We present Searn, an algorithm for integrating search and learning to solve complex structured prediction problems such as those that occur in natural language, speech, computational biology, and vision. Searn is a meta-algorithm that transforms these complex problems into simple classification problems to which any binary classifier may be applied. Unlike current algorithms for structured learning that require decomposition of both the loss function and the feature functions over the predicted structure, Searn is able to learn prediction functions for any loss function and any class of features. Moreover, Searn comes with a strong, natural theoretical guarantee: good performance on the derived classification problems implies good performance on the structured prediction problem.
Mappings to structured output spaces (strings, trees, partitions, etc.) are typically learned using extensions of classification algorithms to simple graphical structures (eg., linear chains) in which search and parameter estimation can be performed exactly. Unfortunately, in many complex problems, it is rare that exact search or parameter estimation is tractable. Instead of learning exact models and searching via heuristic means, we embrace this difficulty and treat the structured output problem in terms of approximate search. We present a framework for learning as search optimization, and tw…
We consider a framework for structured prediction based on search in the space of complete structured outputs. Given a structured input, an output is produced by running a time-bounded search procedure guided by a learned cost function, and then returning the least cost output uncovered during the search. This framework can be instantiated for a wide range of search spaces and search procedures, and easily incorporates arbitrary structured-prediction loss functions. In this paper, we make two main technical contributions. First, we define the limited-discrepancy search space over structured ou…
We describe an adaptation and application of a search-based structured prediction algorithm "Searn" to unsupervised learning problems. We show that it is possible to reduce unsupervised learning to supervised learning and demonstrate a high-quality unsupervised shift-reduce parsing model. We additionally show a close connection between unsupervised Searn and expectation maximization. Finally, we demonstrate the efficacy of a semi-supervised extension. The key idea that enables this is an application of the predict-self idea for unsupervised learning.
Various natural language processing tasks are structured prediction problems where outputs are constructed with multiple interdependent decisions. Past work has shown that domain knowledge, framed as constraints over the output space, can help improve predictive accuracy. However, designing good constraints often relies on domain expertise. In this paper, we study the problem of learning such constraints. We frame the problem as that of training a two-layer rectifier network to identify valid structures or substructures, and show a construction for converting a trained network into a system of…
Microsoft Research New England（微软新英格兰研究院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Microsoft Research（微软研究院）；University of Utah（犹他大学）
IllinoisSL is a Java library for learning structured prediction models. It supports structured Support Vector Machines and structured Perceptron. The library consists of a core learning module and several applications, which can be executed from command-lines. Documentation is provided to guide users. In Comparison to other structured learning libraries, IllinoisSL is efficient, general, and easy to use.
In this paper, we prove that separable negative log-likelihood losses for structured prediction are not necessarily Bayes consistent, or, in other words, minimizing these losses may not result in a model that predicts the most probable structure in the data distribution for a given input. This fact opens the question of whether these losses are well-adapted for structured prediction and, if so, why.
