Meta-learning, or "learning to learn", refers to techniques that infer an inductive bias from data corresponding to multiple related tasks with the goal of improving the sample efficiency for new, previously unobserved, tasks. A key performance measure for meta-learning is the meta-generalization gap, that is, the difference between the average loss measured on the meta-training data and on a new, randomly selected task. This paper presents novel information-theoretic upper bounds on the meta-generalization gap. Two broad classes of meta-learning algorithms are considered that uses either separate within-task training and test sets, like MAML, or joint within-task training and test sets, like Reptile. Extending the existing work for conventional learning, an upper bound on the meta-generalization gap is derived for the former class that depends on the mutual information (MI) between the output of the meta-learning algorithm and its input meta-training data. For the latter, the derived bound includes an additional MI between the output of the per-task learning procedure and corresponding data set to capture within-task uncertainty. Tighter bounds are then developed, under given technical conditions, for the two classes via novel Individual Task MI (ITMI) bounds. Applications of the derived bounds are finally discussed, including a broad class of noisy iterative algorithms for meta-learning.
Chalmers University of Technology（查尔姆斯理工大学）；Kings College London（伦敦国王学院）
Meta-learning optimizes an inductive bias---typically in the form of the hyperparameters of a base-learning algorithm---by observing data from a finite number of related tasks. This paper presents an information-theoretic bound on the generalization performance of any given meta-learner, which builds on the conditional mutual information (CMI) framework of Steinke and Zakynthinou (2020). In the proposed extension to meta-learning, the CMI bound involves a training \textit{meta-supersample} obtained by first sampling $2N$ independent tasks from the task environment, and then drawing $2M$ indepe…
We derive a novel information-theoretic analysis of the generalization property of meta-learning algorithms. Concretely, our analysis proposes a generic understanding of both the conventional learning-to-learn framework and the modern model-agnostic meta-learning (MAML) algorithms. Moreover, we provide a data-dependent generalization bound for a stochastic variant of MAML, which is non-vacuous for deep few-shot learning. As compared to previous bounds that depend on the square norm of gradients, empirical validations on both simulated data and a well-known few-shot benchmark show that our boun…
A Unified Information-Theoretic Framework for Meta-Learning Generalization
School of Computer Science of Technology（科技学院计算机科学系）；School of Biomedical Informatics（生物医学信息学系）；Department of Oncology（肿瘤学系）；Department of Computer Science and Technology（计算机科学与技术系）
We are motivated by the problem of providing strong generalization guarantees in the context of meta-learning. Existing generalization bounds are either challenging to evaluate or provide vacuous guarantees in even relatively simple settings. We derive a probably approximately correct (PAC) bound for gradient-based meta-learning using two different generalization frameworks in order to deal with the qualitatively different challenges of generalization at the "base" and "meta" levels. We employ bounds for uniformly stable algorithms at the base level and bounds from the PAC-Bayes framework at t…
We derive upper bounds on the generalization error of a learning algorithm in terms of the mutual information between its input and output. The bounds provide an information-theoretic understanding of generalization in learning problems, and give theoretical guidelines for striking the right balance between data fit and generalization by controlling the input-output mutual information. We propose a number of methods for this purpose, among which are algorithms that regularize the ERM algorithm with relative entropy or with random noise. Our work extends and leads to nontrivial improvements on …
Tsinghua University（清华大学）；The Ohio State University（俄亥俄州立大学）
Despite the superior empirical success of deep meta-learning, theoretical understanding of overparameterized meta-learning is still limited. This paper studies the generalization of a widely used meta-learning approach, Model-Agnostic Meta-Learning (MAML), which aims to find a good initialization for fast adaptation to new tasks. Under a mixed linear regression model, we analyze the generalization properties of MAML trained with SGD in the overparameterized regime. We provide both upper and lower bounds for the excess risk of MAML, which captures how SGD dynamics affect these generalization bo…
Massachusetts Institute of Technology（麻省理工学院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
