This paper studies empirical risk minimization (ERM) problems for large-scale datasets and incorporates the idea of adaptive sample size methods to improve the guaranteed convergence bounds for first-order stochastic and deterministic methods. In contrast to traditional methods that attempt to solve the ERM problem corresponding to the full dataset directly, adaptive sample size schemes start with a small number of samples and solve the corresponding ERM problem to its statistical accuracy. The sample size is then grown geometrically -- e.g., scaling by a factor of two -- and use the solution of the previous ERM as a warm start for the new ERM. Theoretical analyses show that the use of adaptive sample size methods reduces the overall computational cost of achieving the statistical accuracy of the whole dataset for a broad range of deterministic and stochastic first-order methods. The gains are specific to the choice of method. When particularized to, e.g., accelerated gradient descent and stochastic variance reduce gradient, the computational cost advantage is a logarithm of the number of training samples. Numerical experiments on various datasets confirm theoretical claims and showcase the gains of using the proposed adaptive sample size scheme.
The theoretical and empirical performance of Empirical Risk Minimization (ERM) often suffers when loss functions are poorly behaved with large Lipschitz moduli and spurious sharp minimizers. We propose and analyze a counterpart to ERM called Diametrical Risk Minimization (DRM), which accounts for worst-case empirical risks within neighborhoods in parameter space. DRM has generalization bounds that are independent of Lipschitz moduli for convex as well as nonconvex problems and it can be implemented using a practical algorithm based on stochastic gradient descent. Numerical results illustrate t…
The Chinese University of Hong Kong（香港中文大学）；Tencent AI Lab（腾讯人工智能实验室）；University of California, Davis（加州大学戴维斯分校）；University of Rochester（罗切斯特大学）
Regularized empirical risk minimization problem with linear predictor appears frequently in machine learning. In this paper, we propose a new stochastic primal-dual method to solve this class of problems. Different from existing methods, our proposed methods only require O(1) operations in each iteration. We also develop a variance-reduction variant of the algorithm that converges linearly. Numerical experiments suggest that our methods are faster than existing ones such as proximal SGD, SVRG and SAGA on high-dimensional problems.
Tokyo Institute of Technology（东京工业大学）；Japan Science and Technology Agency（日本科学技术振兴机构）
We consider a composite convex minimization problem associated with regularized empirical risk minimization, which often arises in machine learning. We propose two new stochastic gradient methods that are based on stochastic dual averaging method with variance reduction. Our methods generate a sparser solution than the existing methods because we do not need to take the average of the history of the solutions. This is favorable in terms of both interpretability and generalization. Moreover, our methods have theoretical support for both a strongly and a non-strongly convex regularizer and achie…
University of California, Berkeley（加州大学伯克利分校）；University of Washington, Seattle（西雅图华盛顿大学）
Min-max optimization is emerging as a key framework for analyzing problems of robustness to strategically and adversarially generated data. We propose a random reshuffling-based gradient free Optimistic Gradient Descent-Ascent algorithm for solving convex-concave min-max problems with finite sum structure. We prove that the algorithm enjoys the same convergence rate as that of zeroth-order algorithms for convex minimization problems. We further specialize the algorithm to solve distributionally robust, decision-dependent learning problems, where gradient information is not readily available. T…
We derive bounds on the sample complexity of empirical risk minimization (ERM) in the context of minimizing non-convex risks that admit the strict saddle property. Recent progress in non-convex optimization has yielded efficient algorithms for minimizing such functions. Our results imply that these efficient algorithms are statistically stable and also generalize well. In particular, we derive fast rates which resemble the bounds that are often attained in the strongly convex setting. We specify our bounds to Principal Component Analysis and Independent Component Analysis. Our results and tech…
In this paper, we study the generalization performance of global minima for implementing empirical risk minimization (ERM) on over-parameterized deep ReLU nets. Using a novel deepening scheme for deep ReLU nets, we rigorously prove that there exist perfect global minima achieving almost optimal generalization error bounds for numerous types of data under mild conditions. Since over-parameterization is crucial to guarantee that the global minima of ERM on deep ReLU nets can be realized by the widely used stochastic gradient descent (SGD) algorithm, our results indeed fill a gap between optimiza…
