In the transfer learning paradigm models learn useful representations (or features) during a data-rich pretraining stage, and then use the pretrained representation to improve model performance on data-scarce downstream tasks. In this work, we explore transfer learning with the goal of optimizing downstream performance. We introduce a simple linear model that takes as input an arbitrary pretrained feature transform. We derive exact asymptotics of the downstream risk and its \textit{fine-grained} bias-variance decomposition. We then identify the pretrained representation that optimizes the asymptotic downstream bias and variance averaged over an ensemble of downstream tasks. Our theoretical and empirical analysis uncovers the surprising phenomenon that the optimal featurization is naturally sparse, even in the absence of explicit sparsity-inducing priors or penalties. Additionally, we identify a phase transition where the optimal pretrained representation shifts from hard selection to soft selection of relevant features.
Tsinghua University（清华大学）；University of California Berkeley（加州大学伯克利分校）
Domain Adaptation (DA) enables transferring a learning machine from a labeled source domain to an unlabeled target one. While remarkable advances have been made, most of the existing DA methods focus on improving the target accuracy at inference. How to estimate the predictive uncertainty of DA models is vital for decision-making in safety-critical scenarios but remains the boundary to explore. In this paper, we delve into the open problem of Calibration in DA, which is extremely challenging due to the coexistence of domain shift and the lack of target labels. We first reveal the dilemma that …
Classical learning theory suggests that the optimal generalization performance of a machine learning model should occur at an intermediate model complexity, with simpler models exhibiting high bias and more complex models exhibiting high variance of the predictive function. However, such a simple trade-off does not adequately describe deep learning models that simultaneously attain low bias and variance in the heavily overparameterized regime. A primary obstacle in explaining this behavior is that deep learning algorithms typically involve multiple sources of randomness whose individual contri…
The classical bias-variance trade-off predicts that bias decreases and variance increase with model complexity, leading to a U-shaped risk curve. Recent work calls this into question for neural networks and other over-parameterized models, for which it is often observed that larger models generalize better. We provide a simple explanation for this by measuring the bias and variance of neural networks: while the bias is monotonically decreasing as in the classical theory, the variance is unimodal or bell-shaped: it increases then decreases with the width of the network. We vary the network arch…
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…
In this note, we study the relationship between the variational gap and the variance of the (log) likelihood ratio. We show that the gap can be upper bounded by some form of dispersion measure of the likelihood ratio, which suggests the bias of variational inference can be reduced by making the distribution of the likelihood ratio more concentrated, such as via averaging and variance reduction.
Bias - variance decomposition of the expected error defined for regression and classification problems is an important tool to study and compare different algorithms, to find the best areas for their application. Here the decomposition is introduced for the survival analysis problem. In our experiments, we study bias -variance parts of the expected error for two algorithms: original Cox proportional hazard regression and CoxPath, path algorithm for L1-regularized Cox regression, on the series of increased training sets. The experiments demonstrate that, contrary expectations, CoxPath does not …
The main goal of this thesis is to point out that the bias-variance tradeoff is not always true (e.g. in neural networks). We advocate for this lack of universality to be acknowledged in textbooks and taught in introductory courses that cover the tradeoff. We first review the history of the bias-variance tradeoff, its prevalence in textbooks, and some of the main claims made about the bias-variance tradeoff. Through extensive experiments and analysis, we show a lack of a bias-variance tradeoff in neural networks when increasing network width. Our findings seem to contradict the claims of the l…
University of Oxford（牛津大学）；Facebook AI Research（脸书人工智能研究院）
Gradient-based methods for optimisation of objectives in stochastic settings with unknown or intractable dynamics require estimators of derivatives. We derive an objective that, under automatic differentiation, produces low-variance unbiased estimators of derivatives at any order. Our objective is compatible with arbitrary advantage estimators, which allows the control of the bias and variance of any-order derivatives when using function approximation. Furthermore, we propose a method to trade off bias and variance of higher order derivatives by discounting the impact of more distant causal de…
University of Genova（热那亚大学）；Istituto Italiano di Tecnologia（意大利理工学院）；MIT（麻省理工学院）
We study the behavior of error bounds for multiclass classification under suitable margin conditions. For a wide variety of methods we prove that the classification error under a hard-margin condition decreases exponentially fast without any bias-variance trade-off. Different convergence rates can be obtained in correspondence of different margin assumptions. With a self-contained and instructive analysis we are able to generalize known results from the binary to the multiclass setting.
