Margin Maximization as Lossless Maximal Compression
Department of Physics & Astronomy, UCL（伦敦大学学院（UCL）物理与天文系）；School of Computer Science, University of Birmingham（伯明翰大学计算机学院）；School of Computer Science, University of Manchester（曼彻斯特大学计算机学院）
Nikolaos Nikolaou, Henry Reeve, Gavin Brown
备注：
19 pages Main Paper + 7 pages Supplementary Material, 7 Figures, Submitted to the Machine Learning journal (11/11/19)
The ultimate goal of a supervised learning algorithm is to produce models constructed on the training data that can generalize well to new examples. In classification, functional margin maximization -- correctly classifying as many training examples as possible with maximal confidence --has been known to construct models with good generalization guarantees. This work gives an information-theoretic interpretation of a margin maximizing model on a noiseless training dataset as one that achieves lossless maximal compression of said dataset -- i.e. extracts from the features all the useful information for predicting the label and no more. The connection offers new insights on generalization in supervised machine learning, showing margin maximization as a special case (that of classification) of a more general principle and explains the success and potential limitations of popular learning algorithms like gradient boosting. We support our observations with theoretical arguments and empirical evidence and identify interesting directions for future work.
Borealis AI；University of Tuebingen（图宾根大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
We study adversarial robustness of neural networks from a margin maximization perspective, where margins are defined as the distances from inputs to a classifier's decision boundary. Our study shows that maximizing margins can be achieved by minimizing the adversarial loss on the decision boundary at the "shortest successful perturbation", demonstrating a close connection between adversarial losses and the margins. We propose Max-Margin Adversarial (MMA) training to directly maximize the margins to achieve adversarial robustness. Instead of adversarial training with a fixed $ε$, MMA offers an …
Linear classifiers and leaky ReLU networks trained by gradient flow on the logistic loss have an implicit bias towards solutions which satisfy the Karush--Kuhn--Tucker (KKT) conditions for margin maximization. In this work we establish a number of settings where the satisfaction of these KKT conditions implies benign overfitting in linear classifiers and in two-layer leaky ReLU networks: the estimators interpolate noisy training data and simultaneously generalize well to test data. The settings include variants of the noisy class-conditional Gaussians considered in previous work as well as new…
University of California, San Diego（加利福尼亚大学圣迭戈分校）；Columbia University（哥伦比亚大学）
A basic problem in the design of privacy-preserving algorithms is the private maximization problem: the goal is to pick an item from a universe that (approximately) maximizes a data-dependent function, all under the constraint of differential privacy. This problem has been used as a sub-routine in many privacy-preserving algorithms for statistics and machine-learning. Previous algorithms for this problem are either range-dependent---i.e., their utility diminishes with the size of the universe---or only apply to very restricted function classes. This work provides the first general-purpose, ran…
Boosting algorithms produce a classifier by iteratively combining base hypotheses. It has been observed experimentally that the generalization error keeps improving even after achieving zero training error. One popular explanation attributes this to improvements in margins. A common goal in a long line of research, is to maximize the smallest margin using as few base hypotheses as possible, culminating with the AdaBoostV algorithm by (R{ä}tsch and Warmuth [JMLR'04]). The AdaBoostV algorithm was later conjectured to yield an optimal trade-off between number of hypotheses trained and the minimal…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Weizmann Institute of Science（魏茨曼科学研究所）
The implicit bias of neural networks has been extensively studied in recent years. Lyu and Li [2019] showed that in homogeneous networks trained with the exponential or the logistic loss, gradient flow converges to a KKT point of the max margin problem in the parameter space. However, that leaves open the question of whether this point will generally be an actual optimum of the max margin problem. In this paper, we study this question in detail, for several neural network architectures involving linear and ReLU activations. Perhaps surprisingly, we show that in many cases, the KKT point is not…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Toyota Technical Institute of Chicago（芝加哥丰田技术研究所）
We present and analyze a momentum-based gradient method for training linear classifiers with an exponentially-tailed loss (e.g., the exponential or logistic loss), which maximizes the classification margin on separable data at a rate of $\widetilde{\mathcal{O}}(1/t^2)$. This contrasts with a rate of $\mathcal{O}(1/\log(t))$ for standard gradient descent, and $\mathcal{O}(1/t)$ for normalized gradient descent. This momentum-based method is derived via the convex dual of the maximum-margin problem, and specifically by applying Nesterov acceleration to this dual, which manages to result in a simp…
School of Computer Science Reichman University（Reichman大学计算机学院）；Data Science Institute Reichman University（Reichman大学数据科学研究所）
