Advances in reinforcement learning (RL) have led to its successful application in complex tasks with continuous state and action spaces. Despite these advances in practice, most theoretical work pertains to finite state and action spaces. We propose building a theoretical understanding of continuous state and action spaces by employing a geometric lens to understand the locally attained set of states. The set of all parametrised policies learnt through a semi-gradient based approach induce a set of attainable states in RL. We show that training dynamics of a two layer neural policy induce a low dimensional manifold of attainable states embedded in the high-dimensional nominal state space trained using an actor-critic algorithm. We prove that, under certain conditions, the dimensionality of this manifold is of the order of the dimensionality of the action space. This is the first result of its kind, linking the geometry of the state space to the dimensionality of the action space. We empirically corroborate this upper bound for four MuJoCo environments and also demonstrate the results in a toy environment with varying dimensionality. We also show the applicability of this theoretical result by introducing a local manifold learning layer to the policy and value function networks to improve the performance in control environments with very high degrees of freedom by changing one layer of the neural network to learn sparse representations.

We propose a novel parameterized skill-learning algorithm that aims to learn transferable parameterized skills and synthesize them into a new action space that supports efficient learning in long-horizon tasks. We propose to leverage off-policy Meta-RL combined with a trajectory-centric smoothness term to learn a set of parameterized skills. Our agent can use these learned skills to construct a three-level hierarchical framework that models a Temporally-extended Parameterized Action Markov Decision Process. We empirically demonstrate that the proposed algorithms enable an agent to solve a set of highly difficult long-horizon (obstacle-course and robot manipulation) tasks.

Deep neural networks can approximate functions on different types of data, from images to graphs, with varied underlying structure. This underlying structure can be viewed as the geometry of the data manifold. By extending recent advances in the theoretical understanding of neural networks, we study how a randomly initialized neural network with piecewise linear activation splits the data manifold into regions where the neural network behaves as a linear function. We derive bounds on the density of boundary of linear regions and the distance to these boundaries on the data manifold. This leads to insights into the expressivity of randomly initialized deep neural networks on non-Euclidean data sets. We empirically corroborate our theoretical results using a toy supervised learning problem. Our experiments demonstrate that number of linear regions varies across manifolds and the results hold with changing neural network architectures. We further demonstrate how the complexity of linear regions is different on the low dimensional manifold of images as compared to the Euclidean space, using the MetFaces dataset.

The recently proposed option-critic architecture (Bacon, Harb, and Precup 2017) provides a stochastic policy gradient approach to hierarchical reinforcement learning. Specifically, it provides a way to estimate the gradient of the expected discounted return with respect to parameters that define a finite number of temporally extended actions, called options. In this paper we show how the option-critic architecture can be extended to estimate the natural gradient (Amari 1998) of the expected discounted return. To this end, the central questions that we consider in this paper are: 1) what is the definition of the natural gradient in this context, 2) what is the Fisher information matrix associated with an option’s parameterized policy, 3) what is the Fisher information matrix associated with an option’s parameterized termination function, and 4) how can a compatible function approximation approach be leveraged to obtain natural gradient estimates for both the parameterized policy and parameterized termination functions of an option with per-time-step time and space complexity linear in the total number of parameters. Based on answers to these questions we introduce the natural option critic algorithm. Experimental results showcase improvement over the vanilla gradient approach.