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…
The University of Sydney（悉尼大学）；Data61, CSIRO（联邦科学与工业研究组织Data61）；National Institute of Informatics（国立信息学研究所）
Hierarchical probabilistic models are able to use a large number of parameters to create a model with a high representation power. However, it is well known that increasing the number of parameters also increases the complexity of the model which leads to a bias-variance trade-off. Although it is a classical problem, the bias-variance trade-off between hidden layers and higher-order interactions have not been well studied. In our study, we propose an efficient inference algorithm for the log-linear formulation of the higher-order Boltzmann machine using a combination of Gibbs sampling and anne…
University of California, Berkeley（加州大学伯克利分校）；University of Pennsylvania（宾夕法尼亚大学）
Adversarially trained models exhibit a large generalization gap: they can interpolate the training set even for large perturbation radii, but at the cost of large test error on clean samples. To investigate this gap, we decompose the test risk into its bias and variance components and study their behavior as a function of adversarial training perturbation radii ($\varepsilon$). We find that the bias increases monotonically with $\varepsilon$ and is the dominant term in the risk. Meanwhile, the variance is unimodal as a function of $\varepsilon$, peaking near the interpolation threshold for the…
University of Massachusetts Boston（马萨诸塞大学波士顿分校）；University of Tennesee（田纳西大学）
The study of model bias and variance with respect to decision boundaries is critically important in supervised classification. There is generally a tradeoff between the two, as fine-tuning of the decision boundary of a classification model to accommodate more boundary training samples (i.e., higher model complexity) may improve training accuracy (i.e., lower bias) but hurt generalization against unseen data (i.e., higher variance). By focusing on just classification boundary fine-tuning and model complexity, it is difficult to reduce both bias and variance. To overcome this dilemma, we take a …
This paper builds upon the work of Pfau (2013), which generalized the bias variance tradeoff to any Bregman divergence loss function. Pfau (2013) showed that for Bregman divergences, the bias and variances are defined with respect to a central label, defined as the mean of the label variable, and a central prediction, of a more complex form. We show that, similarly to the label, the central prediction can be interpreted as the mean of a random variable, where the mean operates in a dual space defined by the loss function itself. Viewing the bias-variance tradeoff through operations taken in du…
University College London（伦敦大学学院）；University of Surrey（萨里大学）
Standard bias and variance (B&V) terminologies were originally defined for the regression setting and their extensions to classification have led to several different models / definitions in the literature. In this paper, we aim to provide the link between the commonly used frameworks of Tumer & Ghosh (T&G) and James. By unifying the two approaches, we relate the B&V defined for the 0/1 loss to the standard B&V of the boundary distributions given for the squared error loss. The closed form relationships provide a deeper understanding of classification performance, and their use is demonstrated…
We consider the off-policy evaluation problem in Markov decision processes with function approximation. We propose a generalization of the recently introduced \emph{emphatic temporal differences} (ETD) algorithm \citep{SuttonMW15}, which encompasses the original ETD($λ$), as well as several other off-policy evaluation algorithms as special cases. We call this framework \ETD, where our introduced parameter $β$ controls the decay rate of an importance-sampling term. We study conditions under which the projected fixed-point equation underlying \ETD\ involves a contraction operator, allowing us to…
Ensemble methods for classification and clustering have been effectively used for decades, while ensemble learning for outlier detection has only been studied recently. In this work, we design a new ensemble approach for outlier detection in multi-dimensional point data, which provides improved accuracy by reducing error through both bias and variance. Although classification and outlier detection appear as different problems, their theoretical underpinnings are quite similar in terms of the bias-variance trade-off [1], where outlier detection is considered as a binary classification task with…
