Year: 2019

Abstract:
Balancing exploration and exploitation remains a key challenge in reinforcement learning (RL). State-of-the-art RL algorithms suffer from high sample complexity, particularly in the sparse reward case, where they can do no better than to explore in all directions until the first positive rewards are found. To mitigate this, we propose Rapidly Randomly-exploring Reinforcement Learning (R3L). We formulate exploration as a search problem and leverage widely-used planning algorithms such as Rapidly-exploring Random Tree (RRT) to find initial solutions. These solutions are used as demonstrations to initialize a policy, then refined by a generic RL algorithm, leading to faster and more stable convergence. We provide theoretical guarantees of R3L exploration finding successful solutions, as well as bounds for its sampling complexity. We experimentally demonstrate the method outperforms classic and intrinsic exploration techniques, requiring only a fraction of exploration samples and achieving better asymptotic performance.

heading: 1 INTRODUCTION
text: Reinforcement Learning (RL) studies how agents can learn a desired behaviour by simply using interactions with an environment and a reinforcement signal. Central to RL is the long-standing problem of balancing exploration and exploitation. Agents must first sufficiently explore the environment to identify high-reward behaviours, before this knowledge can be exploited and refined to maximize long-term rewards. Many recent RL successes have been obtained by relying on well-formed reward signals, that provide rich gradient information to guide policy learning. However, designing such informative rewards is challenging, and rewards are often highly specific to the particular task being solved. Sparse rewards, which carry little or no information besides binary success or failure, are much easier to design. This simplicity comes at a cost; most rewards are identical, so that there is little gradient information to guide policy learning. In this setting, the sample complexity of simple exploration strategies was shown to grow exponentially with state dimension in some cases (Osband et al., 2016b). Intuition behind this phenomenon can be gained by inspecting Figure 1a: exploration in regions where the return surface is flat leads to a random walk type search. This inefficient search continues until non-zero gradients are found, which can then be followed to a local optimum.
Planning algorithms can achieve much better exploration performance than random walk by taking search history into account (Lavalle, 1998). These techniques are also often guaranteed to find a solution in finite time if one exists (Karaman & Frazzoli, 2011). In order to leverage the advantages of these methods, we formulate RL exploration as a planning problem in the state space. Solutions found by search algorithms are then used as demonstrations for RL algorithms, initializing them in regions of policy parameter space where the return surface is not flat. Figure 1b shows the importance of such good initialization; surface gradients can be followed, which greatly facilitates learning.
This paper brings the following contributions. We first formulate RL exploration as a planning problem. This yields a simple and effective method for automatically generating demonstrations without the need for an external expert, solving the planning problem by adapting the classic Rapidly-exploring Random Tree algorithm (RRT) (Kuffner & LaValle, 2000). The demonstrations are then used to initialize an RL policy, which can be refined with a classic RL method such as TRPO (Schulman et al., 2015). We call the proposed method Rapidly Randomly-exploring Reinforcement Learning (R3L)1, provide theoretical guarantees for finding successful solutions
1Code will be made available on Github.
and derive bounds for its sampling complexity. Experimentally, we demonstrate R3L improves exploration and outperforms classic and recent exploration techniques, and requires only a fraction of the samples while achieving better asymptotic performance. Lastly, we show that R3L lowers the variance of policy gradient methods such as TRPO, and verify that initializing policies in regions with rich gradient information makes them less sensitive to initial conditions and random seed.
The paper is structured as follows: Section 2 analyzes the limitations of classic RL exploration. Section 3 describes R3L and provides theoretical exploration guarantees. Related work is discussed in Section 4, followed by experimental results and comments in Section 5. Finally, Section 6 concludes and gives directions for future work.

heading: 2 SPARSE-REWARD RL AS RANDOM WALK
text: Many recent RL methods are based on a policy gradient optimization scheme. This approach optimizes policy parameters θ with gradient descent, using a loss function L(θ) (e.g. expected return) and gradient g(θ) ≡ ∇θL(θ). Since computing L(θ) exactly is intractable, it is common to use unbiased empirical estimators L̂(θ) and ĝ(θ), estimated from samples acquired by executing the policy. Optimization of θ then follows the common stochastic gradient descent (SGD) update-rule (Bottou, 2010; Robbins & Monro, 1951): θn+1 = θn − ĝ (θn), where is the learning rate. The SGD update rule defines a discrete-time stochastic process (Mandt et al., 2017). Note that ĝ is the mean of nmb i.i.d. samples. Following the central limit theorem, the distribution over ĝ is approximately ĝ(θ) ∼ N (g(θ), 1nmbC(θ)), meaning ĝ is an unbiased estimator of g with covariance
1 nmb C(θ). Consequently, the update rule can be rewritten as (Mandt et al., 2017):
θn+1 = θn − g(θn) + nmb B∆W, ∆W ∼ N (0, I). (1)
Here we assume that C(θ) = C, i.e. approximately constant w.r.t. θ, and factorizes as C = BBT .
SGD is efficient for high-dimensional problems as it offers almost dimension independent convergence rates (Nesterov, 2018). However, SGD requires non-zero gradients to guide the search towards the optimum θ∗, i.e. |g(θ)| > g , ∀θ 6= θ∗, g ∈ R. In the case of sparse-reward RL problems, such as in Figure 1, much of the loss surface is flat. This leads to inefficient exploration of parameter space Θ, as the drift component in Eq. (1) g ≈ 0, turning the SGD to a random walk in Θ; ∆θ = nmbB∆W . Random walk is guaranteed to wander to infinity when dimensionality dΘ ≥ 3 (Pólya, 1921; Kakutani, 1944). However, the probability of it reaching a desired region in Θ, e.g. where g 6= 0, depends heavily on problem specific parameters. The probability of θn ever reaching a sphere of radius r centered at C such that ‖C − θ0‖ = R > r is (Dvoretzky & Erdős, 1951):
Pr{‖θn − C‖ < r, for some n > 0} = ( r R )dΘ−2 < 1, ∀dΘ ≥ 3. (2)
In sparse RL problems r < R, thus the probability of reaching a desired region by random walk is smaller than 1, i.e. there are no guarantees of finding any solution, even in infinite time. This is in stark contrast with the R3L exploration paradigm, as discussed in Section 3.5.

