We can see that our proposed model can effectively reduce the number of tasks with classification rates of less than 60%. To be our best knowledge, those novel tasks performed poorly by few-shot learning methods usually have the relatively large domain differences with all base classes, where the importance of each base class for novel sample might be similar. Different from Free-lunch, which only selects topw base classes to estimate the distribution of novel sample and might omit some relevant information, we utilizes all base classes by introducing the adaptive weight information over all base classes for each novel sample. It indicates that our proposed H-OT can effectively enhance distribution calibration method when there is a big domain difference between base and novel classes.

Few-shot classification aims to learn a classifier to recognize unseen classes during training, where the learned model can easily become over-fitted based on the biased distribution formed by only a few training examples. A recent solution to this problem is calibrating the distribution of these few sample classes by transferring statistics from the base classes with sufficient examples, where how to decide the transfer weights from base classes to novel classes is the key. However, principled approaches for learning the transfer weights have not been carefully studied. To this end, we propose a novel distribution calibration method by learning the adaptive weight matrix between novel samples and base classes, which is built upon a hierarchical Optimal Transport (H-OT) framework. By minimizing the high-level OT distance between novel samples and base classes, we can view the learned transport plan as the adaptive weight information for transferring the statistics of base classes. The learning of the cost function between a base class and novel class in the high-level OT leads to the introduction of the lowlevel OT, which considers the weights of all the data samples in the base class. Experiments on standard benchmarks demonstrate that our proposed plug-andplay model outperforms competing approaches and owns desired cross-domain generalization ability, proving the effectiveness of the learned adaptive weights. 1

Building trees to represent or to fit distances is a critical component of phylogenetic analysis, metric embeddings, approximation algorithms, geometric graph neural nets, and the analysis of hierarchical data. Much of the previous algorithmic work, however, has focused on generic metric spaces (i.e., those with no a priori constraints). Leveraging several ideas from the mathematical analysis of hyperbolic geometry and geometric group theory, we study the tree fitting problem as finding the relation between the hyperbolicity (ultrametricity) vector and the error of tree (ultrametric) embedding. That is, we define a vector of hyperbolicity (ultrametric) values over all triples of points and compare the ℓp norms of this vector with the ℓq norm of the distortion of the best tree fit to the distances. This formulation allows us to define the average hyperbolicity (ultrametricity) in terms of a normalized ℓ1 norm of the hyperbolicity vector. Furthermore, we can interpret the classical tree fitting result of Gromov as a p = q = result. We present an algorithm HCCROOTEDTREEFIT such that the ℓ1 error of the output embedding is analytically bounded in terms of the ℓ1 norm of the hyperbolicity vector (i.e., p = q = 1) and that this result is tight. Furthermore, this algorithm has significantly different theoretical and empirical performance as compared to Gromov's result and related algorithms.

This appendix contains a proofs of the results in the main text and further analysis on the two FIM estimators ˆI1(θ)and ˆI2(θ). In particular, Appendix C presents an analysis of how the FIM estimators and their covariance tensors change under reparametrization. Appendix D presents element-wise bound alternatives to those presented in Section 3.2. Appendix E explores various results using alternative norms to the Frobenius norm results of the main text. Appendix F presents an analysis on taking a linear combination of the two FIM estimators.

The training is stalled if the size of the replay buffer is smaller than the minibatch size, i.e., if |B|< M. Algorithms 3 and 4 show the critic network update and the actor network and uncertainty parameter sampler update, respectively. Although we write the gradient-based update in the form of a mini-batch stochastic gradient update for simplicity, we employ an adaptive approach such as Adam [16]. The update of pk follows the exponential moving average with the momentum (1/Tlast), where Tlast is the number of steps spent in the last episode (Tlast is set to 1000 for the first episode). The reason behind this design choice is as follows. The short episode is a meaning that a bad uncertainty parameter ω is used in the last episode.