Provable Data Scaling Law for Meta Learning via Complexity Minimization
University of Tsukuba（筑波大学）；RIKEN AIP（理化学研究所人工智能研究中心）；SB Intuitions Corp.（SB Intuitions公司）；Kyoto University（京都大学）；Shiga University（滋贺大学）；Institute of Science Tokyo（东京科学研究所）
Pre-training has become a fundamental paradigm in modern machine learning, with one of its key empirical benefits being reduced downstream sample complexity as the scale of pre-training data increases. However, existing theoretical frameworks for pre-training do not fully explain this phenomenon. In this paper, we introduce complexity minimization, a novel meta-representation learning framework designed to enable theoretical analysis of this scaling behavior, which learns representations by evaluating the downstream model complexity best suited to each domain and minimizing the worst-case such complexity across source domains. Our end-to-end theoretical analysis, spanning pre-training through downstream regression, shows that this framework provably captures this scaling behavior; in particular, we show that the error rate of few-shot adaptation improves as the amount of meta-training data grows. Empirically, we demonstrate that incorporating complexity regularization into existing meta-learning methods consistently improves downstream sample efficiency.
We study the problem of meta-learning through the lens of online convex optimization, developing a meta-algorithm bridging the gap between popular gradient-based meta-learning and classical regularization-based multi-task transfer methods. Our method is the first to simultaneously satisfy good sample efficiency guarantees in the convex setting, with generalization bounds that improve with task-similarity, while also being computationally scalable to modern deep learning architectures and the many-task setting. Despite its simplicity, the algorithm matches, up to a constant factor, a lower boun…
University of California, Berkeley（加州大学伯克利分校）；Princeton University（普林斯顿大学）
Meta-learning, or learning-to-learn, seeks to design algorithms that can utilize previous experience to rapidly learn new skills or adapt to new environments. Representation learning -- a key tool for performing meta-learning -- learns a data representation that can transfer knowledge across multiple tasks, which is essential in regimes where data is scarce. Despite a recent surge of interest in the practice of meta-learning, the theoretical underpinnings of meta-learning algorithms are lacking, especially in the context of learning transferable representations. In this paper, we focus on the …
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Domain generalization asks for models trained over a set of training environments to perform well in unseen test environments. Recently, a series of algorithms such as Invariant Risk Minimization (IRM) has been proposed for domain generalization. However, Rosenfeld et al. (2021) shows that in a simple linear data model, even if non-convexity issues are ignored, IRM and its extensions cannot generalize to unseen environments with less than $d_s+1$ training environments, where $d_s$ is the dimension of the spurious-feature subspace. In this paper, we propose to achieve domain generalization with…
Domain generalization aims at performing well on unseen test environments with data from a limited number of training environments. Despite a proliferation of proposal algorithms for this task, assessing their performance both theoretically and empirically is still very challenging. Distributional matching algorithms such as (Conditional) Domain Adversarial Networks [Ganin et al., 2016, Long et al., 2018] are popular and enjoy empirical success, but they lack formal guarantees. Other approaches such as Invariant Risk Minimization (IRM) require a prohibitively large number of training environme…
In this work we provide provable regret guarantees for an online meta-learning control algorithm in an iterative control setting, where in each iteration the system to be controlled is a linear deterministic system that is different and unknown, the cost for the controller in an iteration is a general additive cost function and the control input is required to be constrained, which if violated incurs an additional cost. We prove (i) that the algorithm achieves a regret for the controller cost and constraint violation that are $O(T^{3/4})$ for an episode of duration $T$ with respect to the best…
Chandra Family Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX, USA（Chandra Family 电子与计算机工程系，德克萨斯大学奥斯汀分校）；Snap, Inc.（Snap公司）；Google Research（Google研究）