heading: 3 R3L: RAPIDLY AND RANDOMLY-EXPLORING REINFORCEMENT LEARNING
text: R3L adapts RRT to the RL setting by formulating exploration as a planning problem in state space. Unlike random walk, RRT encourages uniform coverage of the search space and is probabilistically complete, i.e. guaranteed to find a solution (Kleinbort et al., 2019).
R3L is decomposed into three main steps: (i) exploration is first achieved using RRT to generate a data-set of successful trajectories, described in Sections 3.2 and 3.3, (ii) successful trajectories are converted to a policy using learning from demonstrations (Section 3.4), and (iii) the policy is refined using classic RL methods.

heading: 3.1 DEFINITIONS
text: This work is based on the Markov Decision Process (MDP) framework, defined as a tuple < S,A, T,R, γ >. S andA are spaces of states s and actions a respectively. T : S×A×S → [0, 1] is a transition probability distribution so that T (st, at, st+1) = p(st+1|st, at), where the subscript t indicates the tth discrete timestep. R : S×A×S → R is a reward function defining rewards rt associated with transitions (st, at, st+1). γ ∈ [0, 1) is a discount factor. Solving a MDP is equivalent to finding the optimal policy π∗ maximizing the expected return J(π∗) = ET,π∗ [ ∑H t=0 γ
tR(st, at, st+1)] for some time horizon H , where actions are chosen according to at = π∗(st). Lastly, let S be a Euclidean space, equipped with the standard Euclidean distance metric with an associated norm denoted by ‖·‖. The space of valid states is denoted by F ⊆ S.

heading: 3.2 EXPLORATION AS A PLANNING PROBLEM WITH RRT
text: The RRT algorithm (LaValle & Kuffner, 2001) provides a principled approach for planning in problems that cannot be solved directly (e.g. using inverse kinematics), but where it is possible to sample transitions. RRT builds a tree of valid transitions between states in F , grown from a root s0. As such, the tree T maintains information over valid trajectories. The exploration problem is defined by the pair (F , s0). In RL environments with a known goal set Fgoal ⊆ F (e.g. MountainCar), the exploration problem is defined by (F , s0,Fgoal). The RRT algorithm starts by sampling a random state srand ∈ S , used to encourage exploration in a specific direction in the current iteration. This necessitates the first of two assumptions.
Assumption 1. Random states can be sampled uniformly from the MDP state space S.
Sampled states are not required to be valid, thus sampling a random state is typically equivalent to trivially sampling a hyper-rectangle.
Then, the vertex snear = arg mins∈T‖srand − snear‖ is found. RRT attempts to expand the tree T from snear toward srand by sampling an action a ∈ A according to a steering function Υ : S × S → A. In many planning scenarios, Υ samples randomly from A. A forward step snew = f(snear, a) is then simulated from snear using action a, where f is defined by the transition dynamics. Being able to expand the tree from arbitrary snear relies on another assumption.
Assumption 2. The environment state can be set to a previously visited state s ∈ T.
Although this assumption largely limits the algorithm to simulators, it has previously been used in Florensa et al. (2017); Nair et al. (2018); Ecoffet et al. (2019) for example; see discussion in Section 6 on overcoming the limitation to simulated environments.
snew is added as a new vertex of T, alongside an edge (snear, snew) with edge information a. This process repeats until a sampling budget k is exhausted or the goal set is reached (i.e. snew ∈ Fgoal). Definition 1. A valid trajectory is a sequence τ = [s0, a0, s1, . . . , stτ−1, atτ−1, stτ ] such that (st, at, st+1) is a valid transition and st ∈ F , ∀t ∈ {0, 1, . . . , tτ}. Whenever a goal set is defined, a successful valid trajectory end state satisfies stτ ∈ Fgoal.
Algorithm 1: R3L exploration Input: s0, k: sampling budget
Once planning is finished, a successful valid trajectory can easily be generated from T by retrieving all nodes between the leaf sleaf ∈ Fgoal and the root. Because tree T is grown from valid transitions between states in F , these trajectories are valid by construction.

