Let Y be a f 0; 1 g-v alued random v ariable (lab el) to b e predicted based on an observ ation of another random v ariable . As we mentioned earlier, the risk () is unknown because the true distribution is unknown. Those regularization terms, which correspond to the prior terms in a Bayesian approach, are thus from the point of view of empirical risk minimization a technical tool to make the minimization problem well defined. The main paradigm of predictive learning is Empirical Risk Minimization (ERM in abbreviated form), see e.g. 45, No. In our work, we consider applying TERM to the following problems: For all applications considered, we find that TERM is competitive with or outperforms state-of-the-art, problem-specific tailored baselines. Published. Our hope is that the TERM framework will allow machine learning practitioners to easily modify the ERM objective to handle practical concerns such as enforcing fairness amongst subgroups, mitigating the effect of outliers, and ensuring robust performance on new, unseen data. Empirical risk minimization is a popular technique for statistical estimation where the model, \(\theta \in R^d\), is estimated by minimizing the average empirical loss over data, \(\{x_1, \dots, x_N\}\): $$\overline{R} (\theta) := \frac{1}{N} \sum_{i \in [N]} f(x_i; \theta).$$. Found inside – Page 97̆/-error bounds in regression problems with quadratic loss, given in Examples 1–5 of Sect. 5.1, are well known. ... Empirical risk minimization with convex loss was studied in detail by Blanchard et al. [26] and Bartlett et al. [16]. Found inside – Page 233We now move on to the second ingredient of empirical risk minimization: how to measure how well the predictor fits the training data. 8.2.2 Loss Function for Training Consider the label yn for a particular example, and the corresponding ... Distributed Algorithms in Large-scaled Empirical Risk Minimization: Non-convexity, Adaptive-sampling, and Matrix-free Second-order Methods. TERM recovers ERM when \(t \to 0\). In this scheme, a sample of training instances is drawn from the underlying data distribution, and the PCA is commonly used for dimension reduction while preserving useful information of the original data for downstream tasks. . Local Complexities for Empirical Risk Minimization 271. worst case scenario, while the Rademacher averages are measure dependent and lead to sharper bounds). Critical to the success of such a framework is understanding the implications of the modified objective (i.e., the impact of varying \(t\)), both theoretically and empirically. The performance of empirical risk minimization is measured by the risk of the selected function, m ERM = E[f ERM(X)jX 1;:::;X n] : In particular, the main object of interest for this paper is the excess risk m ERM m. The performance of empirical risk minimization has been thoroughly studied and well understood using tools of empirical process . TERM approximates a popular family of quantile losses (such as median loss, shown in the orange line) with different tilting parameters. 10/25/2017 ∙ by Hongyi Zhang, et al. 1 Empirical Risk Minimization Given a loss function !(.,. US Patent 10,713,566. , 2020. Empirical risk minimization for a classification problem with a 0-1 loss function is known to be an NP-hard problem even for such a relatively simple class of functions as linear classifiers. While the tilted objective used in TERM is not new and is commonly used in other domains,1For instance, this type of exponential smoothing (when \(t>0\)) is commonly used to approximate the max. Please see our paper for full statements and proofs. Risk estimates for a misclassification as a function of the training sample size and other model parameters . sample complexity as findingnas small as possible for achieving Err . In particular, we find: Next, we discuss these properties and interpretations in more detail. Empirical Risk Minimization in Julia. unique solution), $\left.\mathbf{w}=(\mathbf{X}\mathbf{X}^\top)^{-1}\mathbf{X}\mathbf{y}^{\top}\right.$, $\left.\mathbf{X}=[\mathbf{x}_{1}, ..., \mathbf{x}_{n}]\right.$, $\left.\mathbf{y}=[y_{1},...,y_{n}]\right.$, $\left.\mathbf{w}=(\mathbf{X}\mathbf{X}^{\top}+\lambda\mathbb{I})^{-1}\mathbf{X}\mathbf{y}^{\top}\right.$, + sparsity inducing (good for feature selection), - Not strictly convex (no unique solution), $\left.\Pr{(y|x)}=\frac{1}{1+e^{-y(\mathbf{w}^{\top}x+b)}}\right.