Meta-Learning is a family of methods that use a set of interrelated tasks to learn a model that can quickly learn a new query task from a possibly small contextual dataset. In this study, we use a probabilistic framework to formalize what it means for two tasks to be related and reframe the meta-learning problem into the problem of Bayesian risk minimization (BRM). In our formulation, the BRM optimal solution is given by the predictive distribution computed from the posterior distribution of the task-specific latent variable conditioned on the contextual dataset, and this justifies the philoso…
Meta-learning optimizes the hyperparameters of a training procedure, such as its initialization, kernel, or learning rate, based on data sampled from a number of auxiliary tasks. A key underlying assumption is that the auxiliary tasks, known as meta-training tasks, share the same generating distribution as the tasks to be encountered at deployment time, known as meta-test tasks. This may, however, not be the case when the test environment differ from the meta-training conditions. To address shifts in task generating distribution between meta-training and meta-testing phases, this paper introdu…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Google Research（谷歌研究院）；University of Washington（华盛顿大学）
Meta-learning synthesizes and leverages the knowledge from a given set of tasks to rapidly learn new tasks using very little data. Meta-learning of linear regression tasks, where the regressors lie in a low-dimensional subspace, is an extensively-studied fundamental problem in this domain. However, existing results either guarantee highly suboptimal estimation errors, or require $Ω(d)$ samples per task (where $d$ is the data dimensionality) thus providing little gain over separately learning each task. In this work, we study a simple alternating minimization method (MLLAM), which alternately l…
King’s College London（伦敦国王学院）；Chalmers Institute of Technology（查尔姆斯理工大学）
Meta-learning automatically infers an inductive bias by observing data from a number of related tasks. The inductive bias is encoded by hyperparameters that determine aspects of the model class or training algorithm, such as initialization or learning rate. Meta-learning assumes that the learning tasks belong to a task environment, and that tasks are drawn from the same task environment both during meta-training and meta-testing. This, however, may not hold true in practice. In this paper, we introduce the problem of transfer meta-learning, in which tasks are drawn from a target task environme…
Hierarchical reinforcement learning (HRL) has seen widespread interest as an approach to tractable learning of complex modular behaviors. However, existing work either assume access to expert-constructed hierarchies, or use hierarchy-learning heuristics with no provable guarantees. To address this gap, we analyze HRL in the meta-RL setting, where a learner learns latent hierarchical structure during meta-training for use in a downstream task. We consider a tabular setting where natural hierarchical structure is embedded in the transition dynamics. Analogous to supervised meta-learning theory, …
Meta Curvature-Aware Minimization for Domain Generalization
School of Computer Science and Engineering, Northwestern Polytechnical University（西北工业大学计算机科学与工程学院）；Faculty of Engineering, Monash University（莫纳什大学工程学院）；Research & Development Institute of Northwestern Polytechnical University in Shenzhen（西北工业大学深圳研究院）；Ningbo Institute of Northwestern Polytechnical University（西北工业大学宁波研究院）
We study the problem of low-rank tensor factorization in the presence of missing data. We ask the following question: how many sampled entries do we need, to efficiently and exactly reconstruct a tensor with a low-rank orthogonal decomposition? We propose a novel alternating minimization based method which iteratively refines estimates of the singular vectors. We show that under certain standard assumptions, our method can recover a three-mode $n\times n\times n$ dimensional rank-$r$ tensor exactly from $O(n^{3/2} r^5 \log^4 n)$ randomly sampled entries. In the process of proving this result, …
Rensselaer Polytechnic Institute（伦斯勒理工学院）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Model-agnostic meta learning (MAML) is currently one of the dominating approaches for few-shot meta-learning. Albeit its effectiveness, the optimization of MAML can be challenging due to the innate bilevel problem structure. Specifically, the loss landscape of MAML is much more complex with possibly more saddle points and local minimizers than its empirical risk minimization counterpart. To address this challenge, we leverage the recently invented sharpness-aware minimization and develop a sharpness-aware MAML approach that we term Sharp-MAML. We empirically demonstrate that Sharp-MAML and its…
Yahoo Research（雅虎研究院）；JP Morgan & Co（摩根大通公司）；U.S. Bancorp（美国合众银行）；Xiaohongshu（小红书）