We present an approach to structured prediction from bandit feedback, called Bandit Structured Prediction, where only the value of a task loss function at a single predicted point, instead of a correct structure, is observed in learning. We present an application to discriminative reranking in Statistical Machine Translation (SMT) where the learning algorithm only has access to a 1-BLEU loss evaluation of a predicted translation instead of obtaining a gold standard reference translation. In our experiment bandit feedback is obtained by evaluating BLEU on reference translations without revealin…
Over the last several years, the field of Structured prediction in NLP has had seen huge advancements with sophisticated probabilistic graphical models, energy-based networks, and its combination with deep learning-based approaches. This survey provides a brief of major techniques in structured prediction and its applications in the NLP domains like parsing, sequence labeling, text generation, and sequence to sequence tasks. We also deep-dived into energy-based and attention-based techniques in structured prediction, identified some relevant open issues and gaps in the current state-of-the-art…
Monash University（莫纳什大学）；University of Melbourne（墨尔本大学）
Linguistic structures exhibit a rich array of global phenomena, however commonly used Markov models are unable to adequately describe these phenomena due to their strong locality assumptions. We propose a novel hierarchical model for structured prediction over sequences and trees which exploits global context by conditioning each generation decision on an unbounded context of prior decisions. This builds on the success of Markov models but without imposing a fixed bound in order to better represent global phenomena. To facilitate learning of this large and unbounded model, we use a hierarchica…
In structured output prediction tasks, labeling ground-truth training output is often expensive. However, for many tasks, even when the true output is unknown, we can evaluate predictions using a scalar reward function, which may be easily assembled from human knowledge or non-differentiable pipelines. But searching through the entire output space to find the best output with respect to this reward function is typically intractable. In this paper, we instead use efficient truncated randomized search in this reward function to train structured prediction energy networks (SPENs), which provide e…
Conditional random field (CRF) and Structural Support Vector Machine (Structural SVM) are two state-of-the-art methods for structured prediction which captures the interdependencies among output variables. The success of these methods is attributed to the fact that their discriminative models are able to account for overlapping features on the whole input observations. These features are usually generated by applying a given set of templates on labeled data, but improper templates may lead to degraded performance. To alleviate this issue, in this paper, we propose a novel multiple template lea…
Boston University（波士顿大学）；Microsoft Research（微软研究院）
We study the problem of structured prediction under test-time budget constraints. We propose a novel approach applicable to a wide range of structured prediction problems in computer vision and natural language processing. Our approach seeks to adaptively generate computationally costly features during test-time in order to reduce the computational cost of prediction while maintaining prediction performance. We show that training the adaptive feature generation system can be reduced to a series of structured learning problems, resulting in efficient training using existing structured learning …
Many machine learning problems require the prediction of multi-dimensional labels. Such structured prediction models can benefit from modeling dependencies between labels. Recently, several deep learning approaches to structured prediction have been proposed. Here we focus on capturing cardinality constraints in such models. Namely, constraining the number of non-zero labels that the model outputs. Such constraints have proven very useful in previous structured prediction approaches, but it is a challenge to introduce them into a deep learning framework. Here we show how to do this via a novel…
Stochastic zeroth-order (SZO), or gradient-free, optimization allows to optimize arbitrary functions by relying only on function evaluations under parameter perturbations, however, the iteration complexity of SZO methods suffers a factor proportional to the dimensionality of the perturbed function. We show that in scenarios with natural sparsity patterns as in structured prediction applications, this factor can be reduced to the expected number of active features over input-output pairs. We give a general proof that applies sparse SZO optimization to Lipschitz-continuous, nonconvex, stochastic…
Structured prediction is the cornerstone of several machine learning applications. Unfortunately, in structured prediction settings with expressive inter-variable interactions, exact inference-based learning algorithms, e.g. Structural SVM, are often intractable. We present a new way, Decomposed Learning (DecL), which performs efficient learning by restricting the inference step to a limited part of the structured spaces. We provide characterizations based on the structure, target parameters, and gold labels, under which DecL is equivalent to exact learning. We then show that in real world set…
Structured prediction plays a central role in machine learning applications from computational biology to computer vision. These models require significantly more computation than unstructured models, and, in many applications, algorithms may need to make predictions within a computational budget or in an anytime fashion. In this work we propose an anytime technique for learning structured prediction that, at training time, incorporates both structural elements and feature computation trade-offs that affect test-time inference. We apply our technique to the challenging problem of scene underst…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
