention (GLA) Transformers perform competitively against a strong LLaMA-architecture Transformer baseline (Touvron et al., 2023) as well as Mamba (Gu & Dao, 2023), a recently introduced state-space model with a data-dependent state transition mechanism. For training speed, our Triton-based implementation performs comparably to CUDA-optimized FlashAttention-2 (Dao, 2023) under the regular 2048 training length setting, while outperforming FlashAttention-2 when training on longer sequences beyond 4096.
Graphs are important data representations for describing objects and their relationships, which appear in a wide diversity of real-world scenarios. As one of a critical problem in this area, graph generation considers learning the distributions of given graphs and generating more novel graphs. Owing to their wide range of applications, generative models for graphs, which have a rich history, however, are traditionally hand-crafted and only capable of modeling a few statistical properties of graphs. Recent advances in deep generative models for graph generation is an important step towards improving the fidelity of generated graphs and paves the way for new kinds of applications. This article provides an extensive overview of the literature in the field of deep generative models for graph generation. Firstly, the formal definition of deep generative models for the graph generation and the preliminary knowledge are provided. Secondly, taxonomies of deep generative models for both unconditional and conditional graph generation are proposed respectively; the existing works of each are compared and analyzed. After that, an overview of the evaluation metrics in this specific domain is provided. Finally, the applications that deep graph generation enables are summarized and five promising future research directions are highlighted.
Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis, thereby allowing manual manipulation in predicting the final answer.
Aspect based sentiment analysis (ABSA) can provide more detailed information than general sentiment analysis, because it aims to predict the sentiment polarities of the given aspects or entities in text. We summarize previous approaches into two subtasks: aspect-category sentiment analysis (ACSA) and aspect-term sentiment analysis (ATSA). Most previous approaches employ long short-term memory and attention mechanisms to predict the sentiment polarity of the concerned targets, which are often complicated and need more training time. We propose a model based on convolutional neural networks and gating mechanisms, which is more accurate and efficient. First, the novel Gated Tanh-ReLU Units can selectively output the sentiment features according to the given aspect or entity. The architecture is much simpler than attention layer used in the existing models. Second, the computations of our model could be easily parallelized during training, because convolutional layers do not have time dependency as in LSTM layers, and gating units also work independently. The experiments on SemEval datasets demonstrate the efficiency and effectiveness of our models.
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