$. Moreover, estimates which are based on comparing the empirical and the actual structures (for example empirical vs. actual means) uniformly over the By minimizing the empirical risk, we hope to obtain a model with a low value of the risk. In particular, we present a bound on the excess risk incurred by the method. There is an interesting connection between Ordinary Least Squares and the first principal component of PCA (Principal Component Analysis). Found insideIn particular, the material in this text directly supports the mathematical analysis and design of old, new, and not-yet-invented nonlinear high-dimensional machine learning algorithms. In our work, we discuss the properties of the objective in terms of its smoothness and convexity behavior. We apply TERM to this problem, reweighting the gradients based on the loss on each group. Found inside – Page 8Finally, in the problem of pattern recognition, algorithms based on the principle of empirical risk minimization are ... It follows from the given example that necessary and sufficient conditions of consistency must be based not only on ... We consider large scale empirical risk minimization (ERM) problems, where both the problem dimension and variable size is large. Kernelized version can be solved very efficiently with specialized algorithms (e.g. 7 1.2 Excess Risk: Distribution Dependent Bounds We do so by extending the seminal works . Found inside – Page 55Empirical risk minimization aims for minimization of the generalization error. Therefore, one defines the training error for a hypothesis hΘ as the fraction of misclassified training examples. The generalization error is then defined as ... Stepsize strategy 1 Introduction Empirical risk minimization (ERM) is one of the most powerful tools in applied statistics, and is regarded as the canonical approach to regression analysis. Figure 3 demonstrates that our approach performs on par with the oracle method that knows the qualities of annotators in advance. but for the purposes of this class, it is assumed to be fixed. We describe these algorithms as well as their convergence guarantees in our paper. You have to deal with overfitting issues. [instagram-feed num=6 cols=6 imagepadding=0 disablemobile=true showbutton=false showheader=false followtext=”Follow @Mint_Theme”], Legal Info | www.cmu.edu We indicate the cate-gories by labels Y = 1 and Y = −1. Figure 1. Found inside – Page 122This is known as the empirical risk, Re(f): Re.f/D1NNX L.y i ;f.xi//: (4.3) iD1 Empirical Risk Minimization A ... For example, a function may map each training data point exactly to its response but carry no information about all other ... Found inside – Page 265This theorem links the size of the example set for which there is an ε-cover of functions to the ε-error generator set of examples. Definition 8. (Empirical risk minimization is PAC) The ... Though, it can be solved efficiently when the minimal empirical risk is zero, i.e. Variants of tilting have also appeared in other contexts, including importance sampling, decision making, and large deviation theory. Found inside – Page 13For example, naive Bayes and linear discriminant analysis are joint probability models, whereas logistic regression is a conditional probability model. There are two basic approaches to choosing for g empirical risk minimization and ... The loss function L. 3. In practice, machine learning algorithms cope with that either . Found inside – Page 25Straight minimization of the empirical risk in F can be problematic. ... The bounds involve the number of examples l and the capacity h of the function space, a quantity measuring the “complexity” of the space. Lecture 2: Risk Minimization In this lecture we introduce the concepts of empirical risk minimization, overfitting, model complexity and regularization. MotivationGradient MethodStochastic SubgradientFinite-Sum MethodsNon-Smooth Objectives Modern Convex Optimization Methods for Large-Scale Empirical Risk Minimization 7 Stability Properties of Empirical Risk Minimization over Donsker Classes . erms: Structural Risk Minimization, Iterativ e Structural Risk Minimization, Rademac her P enalt y, Oracle Inequalities, Empirical Pro cess, Classi cation 1. A toy linear regression example illustrating Tilted Empirical Risk Minimization (TERM) as a function of the tilt hyperparameter t t. Classical ERM ( t =0 t = 0) minimizes the average loss and is shown in pink. Found insideIn empirical risk minimization this empirical generalization error is used, for example, to determine adaptive (hyper-)parameters of regularization terms. A typical example is a factor multiplying the regularization terms controlling ... TERM with varying \(t\)’s reweights samples to magnify/suppress outliers (as in Figure 1). As illustrated in Figure 1, for positive values of \(t\), TERM will thus magnify outliers (samples with large losses), and for negative t’s, it will suppress outliers by downweighting them. Empirical Risk Minimization for Causally Identifiable Functionals. Only differentiable everywhere with $\left.p=2\right.