heading: 3.3 R3L EXPLORATION: ADAPTING RRT TO RL
text: RRT is not directly applicable to RL problems. This subsection presents the necessary adaptations to the exploration phase of R3L, summed up in Algorithm 1. Figure 2 shows R3L’s typical exploration behaviour.
Local policy learning: In classic planning problems, selecting actions extending snear towards srand is easy, as these have a geometric interpretation or there is a known steering function. RL state spaces do not benefit from the same geometric properties, and properly selecting actions can be challenging. We solve this problem by defining a local policy πl, which models the distribution of actions to transition from a state to a neighbouring goal state. Actions to extend a tree branch are sampled as a ∼ πl(snear, srand − snear). We formulate the problem of learning πl as supervised learning, where inputs are starting states st augmented with the difference st+1 − st, and targets are actions. The model is learned using transition data collected from previous tree expansion iterations. Results in this paper use Bayesian linear regression to represent πl, but any supervised learning model can applied instead.
Unknown dynamics: RRT was designed for problems with known continuous dynamics f , but RL features unknown discrete transition dynamics. In R3L, f is replaced with an environment interaction from snear, with selected action a, resulting in a new state snew. Since (snear, a, snew) is a real transition, it must be valid, and snew can be added to the tree T.
Biasing search with Fgoal: Better exploration efficiency can be achieved if goal information is available. Indeed, the RRT framework allows for biasing exploration towards Fgoal, often resulting in faster exploration. This is achieved by sampling srand from Fgoal instead of F with low probability pg , while the rest of the iteration remains unchanged.
Since R3L uses RRT to explore, the algorithm is most suitable for RL problems that are fully observable, exhibit sparse rewards and have continuous controls. R3L is applicable to other RL problems, but may not perform as well as methods tailored to specific cases.

heading: 3.4 POLICY INITIALIZATION FROM R3L DEMONSTRATIONS
text: Upon completion, R3L exploration yields a successful trajectory τ , which may not be robust to various starting conditions and/or stochastic transitions. Converting successful trajectories into a policy is crucial to achieve robustness and enable further refinement with RL.
Policy initialization is applied to a set of successful trajectories τ = {τi}N1 generated using N runs of R3L exploration with different starting conditions. An imitation policy π0 is learned by supervised
learning on transitions from τ . Policy π0 is then refined using traditional RL algorithms like TRPO. As shown in Figure 1, initializing policy parameters in the vicinity of a local optimum is crucial.

heading: 3.5 EXPLORATION GUARANTEES
text: The RL planning environment defines differential constraints of the form:
ṡ = f(s(t), a(t)), s(t) ∈ F , a(t) ∈ A. (3) Therefore, starting at s0, the trajectory τ can be generated by forward integrating Eq. (3) with the applied actions. As with many RL problems, a(t) is time-discretized resulting in a piecewise constant control function. This means τ is constructed of nτ segments of fixed time duration ∆t such that the overall trajectory duration tτ = nτ ·∆t. Thus, a(t) is defined as a(t) = ai ∈ A where t ∈ [(i − 1) · ∆t, i · ∆t) and 1 ≤ i ≤ nτ . Furthermore, as all transitions between states in τ are known, the trajectory return can be defined as Rτ = ∑nτ t=0 γ tR(st, at, st+1).
R3L explores in state-action/trajectory space instead of policy parameter space. Furthermore, it is an effective exploration framework which provides probabilistic completeness (PC): Definition 2. A probabilistically complete planner finds a feasible solution (if one exists) with a probability approaching 1 in the limit of infinite samples.
With the aforementioned dynamic characteristics, we prove that R3L exploration under the RL setting is PC. This is in stark contrast to the random walk exploration process, discussed in section 2, which is not PC. We begin with the following theorem, a modification of Theorem 2 from Kleinbort et al. (2019), which is applied to kinodynamic RRT where a goal set Fgoal is defined. Theorem 1. Suppose that there exists a valid trajectory τ from s0 to Fgoal as defined in definition 1, with a corresponding piecewise constant control. The probability that R3L exploration fails to reach Fgoal from s0 after k iterations is bounded by ae−bk, for some constants a, b > 0.
The proof, which is a modification of Theorem 2 from Kleinbort et al. (2019), can be found in Appendix S2. It should be noted that R3L exploration does not require an explicit definition for Fgoal in order to explore the space. While in some path planning variants of RRT, Fgoal is used to bias sampling, the main purpose of Fgoal is to indicate that a solution has been found. Therefore, Fgoal can be replaced by another implicit success criterion. In the RL setting, this can be replaced by a return-related criterion. Theorem 2. Suppose that there exists a trajectory with a return Rτ ≥ R̂, R̂ ∈ R. The probability that R3L exploration fails to find a valid trajectory from s0 with Rτ ≥ R̂ after k iterations is bounded by âe−b̂k, for some constants â, b̂ > 0.
Proof. The proof is straightforward. We augment each state in τ with the return to reach it from s0:
s′n = [ sn Rsn ] , ∀n = 1 : nτ , (4)
where Rsn= = ∑n≤nτ t=0 γ
tR(st, at, st+1). For consistency we modify the distance metric by simply adding a reward distance metric. With the above change in notation, we modify the goal set to FRLgoal = {(s,Rs)|s ∈ Fgoal, Rs ≥ R̂}, such that there is an explicit criterion for minimal return as a goal. Consequently, the exploration problem can be written for the augmented representation as (F , sRL0 ,FRLgoal), where sRL0 = [s0, 0]>. Theorem 1 satisfies that R3L exploration can find a feasible solution to this problem within finite time, i.e. PC, and therefore the probability of not reaching FRLgoal after k iterations is upper-bounded by the exponential term âe−b̂k, for some constants â, b̂ > 0
We can now state our main result on the sampling complexity of the exploration process. Theorem 3. If trajectory exploration is probabilistically complete and satisfies an exponential convergence bound, the expected sampling complexity is finite and bounded such that
E[k] ≤ â 4 sinh2 b̂2 , (5)
where â, b̂ > 0.
Proof. Theorem 2 provides an exponential bound for the probability the planner will fail in finding a feasible path. Hence, we can compute a bound for the expected number of iterations needed to find a solution, i.e. sampling complexity:
E[k] ≤ ∞∑ k=1 kâe−b̂k = ∞∑ k=1 −â de −b̂k db̂ = −â d db̂ ∞∑ k=1 e−b̂k = −â d db̂ 1 eb̂ − 1 =
â
4 sinh2 b̂2
, (6)
where we used the relation ∑∞ k=1 e −b̂k = 1 eb̂−1 .
It is worth noting that while the sample complexity is bounded, the above result implies that its complexity varies according to problem-specific properties, which are encapsulated in the value of â and b̂. Intuitively, â depends on the scale of the problem. It grows as |FRLgoal| becomes smaller or as the length of the solution trajectory becomes longer. b̂ depends on the probability of sampling states that will expand the tree in the right direction. It therefore shrinks as the dimensionality of S increases. We refer the reader to Appendix S2 for more details on the meaning of â, b̂ and the derivation of the tail bound in Theorem 1.

