MetaNODE: Prototype Optimization as a Neural ODE for Few-Shot Learning

Few-Shot Learning (FSL) is a challenging task, \emph{i.e.}, how to recognize novel classes with few examples? Pre-training based methods effectively tackle the problem by pre-training a feature extractor and then predicting novel classes via a cosine nearest neighbor classifier with mean-based prototypes. Nevertheless, due to the data scarcity, the mean-based prototypes are usually biased. In this paper, we attempt to diminish the prototype bias by regarding it as a prototype optimization problem. To this end, we propose a novel meta-learning based prototype optimization framework to rectify prototypes, \emph{i.e.}, introducing a meta-optimizer to optimize prototypes. Although the existing meta-optimizers can also be adapted to our framework, they all overlook a crucial gradient bias issue, \emph{i.e.}, the mean-based gradient estimation is also biased on sparse data. To address the issue, we regard the gradient and its flow as meta-knowledge and then propose a novel Neural Ordinary Differential Equation (ODE)-based meta-optimizer to polish prototypes, called MetaNODE. In this meta-optimizer, we first view the mean-based prototypes as initial prototypes, and then model the process of prototype optimization as continuous-time dynamics specified by a Neural ODE. A gradient flow inference network is carefully designed to learn to estimate the continuous gradient flow for prototype dynamics. Finally, the optimal prototypes can be obtained by solving the Neural ODE. Extensive experiments on miniImagenet, tieredImagenet, and CUB-200-2011 show the effectiveness of our method.Comment: Accepted by AAAI 202

Continuum field theory of 3D topological orders with emergent fermions and braiding statistics

Universal topological data of topologically ordered phases can be captured by topological quantum field theory in continuous space time by taking the limit of low energies and long wavelengths. While previous continuum field-theoretical studies of topological orders in $3$D real space focus on either self-statistics, braiding statistics, shrinking rules, fusion rules or quantum dimensions, it is yet to systematically put all topological data together in a unified continuum field-theoretical framework. Here, we construct the topological $BF$ field theory with twisted terms (e.g., $AAdA$ and $AAB$) as well as a $K$-matrix $BB$ term, in order to simultaneously explore all such topological data and reach anomaly-free topological orders. Following the spirit of the famous $K$-matrix Chern-Simons theory of $2$D topological orders, we present general formulas and systematically show how the $K$-matrix $BB$ term confines topological excitations, and how self-statistics of particles is transmuted between bosonic one and fermionic one. In order to reach anomaly-free topological orders, we explore, within the present continuum field-theoretical framework, how the principle of gauge invariance fundamentally influences possible realizations of topological data. More concretely, we present the topological actions of (i) particle-loop braidings with emergent fermions, (ii) multiloop braidings with emergent fermions, and (iii) Borromean-Rings braidings with emergent fermions, and calculate their universal topological data. Together with the previous efforts, our work paves the way toward a more systematic and complete continuum field-theoretical analysis of exotic topological properties of $3$D topological orders. Several interesting future directions are also discussed

Non-Abelian Fusion, Shrinking and Quantum Dimensions of Abelian Gauge Fluxes

Braiding and fusion rules of topological excitations are indispensable topological invariants in topological quantum computation and topological orders. While excitations in 2D are always particle-like anyons, those in 3D incorporate not only particles but also loops -- spatially nonlocal objects -- making it novel and challenging to study topological invariants in higher dimensions. While 2D fusion rules have been well understood from bulk Chern-Simons field theory and edge conformal field theory, it is yet to be thoroughly explored for 3D fusion rules from higher dimensional bulk topological field theory. Here, we perform a field-theoretical study on (i) how loops that carry Abelian gauge fluxes fuse and (ii) how loops are shrunk into particles in the path integral, which generates fusion rules, loop-shrinking rules, and descendent invariants, e.g., quantum dimensions. We first assign a gauge-invariant Wilson operator to each excitation and determine the number of distinct excitations through equivalence classes of Wilson operators. Then, we adiabatically shift two Wilson operators together to observe how they fuse and are split in the path integral; despite the Abelian nature of the gauge fluxes carried by loops, their fusions may be of non-Abelian nature. Meanwhile, we adiabatically deform world-sheets of unknotted loops into world-lines and examine the shrinking outcomes; we find that the resulting loop-shrinking rules are algebraically consistent to fusion rules. Interestingly, fusing a pair of loop and anti-loop may generate multiple vacua, but fusing a pair of anyon and anti-anyon in 2D has one vacuum only. By establishing a field-theoretical ground for fusion and shrinking in 3D, this work leaves intriguing directions, e.g., symmetry enrichment, quantum gates, and physics of braided monoidal 2-category of 2-group.Comment: Title adjusted. Abstract, Intro and Discussions revised. about 30 pages, 5 figures. 9 table

Corruption-Robust Offline Reinforcement Learning with General Function Approximation

We investigate the problem of corruption robustness in offline reinforcement learning (RL) with general function approximation, where an adversary can corrupt each sample in the offline dataset, and the corruption level $\zeta\geq0$ quantifies the cumulative corruption amount over $n$ episodes and $H$ steps. Our goal is to find a policy that is robust to such corruption and minimizes the suboptimality gap with respect to the optimal policy for the uncorrupted Markov decision processes (MDPs). Drawing inspiration from the uncertainty-weighting technique from the robust online RL setting \citep{he2022nearly,ye2022corruptionrobust}, we design a new uncertainty weight iteration procedure to efficiently compute on batched samples and propose a corruption-robust algorithm for offline RL. Notably, under the assumption of single policy coverage and the knowledge of $\zeta$, our proposed algorithm achieves a suboptimality bound that is worsened by an additive factor of $\mathcal O(\zeta \cdot (\text{CC}(\lambda,\hat{\mathcal F},\mathcal Z_n^H))^{1/2} (C(\hat{\mathcal F},\mu))^{-1/2} n^{-1})$ due to the corruption. Here $\text{CC}(\lambda,\hat{\mathcal F},\mathcal Z_n^H)$ is the coverage coefficient that depends on the regularization parameter $\lambda$, the confidence set $\hat{\mathcal F}$, and the dataset $\mathcal Z_n^H$, and $C(\hat{\mathcal F},\mu)$ is a coefficient that depends on $\hat{\mathcal F}$ and the underlying data distribution $\mu$. When specialized to linear MDPs, the corruption-dependent error term reduces to $\mathcal O(\zeta d n^{-1})$ with $d$ being the dimension of the feature map, which matches the existing lower bound for corrupted linear MDPs. This suggests that our analysis is tight in terms of the corruption-dependent term