Online-Within-Online (OWO) meta learning stands for the online multi-task learning paradigm in which both tasks and data within each task become available in a sequential order. In this work, we study the OWO meta learning of the initialization and step size of within-task online algorithms in the non-convex setting, and provide improved regret bounds under mild assumptions of loss functions. Previous work analyzing this scenario has obtained for bounded and piecewise Lipschitz functions an averaged regret bound $O((\frac{\sqrt{m}}{T^{1/4}}+\frac{(\log{m})\log{T}}{\sqrt{T}}+V)\sqrt{m})$ across $T$ tasks, with $m$ iterations per task and $V$ the task similarity. Our first contribution is to modify the existing non-convex OWO meta learning algorithm and improve the regret bound to $O((\frac{1}{T^{1/2-\alpha}}+\frac{(\log{T})^{9/2}}{T}+V)\sqrt{m})$, for any $\alpha \in (0,1/2)$. The derived bound has a faster convergence rate with respect to $T$, and guarantees a vanishing task-averaged regret with respect to $m$ (for any fixed $T$). Then, we propose a new algorithm of regret $O((\frac{\log{T}}{T}+V)\sqrt{m})$ for non-convex OWO meta learning. This regret bound exhibits a better asymptotic performance than previous ones, and holds for any bounded (not necessarily Lipschitz) loss functions. Besides the improved regret bounds, our contributions include investigating how to attain generalization bounds for statistical meta learning via regret analysis. Specifically, by online-to-batch arguments, we achieve a transfer risk bound for batch meta learning that assumes all tasks are drawn from a distribution. Moreover, by connecting multi-task generalization error with task-averaged regret, we develop for statistical multi-task learning a novel PAC-Bayes generalization error bound that involves our regret bound for OWO meta learning.

Hierarchical Bayesian bandit refers to the multi-task bandit problem in which bandit tasks are assumed to be drawn from the same distribution. In this work, we provide improved Bayes regret bounds for hierarchical Bayesian bandit algorithms in the multi-task linear bandit and semi-bandit settings. For the multi-task linear bandit, we first analyze the preexisting hierarchical Thompson sampling (HierTS) algorithm, and improve its gap-independent Bayes regret bound from $O(m\sqrt{n\log{n}\log{(mn)}})$ to $O(m\sqrt{n\log{n}})$ in the case of infinite action set, with $m$ being the number of tasks and $n$ the number of iterations per task. In the case of finite action set, we propose a novel hierarchical Bayesian bandit algorithm, named hierarchical BayesUCB (HierBayesUCB), that achieves the logarithmic but gap-dependent regret bound $O(m\log{(mn)}\log{n})$ under mild assumptions. All of the above regret bounds hold in many variants of hierarchical Bayesian linear bandit problem, including when the tasks are solved sequentially or concurrently. Furthermore, we extend the aforementioned HierTS and HierBayesUCB algorithms to the multi-task combinatorial semi-bandit setting. Concretely, our combinatorial HierTS algorithm attains comparable Bayes regret bound $O(m\sqrt{n}\log{n})$ with respect to the latest one. Moreover, our combinatorial HierBayesUCB yields a sharper Bayes regret bound $O(m\log{(mn)}\log{n})$. Experiments are conducted to validate the soundness of our theoretical results for multi-task bandit algorithms.

PAC-Bayesian error bounds provide a theoretical guarantee on the generalization abilities of meta-learning from training tasks to unseen tasks. However, it is still unclear how tight PAC-Bayesian bounds we can achieve for meta-learning. In this work, we propose a general PAC-Bayesian framework to cope with single-task learning and meta-learning uniformly. With this framework, we generalize the two tightest PAC-Bayesian bounds (i.e., kl-bound and Catoni-bound) from single-task learning to standard meta-learning, resulting in fast convergence rates for PAC-Bayesian meta-learners. By minimizing the derived two bounds, we develop two meta-learning algorithms for classification problems with deep neural networks. For regression problems, by setting Gibbs optimal posterior for each training task, we obtain the closed-form formula of the minimizer of our Catoni-bound, leading to an efficient Gibbs meta-learning algorithm. Although minimizing our kl-bound can not yield a closed-form solution, we show that it can be extended for analyzing the more challenging meta-learning setting where samples from different training tasks exhibit interdependencies. Experiments empirically show that our proposed meta-learning algorithms achieve competitive results with respect to latest works.

The support/query episodic training strategy has been widely applied in modern meta learning algorithms. Supposing the $n$ training episodes and the test episodes are sampled independently from the same environment, previous work has derived a generalization bound of $O(1/\sqrt{n})$ for smooth non-convex functions via algorithmic stability analysis. In this paper, we provide fine-grained analysis of stability and generalization for modern meta learning algorithms by considering more general situations. Firstly, we develop matching lower and upper stability bounds for meta learning algorithms with two types of loss functions: (1) nonsmooth convex functions with $\alpha$-H{\"o}lder continuous subgradients $(\alpha \in [0,1))$; (2) smooth (including convex and non-convex) functions. Our tight stability bounds show that, in the nonsmooth convex case, meta learning algorithms can be inherently less stable than in the smooth convex case. For the smooth non-convex functions, our stability bound is sharper than the existing one, especially in the setting where the number of iterations is larger than the number $n$ of training episodes. Secondly, we derive improved generalization bounds for meta learning algorithms that hold with high probability. Specifically, we first demonstrate that, under the independent episode environment assumption, the generalization bound of $O(1/\sqrt{n})$ via algorithmic stability analysis is near optimal. To attain faster convergence rate, we show how to yield a deformed generalization bound of $O(\ln{n}/n)$ with the curvature condition of loss functions. Finally, we obtain a generalization bound for meta learning with dependent episodes whose dependency relation is characterized by a graph. Experiments on regression problems are conducted to verify our theoretical results.