We introduce a new surrogate loss function called orbit loss in the structured prediction framework, which has good theoretical and practical advantages. While the orbit loss is not convex, it has a simple analytical gradient and a simple perceptron-like learning rule. We analyze the new loss theoretically and state a PAC-Bayesian generalization bound. We also prove that the new loss is consistent in the strong sense; namely, the risk achieved by the set of the trained parameters approaches the infimum risk achievable by any linear decoder over the given features. Methods that are aimed at ris…
University of Michigan, Ann Arbor（密歇根大学安娜堡分校）；Microsoft Research（微软研究院）；Princeton University（普林斯顿大学）
This paper proves, in very general settings, that convex risk minimization is a procedure to select a unique conditional probability model determined by the classification problem. Unlike most previous work, we give results that are general enough to include cases in which no minimum exists, as occurs typically, for instance, with standard boosting algorithms. Concretely, we first show that any sequence of predictors minimizing convex risk over the source distribution will converge to this unique model when the class of predictors is linear (but potentially of infinite dimension). Secondly, we…
University of Alberta（阿尔伯塔大学）；Imperial College London（帝国理工学院）
This paper extends the standard chaining technique to prove excess risk upper bounds for empirical risk minimization with random design settings even if the magnitude of the noise and the estimates is unbounded. The bound applies to many loss functions besides the squared loss, and scales only with the sub-Gaussian or subexponential parameters without further statistical assumptions such as the bounded kurtosis condition over the hypothesis class. A detailed analysis is provided for slope constrained and penalized linear least squares regression with a sub-Gaussian setting, which often proves …
Universitat Pompeu Fabra（庞培法布拉大学）；Barcelona Graduate School of Economics（巴塞罗那经济学院）
Empirical risk minimization is a standard principle for choosing algorithms in learning theory. In this paper we study the properties of empirical risk minimization for time series. The analysis is carried out in a general framework that covers different types of forecasting applications encountered in the literature. We are concerned with 1-step-ahead prediction of a univariate time series generated by a parameter-driven process. A class of recursive algorithms is available to forecast the time series. The algorithms are recursive in the sense that the forecast produced in a given period is a…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We consider the problem of designing uniformly stable first-order optimization algorithms for empirical risk minimization. Uniform stability is often used to obtain generalization error bounds for optimization algorithms, and we are interested in a general approach to achieve it. For Euclidean geometry, we suggest a black-box conversion which given a smooth optimization algorithm, produces a uniformly stable version of the algorithm while maintaining its convergence rate up to logarithmic factors. Using this reduction we obtain a (nearly) optimal algorithm for smooth optimization with converge…
NTT DATA Mathematical Systems Inc.（NTT DATA 数理系统公司）；The University of Tokyo（东京大学）
In this paper, we develop a new accelerated stochastic gradient method for efficiently solving the convex regularized empirical risk minimization problem in mini-batch settings. The use of mini-batches is becoming a golden standard in the machine learning community, because mini-batch settings stabilize the gradient estimate and can easily make good use of parallel computing. The core of our proposed method is the incorporation of our new "double acceleration" technique and variance reduction technique. We theoretically analyze our proposed method and show that our method much improves the min…
We propose a new stochastic optimization framework for empirical risk minimization problems such as those that arise in machine learning. The traditional approaches, such as (mini-batch) stochastic gradient descent (SGD), utilize an unbiased gradient estimator of the empirical average loss. In contrast, we develop a computationally efficient method to construct a gradient estimator that is purposely biased toward those observations with higher current losses. On the theory side, we show that the proposed method minimizes a new ordered modification of the empirical average loss, and is guarante…
We present a Bayesian view of counterfactual risk minimization (CRM) for offline learning from logged bandit feedback. Using PAC-Bayesian analysis, we derive a new generalization bound for the truncated inverse propensity score estimator. We apply the bound to a class of Bayesian policies, which motivates a novel, potentially data-dependent, regularization technique for CRM. Experimental results indicate that this technique outperforms standard $L_2$ regularization, and that it is competitive with variance regularization while being both simpler to implement and more computationally efficient.
