本文在对 f 的温和条件下，给出了带 f-散度正则化的经验风险最小化（ERM-fDR）的解。在这些条件下，证明了最优测度是唯一的。文中给出了函数 f 的若干特定选择所对应的解示例。通过利用 f-散度族的灵活性，得到了常见正则化选择的此前已知解，其中包括带相对熵正则化的经验风险最小化的唯一解（I 型和 II 型）。对该解的分析揭示了 f-散度用于 ERM-fDR 问题时的以下性质：i）f-散度正则化迫使解的支撑集与参考测度的支撑集重合，这引入了一种强归纳偏置，并压倒训练数据提供的证据；以及 ii）任何 f-散度正则化都等价于对经验风险函数进行适当变换后的另一种 f-散度正则化。
英文摘要：
The solution to empirical risk minimization with $f$-divergence regularization (ERM-$f$DR) is presented under mild conditions on $f$. Under such conditions, the optimal measure is shown to be unique. Examples of the solution for particular choices of the function $f$ are presented. Previously known solutions to common regularization choices are obtained by leveraging the flexibility of the family of $f$-divergences. These include the unique solutions to empirical risk minimization with relative entropy regularization (Type-I and Type-II). The analysis of the solution unveils the following properties of $f$-divergences when used in the ERM-$f$DR problem: $i\bigl)$ $f$-divergence regularization forces the support of the solution to coincide with the support of the reference measure, which introduces a strong inductive bias that dominates the evidence provided by the training data; and $ii\bigl)$ any $f$-divergence regularization is equivalent to a different $f$-divergence regularization with an appropriate transformation of the empirical risk function.
CyberAgent, Inc.（赛博代理公司）；The University of Tokyo（东京大学）
Density ratio estimation (DRE) is at the core of various machine learning tasks such as anomaly detection and domain adaptation. In existing studies on DRE, methods based on Bregman divergence (BD) minimization have been extensively studied. However, BD minimization when applied with highly flexible models, such as deep neural networks, tends to suffer from what we call train-loss hacking, which is a source of overfitting caused by a typical characteristic of empirical BD estimators. In this paper, to mitigate train-loss hacking, we propose a non-negative correction for empirical BD estimators…
Univ Lyon（里昂大学）；UJM-Saint-Etienne（圣艾蒂安大学）；CNRS（法国国家科学研究中心）；Institut d’Optique Graduate School（法国高等光学研究所）；Univ. Grenoble Alps（格勒诺布尔阿尔卑斯大学）；Laboratoire d’Informatique de Grenoble（格勒诺布尔信息学实验室）
We tackle the issue of classifier combinations when observations have multiple views. Our method jointly learns view-specific weighted majority vote classifiers (i.e. for each view) over a set of base voters, and a second weighted majority vote classifier over the set of these view-specific weighted majority vote classifiers. We show that the empirical risk minimization of the final majority vote given a multiview training set can be cast as the minimization of Bregman divergences. This allows us to derive a parallel-update optimization algorithm for learning our multiview model. We empiricall…
INRIA（法国国家信息与自动化研究所）；Université de la Polynésie Française（法属波利尼西亚大学）；University of Sheffield（谢菲尔德大学）；National Chao Tung University（国立交通大学）；Princeton University（普林斯顿大学）
The optimality and sensitivity of the empirical risk minimization problem with relative entropy regularization (ERM-RER) are investigated for the case in which the reference is a sigma-finite measure instead of a probability measure. This generalization allows for a larger degree of flexibility in the incorporation of prior knowledge over the set of models. In this setting, the interplay of the regularization parameter, the reference measure, the risk function, and the empirical risk induced by the solution of the ERM-RER problem is characterized. This characterization yields necessary and suf…
Black-box alpha (BB-$α$) is a new approximate inference method based on the minimization of $α$-divergences. BB-$α$ scales to large datasets because it can be implemented using stochastic gradient descent. BB-$α$ can be applied to complex probabilistic models with little effort since it only requires as input the likelihood function and its gradients. These gradients can be easily obtained using automatic differentiation. By changing the divergence parameter $α$, the method is able to interpolate between variational Bayes (VB) ($α\rightarrow 0$) and an algorithm similar to expectation propagat…
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…
We propose Noise-Augmented Privacy-Preserving Empirical Risk Minimization (NAPP-ERM) that solves ERM with differential privacy guarantees. Existing privacy-preserving ERM approaches may be subject to over-regularization with the employment of an l2 term to achieve strong convexity on top of the target regularization. NAPP-ERM improves over the current approaches and mitigates over-regularization by iteratively realizing target regularization through appropriately designed augmented data and delivering strong convexity via a single adaptively weighted dual-purpose l2 regularizer. When the targe…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Wisconsin Institute for Discovery（威斯康星发现研究所）