This paper provides a framework to analyze stochastic gradient algorithms in a mean squared error (MSE) sense using the asymptotic normality result of the stochastic gradient descent (SGD) iterates. We perform this analysis by taking the asymptotic normality result and applying it to the finite iteration case. Specifically, we look at problems where the gradient estimators are biased and have reduced variance and compare the iterates generated by these gradient estimators to the iterates generated by the SGD algorithm. We use the work of Fabian to characterize the mean and the variance of the …
The bias-variance trade-off is a central concept in supervised learning. In classical statistics, increasing the complexity of a model (e.g., number of parameters) reduces bias but also increases variance. Until recently, it was commonly believed that optimal performance is achieved at intermediate model complexities which strike a balance between bias and variance. Modern Deep Learning methods flout this dogma, achieving state-of-the-art performance using "over-parameterized models" where the number of fit parameters is large enough to perfectly fit the training data. As a result, understandi…
Vector embedding is a foundational building block of many deep learning models, especially in natural language processing. In this paper, we present a theoretical framework for understanding the effect of dimensionality on vector embeddings. We observe that the distributional hypothesis, a governing principle of statistical semantics, requires a natural unitary-invariance for vector embeddings. Motivated by the unitary-invariance observation, we propose the Pairwise Inner Product (PIP) loss, a unitary-invariant metric on the similarity between two embeddings. We demonstrate that the PIP loss c…
University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）
Self-paced learning and hard example mining re-weight training instances to improve learning accuracy. This paper presents two improved alternatives based on lightweight estimates of sample uncertainty in stochastic gradient descent (SGD): the variance in predicted probability of the correct class across iterations of mini-batch SGD, and the proximity of the correct class probability to the decision threshold. Extensive experimental results on six datasets show that our methods reliably improve accuracy in various network architectures, including additional gains on top of other popular traini…
Deep reinforcement learning (DRL) on Markov decision processes (MDPs) with continuous action spaces is often approached by directly training parametric policies along the direction of estimated policy gradients (PGs). Previous research revealed that the performance of these PG algorithms depends heavily on the bias-variance tradeoffs involved in estimating and using PGs. A notable approach towards balancing this tradeoff is to merge both on-policy and off-policy gradient estimations. However existing PG merging methods can be sample inefficient and are not suitable to train deterministic polic…
Scalable Gaussian Process methods are computationally attractive, yet introduce modeling biases that require rigorous study. This paper analyzes two common techniques: early truncated conjugate gradients (CG) and random Fourier features (RFF). We find that both methods introduce a systematic bias on the learned hyperparameters: CG tends to underfit while RFF tends to overfit. We address these issues using randomized truncation estimators that eliminate bias in exchange for increased variance. In the case of RFF, we show that the bias-to-variance conversion is indeed a trade-off: the additional…
The Ohio State University（俄亥俄州立大学）；Columbia University（哥伦比亚大学）
Breakthroughs in machine learning are rapidly changing science and society, yet our fundamental understanding of this technology has lagged far behind. Indeed, one of the central tenets of the field, the bias-variance trade-off, appears to be at odds with the observed behavior of methods used in the modern machine learning practice. The bias-variance trade-off implies that a model should balance under-fitting and over-fitting: rich enough to express underlying structure in data, simple enough to avoid fitting spurious patterns. However, in the modern practice, very rich models such as neural n…
