Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGBHT: Gradient Boosting Histogram Transform for Density Estimation
98
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we propose a density estimation algorithm called textit{Gradient Boosting Histogram Transform} (GBHT), where we adopt the textit{Negative Log Likelihood} as the loss function to make the boosting procedure available for the unsupervised tasks. From a learning theory viewpoint, we first prove fast convergence rates for GBHT with the smoothness assumption that the underlying density function lies in the space $C^{0,alpha}$. Then when the target density function lies in spaces $C^{1,alpha}$, we present an upper bound for GBHT which is smaller than the lower bound of its corresponding base learner, in the sense of convergence rates. To the best of our knowledge, we make the first attempt to theoretically explain why boosting can enhance the performance of its base learners for density estimation problems. In experiments, we not only conduct performance comparisons with the widely used KDE, but also apply GBHT to anomaly detection to showcase a further application of GBHT.
rate research
Read More
We investigate an algorithm named histogram transform ensembles (HTE) density estimator whose effectiveness is supported by both solid theoretical analysis and significant experimental performance. On the theoretical side, by decomposing the error term into approximation error and estimation error, we are able to conduct the following analysis: First of all, we establish the universal consistency under $L_1(mu)$-norm. Secondly, under the assumption that the underlying density function resides in the H{o}lder space $C^{0,alpha}$, we prove almost optimal convergence rates for both single and ensemble density estimators under $L_1(mu)$-norm and $L_{infty}(mu)$-norm for different tail distributions, whereas in contrast, for its subspace $C^{1,alpha}$ consisting of smoother functions, almost optimal convergence rates can only be established for the ensembles and the lower bound of the single estimators illustrates the benefits of ensembles over single density estimators. In the experiments, we first carry out simulations to illustrate that histogram transform ensembles surpass single histogram transforms, which offers powerful evidence to support the theoretical results in the space $C^{1,alpha}$. Moreover, to further exert the experimental performances, we propose an adaptive version of HTE and study the parameters by generating several synthetic datasets with diversities in dimensions and distributions. Last but not least, real data experiments with other state-of-the-art density estimators demonstrate the accuracy of the adaptive HTE algorithm.
We propose a novel algorithm for large-scale regression problems named histogram transform ensembles (HTE), composed of random rotations, stretchings, and translations. First of all, we investigate the theoretical properties of HTE when the regression function lies in the H{o}lder space $C^{k,alpha}$, $k in mathbb{N}_0$, $alpha in (0,1]$. In the case that $k=0, 1$, we adopt the constant regressors and develop the na{i}ve histogram transforms (NHT). Within the space $C^{0,alpha}$, although almost optimal convergence rates can be derived for both single and ensemble NHT, we fail to show the benefits of ensembles over single estimators theoretically. In contrast, in the subspace $C^{1,alpha}$, we prove that if $d geq 2(1+alpha)/alpha$, the lower bound of the convergence rates for single NHT turns out to be worse than the upper bound of the convergence rates for ensemble NHT. In the other case when $k geq 2$, the NHT may no longer be appropriate in predicting smoother regression functions. Instead, we apply kernel histogram transforms (KHT) equipped with smoother regressors such as support vector machines (SVMs), and it turns out that both single and ensemble KHT enjoy almost optimal convergence rates. Then we validate the above theoretical results by numerical experiments. On the one hand, simulations are conducted to elucidate that ensemble NHT outperform single NHT. On the other hand, the effects of bin sizes on accuracy of both NHT and KHT also accord with theoretical analysis. Last but not least, in the real-data experiments, comparisons between the ensemble KHT, equipped with adaptive histogram transforms, and other state-of-the-art large-scale regression estimators verify the effectiveness and accuracy of our algorithm.
Automatic machine learning performs predictive modeling with high performing machine learning tools without human interference. This is achieved by making machine learning applications parameter-free, i.e. only a dataset is provided while the complete model selection and model building process is handled internally through (often meta) optimization. Projects like Auto-WEKA and auto-sklearn aim to solve the Combined Algorithm Selection and Hyperparameter optimization (CASH) problem resulting in huge configuration spaces. However, for most real-world applications, the optimization over only a few different key learning algorithms can not only be sufficient, but also potentially beneficial. The latter becomes apparent when one considers that models have to be validated, explained, deployed and maintained. Here, less complex model are often preferred, for validation or efficiency reasons, or even a strict requirement. Automatic gradient boosting simplifies this idea one step further, using only gradient boosting as a single learning algorithm in combination with model-based hyperparameter tuning, threshold optimization and encoding of categorical features. We introduce this general framework as well as a concrete implementation called autoxgboost. It is compared to current AutoML projects on 16 datasets and despite its simplicity is able to achieve comparable results on about half of the datasets as well as performing best on two.
In this paper, we propose a gradient boosting algorithm for large-scale regression problems called textit{Gradient Boosted Binary Histogram Ensemble} (GBBHE) based on binary histogram partition and ensemble learning. From the theoretical perspective, by assuming the H{o}lder continuity of the target function, we establish the statistical convergence rate of GBBHE in the space $C^{0,alpha}$ and $C^{1,0}$, where a lower bound of the convergence rate for the base learner demonstrates the advantage of boosting. Moreover, in the space $C^{1,0}$, we prove that the number of iterations to achieve the fast convergence rate can be reduced by using ensemble regressor as the base learner, which improves the computational efficiency. In the experiments, compared with other state-of-the-art algorithms such as gradient boosted regression tree (GBRT), Breimans forest, and kernel-based methods, our GBBHE algorithm shows promising performance with less running time on large-scale datasets.
Residual Networks (ResNets) have become state-of-the-art models in deep learning and several theoretical studies have been devoted to understanding why ResNet works so well. One attractive viewpoint on ResNet is that it is optimizing the risk in a functional space by combining an ensemble of effective features. In this paper, we adopt this viewpoint to construct a new gradient boosting method, which is known to be very powerful in data analysis. To do so, we formalize the gradient boosting perspective of ResNet mathematically using the notion of functional gradients and propose a new method called ResFGB for classification tasks by leveraging ResNet perception. Two types of generalization guarantees are provided from the optimization perspective: one is the margin bound and the other is the expected risk bound by the sample-splitting technique. Experimental results show superior performance of the proposed method over state-of-the-art methods such as LightGBM.
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