We propose a communication- and computation-efficient distributed optimization algorithm using second-order information for solving ERM problems with a nonsmooth regularization term. Current second-order and quasi-Newton methods for this problem either do not work well in the distributed setting or work only for specific regularizers. Our algorithm uses successive quadratic approximations, and we describe how to maintain an approximation of the Hessian and solve subproblems efficiently in a distributed manner. The proposed method enjoys global linear convergence for a broad range of non-strong…
Stanford University（斯坦福大学）；Microsoft Research, New England（微软新英格兰研究院）；MIT（麻省理工学院）
We develop a family of accelerated stochastic algorithms that minimize sums of convex functions. Our algorithms improve upon the fastest running time for empirical risk minimization (ERM), and in particular linear least-squares regression, across a wide range of problem settings. To achieve this, we establish a framework based on the classical proximal point algorithm. Namely, we provide several algorithms that reduce the minimization of a strongly convex function to approximate minimizations of regularizations of the function. Using these results, we accelerate recent fast stochastic algorith…
We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence t…
University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）
In this paper we empirically evaluate biased methods for alpha-divergence minimization. In particular, we focus on how the bias affects the final solutions found, and how this depends on the dimensionality of the problem. We find that (i) solutions returned by these methods appear to be strongly biased towards minimizers of the traditional "exclusive" KL-divergence, KL(q||p), and (ii) in high dimensions, an impractically large amount of computation is needed to mitigate this bias and obtain solutions that actually minimize the alpha-divergence of interest.
Non-saturating generative adversarial network (GAN) training is widely used and has continued to obtain groundbreaking results. However so far this approach has lacked strong theoretical justification, in contrast to alternatives such as f-GANs and Wasserstein GANs which are motivated in terms of approximate divergence minimization. In this paper we show that non-saturating GAN training does in fact approximately minimize a particular f-divergence. We develop general theoretical tools to compare and classify f-divergences and use these to show that the new f-divergence is qualitatively similar…
While recent state-of-the-art results for adversarial imitation-learning algorithms are encouraging, recent works exploring the imitation learning from observation (ILO) setting, where trajectories \textit{only} contain expert observations, have not been met with the same success. Inspired by recent investigations of $f$-divergence manipulation for the standard imitation learning setting(Ke et al., 2019; Ghasemipour et al., 2019), we here examine the extent to which variations in the choice of probabilistic divergence may yield more performant ILO algorithms. We unfortunately find that $f$-div…
University of Washington（华盛顿大学）；Carnegie Mellon University（卡内基梅隆大学）
We address the problem of imitation learning with multi-modal demonstrations. Instead of attempting to learn all modes, we argue that in many tasks it is sufficient to imitate any one of them. We show that the state-of-the-art methods such as GAIL and behavior cloning, due to their choice of loss function, often incorrectly interpolate between such modes. Our key insight is to minimize the right divergence between the learner and the expert state-action distributions, namely the reverse KL divergence or I-projection. We propose a general imitation learning framework for estimating and minimizi…
Deep energy-based models (EBMs) are very flexible in distribution parametrization but computationally challenging because of the intractable partition function. They are typically trained via maximum likelihood, using contrastive divergence to approximate the gradient of the KL divergence between data and model distribution. While KL divergence has many desirable properties, other f-divergences have shown advantages in training implicit density generative models such as generative adversarial networks. In this paper, we propose a general variational framework termed f-EBM to train EBMs using a…
Several approximate inference algorithms have been proposed to minimize an alpha-divergence between an approximating distribution and a target distribution. Many of these algorithms introduce bias, the magnitude of which becomes problematic in high dimensions. Other algorithms are unbiased. These often seem to suffer from high variance, but little is rigorously known. In this work we study unbiased methods for alpha-divergence minimization through the Signal-to-Noise Ratio (SNR) of the gradient estimator. We study several representative scenarios where strong analytical results are possible, s…
Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）