We show that the average stability notion introduced by \cite{kearns1999algorithmic, bousquet2002stability} is invariant to data preconditioning, for a wide class of generalized linear models that includes most of the known exp-concave losses. In other words, when analyzing the stability rate of a given algorithm, we may assume the optimal preconditioning of the data. This implies that, at least from a statistical perspective, explicit regularization is not required in order to compensate for ill-conditioned data, which stands in contrast to a widely common approach that includes a regularizat…
George Washington University（乔治华盛顿大学）；University of Southern California（南加州大学）；University of North Carolina, Chapel Hill（北卡罗来纳大学教堂山分校）
This paper has two main goals: (a) establish several statistical properties---consistency, asymptotic distributions, and convergence rates---of stationary solutions and values of a class of coupled nonconvex and nonsmoothempirical risk minimization problems, and (b) validate these properties by a noisy amplitude-based phase retrieval problem, the latter being of much topical interest.Derived from available data via sampling, these empirical risk minimization problems are the computational workhorse of a population risk model which involves the minimization of an expected value of a random func…
Empirical risk minimization is the main tool for prediction problems, but its extension to relational data remains unsolved. We solve this problem using recent ideas from graph sampling theory to (i) define an empirical risk for relational data and (ii) obtain stochastic gradients for this empirical risk that are automatically unbiased. This is achieved by considering the method by which data is sampled from a graph as an explicit component of model design. By integrating fast implementations of graph sampling schemes with standard automatic differentiation tools, we provide an efficient turnk…
University of Edinburgh（爱丁堡大学）；University of Cambridge（剑桥大学）
It is difficult to continually update private machine learning models with new data while maintaining privacy. Data incur increasing privacy loss -- as measured by differential privacy -- when they are used in repeated computations. In this paper, we describe regularized empirical risk minimization algorithms that continually release models for a recent window of data. One version of the algorithm uses the entire data history to improve the model for the recent window. The second version uses a sliding window of constant size to improve the model, ensuring more relevant models in case of evolv…
National Key Laboratory for Novel Software Technology（新型软件技术国家重点实验室）；Nanjing University（南京大学）；the University of Iowa（爱荷华大学）；Alibaba Group（阿里巴巴集团）
Although there exist plentiful theories of empirical risk minimization (ERM) for supervised learning, current theoretical understandings of ERM for a related problem---stochastic convex optimization (SCO), are limited. In this work, we strengthen the realm of ERM for SCO by exploiting smoothness and strong convexity conditions to improve the risk bounds. First, we establish an $\widetilde{O}(d/n + \sqrt{F_*/n})$ risk bound when the random function is nonnegative, convex and smooth, and the expected function is Lipschitz continuous, where $d$ is the dimensionality of the problem, $n$ is the num…
Dalian University of Technology（大连理工大学）；University of Technology, Sydney（悉尼科技大学）；The University of Sydney（悉尼大学）
The vicinal risk minimization (VRM) principle, first proposed by \citet{vapnik1999nature}, is an empirical risk minimization (ERM) variant that replaces Dirac masses with vicinal functions. Although there is strong numerical evidence showing that VRM outperforms ERM if appropriate vicinal functions are chosen, a comprehensive theoretical understanding of VRM is still lacking. In this paper, we study the generalization bounds for VRM. Our results support Vapnik's original arguments and additionally provide deeper insights into VRM. First, we prove that the complexity of function classes convolv…
Differential Privacy (DP) is an important privacy-enhancing technology for private machine learning systems. It allows to measure and bound the risk associated with an individual participation in a computation. However, it was recently observed that DP learning systems may exacerbate bias and unfairness for different groups of individuals. This paper builds on these important observations and sheds light on the causes of the disparate impacts arising in the problem of differentially private empirical risk minimization. It focuses on the accuracy disparity arising among groups of individuals in…