The weight decay regularization term is widely used during training to constrain expressivity, avoid overfitting, and improve generalization. Historically, this concept was borrowed from the SVM maximum margin principle and extended to multi-class deep networks. Carefully inspecting this principle reveals that it is not optimal for multi-class classification in general, and in particular when using deep neural networks. In this paper, we explain why this commonly used principle is not optimal and propose a new regularization scheme, called {\em Pairwise Margin Maximization} (PMM), which measur…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Facebook AI Research（脸书人工智能研究院）；University College London（伦敦大学学院）；University of Oxford（牛津大学）
Latent variable models have been successfully applied in lossless compression with the bits-back coding algorithm. However, bits-back suffers from an increase in the bitrate equal to the KL divergence between the approximate posterior and the true posterior. In this paper, we show how to remove this gap asymptotically by deriving bits-back coding algorithms from tighter variational bounds. The key idea is to exploit extended space representations of Monte Carlo estimators of the marginal likelihood. Naively applied, our schemes would require more initial bits than the standard bits-back coder,…
Ensemble models refer to methods that combine a typically large number of classifiers into a compound prediction. The output of an ensemble method is the result of fitting a base-learning algorithm to a given data set, and obtaining diverse answers by reweighting the observations or by resampling them using a given probabilistic selection. A key challenge of using ensembles in large-scale multidimensional data lies in the complexity and the computational burden associated with them. The models created by ensembles are often difficult, if not impossible, to interpret and their implementation re…
UCLA（加州大学洛杉矶分校）；University of California, Irvine（加州大学欧文分校）
Despite extensive progress on image generation, common deep generative model architectures are not easily applied to lossless compression. For example, VAEs suffer from a compression cost overhead due to their latent variables. This overhead can only be partially eliminated with elaborate schemes such as bits-back coding, often resulting in poor single-sample compression rates. To overcome such problems, we establish a new class of tractable lossless compression models that permit efficient encoding and decoding: Probabilistic Circuits (PCs). These are a class of neural networks involving $|p|…
Institute for Artificial Intelligence, Tsinghua University (THUAI)（清华大学人工智能研究院）；State Key Lab of Intelligent Technologies and Systems（智能技术与系统国家重点实验室）；Beijing National Research Center for Information Science and Technology (BNRist)（北京信息科学与技术国家研究中心）；Department of Automation, Tsinghua University（清华大学自动化系）；Bytedance AI Lab（字节跳动人工智能实验室）
The tremendous recent success of deep neural networks (DNNs) has sparked a surge of interest in understanding their predictive ability. Unlike the human visual system which is able to generalize robustly and learn with little supervision, DNNs normally require a massive amount of data to learn new concepts. In addition, research works also show that DNNs are vulnerable to adversarial examples-maliciously generated images which seem perceptually similar to the natural ones but are actually formed to fool learning models, which means the models have problem generalizing to unseen data with certa…
The generalization mystery of overparametrized deep nets has motivated efforts to understand how gradient descent (GD) converges to low-loss solutions that generalize well. Real-life neural networks are initialized from small random values and trained with cross-entropy loss for classification (unlike the "lazy" or "NTK" regime of training where analysis was more successful), and a recent sequence of results (Lyu and Li, 2020; Chizat and Bach, 2020; Ji and Telgarsky, 2020) provide theoretical evidence that GD may converge to the "max-margin" solution with zero loss, which presumably generalize…
The University of Iowa（爱荷华大学）；Washington State University（华盛顿州立大学）
Deep AUC Maximization (DAM) is a new paradigm for learning a deep neural network by maximizing the AUC score of the model on a dataset. Most previous works of AUC maximization focus on the perspective of optimization by designing efficient stochastic algorithms, and studies on generalization performance of large-scale DAM on difficult tasks are missing. In this work, we aim to make DAM more practical for interesting real-world applications (e.g., medical image classification). First, we propose a new margin-based min-max surrogate loss function for the AUC score (named as AUC min-max-margin lo…
Nokia Technologies（诺基亚科技）；Tampere University（坦佩雷大学）
Lossless image compression is an important technique for image storage and transmission when information loss is not allowed. With the fast development of deep learning techniques, deep neural networks have been used in this field to achieve a higher compression rate. Methods based on pixel-wise autoregressive statistical models have shown good performance. However, the sequential processing way prevents these methods to be used in practice. Recently, multi-scale autoregressive models have been proposed to address this limitation. Multi-scale approaches can use parallel computing systems effic…