Structured prediction energy networks (SPENs; Belanger & McCallum 2016) use neural network architectures to define energy functions that can capture arbitrary dependencies among parts of structured outputs. Prior work used gradient descent for inference, relaxing the structured output to a set of continuous variables and then optimizing the energy with respect to them. We replace this use of gradient descent with a neural network trained to approximate structured argmax inference. This "inference network" outputs continuous values that we treat as the output structure. We develop large-margin …
Heidelberg University（海德堡大学）；Tufts University（塔夫茨大学）；Amazon Development Center（亚马逊开发中心）
Stochastic structured prediction under bandit feedback follows a learning protocol where on each of a sequence of iterations, the learner receives an input, predicts an output structure, and receives partial feedback in form of a task loss evaluation of the predicted structure. We present applications of this learning scenario to convex and non-convex objectives for structured prediction and analyze them as stochastic first-order methods. We present an experimental evaluation on problems of natural language processing over exponential output spaces, and compare convergence speed across differe…
University of Salzburg（萨尔茨堡大学）；Purdue University（普渡大学）
Expressive text encoders such as RNNs and Transformer Networks have been at the center of NLP models in recent work. Most of the effort has focused on sentence-level tasks, capturing the dependencies between words in a single sentence, or pairs of sentences. However, certain tasks, such as argumentation mining, require accounting for longer texts and complicated structural dependencies between them. Deep structured prediction is a general framework to combine the complementary strengths of expressive neural encoders and structured inference for highly structured domains. Nevertheless, when the…
The Chinese University of Hong Kong（香港中文大学）；Amazon AWS AI Labs（亚马逊AWS人工智能实验室）；University of Cambridge（剑桥大学）
Recent advance in deep learning has led to the rapid adoption of machine learning-based NLP models in a wide range of applications. Despite the continuous gain in accuracy, backward compatibility is also an important aspect for industrial applications, yet it received little research attention. Backward compatibility requires that the new model does not regress on cases that were correctly handled by its predecessor. This work studies model update regression in structured prediction tasks. We choose syntactic dependency parsing and conversational semantic parsing as representative examples of …
Bandit structured prediction describes a stochastic optimization framework where learning is performed from partial feedback. This feedback is received in the form of a task loss evaluation to a predicted output structure, without having access to gold standard structures. We advance this framework by lifting linear bandit learning to neural sequence-to-sequence learning problems using attention-based recurrent neural networks. Furthermore, we show how to incorporate control variates into our learning algorithms for variance reduction and improved generalization. We present an evaluation on a …
University of Maryland（马里兰大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
Modern neural networks do not always produce well-calibrated predictions, even when trained with a proper scoring function such as cross-entropy. In classification settings, simple methods such as isotonic regression or temperature scaling may be used in conjunction with a held-out dataset to calibrate model outputs. However, extending these methods to structured prediction is not always straightforward or effective; furthermore, a held-out calibration set may not always be available. In this paper, we study ensemble distillation as a general framework for producing well-calibrated structured …
We introduce structured prediction energy networks (SPENs), a flexible framework for structured prediction. A deep architecture is used to define an energy function of candidate labels, and then predictions are produced by using back-propagation to iteratively optimize the energy with respect to the labels. This deep architecture captures dependencies between labels that would lead to intractable graphical models, and performs structure learning by automatically learning discriminative features of the structured output. One natural application of our technique is multi-label classification, wh…
Instituto de Telecomunicações（电信研究所）；Instituto Superior Técnico, University of Lisbon（里斯本大学高等技术学院）；Unbabel
Structured prediction requires manipulating a large number of combinatorial structures, e.g., dependency trees or alignments, either as latent or output variables. Recently, the SparseMAP method has been proposed as a differentiable, sparse alternative to maximum a posteriori (MAP) and marginal inference. SparseMAP returns a combination of a small number of structures, a desirable property in some downstream applications. However, SparseMAP requires a tractable MAP inference oracle. This excludes, e.g., loopy graphical models or factor graphs with logic constraints, which generally require app…
Databases are widespread, yet extracting relevant data can be difficult. Without substantial domain knowledge, multivariate search queries often return sparse or uninformative results. This paper introduces an approach for searching structured data based on probabilistic programming and nonparametric Bayes. Users specify queries in a probabilistic language that combines standard SQL database search operators with an information theoretic ranking function called predictive relevance. Predictive relevance can be calculated by a fast sparse matrix algorithm based on posterior samples from CrossCa…
The Hong Kong University of Science and Technology（香港科技大学）
