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Register a new userA Mathematical Foundation for Robust Machine Learning based on Bias-Variance Trade-off
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Weiyao Zhu
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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A common assumption in machine learning is that samples are independently and identically distributed (i.i.d). However, the contributions of different samples are not identical in training. Some samples are difficult to learn and some samples are noisy. The unequal contributions of samples has a considerable effect on training performances. Studies focusing on unequal sample contributions (e.g., easy, hard, noisy) in learning usually refer to these contributions as robust machine learning (RML). Weighing and regularization are two common techniques in RML. Numerous learning algorithms have been proposed but the strategies for dealing with easy/hard/noisy samples differ or even contradict with different learning algorithms. For example, some strategies take the hard samples first, whereas some strategies take easy first. Conducting a clear comparison for existing RML algorithms in dealing with different samples is difficult due to lack of a unified theoretical framework for RML. This study attempts to construct a mathematical foundation for RML based on the bias-variance trade-off theory. A series of definitions and properties are presented and proved. Several classical learning algorithms are also explained and compared. Improvements of existing methods are obtained based on the comparison. A unified method that combines two classical learning strategies is proposed.
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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 networks are trained to exactly fit (i.e., interpolate) the data. Classically, such models would be considered over-fit, and yet they often obtain high accuracy on test data. This apparent contradiction has raised questions about the mathematical foundations of machine learning and their relevance to practitioners. In this paper, we reconcile the classical understanding and the modern practice within a unified performance curve. This double descent curve subsumes the textbook U-shaped bias-variance trade-off curve by showing how increasing model capacity beyond the point of interpolation results in improved performance. We provide evidence for the existence and ubiquity of double descent for a wide spectrum of models and datasets, and we posit a mechanism for its emergence. This connection between the performance and the structure of machine learning models delineates the limits of classical analyses, and has implications for both the theory and practice of machine learning.
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 architecture, loss function, and choice of dataset and confirm that variance unimodality occurs robustly for all models we considered. The risk curve is the sum of the bias and variance curves and displays different qualitative shapes depending on the relative scale of bias and variance, with the double descent curve observed in recent literature as a special case. We corroborate these empirical results with a theoretical analysis of two-layer linear networks with random first layer. Finally, evaluation on out-of-distribution data shows that most of the drop in accuracy comes from increased bias while variance increases by a relatively small amount. Moreover, we find that deeper models decrease bias and increase variance for both in-distribution and out-of-distribution data.
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 well explored in the field of deep learning and it is even less explored in the area of adversarial machine learning. In this study, we investigate the effect of adversarial machine learning on the bias and variance of a trained deep neural network and analyze how adversarial perturbations can affect the generalization of a network. We derive the bias-variance trade-off for both classification and regression applications based on two main loss functions: (i) mean squared error (MSE), and (ii) cross-entropy. Furthermore, we perform quantitative analysis with both simulated and real data to empirically evaluate consistency with the derived bias-variance tradeoffs. Our analysis sheds light on why the deep neural networks have poor performance under adversarial perturbation from a bias-variance point of view and how this type of perturbation would change the performance of a network. Moreover, given these new theoretical findings, we introduce a new adversarial machine learning algorithm with lower computational complexity than well-known adversarial machine learning strategies (e.g., PGD) while providing a high success rate in fooling deep neural networks in lower perturbation magnitudes.
Intuitively, a scientist might assume that a more complex regression model will necessarily yield a better predictive model of experimental data. Herein, we disprove this notion in the context of extracting the proton charge radius from charge form factor data. Using a Monte Carlo study, we show that a simpler regression model can in certain cases be the better predictive model. This is especially true with noisy data where the complex model will fit the noise instead of the physical signal. Thus, in order to select the appropriate regression model to employ, a clear technique should be used such as the Akaike information criterion or Bayesian information criterion, and ideally selected previous to seeing the results. Also, to ensure a reasonable fit, the scientist should also make regression quality plots, such as residual plots, and not just rely on a single criterion such as reduced chi2. When we apply these techniques to low four-momentum transfer cross section data, we find a proton radius that is consistent with the muonic Lamb shift results. While presented for the case of proton radius extraction, these concepts are applicable in general and can be used to illustrate the necessity of balancing bias and variance when building a regression model and validating results, ideas that are at the heart of modern machine learning algorithms.
Modern Monte Carlo-type approaches to dynamic decision problems are reformulated as empirical loss minimization, allowing direct applications of classical results from statistical machine learning. These computational methods are then analyzed in this framework to demonstrate their effectiveness as well as their susceptibility to generalization error. Standard uses of classical results prove potential overlearning, thus bias-variance trade-off, by connecting over-trained networks to anticipating controls. On the other hand, non-asymptotic estimates based on Rademacher complexity show the convergence of these algorithms for sufficiently large training sets. A numerically studied stylized example illustrates these possibilities, including the importance of problem dimension in the degree of overlearning, and the effectiveness of this approach.
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