heading: 4 RELATED WORK
text: Exploration in RL has been extensively studied. Classic techniques typically rely on adding noise to actions (Mnih et al., 2015; Schulman et al., 2015) or to policy parameters (Plappert et al., 2018). However, these methods perform very poorly in settings with sparse rewards.
Intrinsic motivation tackles this problem by defining a new reward to direct exploration. Many intrinsic reward definitions were proposed, based on information theory (Oudeyer & Kaplan, 2008), state visit count (Lopes et al., 2012; Bellemare et al., 2016; Szita & Lőrincz, 2008; Fox et al., 2018), value function posterior variance (Osband et al., 2016a; Morere & Ramos, 2018), or model prediction error (Stadie et al., 2015; Pathak et al., 2017). Methods extending intrinsic motivation to continuous state and action spaces were recently proposed (Houthooft et al., 2016; Morere & Ramos, 2018). However, these approaches are less interpretable and offer no guarantees for the exploration of the state space.
Offering exploration guarantees, Bayesian optimization was adapted to RL in Wilson et al. (2014), to search over the space of policy parameters in problems with very few parameters. Recent work extends the method to functional policy representations (Vien et al., 2018), but results are still limited to toy problems and specific policy model classes.
Motion planning in robotics is predominantly addressed with sampling-based methods. This type of approach offers a variety of methodologies for exploration and solution space representation (e.g., Probabilistic roadmaps (PRM) (Kavraki et al., 1996), Expansive space trees (ESP) (Hsu et al., 1997) and Rapidly-exploring random tree (RRT) (Kuffner & LaValle, 2000)), which have shown excellent performance in path planning in high-dimensional spaces under dynamic constraints (LaValle & Kuffner, 2001; Hsu et al., 2002; Kavraki et al., 1996).
RL was previously combined with sampling-based planning to replace core elements of planning algorithms, such as PRM’s point-to-point connection (Faust et al., 2018), local RRT steering function (Chiang et al., 2019) or RRT expansion policy (Chen et al., 2019). In contrast, the proposed method bridges the gap in the opposite direction, employing a sampling-based planner to generate demonstrations that kick-start RL algorithms and enhance their performance.
Accelerating RL by learning from demonstration was investigated in Niekum et al. (2015); Bojarski et al. (2016); Torabi et al. (2018). However, these techniques rely on user-generated demonstrations or a-priori knowledge of environment parameters. In contrast, R3L automatically generates demonstrations, with no need of an external expert.