Mila - Quebec AI Institute（米拉-魁北克人工智能研究所）；IBM Research（IBM研究院）；Rensselaer Polytechnic Institute（伦斯勒理工学院）；Google Research（谷歌研究院）
Recently, invariant risk minimization (IRM) was proposed as a promising solution to address out-of-distribution (OOD) generalization. However, it is unclear when IRM should be preferred over the widely-employed empirical risk minimization (ERM) framework. In this work, we analyze both these frameworks from the perspective of sample complexity, thus taking a firm step towards answering this important question. We find that depending on the type of data generation mechanism, the two approaches might have very different finite sample and asymptotic behavior. For example, in the covariate shift se…
Microsoft Research (MSR) Redmond（微软研究院（雷德蒙德））；University of Washington（华盛顿大学）；Shanghai Qi Zhi Institute（上海期智研究院）
We study the differentially private Empirical Risk Minimization (ERM) and Stochastic Convex Optimization (SCO) problems for non-smooth convex functions. We get a (nearly) optimal bound on the excess empirical risk and excess population loss with subquadratic gradient complexity. More precisely, our differentially private algorithm requires $O(\frac{N^{3/2}}{d^{1/8}}+ \frac{N^2}{d})$ gradient queries for optimal excess empirical risk, which is achieved with the help of subsampling and smoothing the function via convolution. This is the first subquadratic algorithm for the non-smooth case when $…
Lehigh University（利哈伊大学）；Industrial and Systems Engineering（工业与系统工程（注：此处为系别，按规则不提取））
In this paper we develop dual free mini-batch SDCA with adaptive probabilities for regularized empirical risk minimization. This work is motivated by recent work of Shai Shalev-Shwartz on dual free SDCA method, however, we allow a non-uniform selection of "dual" coordinates in SDCA. Moreover, the probability can change over time, making it more efficient than fix uniform or non-uniform selection. We also propose an efficient procedure to generate a random non-uniform mini-batch through iterative process. The work is concluded with multiple numerical experiments to show the efficiency of propos…
We consider empirical risk minimization for large-scale datasets. We introduce Ada Newton as an adaptive algorithm that uses Newton's method with adaptive sample sizes. The main idea of Ada Newton is to increase the size of the training set by a factor larger than one in a way that the minimization variable for the current training set is in the local neighborhood of the optimal argument of the next training set. This allows to exploit the quadratic convergence property of Newton's method and reach the statistical accuracy of each training set with only one iteration of Newton's method. We sho…
Recently Kutin and Niyogi investigated several notions of algorithmic stability--a property of a learning map conceptually similar to continuity--showing that training-stability is sufficient for consistency of Empirical Risk Minimization while distribution-free CV-stability is necessary and sufficient for having finite VC-dimension. This paper concerns a phase transition in the training stability of ERM, conjectured by the same authors. Kutin and Niyogi proved that ERM on finite hypothesis spaces containing a unique risk minimizer has training stability that scales exponentially with sample s…
Hong Kong University of Science and Technology（香港科技大学）
In regularized risk minimization, the associated optimization problem becomes particularly difficult when both the loss and regularizer are nonsmooth. Existing approaches either have slow or unclear convergence properties, are restricted to limited problem subclasses, or require careful setting of a smoothing parameter. In this paper, we propose a continuation algorithm that is applicable to a large class of nonsmooth regularized risk minimization problems, can be flexibly used with a number of existing solvers for the underlying smoothed subproblem, and with convergence results on the whole a…
The k-support norm has been recently introduced to perform correlated sparsity regularization. Although Argyriou et al. only reported experiments using squared loss, here we apply it to several other commonly used settings resulting in novel machine learning algorithms with interesting and familiar limit cases. Source code for the algorithms described here is available.