We present a new method for score-based adversarial attack, where the attacker queries the loss-oracle of the target model. Our method employs a parameterized search space with a structure that captures the relationship of the gradient of the loss function. We show that searching over the structured space can be approximated by a time-varying contextual bandits problem, where the attacker takes feature of the associated arm to make modifications of the input, and receives an immediate reward as the reduction of the loss function. The time-varying contextual bandits problem can then be solved b…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Vanderbilt University（范德堡大学）；Suning Commerce R&D Center USA, Inc（苏宁美国研发中心）；Beijing Jiaotong University（北京交通大学）；New York University Abu Dhabi（纽约大学阿布扎比分校）；National Time Service Center, Chinese Academy of Sciences（中国科学院国家授时中心）；Graduate University of Chinese Academy of Sciences（中国科学院研究生院）
We propose a novel semi-supervised structured output prediction method based on local linear regression in this paper. The existing semi-supervise structured output prediction methods learn a global predictor for all the data points in a data set, which ignores the differences of local distributions of the data set, and the effects to the structured output prediction. To solve this problem, we propose to learn the missing structured outputs and local predictors for neighborhoods of different data points jointly. Using the local linear regression strategy, in the neighborhood of each data point…
Shanghai University（上海大学）；Tongji University（同济大学）；The 3rd Research Institute of Ministry of Public Security（公安部第三研究所）；Kansas State University（堪萨斯州立大学）
Structured output prediction aims to learn a predictor to predict a structured output from a input data vector. The structured outputs include vector, tree, sequence, etc. We usually assume that we have a training set of input-output pairs to train the predictor. However, in many real-world appli- cations, it is difficult to obtain the output for a input, thus for many training input data points, the structured outputs are missing. In this paper, we dis- cuss how to learn from a training set composed of some input-output pairs, and some input data points without outputs. This problem is called…
INRIA（法国国家信息与自动化研究所）；Ecole Normale Supérieure - PSL Research University（巴黎高等师范学院-巴黎文理研究大学）；University College London（伦敦大学学院）；Istituto Italiano di Tecnologia（意大利技术研究院）；Massachusetts Institute of Technology（麻省理工学院）；Universitá degli Studi di Genova（热那亚大学）
Structured prediction provides a general framework to deal with supervised problems where the outputs have semantically rich structure. While classical approaches consider finite, albeit potentially huge, output spaces, in this paper we discuss how structured prediction can be extended to a continuous scenario. Specifically, we study a structured prediction approach to manifold valued regression. We characterize a class of problems for which the considered approach is statistically consistent and study how geometric optimization can be used to compute the corresponding estimator. Promising exp…
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…
LTCI, Télécom Paris, IP Paris（LTCI，巴黎电信，巴黎理工学院）；INRIA - Département d’Informatique de l’ENS, PSL（法国国家信息与自动化研究所 - 巴黎高等师范学院计算机系，巴黎文理研究大学）；INRAE, MIAT, Université de Toulouse（法国国家农业食品与环境研究院，图卢兹大学）；Department of Computer Science, Aalto University（阿尔托大学计算机科学系）
A powerful and flexible approach to structured prediction consists in embedding the structured objects to be predicted into a feature space of possibly infinite dimension by means of output kernels, and then, solving a regression problem in this output space. A prediction in the original space is computed by solving a pre-image problem. In such an approach, the embedding, linked to the target loss, is defined prior to the learning phase. In this work, we propose to jointly learn a finite approximation of the output embedding and the regression function into the new feature space. For that purp…
Margin-based structured prediction commonly uses a maximum loss over all possible structured outputs \cite{Altun03,Collins04b,Taskar03}. In natural language processing, recent work \cite{Zhang14,Zhang15} has proposed the use of the maximum loss over random structured outputs sampled independently from some proposal distribution. This method is linear-time in the number of random structured outputs and trivially parallelizable. We study this family of loss functions in the PAC-Bayes framework under Gaussian perturbations \cite{McAllester07}. Under some technical conditions and up to statistical…
University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）；Carnegie Mellon University（卡内基梅隆大学）
Structured Prediction Energy Networks (SPENs) are a simple, yet expressive family of structured prediction models (Belanger and McCallum, 2016). An energy function over candidate structured outputs is given by a deep network, and predictions are formed by gradient-based optimization. This paper presents end-to-end learning for SPENs, where the energy function is discriminatively trained by back-propagating through gradient-based prediction. In our experience, the approach is substantially more accurate than the structured SVM method of Belanger and McCallum (2016), as it allows us to use more …
We propose Nester, a method for injecting neural networks into constrained structured predictors. The job of the neural network(s) is to compute an initial, raw prediction that is compatible with the input data but does not necessarily satisfy the constraints. The structured predictor then builds a structure using a constraint solver that assembles and corrects the raw predictions in accordance with hard and soft constraints. In doing so, Nester takes advantage of the features of its two components: the neural network learns complex representations from low-level data while the constraint prog…