We consider the least-squares regression problem and provide a detailed asymptotic analysis of the performance of averaged constant-step-size stochastic gradient descent (a.k.a. least-mean-squares). In the strongly-convex case, we provide an asymptotic expansion up to explicit exponentially decaying terms. Our analysis leads to new insights into stochastic approximation algorithms: (a) it gives a tighter bound on the allowed step-size; (b) the generalization error may be divided into a variance term which is decaying as O(1/n), independently of the step-size $γ$, and a bias term that decays as…
Uncertainty Estimates of Predictions via a General Bias-Variance Decomposition
German Cancer Research Center (DKFZ)（德国癌症研究中心（DKFZ））；German Cancer Consortium (DKTK)（德国癌症联盟（DKTK））；Goethe University Frankfurt（法兰克福歌德大学）；Frankfurt Cancer Institute（法兰克福癌症研究所）
本文提出 proper scores 的通用偏差-方差分解，以 Bregman Information 刻画方差项，并将其用于模型集成、置信区域及各程度域漂移下的分布外检测。
Department of Electrical Engineering and Computer Science, University of Michigan（密歇根大学电气工程与计算机科学系）
This paper presents a bias-variance tradeoff of graph Laplacian regularizer, which is widely used in graph signal processing and semi-supervised learning tasks. The scaling law of the optimal regularization parameter is specified in terms of the spectral graph properties and a novel signal-to-noise ratio parameter, which suggests selecting a mediocre regularization parameter is often suboptimal. The analysis is applied to three applications, including random, band-limited, and multiple-sampled graph signals. Experiments on synthetic and real-world graphs demonstrate near-optimal performance of…
Instacart（Instacart公司）；University of Illinois at Urbana Champaign（伊利诺伊大学厄巴纳-香槟分校）
Federated learning learns a neural network model by aggregating the knowledge from a group of distributed clients under the privacy-preserving constraint. In this work, we show that this paradigm might inherit the adversarial vulnerability of the centralized neural network, i.e., it has deteriorated performance on adversarial examples when the model is deployed. This is even more alarming when federated learning paradigm is designed to approximate the updating behavior of a centralized neural network. To solve this problem, we propose an adversarially robust federated learning framework, named…
Waterloo AI Institute, University of Waterloo（滑铁卢大学滑铁卢人工智能研究所）；ADC Automotive Distance Control Systems GmbH, Continental（大陆集团ADC汽车距离控制系统有限公司）
Prior studies have unveiled the vulnerability of the deep neural networks in the context of adversarial machine learning, leading to great recent attention into this area. One interesting question that has yet to be fully explored is the bias-variance relationship of adversarial machine learning, which can potentially provide deeper insights into this behaviour. The notion of bias and variance is one of the main approaches to analyze and evaluate the generalization and reliability of a machine learning model. Although it has been extensively used in other machine learning models, it is not wel…
Randomization Can Reduce Both Bias and Variance: A Case Study in Random Forests
Operations Research Center Massachusetts Institute of Technology（麻省理工学院运筹学研究中心）；Sloan School of Management Massachusetts Institute of Technology（麻省理工学院斯隆管理学院）
A Bias-Variance-Covariance Decomposition of Kernel Scores for Generative Models
German Cancer Research Center (DKFZ)（德国癌症研究中心）；German Cancer Consortium (DKTK)（德国癌症联盟）；Goethe University Frankfurt（法兰克福大学）；Frankfurt Cancer Institute (FCI)（法兰克福癌症研究所）
In this paper we propose using the principle of boosting to reduce the bias of a random forest prediction in the regression setting. From the original random forest fit we extract the residuals and then fit another random forest to these residuals. We call the sum of these two random forests a \textit{one-step boosted forest}. We show with simulated and real data that the one-step boosted forest has a reduced bias compared to the original random forest. The paper also provides a variance estimate of the one-step boosted forest by an extension of the infinitesimal Jackknife estimator. Using thi…
Societal bias towards certain communities is a big problem that affects a lot of machine learning systems. This work aims at addressing the racial bias present in many modern gender recognition systems. We learn race invariant representations of human faces with an adversarially trained autoencoder model. We show that such representations help us achieve less biased performance in gender classification. We use variance in classification accuracy across different races as a surrogate for the racial bias of the model and achieve a drop of over 40% in variance with race invariant representations.
