1 Problem description
We consider discrete-time infinite horizon discounted Markov Decision Process (MDP), with states, admissible actions , discount factor , reward function
, and transition probability density. Solving the MDP means finding a policy that maximizes the accumulated discounted rewards with states and actions distribution induced by applying that policy:
Policy evaluation is a key component of policy iteration and policy gradient approaches to (1), with the objective of finding the Q-function111Throughout this work, we consider Q-function, instead of state value function, in order to cover the model-free setting. associated with a stationary deterministic policy , which is the fixed point of the Bellman operator or the solution of the Bellman equation:
When the state space is large or infinite, solving (2) exactly becomes intractable. Least-Squares Temporal Difference (LSTD) is a widely used simulation-based algorithm for approximately solving the projected Bellman equation with linear representation of value function . In order to apply LSTD to policy iteration, Lagoudakis and Parr  proposed a Q-function extension, and showed that the resulting policy iteration algorithm can be successfully applied to control problems.
In this work, we study LSTD-based approximate policy evaluation methods that benefit from manifold regularized learning, with the intuition that the Q-function is smooth on the manifold where states lie and not necessary in the ambient Euclidean space. Such manifold structure naturally arises in many robotics tasks due to constraints in the state space. For example, contact, e.g., between foot and the ground or a hand and an object 
, restricts feasible states to lie along a manifold or a union of manifolds. Other examples include the cases when state variables belong to Special Euclidean group SE(3) to encode 3D poses, and obstacle avoidance settings where geodesic paths between state vectors are naturally better motivated than geometrically infeasible straight-line paths in the ambient space.
2.1 Manifold regularized learning
Manifold regularization has previously been studied in semi-supervised learning. This data-dependent regularization exploits the geometry of the input space, thus achieving better generalization error. Given a labeled dataset , Laplacian Regularized Least-Squares method (LapRLS) , finds a function in a reproducing kernel Hilbert space that fits the data with proper complexity for generalization:
where data matrices and
with and ,
is the norm in and is a penalty that reflects the intrinsic structure of (see §2.2 in  for choices of ),
and scalars , are regularization parameters222 denotes the marginal probability distribution of inputs, and
denotes the marginal probability distribution of inputs, and.. A natural choice of is , where is the support of which is assumed to be a compact manifold  and is the gradient along . When is unknown as in most learning settings,
can be estimated empirically, and the optimization problem (3) becomes
where and matrix is a graph Laplacian333 denotes the set of symmetric positive semidefinite matrices. (different ways of constructing graph Laplacian from data can be found in ). Note that problem (4) is still convex, since graph Laplacian
is positive semidefinite, and the multiplicity of eigenvalueis the number of connected components in the graph. In fact, a broad family of graph regularizers can be used in this context [6, 7]. This includes the iterated graph Laplacian which is theoretically better than the standard Laplacian , or the diffusion kernel on graphs.
Based on Representer Theorem , the solution of (4) has the form , where , and is the kernel associated with . After substituting the form of in (4), and solving the resulting least-squares problem, we can derive
where matrix is the gram matrix, and matrix
is an identity matrix of matching size.
2.2 Kernelized LSTD with regularization
Farahmand et al.  introduce an regularization extension to kernelized LSTD , termed Regularized LSTD (REG-LSTD), featuring better control of the complexity of the function approximator through regularization, and mitigating the burden of selecting basis functions through kernel machinery. REG-LSTD is formulated by adding regularization terms to both steps in a nested minimization problem that is equivalent to the original LSTD :
can be selected as a tensor product kernel, where (note that the multiplication of two kernels is a kernel ). Furthermore, can be the Kronecker delta function, if the admissible action set is finite.
where , , , , , and .
3 Our approach
) that offers better control over function complexity and eases feature selection.