$. As \(t\) goes from 0 to \(+\infty\), the average loss will increase, and the max-loss will decrease (going from the pink star to the red star in Figure 2), smoothly trading average-loss for max-loss. Classical ERM (\(t=0\)) minimizes the average loss and is shown in pink. Our main motivation is the . We propose a new algorithm for minimizing regularized empirical loss: Stochastic Dual Newton Ascent (SDNA). To promote fairness, previous methods have proposed to solve a min-max problem via semidefinite programming, which scales poorly with the problem dimension. In this work, we analyze both these frameworks from . It is also natural to perform tilting at the group level to upweight underrepresented groups. empirical risk minimization (ERM) problems, and develops a randomized algorithm that can provide di erential privacy [8, 17] while keeping the learning procedure accurate. For instance, this type of exponential smoothing (when \(t>0\)) is commonly used to approximate the max. . . ADVANTAGE: Similar to Huber Loss, but twice differentiable everywhere, (Differentiable) Squared Hingeless SVM ($\left.p=2\right.$). However, we empirically observe competitive performance when applying TERM to broader classes of objectives, including deep neural networks. Parametrized predictors . Here, and throughout, (A) denotes the indicator function of a set A. TERM is a general framework applicable to a variety of real-world machine learning problems. Found inside – Page 7This principle is called empirical risk minimization (ERM), and it is the source of many algorithms in machine learning. ... is according to the ERM principle, is it possible to find a prediction function from a finite set of examples, ... CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): In this paper, we study a two-category classification problem. Vapnik and Chervonenkis (1971, 1991) showed necessary and sufficient conditions for its consistency. We see that TERM with a large \(t\) can recover the min-max results where the resulting losses on two groups are almost identical. 2 Local Risk-Minimization and Stochastic Volatility Ridge Regression is just 1 line of Julia / Python. Found inside – Page 273... empirical risk minimization with K thresholds and a penalty of the form ~ VK / n ( again apart from logarithmic factors ) . 5. Conclusion We studied the problem of estimating a classification rule from a set of training examples . Found inside – Page 513.3.1 EMPIRICAL RISK MINIMIZATION Ideally, the parameters in a learning algorithm should be inferred based on the ... For example, K-lines clustering can be posed as an ERM problem over the distortion function class LK ( lD ) ; (3.41) D ... Section 3 studies model risk. Suppose our goal is to learn a predictive model in terms of parameters $\theta_t$ for the target domain, based on the learning framework of empirical risk minimization (Vapnik, 1998), the optimal solution of $\theta_t$ can be learned by solving the following optimization problem. Found inside – Page 2Empirical risk minimization (ERM) is a widely adopted principle in discriminative supervised learning, for example, neural networks [3] and logistic regression [2]. As opposed to probabilistic methods, the decision boundary is modeled ... We exemplify the difficulties of this marriage for both spouses (WERM and ID) through a simple example. However, the quality of annotators varies significantly as annotators may be unskilled or even malicious. X He. Below we explore properties of TERM with varying \(t\) to better understand the potential benefits of \(t\)-tilted losses. One of the most popular loss functions in Machine Learning, since its outputs are well-calibrated probabilities. . Introduction The empirical risk minimization (ERM) algorithm has been studied in learning theory to a great extent. These algorithms are private under the ε-differential privacy definition due to Dwork et al. Example: Diabetes 20 30 40 50 60 70 80 AGE 50 100 150 200 250 300 350 1.0 1.2 1.4 1.6 1.8 2.0 SEX 50 100 150 200 250 300 350 20 25 30 35 40 BMI 50 100 150 200 250 300 350 Introduction Empirical risk minimization (ERM) algorithm has been studied in learning theory to a great extent. Applying standard PCA can be unfair to underrepresented groups. 1 Introduction Additionally, we find that the accuracy of TERM alone is 5% higher than that reported by previous approaches which are specifically designed for this problem. Keywords: empirical risk minimization, empirical processes, stability, Donsker classes 1. Found inside – Page 152Empirical risk minimization is one of the most commonly used techniques where the goal is to find a parameter setting ... For example, Fig.2 shows a two-class problem and the corresponding decision regions in the form of hyperplanes. 43, No. . Tomaso Poggio The Learning Problem and Regularization DISADVANTAGE: Uses weights on all features, i.e. Tran—This work was done when he was a master's student in the Department of Computer Science, Graduated School of SIE, University of Tsukuba. This process involves a trade-off: Our method is dual in nature: in each iteration we update a random subset of the dual variables. However, it is unclear when IRM should be preferred over the widely-employed empirical risk minimization (ERM) framework. Found inside – Page 69However, it has been argued that ERM is an incomplete inductive principle [295] since it is does not guarantee high generalization ability: an out-of-sample example generated by the same probability distribution does not necessarily lie ... . (2006). 