It is a common phenomenon that for high-dimensional and nonparametric statistical models, rate-optimal estimators balance squared bias and variance. Although this balancing is widely observed, little is known whether methods exist that could avoid the trade-off between bias and variance. We propose a general strategy to obtain lower bounds on the variance of any estimator with bias smaller than a prespecified bound. This shows to which extent the bias-variance trade-off is unavoidable and allows to quantify the loss of performance for methods that do not obey it. The approach is based on a num…
In classical statistics, the bias-variance trade-off describes how varying a model's complexity (e.g., number of fit parameters) affects its ability to make accurate predictions. According to this trade-off, optimal performance is achieved when a model is expressive enough to capture trends in the data, yet not so complex that it overfits idiosyncratic features of the training data. Recently, it has become clear that this classic understanding of the bias-variance must be fundamentally revisited in light of the incredible predictive performance of "overparameterized models" -- models that avoi…
Deep neural networks can achieve remarkable generalization performances while interpolating the training data perfectly. Rather than the U-curve emblematic of the bias-variance trade-off, their test error often follows a "double descent" - a mark of the beneficial role of overparametrization. In this work, we develop a quantitative theory for this phenomenon in the so-called lazy learning regime of neural networks, by considering the problem of learning a high-dimensional function with random features regression. We obtain a precise asymptotic expression for the bias-variance decomposition of …
With the purpose of examining biased updates in variance-reduced stochastic gradient methods, we introduce SVAG, a SAG/SAGA-like method with adjustable bias. SVAG is analyzed in a cocoercive root-finding setting, a setting which yields the same results as in the usual smooth convex optimization setting for the ordinary proximal-gradient method. We show that the same is not true for SVAG when biased updates are used. The step-size requirements for when the operators are gradients are significantly less restrictive compared to when they are not. This highlights the need to not rely solely on coc…
There exist a number of reinforcement learning algorithms which learnby climbing the gradient of expected reward. Their long-runconvergence has been proved, even in partially observableenvironments with non-deterministic actions, and without the need fora system model. However, the variance of the gradient estimator hasbeen found to be a significant practical problem. Recent approacheshave discounted future rewards, introducing a bias-variance trade-offinto the gradient estimate. We incorporate a reward baseline into thelearning system, and show that it affects variance without introducingfurt…
Adaptive importance sampling is a widely spread Monte Carlo technique that uses a re-weighting strategy to iteratively estimate the so-called target distribution. A major drawback of adaptive importance sampling is the large variance of the weights which is known to badly impact the accuracy of the estimates. This paper investigates a regularization strategy whose basic principle is to raise the importance weights at a certain power. This regularization parameter, that might evolve between zero and one during the algorithm, is shown (i) to balance between the bias and the variance and (ii) to …
Peking University（北京大学）；University of Pennsylvania（宾夕法尼亚大学）
Modern machine learning methods are often overparametrized, allowing adaptation to the data at a fine level. This can seem puzzling; in the worst case, such models do not need to generalize. This puzzle inspired a great amount of work, arguing when overparametrization reduces test error, in a phenomenon called "double descent". Recent work aimed to understand in greater depth why overparametrization is helpful for generalization. This leads to discovering the unimodality of variance as a function of the level of parametrization, and to decomposing the variance into that arising from label nois…
The Doubly Robust (DR) estimation of ATE can be carried out in 2 steps, where in the first step, the treatment and outcome are modeled, and in the second step the predictions are inserted into the DR estimator. The model misspecification in the first step has led researchers to utilize Machine Learning algorithms instead of parametric algorithms. However, existence of strong confounders and/or Instrumental Variables (IVs) can lead the complex ML algorithms to provide perfect predictions for the treatment model which can violate the positivity assumption and elevate the variance of DR estimator…