More concretely, besides the regularization term in (9) that controls the complexity of in the ambient space, we augment the objective function with a manifold regularization term that enforces to be smooth on the manifold that supports . In particular, if the manifold regularization term is chosen as in §2.1, large penalty is imposed if varying too fast along the manifold. With the empirically estimated manifold regularization term, optimization problem (9) becomes (cf., (4))
which admits the optimal weight vector (cf., (11)):
4 Experimental results
We present experimental results on two standard RL benchmarks: two-room navigation and cart-pole balancing. REG-LSTD with manifold regularization (MR-LSTD) (§3) is compared against REG-LSTD without manifold regularization (§2.2) and LSTD with three commonly used basis function construction mechanisms: polynomial [13, 2]
, radial basis functions (RBF)[2, 14], and Laplacian eigenmap [15, 16], in terms of the quality of the policy produced by Least-Squares Policy Iteration (LSPI) . The kernel used in the experiments is
with hyperparameter. We use combinatorial Laplacian with adjacency matrix computed from -neighborhood with weights .
4.1 Two-room navigation
The two-room navigation problem is a classic benchmark for RL algorithms that cope with either discrete [16, 17, 15] or continuous state space [14, 18]. In the vanilla two-room problem, the state space is a discrete grid world, and admissible actions are stepping in one of the four cardinal directions, i.e., up, right, down, and left. The dynamics is stochastic: each action succeeds with probability when movement is not blocked by obstacle or border, otherwise leaves the agent in the same location. The goal is to navigate to the diagonally opposite corner in the other room, with a single-cell doorway connecting the two rooms. Reward is at the goal location otherwise , and the discount factor is set to . Data are collected beforehand by uniformly sampling states and actions, and used throughout LSPI iterations. Seen from Table 1, Laplacian eigenmap which exploits intrinsic state geometry outperforms parametric basis functions: polynomial and RBF. The MR-LSTD method we propose achieves the best performance.
|# of samples||polynomial||RBF||eigenmap||REG-LSTD||MR-LSTD|
4.2 Cart-pole Balancing
The cart-pole balancing task is to balance a pole upright by applying force to the cart to which it’s attached444The cart-pole environment in OpenAI Gym package is used in our implementation. . The agent constantly receives reward until trial ends with reward when it’s away from the upright posture. On the contrary to the two-room navigation task, the state space is continuous, consisting of angle and angular velocity of the pole. Admissible actions are finite: pushing the cart to left or right. The discount factor . Data are collected from random episodes, i.e., starting from a perturbed upright posture and applying uniformly sampled actions. Results are reported in Table 2, which shows that REG-LSTD achieves significantly better performance than parametric basis functions, and performance is even improved further with manifold regularization.
|# of samples||polynomial||RBF||REG-LSTD||MR-LSTD|
5 Related work
The closest work to this paper was recently introduced by Li et al. 
, which utilized manifold regularization by learning state representation through unsupervised learning, and then adopting the learned representation in policy iteration. In contrast to this work, we naturally blend manifold regularization with policy evaluation with possibly provable performance guarantee (left for future work). There is also work on constructing basis functions directly from geometry,e.g., Laplacian methods [15, 16], and geodesic Gaussian kernels . Furthermore, different regularization mechanisms to LSTD have been proposed, including regularization for feature selection , and nested and penalization to avoid overfitting .
6 Conclusion and future work
We propose manifold regularization for a kernelized LSTD approach in order to exploit the intrinsic geometry of the state space for better sample efficiency and Q-function approximation, and demonstrate superior performance on two standard RL benchmarks. Future work directions include 1) accelerating by structured random matrices for kernel machinery [23, 24], and graph sketching for graph regularizer construction to scale up to large datasets and rich observations, e.g., images, 2) providing theoretical justification, and combining manifold regularization with deep neural nets [25, 26] and other policy evaluation, e.g., [27, 10] and policy iteration algorithms, 3) learning with a data-dependent kernel that capturing the geometry (equivalent to the manifold regularized solution ) that makes it easier to derive new algorithms, and 4) extension to continuous action spaces by constructing kernels such that policy improvement (optimize over actions) is tractable .
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