Entropic regularization provides a generalization of the original optimal transport problem. It introduces a penalty term defined by the Kullback-Leibler divergence, making the problem more tractable via the celebrated Sinkhorn algorithm. Replacing the Kullback-Leibler divergence with a general $f$-divergence leads to a natural generalization. The case of divergences defined by superlinear functions was recently studied by Di Marino and Gerolin. Using convex analysis, we extend the theory developed so far to include all $f$-divergences defined by functions of Legendre type, and prove that unde…
Generative neural samplers are probabilistic models that implement sampling using feedforward neural networks: they take a random input vector and produce a sample from a probability distribution defined by the network weights. These models are expressive and allow efficient computation of samples and derivatives, but cannot be used for computing likelihoods or for marginalization. The generative-adversarial training method allows to train such models through the use of an auxiliary discriminative neural network. We show that the generative-adversarial approach is a special case of an existing…
Microsoft Research（微软研究院）；Ochanomizu University（御茶水女子大学）；University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）
We address the problem of enhancing model robustness through regularization. Specifically, we focus on methods that regularize the model posterior difference between clean and noisy inputs. Theoretically, we provide a connection of two recent methods, Jacobian Regularization and Virtual Adversarial Training, under this framework. Additionally, we generalize the posterior differential regularization to the family of $f$-divergences and characterize the overall regularization framework in terms of Jacobian matrix. Empirically, we systematically compare those regularizations and standard BERT tra…
We consider the nonlinear Kalman filtering problem using Kullback-Leibler (KL) and $α$-divergence measures as optimization criteria. Unlike linear Kalman filters, nonlinear Kalman filters do not have closed form Gaussian posteriors because of a lack of conjugacy due to the nonlinearity in the likelihood. In this paper we propose novel algorithms to optimize the forward and reverse forms of the KL divergence, as well as the alpha-divergence which contains these two as limiting cases. Unlike previous approaches, our algorithms do not make approximations to the divergences being optimized, but us…
Recruit Technologies Co., Ltd.（Recruit Technologies株式会社）；National Institute of Advanced Industrial Science and Technology (AIST)（日本产业技术综合研究所）；Kyoto University（京都大学）；Preferred Networks, Inc.（Preferred Networks公司）；ATR Cognitive Mechanisms Laboratories（ATR认知机制实验室）
We propose a new neural sequence model training method in which the objective function is defined by $α$-divergence. We demonstrate that the objective function generalizes the maximum-likelihood (ML)-based and reinforcement learning (RL)-based objective functions as special cases (i.e., ML corresponds to $α\to 0$ and RL to $α\to1$). We also show that the gradient of the objective function can be considered a mixture of ML- and RL-based objective gradients. The experimental results of a machine translation task show that minimizing the objective function with $α> 0$ outperforms $α\to 0$, which …
Generative adversarial networks (GANs) can be interpreted as an adversarial game between two players, a discriminator D and a generator G, in which D learns to classify real from fake data and G learns to generate realistic data by "fooling" D into thinking that fake data is actually real data. Currently, a dominating view is that G actually learns by minimizing a divergence given that the general objective function is a divergence when D is optimal. However, this view has been challenged due to inconsistencies between theory and practice. In this paper, we discuss of the properties associated…
We study the robustness properties of $\ell_1$ norm minimization for the classical linear regression problem with a given design matrix and contamination restricted to the dependent variable. We perform a fine error analysis of the $\ell_1$ estimator for measurements errors consisting of outliers coupled with noise. We introduce a new estimation technique resulting from a regularization of $\ell_1$ minimization by inf-convolution with the $\ell_2$ norm. Concerning robustness to large outliers, the proposed estimator keeps the breakdown point of the $\ell_1$ estimator, and reduces to least squa…
Given an empirical distribution $f(x)$ of sensitive data $x$, we consider the task of minimizing $F(y) = D_{\text{KL}} (f(x)\Vert y)$ over a probability simplex, while protecting the privacy of $x$. We observe that, if we take the exponential mechanism and use the KL divergence as the loss function, then the resulting algorithm is the Dirichlet mechanism that outputs a single draw from a Dirichlet distribution. Motivated by this, we propose a Rényi differentially private (RDP) algorithm that employs the Dirichlet mechanism to solve the KL divergence minimization task. In addition, given $f(x)$…
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.