heading: 5 EXPERIMENTS
text: In this section, we investigate (i) how learning a local policy πl and biasing search towards Fgoal with probability pg affects R3L exploration, (ii) whether separating exploration from policy refinement
is a viable and robust methodology in RL, (iii) whether R3L reduces the number of exploration samples needed to find good policies, compared with methods using classic and intrinsic exploration, and (iv) how R3L exploration can reduce the variance associated with policy gradient methods. All experiments make use of the Garage (Duan et al., 2016) and Gym (Brockman et al., 2016) frameworks. The experimental setup features the following tasks with sparse rewards: Cartpole Swingup (S ⊆ R4,A ⊆ R), MountainCar (S ⊆ R2,A ⊆ R), Acrobot (S ⊆ R4,A ⊆ R), Pendulum (S ⊆ R2,A ⊆ R), Reacher (S ⊆ R6,A ⊆ R2) Fetch Reach (S ⊆ R13,A ⊆ R4), and Hand Reach (S ⊆ R78,A ⊆ R20). The exact environment and reward definitions are described in Appendix S3. R3L exploration analysis We first analyze the exploration performance of R3L in a limited set of RL environments, to determine the impact that learning policy πl has on exploration speed. We also investigate whether R3L exploration is viable in environments where no goal information is available. Table 1 shows the results of this analysis. Learning πl seems to greatly decrease the number of exploration timesteps needed on most environments. However, it significantly increases the number of timesteps on the acrobot and reacher environments. Results also suggest that learning πl helps R3L to find shorter trajectories on the same environments, which is a desirable property in many RL problems. Biasing R3L exploration towards the goal set Fgoal helps finding successful trajectories faster, as well as reducing their length. However, R3L exploration without goal bias is still viable in all cases. Although goal information is not given in the classic MDP framework, it is often available in real-world problems and can be easily utilized by R3L. Lastly, successful trajectory lengths have low variance, which suggests R3L finds consistent solutions.
Comparison to classic and intrinsic exploration on RL benchmarks We examine the rates at which R3L learns to solve several RL benchmarks, and compare them with state-of-the-art RL algorithms. Performance is measured in terms of undiscounted returns and aggregated over 10 random seeds, sampled at random for each environment. We focus on domains with sparse rewards, which are notoriously difficult to explore for traditional RL methods. Our experiments focus on the widely-used methods TRPO (Schulman et al., 2015) and DDPG (Lillicrap et al., 2015). R3L-TRPO and R3L-DDPG are compared to the baseline algorithms with Gaussian action noise. As an additional baseline we include VIME-TRPO (Houthooft et al., 2016). VIME is an exploration strategy based on maximizing information gain about the agent’s belief of the environment dynamics. It is included to show that R3L can improve on state-of-the-art exploration methods as well as naive ones, even though the return surface for VIME-TRPO is no longer flat, unlike Figure 1. The exact experimental setup is described in Appendix S3.2. The R3L exploration phase is first run to generate training trajectories for all environments. The number of environment interactions during this phase is accounted for in the results, displayed as an offset with a vertical dashed black line. The average performance achieved by these trajectories is also reported as a guideline, with the exception of Cartpole Swingup where doing so does not make sense. RL is then used to refine a policy pretrained with these trajectories.
Figure 3 shows the median and interquartile range for all methods. R3L is very competitive with both vanilla and VIME baselines. It converges faster and achieves higher performance at the end of the experiment. In most cases, the upper quartile for our method begins well above the minimum return, indicating that R3L exploration and pre-training are able to produce successful though not optimal policies. R3L-DDPG performance, for the majority of problems, starts significantly above the minimum return, plunges due to the inherent instability of DDPG, but eventually recovers, indicating that R3L pre-training can help mitigate the instability. It is worth noting that R3L’s lower quartile is considerably higher than that of baselines. Indeed, for many of the baselines the lower quartile
takes a long time to improve on the minimum return, and in some cases it never manages to do so at all. This is a common problem in sparse reward RL, where there is no alternative but to search the parameter space randomly until the first successful trajectory is found, as explained in Section 2. While a few random seeds will by chance find a successful trajectory quickly (represented by the quickly rising upper quartile), others take a long time (represented by the much slower rise of the median and lower quartile). In other words, R3L-TRPO/DDPG is much more robust to random policy initialization and to the random seed than standard RL methods. This is because R3L is able to use automatically generated demonstrations to initialize RL policy parameters to a region with informative return gradients.

heading: 6 CONCLUSION
text: We proposed Rapidly Randomly-exploring Reinforcement Learning (R3L), an exploration paradigm for leveraging planing algorithms to automatically generate successful demonstrations, which can be converted to policies then refined by classic RL methods. We provided theoretical guarantees of R3L finding solutions, as well as sampling complexity bounds. Empirical results show that R3L outperforms classic and intrinsic exploration techniques, requiring only a fraction of exploration samples and achieving better asymptotic performance.
As future work, R3L could be extended to real-world problems by leveraging recent advances on bridging the gap between simulation and reality (Peng et al., 2018). Respecting Assumption 2, a policy would first be trained on a simulator and then transferred to the real-world. Exploration in high-dimensional tasks is also challenging as stated in Theorem 3 and confirmed experimentally by increased R3L exploration timesteps. Exploiting external prior knowledge and/or the structure of the problem can benefit exploration in high-dimensional tasks, and help make R3L practical for problems such as Atari games. Lastly, recent advances in RRT (Chiang et al., 2019) and learning from demonstration (Torabi et al., 2018) could also improve R3L.

heading: S1 APPENDIX A: RRT ALGORITHM PSUEDO-CODE
text: In this section, we provide pseudo-code of the classic RRT algorithm.
Algorithm S1: Rapidly-exploring Random Trees (RRT) Input: sinit
1 k: sampling budget 2 δt: Euler integration time interval
Output: T 3 T.init(sinit) 4 for i = 1 : k do 5 srand ← RANDOM_UNIFORM(S) 6 if srand /∈ F then 7 pass 8 end 9 snear ← arg mins∈T‖srand − snear‖ /* Find nearest vertex */
10 a← Υ(snear, srand) /* Sample action */ 11 snew ← snear + ∆t · f(snear, a) /* Propagate to new state, Eq. (3) */ 12 if VALID_TRANSITION(snear, snew) then 13 T.add_vertex(snew) 14 T.add_edge(snear, snew) 15 end 16 end