2020. In this work, we propose mixup, a simple learning principle to . empirical risk minimization (ERM). Found inside – Page 83... as far as possible, a process referred to as empirical risk minimization (ERM), where the error is termed risk (Vapnik, 2000). ... For example, the perceptron learning procedure (Rosenblatt, 1958) can be summarized as follows. In this paper, we study differentially private Empirical Risk Minimization in the non-interactive local model. Are Sixteen Heads Really Better than One? Our work rigorously explores these effects—demonstrating the potential benefits of tilted objectives across a wide range of applications in machine learning. Then we propose a new metho d, objective perturbation, for privacy-preserving machine learning algorithm design. [Note] All discussions above assume that the loss functions belong to generalized linear models. The importance-weighted empirical risk is an unbiased estimator for the risk with respect to the target distribution, for any N. But it's difficult to say anything about the minimizer of the importance-weighted empirical risk for finite sample sizes. For example, we can perform negative tilting at the sample level within each group to mitigate outlier samples, and perform positive tilting across all groups to promote fairness. In addition, with moderate values of \(t\), TERM offers more flexible tradeoffs between performance and fairness by reducing the performance gap less aggressively. Also presented is the algorithms which efficiently compute the gradient and solve the model. [Solving TERM] Wondering how to solve TERM? %PDF-1.3 mixup: Beyond Empirical Risk Minimization. Therefore, if empirical risk minimization over noisy samples has to work, we necessarily have to change the loss used to calculate the empirical risk. Here we consider applying TERM (\(t<0\)) to the application of mitigating noisy annotators. • the probability of the empirical risk on a sample of n points differing from the risk by more that εcan be bounded by • twice the probability that it differs from the empirical risk on a second sample of size 2n by more than ε/2 Theorem: for nε2 > 2 where • the 1st P refers to sample of size n and the 2nd to that of size 2n. TERM considers a modification to ERM that can be used for diverse applications such as enforcing fairness between subgroups, mitigating the effect of outliers, and addressing class imbalance—all in one unified framework. empirical risk minimization A learning rule that minimizes the empirical . An alternative and widely-used learning paradigm is that of Empirical Risk Minimization (ERM), which is often regarded as the strategy of choice due to its generality and its statistical efficiency. As \(t \to -\infty \) (blue), TERM finds a line of best fit while ignoring outliers. Empirical Risk Minimization: Abstract risk bounds and Rademacher averages Maxim Raginsky September 15, 2015 In the last lecture, we have left off with a theorem that gave a sufficient condition for the Empirical Risk Minimization (ERM) algorithm fb n ˘argmin f 2F PZn ('f) (1) ˘argmin X Z Y (a) Back-door X W R Y (b) Example 1 Figure 1: Causal graphs corresponding to BD and Example 1. Quantile losses have nice properties but can be hard to directly optimize. 1 Introduction Empirical risk minimization (ERM) on a class of functions H, called the hypoth- esis space, is the classical approach to the problem of learning from examples. In Table 1 below, we find that TERM is superior to all baselines which perform well in their respective problem settings (only showing a subset here) when considering noisy samples and class imbalance simultaneously. The book is intended for graduate students and researchers in machine learning, statistics, and related areas; it can be used either as a textbook or as a reference text for a research seminar. Found inside – Page 49For example, we saw in the introduction that SVMs use the convex hinge loss instead of the discontinuous ... However, this approach may often be flawed, as the following examples illustrate: • ERM optimization problems based on the ... 1, 7-57 DOI: 10.1214/07-AIHP146 © Association des . Found inside – Page 61Each algorithm corresponds to a specific filter function and in general there is no natural interpretation in terms of penalized empirical risk minimization. ... For example, in the case of Tikhonov regularization gλ(σ) = 1σ+nλ. In this post, we discuss three examples on robust classification with \(t<0\), fair PCA with \(t>0\), and hierarchical tilting. By doing so, mixup regularizes the neural network to favor simple linear behavior in-between training examples. Finally, we note that in practice, multiple issues can co-exist in the data, e.g., we may have issues with both class imbalance and label noise. regularization to make empirical risk minimization generalize well to new data. Crowdsourcing is a popular technique for obtaining data