Accurate understanding of users in terms of predicative segments play an essential role in the day to day operation of modern internet enterprises. Nevertheless, there are significant challenges that limit the quality of data, especially on long tail predictive tasks. In this work, we present MetaCon, our unified predicative segments system with scalable, trillion concepts meta learning that addresses these challenges. It builds on top of a flat concept representation that summarizes entities' heterogeneous digital footprint, jointly considers the entire spectrum of predicative tasks as a sing…
Parameter-transfer is a well-known and versatile approach for meta-learning, with applications including few-shot learning, federated learning, and reinforcement learning. However, parameter-transfer algorithms often require sharing models that have been trained on the samples from specific tasks, thus leaving the task-owners susceptible to breaches of privacy. We conduct the first formal study of privacy in this setting and formalize the notion of task-global differential privacy as a practical relaxation of more commonly studied threat models. We then propose a new differentially private alg…
In this paper we provide provable regret guarantees for an online meta-learning receding horizon control algorithm in an iterative control setting. We consider the setting where, in each iteration the system to be controlled is a linear deterministic system that is different and unknown, the cost for the controller in an iteration is a general additive cost function and there are affine control input constraints. By analysing conditions under which sub-linear regret is achievable, we prove that the meta-learning online receding horizon controller achieves an average of the dynamic regret for t…
We study conditional risk minimization (CRM), i.e. the problem of learning a hypothesis of minimal risk for prediction at the next step of sequentially arriving dependent data. Despite it being a fundamental problem, successful learning in the CRM sense has so far only been demonstrated using theoretical algorithms that cannot be used for real problems as they would require storing all incoming data. In this work, we introduce MACRO, a meta-algorithm for CRM that does not suffer from this shortcoming, but nevertheless offers learning guarantees. Instead of storing all data it maintains and ite…
Developing classification algorithms that are fair with respect to sensitive attributes of the data has become an important problem due to the growing deployment of classification algorithms in various social contexts. Several recent works have focused on fairness with respect to a specific metric, modeled the corresponding fair classification problem as a constrained optimization problem, and developed tailored algorithms to solve them. Despite this, there still remain important metrics for which we do not have fair classifiers and many of the aforementioned algorithms do not come with theore…
Unsupervised pretraining, which learns a useful representation using a large amount of unlabeled data to facilitate the learning of downstream tasks, is a critical component of modern large-scale machine learning systems. Despite its tremendous empirical success, the rigorous theoretical understanding of why unsupervised pretraining generally helps remains rather limited -- most existing results are restricted to particular methods or approaches for unsupervised pretraining with specialized structural assumptions. This paper studies a generic framework, where the unsupervised representation le…
National University of Defense Technology（国防科技大学）；Beihang University（北京航空航天大学）；Imperial College London（帝国理工学院）；University of Modena and Reggio Emilia (UNIMORE)（摩德纳-雷焦艾米利亚大学（UNIMORE））
In this paper, we propose a novel solution for non-convex problems of multiple variables, especially for those typically solved by an alternating minimization (AM) strategy that splits the original optimization problem into a set of sub-problems corresponding to each variable, and then iteratively optimize each sub-problem using a fixed updating rule. However, due to the intrinsic non-convexity of the original optimization problem, the optimization can usually be trapped into spurious local minimum even when each sub-problem can be optimally solved at each iteration. Meanwhile, learning-based …
In this work we propose a graph-based learning framework to train models with provable robustness to adversarial perturbations. In contrast to regularization-based approaches, we formulate the adversarially robust learning problem as one of loss minimization with a Lipschitz constraint, and show that the saddle point of the associated Lagrangian is characterized by a Poisson equation with weighted Laplace operator. Further, the weighting for the Laplace operator is given by the Lagrange multiplier for the Lipschitz constraint, which modulates the sensitivity of the minimizer to perturbations. …
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…