Predicting structured outputs can be computationally onerous due to the combinatorially large output spaces. In this paper, we focus on reducing the prediction time of a trained black-box structured classifier without losing accuracy. To do so, we train a speedup classifier that learns to mimic a black-box classifier under the learning-to-search approach. As the structured classifier predicts more examples, the speedup classifier will operate as a learned heuristic to guide search to favorable regions of the output space. We present a mistake bound for the speedup classifier and identify infer…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
For joint inference over multiple variables, a variety of structured prediction techniques have been developed to model correlations among variables and thereby improve predictions. However, many classical approaches suffer from one of two primary drawbacks: they either lack the ability to model high-order correlations among variables while maintaining computationally tractable inference, or they do not allow to explicitly model known correlations. To address this shortcoming, we introduce `Graph Structured Prediction Energy Networks,' for which we develop inference techniques that allow to bo…
ETH Zurich（苏黎世联邦理工学院）；Toyota Technological Institute at Chicago（芝加哥丰田技术学院）
In this paper we propose a unified framework for structured prediction with latent variables which includes hidden conditional random fields and latent structured support vector machines as special cases. We describe a local entropy approximation for this general formulation using duality, and derive an efficient message passing algorithm that is guaranteed to converge. We demonstrate its effectiveness in the tasks of image segmentation as well as 3D indoor scene understanding from single images, showing that our approach is superior to latent structured support vector machines and hidden cond…
Chain-structured neural architecture search for financial time series forecasting
School of Engineering and Management Vaud (HEIG-VD), University of Applied Sciences and Arts Western Switzerland (HES-SO)（沃州工程与管理学院（HEIG-VD），瑞士西部应用科学与艺术大学（HES-SO））；Predictive Layer SA（Predictive Layer SA公司）
We introduce a conceptually novel structured prediction model, GPstruct, which is kernelized, non-parametric and Bayesian, by design. We motivate the model with respect to existing approaches, among others, conditional random fields (CRFs), maximum margin Markov networks (M3N), and structured support vector machines (SVMstruct), which embody only a subset of its properties. We present an inference procedure based on Markov Chain Monte Carlo. The framework can be instantiated for a wide range of structured objects such as linear chains, trees, grids, and other general graphs. As a proof of conc…
Laboratory for Computational and Statistical Learning - Istituto Italiano di Tecnologia（意大利理工学院计算与统计学习实验室）；Massachusetts Institute of Technology（麻省理工学院）；Università degli Studi di Genova（热那亚大学）
We propose and analyze a regularization approach for structured prediction problems. We characterize a large class of loss functions that allows to naturally embed structured outputs in a linear space. We exploit this fact to design learning algorithms using a surrogate loss approach and regularization techniques. We prove universal consistency and finite sample bounds characterizing the generalization properties of the proposed methods. Experimental results are provided to demonstrate the practical usefulness of the proposed approach.
Mines ParisTech – PSL（巴黎高等矿业学院——巴黎文理研究大学）；Inria（法国国家信息与自动化研究所）；Sorbonne Université（索邦大学）；University College London（伦敦大学学院）
Many practical machine learning tasks can be framed as Structured prediction problems, where several output variables are predicted and considered interdependent. Recent theoretical advances in structured prediction have focused on obtaining fast rates convergence guarantees, especially in the Implicit Loss Embedding (ILE) framework. PAC-Bayes has gained interest recently for its capacity of producing tight risk bounds for predictor distributions. This work proposes a novel PAC-Bayes perspective on the ILE Structured prediction framework. We present two generalization bounds, on the risk and e…
While there are many studies on weight regularization, the study on structure regularization is rare. Many existing systems on structured prediction focus on increasing the level of structural dependencies within the model. However, this trend could have been misdirected, because our study suggests that complex structures are actually harmful to generalization ability in structured prediction. To control structure-based overfitting, we propose a structure regularization framework via \emph{structure decomposition}, which decomposes training samples into mini-samples with simpler structures, de…
Active search is the process of identifying high-value data points in a large and often high-dimensional parameter space that can be expensive to evaluate. Traditional active search techniques like Bayesian optimization trade off exploration and exploitation over consecutive evaluations, and have historically focused on single or small (<5) numbers of examples evaluated per round. As modern data sets grow, so does the need to scale active search to large data sets and batch sizes. In this paper, we present a general hierarchical framework based on bandit algorithms to scale active search to la…
Markov Chain Monte Carlo (MCMC) algorithms are often used for approximate inference inside learning, but their slow mixing can be difficult to diagnose and the approximations can seriously degrade learning. To alleviate these issues, we define a new model family using strong Doeblin Markov chains, whose mixing times can be precisely controlled by a parameter. We also develop an algorithm to learn such models, which involves maximizing the data likelihood under the induced stationary distribution of these chains. We show empirical improvements on two challenging inference tasks.