labels from a large crowd of annotators. Found inside – Page 85Let Z = X ×Y be the domain of examples. ... Learning algorithms following the empirical risk minimization principle choose the model that minimizes the empirical risk: J(w;D) = n 1 n∑ l(w,(xi,yi))+ n1Ω(w), (1) i=1 where Ω(w) is a ... Efficient calculations of negative curvature in a hessian free deep learning framework. We give conditions under which sample variance penalization is effective. Found inside – Page 678For example, a necessary and sufficient condition for consistency of ERM is that log N(F,n)/n → 0 (cf. ... That is, instead of fixing ε and then computing the probability that the empirical risk deviates from the true risk by more than ... Found inside – Page 64Ideally , f shall correctly describe both the training samples and the unseen examples from the population . Traditionally , classical ML methods are based on the Empirical Risk Minimization ( ERM ) principle ! “ ) . 1 Introduction Despite its popularity, ERM is known to perform poorly in situations where average performance is not an appropriate surrogate for the problem of interest. PCA also minimizes square loss, but looks at perpendicular loss (the horizontal distance between each point and the regression line) instead. Introduces machine learning and its algorithmic paradigms, explaining the principles behind automated learning approaches and the considerations underlying their usage. Commonly Used Binary Classification Loss Functions Different Machine Learning algorithms employ their own loss functions; Table 4.1 shows just a few: Adding for example a regularizing term as in gives (241) (242) (243) with . CiteSeerX - Scientific documents that cite the following paper: Statistical learning: Stability is sufficient for generalization and necessary and sufficient for consistency of empirical risk minimization . First, we take a closer look at the gradient of the t-tilted loss, and observe that the gradients of the t-tilted objective \(\widetilde{R}(t; \theta)\) are of the form: $$\nabla_{\theta} \widetilde{R}(t; \theta) = \sum_{i \in [N]} w_i(t; \theta) \nabla_{\theta} f(x_i; \theta), \text{where } w_i \propto e^{tf(x_i; \theta)}.$$. $\left.y\in\mathbb{R}\right.$, Figure 4.2: Plots of Common Regression Loss Functions - x-axis: $\left.h(\mathbf{x}_{i})y_{i}\right.$, or As (t to -infty ) (blue), TERM finds a line of best fit while ignoring outliers. In this work, we propose mixup, a simple learning principle to alleviate these issues. We will talk about these things much more later on, when discussing specific classifiers. Non-continuous and thus impractical to optimize. data is linearly separable.. The solution is searched for in the form of a finite linear combination of kernel support functions (Vapnik's support vector machines). . Recently, invariant risk minimization (IRM) was proposed as a promising solution to address out-of-distribution (OOD) generalization. Home Collections Hosted Content The Journal of Machine Learning Research Vol. Understanding ERM is essential to understanding the limits of machine learning algorithms and to form a good basis for practical problem-solving skills. 19 0 obj . This book honours the outstanding contributions of Vladimir Vapnik, a rare example of a scientist for whom the following statements hold true simultaneously: his work led to the inception of a new field of research, the theory of ... A key property required for learnability is consistency, which is equivalent to generalization that is convergence (as the number of training examples goes to infinity) of the empirical risk to the expected risk for the . SGD with Variance Reduction beyond Empirical Risk Minimization. Here the empirical distribution function = is given by = (or, including the variable, = for which could be extended to a linear = for = ). Given a training set S and a function space H, empirical risk minimization (Vapnik introduced the term) is the class of algorithms that look at S and select f S as f S = argmin f2H I S[f]. (1996). In this work, we establish risk bounds for the Empirical Risk Minimization (ERM) with both dependent and heavy-tailed data-generating processes. As an alternative method to maximum likelihood, we can calculate an Empirical Risk function by averaging the loss on the training set: 1 Introduction Prediction problems are of major importance in statistical learning. In machine learning, models are commonly estimated via empirical risk minimization (ERM), a principle that considers minimizing the average empirical loss on observed data. Specifically, we explore a common benchmark—taking the CIFAR10 dataset and simulating 100 annotators where 20 of them are always correct and 80 of them assign labels uniformly at random. First we apply the outpu t perturbation ideas of Dwork et al. . Structural risk minimization includes a penalty function that controls the bias/variance tradeoff. Title: mixup: Beyond Empirical Risk Minimization. Depending on the application, one