heading: S2 APPENDIX B: PROOF OF THEOREM 1
text: This appendix proves Theorem 1 which shows that planning using RRT under differential constraints is probabilistically complete. The following proof is a modification of Theorem 2 from Kleinbort et al. (2019), where completeness of RRT in the RL setting is maintained without the need to explicitly sample a duration for every action.
Equation 3 defines the environment’s differential constraints. In practice, Eq. (3) is approximated by an Euler integration step. With the interval [0, tτ ] divided into l >> nτ equal time intervals of duration h with tτ = l · h. Eq. (3) is then approximated by an Euler integration step, where the transition between consecutive time steps is given by:
sn+1 = sn + f(sn, an) · h, sn, sn+1 ∈ τ, s.t. lim
l→∞,h→0 ‖sn − τ(n · h)‖ = 0. (S1)
Furthermore, we define Br(s) as a ball with a radius r centered at s for any given state s ∈ S. We assume that the planning environment is Lipschitz continuous in both state and action, constraining the rate of change of Eq. (S1). Formally, there exists two positive constants Ks,Ka > 0, such that ∀s0, s1 ∈ F , a0, a1 ∈ A :
‖f(s0, a0)− f(s0, a1)‖ ≤ Ka‖a0 − a1‖, (S2) ‖f(s0, a0)− f(s1, a0)‖ ≤ Ks‖s0 − s1‖. (S3)
Lemma 1. For two trajectories τ , τ ′, where s0 = τ(0) and s′0 = τ ′(0) such that ‖s0 − s′0‖ ≤ δs with δs a positive constant. Suppose that for each trajectory a piecewise constant action is applied, so that Υ(t) = a and Υ′(t) = a′ is fixed during a time period T ≥ 0. Then ‖τ(T ) − τ ′(T )‖ ≤ eKsT δs + TKae KsT ‖a− a′‖.
The proof for Lemma 1 is given in Lemma 2 of (Kleinbort et al., 2019). Intuitively, this bound is derived from compounding worst-case divergence between τ and τ ′ at every Euler step along T which leads to an overall exponential dependence.
Using Lemma 1, we want to provide a lower bound on the probability of choosing an action that will expand the tree successfully. We note that this scenario assumes that actions are drawn uniformly from A, i.e. there is no steering function 2. When better estimations of the steering function are available, as described in 3.3, the performance of RRT significantly improves. Definition 3. A trajectory τ is defined as δ-clear if for δclear > 0, Bδclear (τ(t)) ∈ F for all t ∈ [0, tτ ]. Lemma 2. Suppose that τ is a valid trajectory from τ(0) = s0 to τ(tτ ) = sgoal with a duration of tτ and a clearance of δ. Without loss of generality, we assume that actions are fixed for all t ∈ [0, tτ ], such that Υ(t) = a ∈ A. Suppose that RRT expands the tree from a state s′0 ∈ B(κδ− )(s0) to a state s′goal, for any κ ∈ (0, 1] and ∈ (0, κδ) we can define the following bound:
Pr[s′goal ∈ Bκδ(sgoal))] ≥ ζ|S| · κδ−e Kstτ (κδ− ) KatτeKstτ
|A| .
Here, ζ|S| = |B1(·)| is the Lebesgue measure for a unit circle in S.
Proof. We denote τ ′ a trajectory that starts from s′0 and is expanded with an applied random action arand. According to Lemma 1,
‖τ(t)−τ ′(t)‖ ≤ eKstδs+tKaeKst‖a−arand‖ ≤ eKst(κδ− )+tKaeKst‖a−arand‖, ∀t ∈ [0, tτ ], where δs ≤ κδ − since s′0 ∈ Bκδ− (s0). Now, we want to find ‖a− arand‖ such that the distance between the goal points of these trajectories, i.e. in the worst-case scenario, is bounded:
eKstτ (κδ − ) + tτKaeKstτ ‖a− arand‖ < κδ. After rearranging this formula, we can obtain a bound for ‖a− arand‖:
∆a = ‖a− arand‖ < κδ − eKstτ (κδ − )
tτKaeKstτ .
Assuming that arand is drawn out of a uniform distribution, the probability of choosing the proper action is
pa = ζ|S| · κδ−e Kstτ (κδ− ) tτKaeKstτ
|A| , (S4)
where ζ|S| is used to account for the degeneracy in action selection due to the dimensionality of S. We note that ∈ (0, κδ) guarantees pa ≥ 0, thus a valid probability>
Equation S4 provides a lower bound for the probability of choosing the suitable action. The following lemma provides a bound on the probability of randomly drawing a state that will expand the tree toward the goal. Lemma 3. Let s ∈ S be a state with clearance δ, i.e. Bδ(s) ∈ F . Suppose that for an RRT tree T there exist a vertex v ∈ T such that v ∈ B2δ/5(s). Following the definition in Section 3.2, we denote snear ∈ T as the closest vertex to srand. Then, the probability that snear ∈ Bδ(s) is at least |Bδ/5(s)|/|S|.
Proof. Let srand ∈ Bδ/5(s). Therefore the distance between srand and v is upper-bounded by ‖srand − v‖ ≤ 3δ/5. If there exists a vertex snear such that snear 6= v and ‖srand − snear‖ ≤ ‖srand − v‖, then snear ∈ B3δ/5(srand) ⊂ Bδ(s). Hence, by choosing srand ∈ Bδ/5(s), we are guaranteed snear ∈ Bδ(s). As srand is drawn uniformly, the probability for srand ∈ Bδ/5(s) is |Bδ/5(s)|/|S|.