$f$-Divergence Regularized RLHF: Two Tales of Sampling and Unified Analyses
Department of Electrical and Computer Engineering, University of Virginia, Virginia, United States（弗吉尼亚大学电气与计算机工程系）；Princeton Language and Intelligence, Princeton University, New Jersey, United States（普林斯顿大学语言与智能）
Variational Autoencoder (VAE), a simple and effective deep generative model, has led to a number of impressive empirical successes and spawned many advanced variants and theoretical investigations. However, recent studies demonstrate that, when equipped with expressive generative distributions (aka. decoders), VAE suffers from learning uninformative latent representations with the observation called KL Varnishing, in which case VAE collapses into an unconditional generative model. In this work, we introduce mutual posterior-divergence regularization, a novel regularization that is able to cont…
Generative adversarial networks (GANs) are a family of generative models that do not minimize a single training criterion. Unlike other generative models, the data distribution is learned via a game between a generator (the generative model) and a discriminator (a teacher providing training signal) that each minimize their own cost. GANs are designed to reach a Nash equilibrium at which each player cannot reduce their cost without changing the other players' parameters. One useful approach for the theory of GANs is to show that a divergence between the training distribution and the model distr…
As the emergence and the thriving development of social networks, a huge number of short texts are accumulated and need to be processed. Inferring latent topics of collected short texts is useful for understanding its hidden structure and predicting new contents. Unlike conventional topic models such as latent Dirichlet allocation (LDA), a biterm topic model (BTM) was recently proposed for short texts to overcome the sparseness of document-level word co-occurrences by directly modeling the generation process of word pairs. Stochastic inference algorithms based on collapsed Gibbs sampling (CGS)…
Google Brain（谷歌大脑）；DeepMind（深度思维）；University of Toronto（多伦多大学）；University College London（伦敦大学学院）
To learn directed behaviors in complex environments, intelligent agents need to optimize objective functions. Various objectives are known for designing artificial agents, including task rewards and intrinsic motivation. However, it is unclear how the known objectives relate to each other, which objectives remain yet to be discovered, and which objectives better describe the behavior of humans. We introduce the Action Perception Divergence (APD), an approach for categorizing the space of possible objective functions for embodied agents. We show a spectrum that reaches from narrow to general ob…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Google Brain（谷歌大脑）
In many settings, it is desirable to learn decision-making and control policies through learning or bootstrapping from expert demonstrations. The most common approaches under this Imitation Learning (IL) framework are Behavioural Cloning (BC), and Inverse Reinforcement Learning (IRL). Recent methods for IRL have demonstrated the capacity to learn effective policies with access to a very limited set of demonstrations, a scenario in which BC methods often fail. Unfortunately, due to multiple factors of variation, directly comparing these methods does not provide adequate intuition for understand…
This work elaborates on the important problem of (1) designing optimal randomized routing policies for reaching a target node t from a source note s on a weighted directed graph G and (2) defining distance measures between nodes interpolating between the least cost (based on optimal movements) and the commute-cost (based on a random walk on G), depending on a temperature parameter T. To this end, the randomized shortest path formalism (RSP, [2,99,124]) is rephrased in terms of Tsallis divergence regularization, instead of Kullback-Leibler divergence. The main consequence of this change is that…
Queen Mary University of London（伦敦玛丽女王大学）；Aarhus University（奥胡斯大学）；University of Oxford（牛津大学）；University of Lübeck（吕贝克大学）；KU Leuven（鲁汶大学）
Brain waves vary between people. An obvious way to improve automatic sleep staging for longitudinal sleep monitoring is personalization of algorithms based on individual characteristics extracted from the first night of data. As a single night is a very small amount of data to train a sleep staging model, we propose a Kullback-Leibler (KL) divergence regularized transfer learning approach to address this problem. We employ the pretrained SeqSleepNet (i.e. the subject independent model) as a starting point and finetune it with the single-night personalization data to derive the personalized mod…
University of California San Diego（加州大学圣地亚哥分校）；Academia Sinica（中央研究院）；Taiwan AI Labs（台湾人工智能实验室）
