From the sampling of data to the initialisation of parameters, randomness is ubiquitous in modern Machine Learning practice. Understanding the statistical fluctuations engendered by the different sources of randomness in prediction is therefore key to understanding robust generalisation. In this manuscript we develop a quantitative and rigorous theory for the study of fluctuations in an ensemble of generalised linear models trained on different, but correlated, features in high-dimensions. In particular, we provide a complete description of the asymptotic joint distribution of the empirical risk minimiser for generic convex loss and regularisation in the high-dimensional limit. Our result encompasses a rich set of classification and regression tasks, such as the lazy regime of overparametrised neural networks, or equivalently the random features approximation of kernels. While allowing to study directly the mitigating effect of ensembling (or bagging) on the bias-variance decomposition of the test error, our analysis also helps disentangle the contribution of statistical fluctuations, and the singular role played by the interpolation threshold that are at the roots of the "double-descent" phenomenon.
Université de Paris（巴黎大学）；Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）
Gradient boosting is a prediction method that iteratively combines weak learners to produce a complex and accurate model. From an optimization point of view, the learning procedure of gradient boosting mimics a gradient descent on a functional variable. This paper proposes to build upon the proximal point algorithm, when the empirical risk to minimize is not differentiable, in order to introduce a novel boosting approach, called proximal boosting. It comes with a companion algorithm inspired by [1] and called residual proximal boosting, which is aimed at better controlling the approximation er…
Hunan University（湖南大学）；University of Sydney（悉尼大学）；University of Science and Technology of China（中国科学技术大学）；Hong Kong Baptist University（香港浸会大学）；University of Melbourne（墨尔本大学）
Algorithms which minimize the averaged loss have been widely designed for dealing with noisy labels. Intuitively, when there is a finite training sample, penalizing the variance of losses will improve the stability and generalization of the algorithms. Interestingly, we found that the variance should be increased for the problem of learning with noisy labels. Specifically, increasing the variance will boost the memorization effects and reduce the harmfulness of incorrect labels. By exploiting the label noise transition matrix, regularizers can be easily designed to reduce the variance of losse…
We consider the problem of online combinatorial optimization under semi-bandit feedback, where a learner has to repeatedly pick actions from a combinatorial decision set in order to minimize the total losses associated with its decisions. After making each decision, the learner observes the losses associated with its action, but not other losses. For this problem, there are several learning algorithms that guarantee that the learner's expected regret grows as $\widetilde{O}(\sqrt{T})$ with the number of rounds $T$. In this paper, we propose an algorithm that improves this scaling to $\widetild…
We introduce a novel bias-variance decomposition for a range of strictly convex margin losses, including the logistic loss (minimized by the classic LogitBoost algorithm), as well as the squared margin loss and canonical boosting loss. Furthermore, we show that, for all strictly convex margin losses, the expected risk decomposes into the risk of a "central" model and a term quantifying variation in the functional margin with respect to variations in the training data. These decompositions provide a diagnostic tool for practitioners to understand model overfitting/underfitting, and have implica…
IIIS, Tsinghua University（清华大学交叉信息研究院）；California Institute of Technology（加州理工学院）
We study online optimization in a setting where an online learner seeks to optimize a per-round hitting cost, which may be non-convex, while incurring a movement cost when changing actions between rounds. We ask: \textit{under what general conditions is it possible for an online learner to leverage predictions of future cost functions in order to achieve near-optimal costs?} Prior work has provided near-optimal online algorithms for specific combinations of assumptions about hitting and switching costs, but no general results are known. In this work, we give two general sufficient conditions t…
We study the problem of online learning with non-convex losses, where the learner has access to an offline optimization oracle. We show that the classical Follow the Perturbed Leader (FTPL) algorithm achieves optimal regret rate of $O(T^{-1/2})$ in this setting. This improves upon the previous best-known regret rate of $O(T^{-1/3})$ for FTPL. We further show that an optimistic variant of FTPL achieves better regret bounds when the sequence of losses encountered by the learner is `predictable'.