2The function steer : S × S → A returns an action asteer given two states srand and snear such that asteer = arg mina∈A‖srand − (snear + ∆t · f(snear, a))‖ s.t. ‖∆t · f(snear, a)‖ < η, for a prespecified η > 0 (Karaman & Frazzoli, 2011).
We can now prove the main theorem.
Theorem 1. Suppose that there exists a valid trajectory τ from s0 to Fgoal as defined in definition 1, with a corresponding piecewise constant control. The probability that RRT fails to reach Fgoal from s0 after k iterations is bounded by ae−bk, for some constants a, b > 0.
Proof. Lemma 2 puts bound on the probability to find actions that expand the tree from one state to another in a given time. As we assume that a valid trajectory exists, we can assume that the probability defined in Lemma 2 is non-zero, i.e. pa > 0, hence:
κδ − eKs∆t(κδ − ) > 0, (S5)
where we set κ = 2/5 and = 5−2 as was also done in (Kleinbort et al., 2019). We additionally require that ∆t, which is typically defined as an RL environment parameter, is chosen accordingly so to ensure that Eq. (S5) holds, i.e. Ks∆t < log ( κδ κδ− ) .
We cover τ with balls of radius δ = min{δgoal, δclear}, where Bδgoal ⊆ Fgoal. The balls are spaced equally in time with the center of the ith ball is in ci = τ(∆t · i),∀i = 0 : m, where m = tτ/∆t. Therefore, c0 = s0 and cm = sgoal. We now examine the probability of RRT propagating along τ . Suppose that there exists a vertex v ∈ B2δ/5(ci), we need to bound the probability p that by taking a random sample srand, there will be a vertex snear ∈ Bδ(ci) such that snew ∈ B2δ/5(ci+1). Lemma 3 provides a lower bound for the probability that snear ∈ Bδ(ci), given that there exists a vertex v ∈ B2δ/5(ci), of |Bδ/5(s)|/|S|. Lemma 2 provide a lower bound for the probability of
choosing an an action from snear to snew of ρ ≡ ζ|S|·
κδ−eKs∆t(κδ− ) ∆tKae Ks∆t
|A| > 0, where we have substituted tτ with ∆t. Consequently, p ≥ (|Bδ/5(s)| · ρ)/|S|. For RRT to recover τ , the transition between consecutive circles must be repeated m times. This stochastic process can be described as a binomial distribution, where we perform k trials (randomly choosing srand), with m successes (transition between circles) and a transition success probability p. The probability mass function of a binomial distribution is Pr(X = m) = Pr(m; k, p) =( k m ) pm(1− p)k−m. We use the cumulative distribution function (CDF) to represent the upper bound for failure, i.e. the process was unable to perform m steps, which can be expressed as:
Pr(X < m) = m−1∑ i=0 ( k i ) pi(1− p)k−i. (S6)
Using Chernoff’s inequality we derive the tail bounds of the CDF when m ≤ p · k:
Pr(X < m) ≤ exp ( − 1
2p
(kp−m)2
k
) (S7)
= exp ( −1
2 kp+m− m
2
kp
) (S8)
≤ eme− 12pk = ae−bk. (S9)
In the other case, where p < m/k < 1, the upper bound is given by (Arratia & Gordon, 1989):
Pr(X < m) ≤ exp ( −kD (m k ‖ p )) , (S10)
where D is the relative entropy such that
D (m k ‖ p ) = m k log m k p + (1− m k ) log 1− mk 1− p .
Rearranging D, we can rewrite S10 as follows: Pr(X < m) ≤ exp ( −k ( m
k log
( m
kp ) + k −m k log ( 1− mk 1− p ))) (S11)
= exp ( −m log ( m
kp
)) exp ( −k log ( 1− mk 1− p )) exp ( m log ( 1− mk 1− p )) (S12)
= exp ( −m log ( m(1− p) kp(1− mk ) )) exp ( −k log ( 1− mk 1− p )) (S13)
≤ exp ( −k log ( 1− mk 1− p )) (S14)
≤ exp ( −k log ( 0.5
1− p
)) (S15)
≤ e−kp = ae−bk, (S16) where (S14) is justified for worst-case scenario where p = m/k, (S15) uses the fact that p < m/k < 0.5, hence 1−m/k > 0.5. The last step, (S16) is derived from the first term of the Taylor expansion of log ( 1
1−p ) = ∑∞ j=1 pj j .
As p and m are fixed and independent of k, we show that the expression for Pr(X < m) decays to zero exponentially with k, therefore RRT is probabilistically complete.
It worth noting that as expected the failure probability Pr(X < m) depends on problem-specific properties, which give rise to the values of a and b. Intuitively, a depends on the scale of the problem such as volume of the goal set |FRLgoal| and how complex and long the solution needs to be, as evident in Eq. (S9). More importantly, b depends on the probability p. Therefore, it is a function of the dimensionality of S (through the probability of sampling srand) and other environment parameters such as clearance (defined by δ) and dynamics (via Ks, Ka), as specified in Eq. (S4).