Tsinghua University（清华大学）；National University of Singapore（新加坡国立大学）
The ability to generalize under distributional shifts is essential to reliable machine learning, while models optimized with empirical risk minimization usually fail on non-$i.i.d$ testing data. Recently, invariant learning methods for out-of-distribution (OOD) generalization propose to find causally invariant relationships with multi-environments. However, modern datasets are frequently multi-sourced without explicit source labels, rendering many invariant learning methods inapplicable. In this paper, we propose Kernelized Heterogeneous Risk Minimization (KerHRM) algorithm, which achieves bot…
Indian Institute of Science（印度科学学院）；GE Global Research（通用电气全球研究中心）
In many applications, the training data, from which one needs to learn a classifier, is corrupted with label noise. Many standard algorithms such as SVM perform poorly in presence of label noise. In this paper we investigate the robustness of risk minimization to label noise. We prove a sufficient condition on a loss function for the risk minimization under that loss to be tolerant to uniform label noise. We show that the $0-1$ loss, sigmoid loss, ramp loss and probit loss satisfy this condition though none of the standard convex loss functions satisfy it. We also prove that, by choosing a suf…
Differential privacy is concerned about the prediction quality while measuring the privacy impact on individuals whose information is contained in the data. We consider differentially private risk minimization problems with regularizers that induce structured sparsity. These regularizers are known to be convex but they are often non-differentiable. We analyze the standard differentially private algorithms, such as output perturbation, Frank-Wolfe and objective perturbation. Output perturbation is a differentially private algorithm that is known to perform well for minimizing risks that are str…
We introduce Invariant Risk Minimization (IRM), a learning paradigm to estimate invariant correlations across multiple training distributions. To achieve this goal, IRM learns a data representation such that the optimal classifier, on top of that data representation, matches for all training distributions. Through theory and experiments, we show how the invariances learned by IRM relate to the causal structures governing the data and enable out-of-distribution generalization.
The University of Iowa（爱荷华大学）；Nanjing University（南京大学）
In this paper, we present a simple analysis of {\bf fast rates} with {\it high probability} of {\bf empirical minimization} for {\it stochastic composite optimization} over a finite-dimensional bounded convex set with exponential concave loss functions and an arbitrary convex regularization. To the best of our knowledge, this result is the first of its kind. As a byproduct, we can directly obtain the fast rate with {\it high probability} for exponential concave empirical risk minimization with and without any convex regularization, which not only extends existing results of empirical risk mini…
Machine learning algorithms with empirical risk minimization usually suffer from poor generalization performance due to the greedy exploitation of correlations among the training data, which are not stable under distributional shifts. Recently, some invariant learning methods for out-of-distribution (OOD) generalization have been proposed by leveraging multiple training environments to find invariant relationships. However, modern datasets are frequently assembled by merging data from multiple sources without explicit source labels. The resultant unobserved heterogeneity renders many invariant…
In this paper we explore noise tolerant learning of classifiers. We formulate the problem as follows. We assume that there is an ${\bf unobservable}$ training set which is noise-free. The actual training set given to the learning algorithm is obtained from this ideal data set by corrupting the class label of each example. The probability that the class label of an example is corrupted is a function of the feature vector of the example. This would account for most kinds of noisy data one encounters in practice. We say that a learning method is noise tolerant if the classifiers learnt with the i…
In this paper we consider the problem of Learning from Satisfying Assignments introduced by \cite{1} of finding a distribution that is a close approximation to the uniform distribution over the satisfying assignments of a low complexity Boolean function $f$. In a later work \cite{2} consider the same problem but with the knowledge of some continuous distribution $D$ and the objective being to estimate $D_f$, which is $D$ restricted to the satisfying assignments of an unknown Boolean function $f$. We consider these problems from the point of view of parameter estimation techniques in statistica…