Existing few-shot learning (FSL) methods make the implicit assumption that the few target class samples are from the same domain as the source class samples. However, in practice, this assumption is often invalid -the target classes could come from a different domain. This poses an additional challenge of domain adaptation (DA) with few training samples. In this paper, the problem of domain-adaptive few-shot learning (DA-FSL) is tackled, which is expected to have wide use in real-world scenarios and requires solving FSL and DA in a unified framework. To this end, we propose a novel domain-adversarial prototypical network (DAPN) model. It is designed to address a specific challenge in DA-FSL: the DA objective means that the source and target data distributions need to be aligned, typically through a shared domain-adaptive feature embedding space; but the FSL objective dictates that the target domain per class distribution must be different from that of any source domain class, meaning aligning the distributions across domains may harm the FSL performance. How to achieve global domain distribution alignment whilst maintaining source/target per-class discriminativeness thus becomes the key. Our solution is to explicitly enhance the source/target per-class separation before domain-adaptive feature embedding learning, to alleviate the negative effect of domain alignment on FSL. Extensive experiments show that our DAPN outperforms the state-of-the-arts. The code is available at https://github.com/dingmyu/DAPN.

Zero-shot learning (ZSL) is made possible by learning a projection function between a feature space and a semantic space (e.g., an attribute space). Key to ZSL is thus to learn a projection that is robust against the often large domain gap between the seen and unseen class domains. In this work, this is achieved by unseen class data synthesis and robust projection function learning. Specifically, a novel semantic data synthesis strategy is proposed, by which semantic class prototypes (e.g., attribute vectors) are used to simply perturb seen class data for generating unseen class ones. As in any data synthesis/hallucination approach, there are ambiguities and uncertainties on how well the synthesised data can capture the targeted unseen class data distribution. To cope with this, the second contribution of this work is a novel projection learning model termed competitive bidirectional projection learning (BPL) designed to best utilise the ambiguous synthesised data. Specifically, we assume that each synthesised data point can belong to any unseen class; and the most likely two class candidates are exploited to learn a robust projection function in a competitive fashion. As a third contribution, we show that the proposed ZSL model can be easily extended to few-shot learning (FSL) by again exploiting semantic (class prototype guided) feature synthesis and competitive BPL. Extensive experiments show that our model achieves the state-of-the-art results on both problems.

Zero-shot learning (ZSL) is originally designed to address the small sample size problem often encountered in computer vision by recognizing unseen object classes without any training samples. Existing ZSL models (particularly deep ones) often assume that hundreds of labelled samples are collected from each seen class. In real-world applications, this assumption tends to become invalid. Therefore, a new ZSL setting is concerned in this paper: each seen class only has few labelled samples, while each unseen class still has no samples. This is more challenging yet more useful/practical than the conventional ZSL setting. To overcome the extreme label scarcity, we choose to obtain more training samples from image search engine for data augmentation: the name of each seen class is used as the query of Google, and the top returned images can be viewed as the noisy labelled samples for this seen class. With the augmented but noisy labelled training data, a novel inductive ZSL model is proposed by formulating label noise reduction (LNR) and semantic projection learning (SPL) within a unified framework: (1) LNR aims to refine the noisy labelled samples for projection learning; (2) SPL aims to learn the projection function with the refined training data. Extensive experiments show that our ZSL model outperforms the state-of-the-art alternatives.

Zero-shot learning (ZSL) typically learns a projection function between a feature space and a semantic space (e.g. attribute space) to recognize unseen classes without any training samples. However, when the projection function learned only with seen class training samples is applied to unseen class test samples, most existing ZSL models suffer from projection domain shift. In this paper, we propose a new ZSL model, termed joint inductive learning (JIL) of bidirectional projections and shared subspace, to learn more generalizable projections and thus algin the seen and unseen class domains. By inducing the two tasks of bidirectional projection learning (BPL) and shared subspace learning (SSL), we integrate three forms of projection learning into a unified framework, which makes our JIL model become more generalizable across the two domains. Moreover, since a shared subspace is actually learned between the feature and semantic spaces, our JIL model can be naturally applied to cross-modal retrieval (CMR). Extensive experiments demonstrate that our JIL model generally outperforms the state-of-the-art alternatives in the two challenging tasks (i.e. ZSL and CMR).

Zero-shot learning (ZSL) aims to recognize unseen object classes without any training samples, which can be regarded as a form of transfer learning from seen classes to unseen ones. This is made possible by learning a projection between a feature space and a semantic space (e.g. attribute space). Key to ZSL is thus to learn a projection function that is robust against the often large domain gap between the seen and unseen classes. In this paper, we propose a novel ZSL model termed domain-invariant projection learning (DIPL). Our model has two novel components: (1) A domain-invariant feature self-reconstruction task is introduced to the seen/unseen class data, resulting in a simple linear formulation that casts ZSL into a min-min optimization problem. Solving the problem is non-trivial, and a novel iterative algorithm is formulated as the solver, with rigorous theoretic algorithm analysis provided. (2) To further align the two domains via the learned projection, shared semantic structure among seen and unseen classes is explored via forming superclasses in the semantic space. Extensive experiments show that our model outperforms the state-of-the-art alternatives by significant margins.