Yandex（扬德克斯）；Higher School of Economics（高等经济学院）；University of Cambridge（剑桥大学）
Uncertainty estimation is important for ensuring safety and robustness of AI systems. While most research in the area has focused on un-structured prediction tasks, limited work has investigated general uncertainty estimation approaches for structured prediction. Thus, this work aims to investigate uncertainty estimation for autoregressive structured prediction tasks within a single unified and interpretable probabilistic ensemble-based framework. We consider: uncertainty estimation for sequence data at the token-level and complete sequence-level; interpretations for, and applications of, vari…
We present a general theoretical analysis of structured prediction with a series of new results. We give new data-dependent margin guarantees for structured prediction for a very wide family of loss functions and a general family of hypotheses, with an arbitrary factor graph decomposition. These are the tightest margin bounds known for both standard multi-class and general structured prediction problems. Our guarantees are expressed in terms of a data-dependent complexity measure, factor graph complexity, which we show can be estimated from data and bounded in terms of familiar quantities. We …
Semi-supervised learning aims to learn prediction models from both labeled and unlabeled samples. There has been extensive research in this area. Among existing work, generative mixture models with Expectation-Maximization (EM) is a popular method due to clear statistical properties. However, existing literature on EM-based semi-supervised learning largely focuses on unstructured prediction, assuming that samples are independent and identically distributed. Studies on EM-based semi-supervised approach in structured prediction is limited. This paper aims to fill the gap through a comparative st…
Motivated by Supervised Opinion Analysis, we propose a novel framework devoted to Structured Output Learning with Abstention (SOLA). The structure prediction model is able to abstain from predicting some labels in the structured output at a cost chosen by the user in a flexible way. For that purpose, we decompose the problem into the learning of a pair of predictors, one devoted to structured abstention and the other, to structured output prediction. To compare fully labeled training data with predictions potentially containing abstentions, we define a wide class of asymmetric abstention-aware…
We present a framework to train a structured prediction model by performing smoothing on the inference algorithm it builds upon. Smoothing overcomes the non-smoothness inherent to the maximum margin structured prediction objective, and paves the way for the use of fast primal gradient-based optimization algorithms. We illustrate the proposed framework by developing a novel primal incremental optimization algorithm for the structural support vector machine. The proposed algorithm blends an extrapolation scheme for acceleration and an adaptive smoothing scheme and builds upon the stochastic vari…
Many predicted structured objects (e.g., sequences, matchings, trees) are evaluated using the F-score, alignment error rate (AER), or other multivariate performance measures. Since inductively optimizing these measures using training data is typically computationally difficult, empirical risk minimization of surrogate losses is employed, using, e.g., the hinge loss for (structured) support vector machines. These approximations often introduce a mismatch between the learner's objective and the desired application performance, leading to inconsistency. We take a different approach: adversarially…
MAP perturbation models have emerged as a powerful framework for inference in structured prediction. Such models provide a way to efficiently sample from the Gibbs distribution and facilitate predictions that are robust to random noise. In this paper, we propose a provably polynomial time randomized algorithm for learning the parameters of perturbed MAP predictors. Our approach is based on minimizing a novel Rademacher-based generalization bound on the expected loss of a perturbed MAP predictor, which can be computed in polynomial time. We obtain conditions under which our randomized learning …