In this paper, we study the generalization properties of Model-Agnostic Meta-Learning (MAML) algorithms for supervised learning problems. We focus on the setting in which we train the MAML model over $m$ tasks, each with $n$ data points, and characterize its generalization error from two points of view: First, we assume the new task at test time is one of the training tasks, and we show that, for strongly convex objective functions, the expected excess population loss is bounded by ${\mathcal{O}}(1/mn)$. Second, we consider the MAML algorithm's generalization to an unseen task and show that th…
The goal of domain generalization is to learn from multiple source domains to generalize to unseen target domains under distribution discrepancy. Current state-of-the-art methods in this area are fully supervised, but for many real-world problems it is hardly possible to obtain enough labeled samples. In this paper, we propose the first method of domain generalization to leverage unlabeled samples, combining of meta learning's episodic training and semi-supervised learning, called DGSML. DGSML employs an entropy-based pseudo-labeling approach to assign labels to unlabeled samples and then util…
Generalization error bounds are critical to understanding the performance of machine learning models. In this work, building upon a new bound of the expected value of an arbitrary function of the population and empirical risk of a learning algorithm, we offer a more refined analysis of the generalization behaviour of a machine learning models based on a characterization of (bounds) to their generalization error moments. We discuss how the proposed bounds -- which also encompass new bounds to the expected generalization error -- relate to existing bounds in the literature. We also discuss how t…
Identifying anomaly multimedia traffic in cyberspace is a big challenge in distributed service systems, multiple generation networks and future internet of everything. This letter explores meta-generalization for a multiparty privacy learning model in graynet to improve the performance of anomaly multimedia traffic identification. The multiparty privacy learning model in graynet is a globally shared model that is partitioned, distributed and trained by exchanging multiparty parameters updates with preserving private data. The meta-generalization refers to discovering the inherent attributes of…
By leveraging experience from previous tasks, meta-learning algorithms can achieve effective fast adaptation ability when encountering new tasks. However it is unclear how the generalization property applies to new tasks. Probably approximately correct (PAC) Bayes bound theory provides a theoretical framework to analyze the generalization performance for meta-learning. We derive three novel generalisation error bounds for meta-learning based on PAC-Bayes relative entropy bound. Furthermore, using the empirical risk minimization (ERM) method, a PAC-Bayes bound for meta-learning with data-depend…
Meta learning automatically infers an inductive bias, that includes the hyperparameter of the base-learning algorithm, by observing data from a finite number of related tasks. This paper studies PAC-Bayes bounds on meta generalization gap. The meta-generalization gap comprises two sources of generalization gaps: the environment-level and task-level gaps resulting from observation of a finite number of tasks and data samples per task, respectively. In this paper, by upper bounding arbitrary convex functions, which link the expected and empirical losses at the environment and also per-task level…
This paper follows up on a recent work of Neu et al. (2021) and presents some new information-theoretic upper bounds for the generalization error of machine learning models, such as neural networks, trained with SGD. We apply these bounds to analyzing the generalization behaviour of linear and two-layer ReLU networks. Experimental study of these bounds provide some insights on the SGD training of neural networks. They also point to a new and simple regularization scheme which we show performs comparably to the current state of the art.
Some of the tightest information-theoretic generalization bounds depend on the average information between the learned hypothesis and a single training example. However, these sample-wise bounds were derived only for expected generalization gap. We show that even for expected squared generalization gap no such sample-wise information-theoretic bounds exist. The same is true for PAC-Bayes and single-draw bounds. Remarkably, PAC-Bayes, single-draw and expected squared generalization gap bounds that depend on information in pairs of examples exist.
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Gradient-based meta-learning (GBML) with deep neural nets (DNNs) has become a popular approach for few-shot learning. However, due to the non-convexity of DNNs and the bi-level optimization in GBML, the theoretical properties of GBML with DNNs remain largely unknown. In this paper, we first aim to answer the following question: Does GBML with DNNs have global convergence guarantees? We provide a positive answer to this question by proving that GBML with over-parameterized DNNs is guaranteed to converge to global optima at a linear rate. The second question we aim to address is: How does GBML a…
USC Information Sciences Institute（南加州大学信息科学研究所）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.