Recent years have seen adversarial losses been applied to many fields. Their applications extend beyond the originally proposed generative modeling to conditional generative and discriminative settings. While prior work has proposed various output activation functions and regularization approaches, some open questions still remain unanswered. In this paper, we aim to study the following two research questions: 1) What types of output activation functions form a well-behaved adversarial loss? 2) How different combinations of output activation functions and regularization approaches perform empi…
We propose an inference method to estimate sparse interactions and biases according to Boltzmann machine learning. The basis of this method is $L_1$ regularization, which is often used in compressed sensing, a technique for reconstructing sparse input signals from undersampled outputs. $L_1$ regularization impedes the simple application of the gradient method, which optimizes the cost function that leads to accurate estimations, owing to the cost function's lack of smoothness. In this study, we utilize the majorizer minimization method, which is a well-known technique implemented in optimizati…
Technische Universität Darmstadt（达姆施塔特工业大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
An optimal feedback controller for a given Markov decision process (MDP) can in principle be synthesized by value or policy iteration. However, if the system dynamics and the reward function are unknown, a learning agent must discover an optimal controller via direct interaction with the environment. Such interactive data gathering commonly leads to divergence towards dangerous or uninformative regions of the state space unless additional regularization measures are taken. Prior works proposed bounding the information loss measured by the Kullback-Leibler (KL) divergence at every policy improv…
Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）；McGill University（麦吉尔大学）；Facebook AI Research（脸书人工智能研究院）
Trust Region Policy Optimization (TRPO) and Proximal Policy Optimization (PPO) are among the most successful policy gradient approaches in deep reinforcement learning (RL). While these methods achieve state-of-the-art performance across a wide range of challenging tasks, there is room for improvement in the stabilization of the policy learning and how the off-policy data are used. In this paper we revisit the theoretical foundations of these algorithms and propose a new algorithm which stabilizes the policy improvement through a proximity term that constrains the discounted state-action visita…
Institute of Mathematical Sciences (ICMAT-CSIC)（数学科学研究所（ICMAT-CSIC））；Escuela Politécnica Superior, Universidad Autónoma de Madrid（马德里自治大学高等理工学院）
Neural networks are popular state-of-the-art models for many different tasks.They are often trained via back-propagation to find a value of the weights that correctly predicts the observed data. Although back-propagation has shown good performance in many applications, it cannot easily output an estimate of the uncertainty in the predictions made. Estimating the uncertainty in the predictions is a critical aspect with important applications, and one method to obtain this information is following a Bayesian approach to estimate a posterior distribution on the model parameters. This posterior di…
A regularized risk minimization procedure for regression function estimation is introduced that achieves near optimal accuracy and confidence under general conditions, including heavy-tailed predictor and response variables. The procedure is based on median-of-means tournaments, introduced by the authors in [8]. It is shown that the new procedure outperforms standard regularized empirical risk minimization procedures such as lasso or slope in heavy-tailed problems.
Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may apply, and a standard test-bed for studying this prospect is matrix factorization (matrix completion via linear neural networks). It is an open question whether norms can explain the implicit regularization in matrix factorization. The current paper resolves this open question in the negative, by proving that there exist natural matrix factorization problems on w…
We propose a new optimization algorithm for Multiple Kernel Learning (MKL) called SpicyMKL, which is applicable to general convex loss functions and general types of regularization. The proposed SpicyMKL iteratively solves smooth minimization problems. Thus, there is no need of solving SVM, LP, or QP internally. SpicyMKL can be viewed as a proximal minimization method and converges super-linearly. The cost of inner minimization is roughly proportional to the number of active kernels. Therefore, when we aim for a sparse kernel combination, our algorithm scales well against increasing number of …
We focus on the maximum regularization parameter for anisotropic total-variation denoising. It corresponds to the minimum value of the regularization parameter above which the solution remains constant. While this value is well know for the Lasso, such a critical value has not been investigated in details for the total-variation. Though, it is of importance when tuning the regularization parameter as it allows fixing an upper-bound on the grid for which the optimal parameter is sought. We establish a closed form expression for the one-dimensional case, as well as an upper-bound for the two-dim…