Contrastive learning (CL) has been successful as a powerful representation learning method. In this work we propose CLIM: Contrastive Learning with mutual Information Maximization, to explore the potential of CL on cross-domain sentiment classification. To the best of our knowledge, CLIM is the first to adopt contrastive learning for natural language processing (NLP) tasks across domains. Due to scarcity of labels on the target domain, we introduce mutual information maximization (MIM) apart from CL to exploit the features that best support the final prediction. Furthermore, MIM is able to mai…
Faster Margin Maximization Rates for Generic and Adversarially Robust Optimization Methods
College of Computing, Georgia Institute of Technology（佐治亚理工学院计算学院）；Google Research（谷歌研究院）；School of Electrical and Computer Engineering, Georgia Institute of Technology（佐治亚理工学院电气与计算机工程学院）；School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院工业与系统工程学院）
In this work, we study a new approach to optimizing the margin distribution realized by binary classifiers. The classical approach to this problem is simply maximization of the expected margin, while more recent proposals consider simultaneous variance control and proxy objectives based on robust location estimates, in the vein of keeping the margin distribution sharply concentrated in a desirable region. While conceptually appealing, these new approaches are often computationally unwieldy, and theoretical guarantees are limited. Given this context, we propose an algorithm which searches the h…
BMW Group（宝马集团）；Technical University of Dortmund（多特蒙德工业大学）
In this work we present a rigorous application of the Expectation Maximization algorithm to determine the marginal distributions and the dependence structure in a Gaussian copula model with missing data. We further show how to circumvent a priori assumptions on the marginals with semiparametric modelling. The joint distribution learned through this algorithm is considerably closer to the underlying distribution than existing methods.
Multinoulli Extension: A Lossless Continuous Relaxation for Partition-Constrained Subset Selection
college of computing and data science, Nanyang Technological University, Singapore（南洋理工大学计算与数据科学学院，新加坡）；Hong Kong University of Science and Technology（香港科学大学）
In this work we consider the problem of online submodular maximization under a cardinality constraint with differential privacy (DP). A stream of $T$ submodular functions over a common finite ground set $U$ arrives online, and at each time-step the decision maker must choose at most $k$ elements of $U$ before observing the function. The decision maker obtains a payoff equal to the function evaluated on the chosen set, and aims to learn a sequence of sets that achieves low expected regret. In the full-information setting, we develop an $(\varepsilon,δ)$-DP algorithm with expected $(1-1/e)$-regr…
Institute for Basic Science（基础科学研究院）；London School of Economics（伦敦政治经济学院）；KAIST（韩国科学技术院）
Motivated by recent developments in designing algorithms based on individual item scores for solving utility maximization problems, we study the framework of using test scores, defined as a statistic of observed individual item performance data, for solving the budgeted stochastic utility maximization problem. We extend an existing scoring mechanism, namely the replication test scores, to incorporate heterogeneous item costs as well as item values. We show that a natural greedy algorithm that selects items solely based on their replication test scores outputs solutions within a constant factor…
California Institute of Technology（加州理工学院）；Carnegie Mellon University（卡内基梅隆大学）；Google, Inc.（谷歌公司）；ETH Zurich（苏黎世联邦理工学院）；Duolingo（多邻国）
Which ads should we display in sponsored search in order to maximize our revenue? How should we dynamically rank information sources to maximize the value of the ranking? These applications exhibit strong diminishing returns: Redundancy decreases the marginal utility of each ad or information source. We show that these and other problems can be formalized as repeatedly selecting an assignment of items to positions to maximize a sequence of monotone submodular functions that arrive one by one. We present an efficient algorithm for this general problem and analyze it in the no-regret model. Our …
The typical algorithmic problem in viral marketing aims to identify a set of influential users in a social network, who, when convinced to adopt a product, shall influence other users in the network and trigger a large cascade of adoptions. However, the host (the owner of an online social platform) often faces more constraints than a single product, endless user attentions, unlimited budget and unbounded time; in reality, multiple products need to be advertised, each user can tolerate only a small number of recommendations, influencing user has a cost and advertisers have only limited budgets,…
Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization
Institute for Advanced Study（高级研究院）；National Research Council Canada（加拿大国家研究理事会）；University of Toronto（多伦多大学）；St. Petersburg State University（圣彼得格勒国立大学）；Skolkovo Institute of Science and Technology（斯克罗夫诺技术研究所）；Kazan Federal University（卡兹兰卡联邦大学）
Deep neural networks (DNNs) for the semantic segmentation of images are usually trained to operate on a predefined closed set of object classes. This is in contrast to the "open world" setting where DNNs are envisioned to be deployed to. From a functional safety point of view, the ability to detect so-called "out-of-distribution" (OoD) samples, i.e., objects outside of a DNN's semantic space, is crucial for many applications such as automated driving. A natural baseline approach to OoD detection is to threshold on the pixel-wise softmax entropy. We present a two-step procedure that significant…