heading: S3 APPENDIX C: EXPERIMENTAL SETUP
text: All experiments were run using a single 2.2GHz core and a GeForce GTX 1080 Ti GPU.
S3.1 ENVIRONMENTS
All environments are made available in supplementary code. Environments are based on Gym (Brockman et al., 2016), with modified sparse reward functions and state spaces. All environments emit a −1 reward per timestep unless noted otherwise. The environments have been further changed from Gym as follows:
• Cartpole Swingup- The state space S ⊆ R4 consists of states s = [ x, θ, ẋ, θ̇ ] where x is
cart position, ẋ is cart linear velocity, θ is pole angle (measuring from the y-axis) and θ̇ pole angular velocity. Actions A ⊆ R are force applied on the cart along the x-axis. The goal space Fgoal is {s ∈ S | cos θ > 0.9}. Note that reaching the goal space does not terminate an episode, but yields a reward of cos θ. Time horizon is H = 500. Reaching the bounds of the rail does not cause failure but arrests the linear movement of the cart. • MountainCar- The state space S ⊆ R2 consists of states s = [x, θ] where x is car position
and ẋ is car velocity. Actions A ⊆ R are force applied by the car engine. The goal space Fgoal is {s ∈ S | x ≥ 0.45}. Time horizon is H = 200.
• Acrobot- The state space S ⊆ R4 consists of states s = [ θ0, θ1, θ̇0, θ̇1 ] where θ0, θ1 are the
angles of the joints (measuring from the y-axis and from the vector parallel to the 1st link, respectively) and θ̇0, θ̇1 are their angular velocities. Actions A ⊆ R are torque applied on the 2nd joint. The goal space Fgoal is {s ∈ S | − cos θ0 − cos (θ0 + θ1) > 1.9}. In other words, the set of states where the end of the second link is at a height y > 1.9. Time horizon is H = 500.
• Pendulum- The state space S ⊆ R2 consists of states s = [ θ, θ̇ ] where θ is the joint angle
(measured from the y-axis) and θ̇ is the joint angular velocity. Actions A ⊆ R are torque applied on the joint. The goal space Fgoal is {s ∈ S | cos θ > 0.99}. Note that reaching the goal space does not terminate an episode, but yields a reward of cos θ. Time horizon is H = 100.
• Reacher- The state space S ⊆ R6 consists of states s = [ θ0, θ1, x, yθ̇0, θ̇1 ] where θ0, θ1
are the angles of the joints, (x, y) are the coordinates of the target and θ̇0, θ̇1 are the joint angular velocities. ActionsA ⊆ R2 are torques applied at the 2 joints. The goal space Fgoal is the set of states where the end-effector is within a distance of 0.01 from the target. Time horizon is H = 50. • Fetch Reach- A high-dimensional robotic task where the state space S ⊆ R13 con-
sists of states s = [gripper_pos, finger_pos, gripper_state, finger_state, goal_pos] where gripper_pos, gripper_vel are the Cartesian coordinates and velocities of the Fetch robot’s gripper, finger_state, finger_vel are the two-dimensional position and velocity of the gripper fingers, and goal_pos are the Cartesian coordinates of the goal. Actions A ⊆ R4 are relative target positions of the gripper and fingers, which the MuJoCo controller will try to achieve. The goal space Fgoal is the set of states where the end-effector is within a distance of 0.05 from goal_pos. Time horizon is H = 50. Note that this problem is harder than the original version in OpenAI Gym, as we only sample goal_pos that are far from the gripper’s initial position. • Hand Reach- A high-dimensional robotic task where the state space S ⊆ R78 consists of
states s = [joint_pos, joint_vel, fingertip_pos, goal_pos] where joint_pos, joint_vel are the angles and angular velocities of the Shadow hand’s 24 joints, fingertip_pos are the Cartesian coordinates of the 5 fingertips, and goal_pos are the Cartesian coordinates of the goal positions for each fingertip. Actions A ⊆ R20 are absolute target angles of the 20 controllable joints, which the MuJoCo controller will try to achieve. The goal space Fgoal is the set of states where all fingertips are simultaneously within a distance of 0.02 from their respective goals. Time horizon is H = 50.
S3.2 EXPERIMENTAL SETUP AND HYPER-PARAMETER CHOICES
All experiments feature a policy with 2 fully-connected hidden layers of 32 units each with tanh activation, with the exception of Reacher, for which a policy network of 4 fully-connected hidden layers of 128 units each with relu activation is used. For all environments we use a linear feature baseline for TRPO.
Default values are used for most hyperparameters. A discount factor of γ = 0.99 is used in all environments. For VIME, hyperparameters values reported in the original paper are used, and the implementation published by the authors was used.
For TRPO, default hyperparameter values and implementation from Garage are used: KL divergence constraint δ = 10−2, and Gaussian action noise N (0, 0.32). In comparisons with VIME-TRPO and vanilla TRPO, the R3L goal sampling probability pg is set to 0.05, as proposed in Urmson & Simmons (2003). Goal sets Fgoal are defined in Appendix S3.1 for each environment. In all experiments, the local policy πl learned during R3L exploration uses Bayesian linear regression with prior precision 0.1 and noise precision 1.0, as well as 300 random Fourier features (Rahimi & Recht, 2008) approximating a square exponential kernel with lengthscale 0.3.

Review summaries:
This paper proposes a method, R3L, for exploration in reinforcement learning. R3L performs an exploration procedure before policy optimization. In R3L exploration, it considers the task as a planning problem, and applies RRT to find feasible solutions (trajectories). Then it applies a warm start procedure for policy optimization, where the feasible solutions found by RRT are used to initialize the policy by supervised learning. The paper provides theoretical guarantees of R3L exploration finding feasible solutions. Empirically, the algorithmic contribution is demonstrated by comparing with information theoretical exploration methods in benchmark control domains.