University of Chicago（芝加哥大学）；Toyota Technological Institute at Chicago（芝加哥丰田工业大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We propose an adaptive (stochastic) gradient perturbation method for differentially private empirical risk minimization. At each iteration, the random noise added to the gradient is optimally adapted to the stepsize; we name this process adaptive differentially private (ADP) learning. Given the same privacy budget, we prove that the ADP method considerably improves the utility guarantee compared to the standard differentially private method in which vanilla random noise is added. Our method is particularly useful for gradient-based algorithms with time-varying learning rates, including variant…
Université Paris-Saclay（巴黎萨克雷大学）；Univ. Paris-Sud（巴黎南大学）；CNRS（法国国家科学研究中心）；Inria Saclay - Ile-de-France（法国国家信息与自动化研究所萨克莱-法兰西岛分院）；University of Southern California（南加州大学）
This paper investigates robust versions of the general empirical risk minimization algorithm, one of the core techniques underlying modern statistical methods. Success of the empirical risk minimization is based on the fact that for a "well-behaved" stochastic process $\left\{ f(X), \ f\in \mathcal F\right\}$ indexed by a class of functions $f\in \mathcal F$, averages $\frac{1}{N}\sum_{j=1}^N f(X_j)$ evaluated over a sample $X_1,\ldots,X_N$ of i.i.d. copies of $X$ provide good approximation to the expectations $\mathbb E f(X)$ uniformly over large classes $f\in \mathcal F$. However, this might…
Pairwise similarities and dissimilarities between data points might be easier to obtain than fully labeled data in real-world classification problems, e.g., in privacy-aware situations. To handle such pairwise information, an empirical risk minimization approach has been proposed, giving an unbiased estimator of the classification risk that can be computed only from pairwise similarities and unlabeled data. However, this direction cannot handle pairwise dissimilarities so far. On the other hand, semi-supervised clustering is one of the methods which can use both similarities and dissimilaritie…
We study conditional risk minimization (CRM), i.e. the problem of learning a hypothesis of minimal risk for prediction at the next step of sequentially arriving dependent data. Despite it being a fundamental problem, successful learning in the CRM sense has so far only been demonstrated using theoretical algorithms that cannot be used for real problems as they would require storing all incoming data. In this work, we introduce MACRO, a meta-algorithm for CRM that does not suffer from this shortcoming, but nevertheless offers learning guarantees. Instead of storing all data it maintains and ite…
We consider a general statistical learning problem where an unknown fraction of the training data is corrupted. We develop a robust learning method that only requires specifying an upper bound on the corrupted data fraction. The method minimizes a risk function defined by a non-parametric distribution with unknown probability weights. We derive and analyse the optimal weights and show how they provide robustness against corrupted data. Furthermore, we give a computationally efficient coordinate descent algorithm to solve the risk minimization problem. We demonstrate the wide range applicabilit…
We design simple screening tests to automatically discard data samples in empirical risk minimization without losing optimization guarantees. We derive loss functions that produce dual objectives with a sparse solution. We also show how to regularize convex losses to ensure such a dual sparsity-inducing property, and propose a general method to design screening tests for classification or regression based on ellipsoidal approximations of the optimal set. In addition to producing computational gains, our approach also allows us to compress a dataset into a subset of representative points.
Deep neural networks have found widespread adoption in solving complex tasks ranging from image recognition to natural language processing. However, these networks make confident mispredictions when presented with data that does not belong to the training distribution, i.e. out-of-distribution (OoD) samples. In this paper we explore whether the property of Vicinal Risk Minimization (VRM) to smoothly interpolate between different class boundaries helps to train better OoD detectors. We apply VRM to existing OoD detection techniques and show their improved performance. We observe that existing O…
Istituto Italiano di Tecnologia（意大利理工学院）；University of Genova（热那亚大学）；University of Waterloo（滑铁卢大学）；University College London（伦敦大学学院）
We address the problem of algorithmic fairness: ensuring that sensitive variables do not unfairly influence the outcome of a classifier. We present an approach based on empirical risk minimization, which incorporates a fairness constraint into the learning problem. It encourages the conditional risk of the learned classifier to be approximately constant with respect to the sensitive variable. We derive both risk and fairness bounds that support the statistical consistency of our approach. We specify our approach to kernel methods and observe that the fairness requirement implies an orthogonali…