can choose whether to apply tilting at each level (e.g., possibly more than two levels of hierarchies exist in the data), and at either direction (\(t>0\) or \(t<0\)). Found inside – Page 488RS ( f) = 1 n n∑ l( f(x i ),y i). i=1 Empirical risk minimization seeks to find a predictor f ∗ in a specified ... However, this may not be the case, since the risk R(f) captures loss on unseen examples, while the empirical risk RS ... ∙ UPMC ∙ 0 ∙ share. The optimal element S* is then selected to minimize the guaranteed risk, defined as the sum of the empirical risk and the confidence interval. <> 4, 1617-1646 DOI: 10.1214/15-AOS1318 © Institute of Mathematical Statistics, 2015 BANDWIDTH SELECTION IN KERNEL . In some applications, these 'outliers' may […] The book is suitable for upper-level undergraduates with an introductory-level college math background and beginning graduate students. For example, optimizing for the median loss instead of mean may be desirable for applications in robustness, and the max-loss is an extreme of the quantile loss which can be used to enforce fairness. For instance, one can tilt at the sample level, as in the linear regression toy example. The TERM objective offers an upper bound on the given quantile of the losses, and the solutions of TERM can provide close approximations to the solutions of the quantile loss optimization problem. In this paper, we propose a simple reduction scheme for empirical risk minimization (ERM) that preserves empirical Rademacher complexity. In some applications, these ‘outliers’ may correspond to minority samples that should not be ignored. Takes on behavior of Squared-Loss when loss is small, and Absolute Loss when loss is large. This treatise by an acknowledged expert includes several topics not found in any previous book. Moreover, estimates which are based on comparing the empirical and the actual structures (for example empirical vs. actual means) uniformly over the class are loose, because this condition is stronger than necessary. In this work, we propose mixup, a simple learning principle to alleviate these issues. Thus, handling a large amount of noise is essential for the crowdsourcing setting. . Ridge Regression is very fast if data isn't too high dimensional. However, minimizing such objectives can be challenging especially in large-scale settings, as they are non-smooth (and generally non-convex). In essence, mixup trains a neural network on convex combinations of pairs of examples and their labels. In the unrealistic case where the likelihood ratio $\Phi(z)=dP/dP'(z)$ is known, one may straightforwardly extends the Empirical Risk Minimization (ERM) approach to this specific \textit{transfer learning} setup using the same idea as that behind Importance Sampling, by minimizing a weighted version of the empirical risk functional computed . Figure 1. Empirical Risk Minimization. Found inside – Page 269The training process based on minimizing this average training error is known as empirical risk minimization. ... For example, exactly minimizing expected 0-1 loss is typically intractable (exponential in the input dimension), ... Before showing our contributions and discussing comparisons with previous works, we first . In this paper, we propose a novel adaptive sample size second-order method, which reduces the . Variants of tilting have also appeared in other contexts, including importance sampling, decision making, and large deviation theory. Another perspective on TERM is that it offers a continuum of solutions between the min and max losses. It also recovers other popular alternatives such as the max-loss (\(t \to +\infty\)) and min-loss (\(t \to -\infty\)). While the previous application explored TERM with \(t<0\), here we consider an application of TERM with positive \(t\)’s to fair PCA. 10/16/2015 ∙ by Massil Achab, et al. As t→ −∞ t → − ∞ (blue), TERM finds a line of best fit while ignoring outliers. Given the modifications that TERM makes to ERM, the first question we ask is: What happens to the TERM objective when we vary \(t\)? Based on that, we develop both batch and (scalable) stochastic solvers for TERM, where the computation cost is within 2\(\times\) of standard ERM solvers. excess risk The di erence between the risk of a given function and the mini-mum possible risk over a function class. www.imstat.org/aihp Annales de l'Institut Henri Poincaré - Probabilités et Statistiques 2009, Vol. Our work explores tilted empirical risk minimization (TERM), a simple and general alternative to ERM, which is ubiquitous throughout machine learning. The Annals of Statistics 2015, Vol. that, if all the M classifiers are binary, the (penalized) Empirical Risk Minimization procedures are suboptimal (even under the margin/low noise condition) when the loss function is somewhat more than convex, whereas, in that case, aggregation procedures with exponential weights achieve the optimal rate of aggregation.
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