Power linear keller maps of rank two are linearly triangularizable

AbstractLet R be a field with characteristic zero. In this paper it is proved that power linear Keller maps Rn→Rn with rank at most two are linearly triangularizable

DCC categories of cohomological dimension one

AbstractA characterization of dcc categories of cohomological dimension one is given from which a characterization of dcc posets of cohomological dimension one is deduced

Provable Reset-free Reinforcement Learning by No-Regret Reduction

Reinforcement learning (RL) so far has limited real-world applications. One key challenge is that typical RL algorithms heavily rely on a reset mechanism to sample proper initial states; these reset mechanisms, in practice, are expensive to implement due to the need for human intervention or heavily engineered environments. To make learning more practical, we propose a generic no-regret reduction to systematically design reset-free RL algorithms. Our reduction turns the reset-free RL problem into a two-player game. We show that achieving sublinear regret in this two-player game would imply learning a policy that has both sublinear performance regret and sublinear total number of resets in the original RL problem. This means that the agent eventually learns to perform optimally and avoid resets. To demonstrate the effectiveness of this reduction, we design an instantiation for linear Markov decision processes, which is the first provably correct reset-free RL algorithm.Comment: Full version of the paper accepted to ICML 2023. Also spotlighted in AAAI 2023 RL4PROD Worksho

Orthogonally Decoupled Variational Gaussian Processes

Gaussian processes (GPs) provide a powerful non-parametric framework for reasoning over functions. Despite appealing theory, its superlinear computational and memory complexities have presented a long-standing challenge. State-of-the-art sparse variational inference methods trade modeling accuracy against complexity. However, the complexities of these methods still scale superlinearly in the number of basis functions, implying that that sparse GP methods are able to learn from large datasets only when a small model is used. Recently, a decoupled approach was proposed that removes the unnecessary coupling between the complexities of modeling the mean and the covariance functions of a GP. It achieves a linear complexity in the number of mean parameters, so an expressive posterior mean function can be modeled. While promising, this approach suffers from optimization difficulties due to ill-conditioning and non-convexity. In this work, we propose an alternative decoupled parametrization. It adopts an orthogonal basis in the mean function to model the residues that cannot be learned by the standard coupled approach. Therefore, our method extends, rather than replaces, the coupled approach to achieve strictly better performance. This construction admits a straightforward natural gradient update rule, so the structure of the information manifold that is lost during decoupling can be leveraged to speed up learning. Empirically, our algorithm demonstrates significantly faster convergence in multiple experiments.Comment: Appearing NIPS 201