In this section, we demonstrate the effectiveness of low-rank matrix factorization in recovering the label relationship matrix. We first present four important facts: f1: the rank of the matrix is equivalent to the number of classes. Specifically, this also means that if ˆZi,k = 1, then ˆZj,k = 1. We consider a toy example (without self-loops), ˆZ = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A = 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 (14) In a standard LRMF problem, it is not possible to recover ˆZ from A since no entries are observed for the third and fourth rows. However, we can demonstrate how LRMF effectively performs in this situation. Recovery: We begin by assuming v1 is in class 1, resulting in U1,: = [1, 1, 1] and V1,: = [1,0,0]. By observing A1,4, we know that v4 is also in class 1, resulting in U4,: = [1, 1, 1]and V4,: = [1,0,0](f2). By analyzing A1,2 and A1,3, we determine that v2 and v3 do not belong to class 1.

Empirically, we verify the computational efficiency of our methods.

A.1 Hyper-Parameters For all datasets, the surrogate gradient function isσ(x) = 1π arctan(π2αx) + 12, thus σ0(x) = The results on the three networks are consistent, indicating that RTD is a general sequential data augmentationmethod. We compare different surrogate functions, including Rectangular (σ0(x) = sign(|x| < 12)),ArcTan(σ0(x) = 11+(πx)2)and Constant 1(σ0(x) 1),intheSNNs on CIFAR-10. The results are shown in Tab.9. Tab.9 indicates that the choice of surrogate function has a considerable influence on the SNN's performance. Although Rectangular and Constant 1 can avoid the gradient exploding/vanishing problems in Eq.(8), they still cause lower accuracy or even make the optimization not converges.

Recurrent spiking neural networks (RSNNs) hold great potential for advancing artificial general intelligence, as they draw inspiration from the biological nervous system and show promise in modeling complex dynamics.

Humans learn efficiently from their environment by engaging multiple interacting neural systems that support distinct yet complementary forms of control, including model-based (goal-directed) planning, model-free (habitual) responding, and episodic memory-based learning. Model-based mechanisms compute prospective action values using an internal model of the environment, supporting flexible but computationally costly planning; model-free mechanisms cache value estimates and build heuristics that enable fast, efficient habitual responding; and memory-based mechanisms allow rapid adaptation from individual experience. In this work, we aim to elucidate the computational principles underlying this biological efficiency and translate them into a sampling algorithm for scalable Bayesian inference through effective exploration of the posterior distribution. More specifically, our proposed algorithm comprises three components: a model-based module that uses the target distribution for guided but computationally slow sampling; a model-free module that uses previous samples to learn patterns in the parameter space, enabling fast, reflexive sampling without directly evaluating the expensive target distribution; and an episodic-control module that supports rapid sampling by recalling specific past events (i.e., samples). We show that this approach advances Bayesian methods and facilitates their application to large-scale statistical machine learning problems. In particular, we apply our proposed framework to Bayesian deep learning, with an emphasis on proper and principled uncertainty quantification.

Safe reinforcement learning (RL) studies problems where an intelligent agent has to not only maximize reward but also avoid exploring unsafe areas. In this study, we propose CUP, a novel policy optimization method based on Constrained Update Projection framework that enjoys rigorous safety guarantee. Central to our CUP development is the newly proposed surrogate functions along with the performance bound. Compared to previous safe reinforcement learning methods, CUP enjoys the benefits of 1) CUP generalizes the surrogate functions to generalized advantage estimator (GAE), leading to strong empirical performance.