Using information-theoretic principles, we consider the generalization error (gen-error) of iterative semi-supervised learning (SSL) algorithms that iteratively generate pseudo-labels for a large amount of unlabelled data to progressively refine the model parameters. In contrast to most previous works that {\em bound} the gen-error, we provide an {\em exact} expression for the gen-error and particularize it to the binary Gaussian mixture model. Our theoretical results suggest that when the class conditional variances are not too large, the gen-error decreases with the number of iterations, but…
University of Science & Technology of China（中国科学技术大学）；Microsoft Research Asia（微软亚洲研究院）；University of Washington（华盛顿大学）
Recently, the information-theoretical framework has been proven to be able to obtain non-vacuous generalization bounds for large models trained by Stochastic Gradient Langevin Dynamics (SGLD) with isotropic noise. In this paper, we optimize the information-theoretical generalization bound by manipulating the noise structure in SGLD. We prove that with constraint to guarantee low empirical risk, the optimal noise covariance is the square root of the expected gradient covariance if both the prior and the posterior are jointly optimized. This validates that the optimal noise is quite close to the…
Universitat Pompeu Fabra（庞培法布拉大学）；Element AI（Element AI公司）；Mila（米拉实验室）；University of Toronto（多伦多大学）；Vector Institute（矢量研究院）
We study the generalization properties of the popular stochastic optimization method known as stochastic gradient descent (SGD) for optimizing general non-convex loss functions. Our main contribution is providing upper bounds on the generalization error that depend on local statistics of the stochastic gradients evaluated along the path of iterates calculated by SGD. The key factors our bounds depend on are the variance of the gradients (with respect to the data distribution) and the local smoothness of the objective function along the SGD path, and the sensitivity of the loss function to pert…
University of Illinois（伊利诺伊大学）；University of Chicago, Booth School of Business（芝加哥大学布斯商学院）
We study the detailed path-wise behavior of the discrete-time Langevin algorithm for non-convex Empirical Risk Minimization (ERM) through the lens of metastability, adopting some techniques from Berglund and Gentz (2003. For a particular local optimum of the empirical risk, with an arbitrary initialization, we show that, with high probability, at least one of the following two events will occur: (1) the Langevin trajectory ends up somewhere outside the $\varepsilon$-neighborhood of this particular optimum within a short recurrence time; (2) it enters this $\varepsilon$-neighborhood by the recu…
Rensselaer Polytechnic Institute（伦斯勒理工学院）；Michigan State University（密歇根州立大学）；MIT-IBM Watson AI Lab, IBM Research（麻省理工学院-IBM沃森人工智能实验室（IBM研究院））；IBM Research（IBM研究院）；University at Buffalo（布法罗大学）
Self-training, a semi-supervised learning algorithm, leverages a large amount of unlabeled data to improve learning when the labeled data are limited. Despite empirical successes, its theoretical characterization remains elusive. To the best of our knowledge, this work establishes the first theoretical analysis for the known iterative self-training paradigm and proves the benefits of unlabeled data in both training convergence and generalization ability. To make our theoretical analysis feasible, we focus on the case of one-hidden-layer neural networks. However, theoretical understanding of it…
Information-Theoretic Generalization Bounds of Replay-based Continual Learning
School of Computer Science and Technology, Xi’an Jiaotong University（西安交通大学计算机科学与技术学院）；Department of Oncology, University of Cambridge（剑桥大学肿瘤学系）；China Telecom, China（中国电信）；Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系）
Tencent AI Lab（腾讯人工智能实验室）；HongKong University of Science and Technology（香港科技大学）；Zhejiang University（浙江大学）
Recently, generalization bounds of the non-convex empirical risk minimization paradigm using Stochastic Gradient Langevin Dynamics (SGLD) have been extensively studied. Several theoretical frameworks have been presented to study this problem from different perspectives, such as information theory and stability. In this paper, we present a unified view from privacy leakage analysis to investigate the generalization bounds of SGLD, along with a theoretical framework for re-deriving previous results in a succinct manner. Aside from theoretical findings, we conduct various numerical studies to emp…