University of California, Berkeley（加州大学伯克利分校）；University of Chicago（芝加哥大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
Ensemble learning is widely applied in Machine Learning (ML) to improve model performance and to mitigate decision risks. In this approach, predictions from a diverse set of learners are combined to obtain a joint decision. Recently, various methods have been explored in literature for estimating decision uncertainties using ensemble learning; however, determining which metrics are a better fit for certain decision-making applications remains a challenging task. In this paper, we study the following key research question in the selection of uncertainty metrics: when does an uncertainty metric …
The infinitesimal jackknife (IJ) has recently been applied to the random forest to estimate its prediction variance. These theorems were verified under a traditional random forest framework which uses classification and regression trees (CART) and bootstrap resampling. However, random forests using conditional inference (CI) trees and subsampling have been found to be not prone to variable selection bias. Here, we conduct simulation experiments using a novel approach to explore the applicability of the IJ to random forests using variations on the resampling method and base learner. Test data p…
We consider the problem of the Zinkevich (2003)-style dynamic regret minimization in online learning with exp-concave losses. We show that whenever improper learning is allowed, a Strongly Adaptive online learner achieves the dynamic regret of $\tilde O^*(n^{1/3}C_n^{2/3} \vee 1)$ where $C_n$ is the total variation (a.k.a. path length) of the an arbitrary sequence of comparators that may not be known to the learner ahead of time. Achieving this rate was highly nontrivial even for squared losses in 1D where the best known upper bound was $O(\sqrt{nC_n} \vee \log n)$ (Yuan and Lamperski, 2019). …
Department of Operations Research and Financial Engineering, Princeton University（运筹学与金融工程系，普林斯顿大学）；Department of Statistics and Data Science, National University of Singapore（统计与数据科学系，新加坡国立大学）
Aggregating multiple learners through an ensemble of models aim to make better predictions by capturing the underlying distribution of the data more accurately. Different ensembling methods, such as bagging, boosting, and stacking/blending, have been studied and adopted extensively in research and practice. While bagging and boosting focus more on reducing variance and bias, respectively, stacking approaches target both by finding the optimal way to combine base learners. In stacking with the weighted average, ensembles are created from weighted averages of multiple base learners. It is known …
Relating the Partial Dependence Plot and Permutation Feature Importance to the Data Generating Process
Ludwig-Maximilian University Munich（慕尼黑路德维希-马克西米利安大学）；University of Vienna（维也纳大学）；Leibniz Institute for Prevention Research and Epidemiology – BIPS（莱布尼茨预防研究与流行病学研究所 – BIPS）；University of Bremen（不来梅大学）
Ensembles are a straightforward, remarkably effective method for improving the accuracy,calibration, and robustness of models on classification tasks; yet, the reasons that underlie their success remain an active area of research. We build upon the extension to the bias-variance decomposition by Pfau (2013) in order to gain crucial insights into the behavior of ensembles of classifiers. Introducing a dual reparameterization of the bias-variance tradeoff, we first derive generalized laws of total expectation and variance for nonsymmetric losses typical of classification tasks. Comparing conditi…
Variance reduction has emerged in recent years as a strong competitor to stochastic gradient descent in non-convex problems, providing the first algorithms to improve upon the converge rate of stochastic gradient descent for finding first-order critical points. However, variance reduction techniques typically require carefully tuned learning rates and willingness to use excessively large "mega-batches" in order to achieve their improved results. We present a new algorithm, STORM, that does not require any batches and makes use of adaptive learning rates, enabling simpler implementation and les…
Covariate shift relaxes the widely-employed independent and identically distributed (IID) assumption by allowing different training and testing input distributions. Unfortunately, common methods for addressing covariate shift by trying to remove the bias between training and testing distributions using importance weighting often provide poor performance guarantees in theory and unreliable predictions with high variance in practice. Recently developed methods that construct a predictor that is inherently robust to the difficulties of learning under covariate shift are restricted to minimizing l…
Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；Singapore Management University（新加坡管理大学）
Few-shot learning aims to train efficient predictive models with a few examples. The lack of training data leads to poor models that perform high-variance or low-confidence predictions. In this paper, we propose to meta-learn the ensemble of epoch-wise empirical Bayes models (E3BM) to achieve robust predictions. "Epoch-wise" means that each training epoch has a Bayes model whose parameters are specifically learned and deployed. "Empirical" means that the hyperparameters, e.g., used for learning and ensembling the epoch-wise models, are generated by hyperprior learners conditional on task-speci…
Cornell University（康奈尔大学）；University of Pittsburgh（匹兹堡大学）；Australian National University（澳大利亚国立大学）
This paper develops a general framework for analyzing asymptotics of $V$-statistics. Previous literature on limiting distribution mainly focuses on the cases when $n \to \infty$ with fixed kernel size $k$. Under some regularity conditions, we demonstrate asymptotic normality when $k$ grows with $n$ by utilizing existing results for $U$-statistics. The key in our approach lies in a mathematical reduction to $U$-statistics by designing an equivalent kernel for $V$-statistics. We also provide a unified treatment on variance estimation for both $U$- and $V$-statistics by observing connections to e…
A Geometric Approach to Constrained Online Learning
Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur（计算机科学与工程系，印度理工学院Kharagpur分校）；School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India（技术与计算机科学学院，印度基础研究科学研究院，孟买，印度）
Carnegie Mellon University（卡内基梅隆大学）；Microsoft Research Asia（微软亚洲研究院）
Stochastic variance reduced methods have gained a lot of interest recently for empirical risk minimization due to its appealing run time complexity. When the data size is large and disjointly stored on different machines, it becomes imperative to distribute the implementation of such variance reduced methods. In this paper, we consider a general framework that directly distributes popular stochastic variance reduced methods in the master/slave model, by assigning outer loops to the parameter server, and inner loops to worker machines. This framework is natural and friendly to implement, but it…