For Internet applications like sponsored search, cautions need to be taken when using machine learning to optimize their mechanisms (e.g., auction) since self-interested agents in these applications may change their behaviors (and thus the data distribution) in response to the mechanisms. To tackle this problem, a framework called game-theoretic machine learning (GTML) was recently proposed, which first learns a Markov behavior model to characterize agents' behaviors, and then learns the optimal mechanism by simulating agents' behavior changes in response to the mechanism. While GTML has demon…
Pre-Training and Personalized Fine-Tuning via Over-the-Air Federated Meta-Learning: Convergence-Generalization Trade-Offs
The Hong Kong University of Science and Technology (Guangzhou)（香港科学与技术大学（广州））；Department of ECE, The Hong Kong University of Science and Technology（香港科学与技术大学电子工程系）；Department of Engineering, King’s College London（伦敦国王学院工程系）
Mila；Université de Montréal（蒙特利尔大学）；Aalto University（阿尔托大学）；École Polytechnique de Montréal（蒙特利尔高等理工学院）；ElementAI
In this work we study generalization of neural networks in gradient-based meta-learning by analyzing various properties of the objective landscapes. We experimentally demonstrate that as meta-training progresses, the meta-test solutions, obtained after adapting the meta-train solution of the model, to new tasks via few steps of gradient-based fine-tuning, become flatter, lower in loss, and further away from the meta-train solution. We also show that those meta-test solutions become flatter even as generalization starts to degrade, thus providing an experimental evidence against the correlation…
Stanford University（斯坦福大学）；Tencent AI Lab（腾讯人工智能实验室）；Rutgers University（罗格斯大学）；City University of Hong Kong（香港城市大学）；Pennsylvania State University（宾夕法尼亚州立大学）
Meta-learning has proven to be a powerful paradigm for transferring the knowledge from previous tasks to facilitate the learning of a novel task. Current dominant algorithms train a well-generalized model initialization which is adapted to each task via the support set. The crux lies in optimizing the generalization capability of the initialization, which is measured by the performance of the adapted model on the query set of each task. Unfortunately, this generalization measure, evidenced by empirical results, pushes the initialization to overfit the meta-training tasks, which significantly i…
A machine learning model that generalizes well should obtain low errors on unseen test examples. Thus, if we know how to optimally perturb training examples to account for test examples, we may achieve better generalization performance. However, obtaining such perturbation is not possible in standard machine learning frameworks as the distribution of the test data is unknown. To tackle this challenge, we propose a novel regularization method, meta-dropout, which learns to perturb the latent features of training examples for generalization in a meta-learning framework. Specifically, we meta-lea…
Meta-learning aims at optimizing the hyperparameters of a model class or training algorithm from the observation of data from a number of related tasks. Following the setting of Baxter [1], the tasks are assumed to belong to the same task environment, which is defined by a distribution over the space of tasks and by per-task data distributions. The statistical properties of the task environment thus dictate the similarity of the tasks. The goal of the meta-learner is to ensure that the hyperparameters obtain a small loss when applied for training of a new task sampled from the task environment…
Mila - McGill University（米拉 - 麦吉尔大学）；Google DeepMind（谷歌DeepMind）
Due to the realization that deep reinforcement learning algorithms trained on high-dimensional tasks can strongly overfit to their training environments, there have been several studies that investigated the generalization performance of these algorithms. However, there has been no similar study that evaluated the generalization performance of algorithms that were specifically designed for generalization, i.e. meta-reinforcement learning algorithms. In this paper, we assess the generalization performance of these algorithms by leveraging high-dimensional, procedurally generated environments. W…
Modular meta-learning is a new framework that generalizes to unseen datasets by combining a small set of neural modules in different ways. In this work we propose abstract graph networks: using graphs as abstractions of a system's subparts without a fixed assignment of nodes to system subparts, for which we would need supervision. We combine this idea with modular meta-learning to get a flexible framework with combinatorial generalization to new tasks built in. We then use it to model the pushing of arbitrarily shaped objects from little or no training data.
The Swiss AI Lab IDSIA（瑞士人工智能实验室IDSIA）；USI（瑞士意大利语区大学）；SUPSI（瑞士南方应用科技大学）
Biological evolution has distilled the experiences of many learners into the general learning algorithms of humans. Our novel meta reinforcement learning algorithm MetaGenRL is inspired by this process. MetaGenRL distills the experiences of many complex agents to meta-learn a low-complexity neural objective function that decides how future individuals will learn. Unlike recent meta-RL algorithms, MetaGenRL can generalize to new environments that are entirely different from those used for meta-training. In some cases, it even outperforms human-engineered RL algorithms. MetaGenRL uses off-policy…
Computer Science Department State University of New York at Binghamton（纽约州立大学宾汉姆顿分校计算机科学系）
Substantial efforts have been made on improving the generalization abilities of deep neural networks (DNNs) in order to obtain better performances without introducing more parameters. On the other hand, meta-learning approaches exhibit powerful generalization on new tasks in few-shot learning. Intuitively, few-shot learning is more challenging than the standard supervised learning as each target class only has a very few or no training samples. The natural question that arises is whether the meta-learning idea can be used for improving the generalization of DNNs on the standard supervised lear…
$μ$LO: Compute-Efficient Meta-Generalization of Learned Optimizers
Université de Montréal（蒙特利尔大学）；Mila – Quebec AI Institute（魁北克人工智能研究所）；Concordia University（康科迪亚大学）；Samsung AI Lab（三星AI实验室）；ISIR, Sorbonne University, CNRS, Paris, France（巴黎法国索邦大学ISIR研究所、法国国家科学研究中心）
The generalization ability of most meta-reinforcement learning (meta-RL) methods is largely limited to test tasks that are sampled from the same distribution used to sample training tasks. To overcome the limitation, we propose Latent Dynamics Mixture (LDM) that trains a reinforcement learning agent with imaginary tasks generated from mixtures of learned latent dynamics. By training a policy on mixture tasks along with original training tasks, LDM allows the agent to prepare for unseen test tasks during training and prevents the agent from overfitting the training tasks. LDM significantly outp…
Neural networks have been successfully applied in applications with a large amount of labeled data. However, the task of rapid generalization on new concepts with small training data while preserving performances on previously learned ones still presents a significant challenge to neural network models. In this work, we introduce a novel meta learning method, Meta Networks (MetaNet), that learns a meta-level knowledge across tasks and shifts its inductive biases via fast parameterization for rapid generalization. When evaluated on Omniglot and Mini-ImageNet benchmarks, our MetaNet models achie…
Meta-learning provides a promising way for learning to efficiently learn and achieves great success in many applications. However, most meta-learning literature focuses on dealing with tasks from a same domain, making it brittle to generalize to tasks from the other unseen domains. In this work, we address this problem by simulating tasks from the other unseen domains to improve the generalization and robustness of meta-learning method. Specifically, we propose a model-agnostic shift layer to learn how to simulate the domain shift and generate pseudo tasks, and develop a new adversarial learni…
The generalization capability of machine learning models, which refers to generalizing the knowledge for an "unseen" domain via learning from one or multiple seen domain(s), is of great importance to develop and deploy machine learning applications in the real-world conditions. Domain Generalization (DG) techniques aim to enhance such generalization capability of machine learning models, where the learnt feature representation and the classifier are two crucial factors to improve generalization and make decisions. In this paper, we propose Discriminative Adversarial Domain Generalization (DADG…
Shanghai Jiao Tong University（上海交通大学）；The Pennsylvania State University（宾夕法尼亚州立大学）
The heavy traffic congestion problem has always been a concern for modern cities. To alleviate traffic congestion, researchers use reinforcement learning (RL) to develop better traffic signal control (TSC) algorithms in recent years. However, most RL models are trained and tested in the same traffic flow environment, which results in a serious overfitting problem. Since the traffic flow environment in the real world keeps varying, these models can hardly be applied due to the lack of generalization ability. Besides, the limited number of accessible traffic flow data brings extra difficulty in …
