LS 3 : Latent Space Safe Sets for Long-Horizon Visuomotor Control of Sparse Reward Iterative Tasks
Anonymous Author(s)
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Abstract:
Reinforcement learning (RL) has shown impressive success in exploring high-dimensional environments to learn complex tasks, but can often exhibitunsafe behaviors and require extensive environment interaction when explorationis unconstrained.
A promising strategy for learning in dynamically uncertainenvironments is requiring that the agent can robustly return to learned safe sets,where task success (and therefore safety) can be guaranteed.
While this approachhas been successful in low-dimensions, enforcing this constraint in environmentswith visual observation spaces is exceedingly challenging.
We present a novelcontinuous representation for safe sets by framing it as a binary classiﬁcationproblem in a learned latent space, which ﬂexibly scales to high-dimensional imageobservations.
We then present a new algorithm, Latent Space Safe Sets (LS 3 ),which uses this representation for long-horizon tasks with sparse rewards.
Weevaluate LS 3 on 4 domains, including a challenging sequential pushing task insimulation and a physical cable routing task.
We ﬁnd that LS 3 can use prior tasksuccesses to restrict exploration and learn more efﬁciently than prior algorithmswhile satisfying constraints.
See https://tinyurl.com/latent-safe-setsfor supplementary material.
Introduction
Visual planning over learned forward dynamics models is a popular area of research in roboticcontrol from images [1, 2, 3, 4, 5, 6, 7], as it enables closed-loop, model-based control for taskswhere the state of the system is not directly observable or difﬁcult to analytically model, such as theconﬁguration of a sheet of fabric or segment of cable.
These methods learn predictive models overeither images or a learned latent space, which can then be used to plan actions which maximize sometask reward.
While these approaches have signiﬁcant promise, there are several open challenges inlearning policies from visual observations.
First, reward speciﬁcation is particularly challenging forvisuomotor control tasks, because high-dimensional observations often do not expose the necessaryfeatures required to design dense, informative reward functions [8], especially for long-horizontasks.
Second, while many prior reinforcement learning methods have been successfully applied toimage-based control tasks [9, 10, 11, 12, 13], learning policies from image observations often requiresextensive exploration due to the high dimensionality of the observation space and the difﬁculties inreward speciﬁcation, making safe and efﬁcient learning exceedingly challenging.
One promising strategy for efﬁciently learning safe control policies is to learn a safe set [14, 15],which captures the set of states from which the agent is known to behave safely, which is oftenreformulated as the set of states where it has previously completed the task.
When used to restrictexploration, this safe set can be used to enable highly efﬁcient and safe learning [16, 17, 14], asexploration is restricted to states in which the agent is conﬁdent in task success.
However, whilethese safe sets can give rise to algorithms with a number of appealing theoretical properties such asconvergence to a goal set, constraint satisfaction, and iterative improvement [14, 18, 16], using themfor controller design for practical problems requires developing continuous approximations at theexpense of maintaining theoretical guarantees [17].
This choice of continuous approximation is a keyelement in determining the applications to which these safe sets can be used for control
[Figure]
Prior works have presented approaches which collect a discrete safe set of states from previouslysuccessful trajectories and represent a continuous relaxation of this set by constructing a convex hullof these states [14] or via kernel density estimation with a tophat kernel function [17].
While theseapproaches have been successful for control tasks with low-dimensional states, extending them tohigh-dimensional observations presents two key challenges: (1) scalability: these prior methodscannot be efﬁciently applied when the number of observations in prior successful trajectories is large,as querying safe set inclusion scales linearly with number of samples it contains and (2) representationcapacity: both of these prior approaches do not scale well to high dimensional observations and arelimited in the space of continuous sets that they can efﬁciently represent.
Applying these ideas tovisuomotor control is even more challenging, since images do not directly expose details about thesystem state or dynamics that are typically needed for formal controller analysis [16, 14, 19].
This work makes several contributions.
First, we introduce a scalable continuous approximationmethod which makes it possible to leverage safe sets for visuomotor policy learning.
The key idea is toreframe the safe set approximation as a binary classiﬁcation problem in a learned latent space, wherethe objective is to distinguish states from successful trajectories from those in unsuccessful trajectories.
Second, we present Latent Space Safe Sets (LS 3 ), a model-based RL algorithm which encouragesthe agent to maintain plans back to regions in which it is conﬁdent in task completion, even whenlearning in high dimensional spaces.
This constraint makes it possible to deﬁne a control strategy to(1) improve safely by encouraging consistent task completion (and therefore avoid unsafe behavior)and (2) learn efﬁciently since the agent only explores promising states in the immediate neighborhoodof those in which it was previously successful.
Third, we present simulation experiments on 3visuomotor control tasks which suggest that LS 3 can learn to improve upon demonstrations moresafely and efﬁciently than prior algorithms.
Fourth, we conduct physical experiments on a visionbased cable routing task which suggest that LS 3 can learn more efﬁciently than prior algorithmswhile consistently completing the task and satisfying constraints during learning.
Related Work
2.1 Safe, Iterative Learning Control
In iterative learning control (ILC), the agent tracks a reference trajectory and uses data from controllerrollouts to reﬁne tracking performance [20].
Rosolia et al.
[21], Rosolia and Borrelli [18, 14] presenta new class of algorithms, known as Learning Model Predictive Control (LMPC), which are referencefree and instead iteratively improve upon the performance of an initial feasible trajectory.
To achievethis, Rosolia et al.
[21], Rosolia and Borrelli [18, 14] use data from controller rollouts to learna safe set and value function, with which recursive feasibility, stability, and local optimality canbe guaranteed given a known, deterministic nonlinear system or stochastic linear system undercertain regularity assumptions.
However, a core challenge with these algorithms is that they assumeknown system dynamics, and cannot be applied to high-dimensional control problems
et al.
[17] extends the LMPC framework to higher dimensional settings in which system dynamicsare unknown and must be learned, but the visuomotor control setting introduces a number of new challenges as learned system dynamics, safe sets, and value functions must ﬂexibly scale to visualinputs.
Richards et al.
[15] designs expressive safe sets for ﬁxed policies using neural networkclassiﬁers with Lyapunov constraints.
In contrast, LS 3 constructs a safe set for an improving policyby optimizing a task cost function instead of uniformly expanding across the state space.
2.2 Model Based Reinforcement Learning
There has been signiﬁcant recent progress in algorithms which combine ideas from model-basedplanning and control with deep learning [22, 23, 24, 25, 26, 27].
These algorithms are gainingpopularity in the robotics community as they enable leaning complex policies from data whilemaintaining some of the sample efﬁciency and safety beneﬁts of classical model-based controltechniques.
However, these algorithms typically require hand-engineered dense cost functions fortask speciﬁcation, which can often be difﬁcult to provide, especially in high-dimensional spaces.
This motivates leveraging demonstrations (possibly suboptimal) to provide an initial signal regardingdesirable agent behavior.
There has been some prior work on leveraging demonstrations in modelbased algorithms such as Quinlan and Khatib [28] and Ichnowski et al.
[29], which use modelbased control with known dynamics to reﬁne initially suboptimal motion plans, and Fu et al.
[24],which uses demonstrations to seed a learned dynamics model for fast online adaptation usingiLQR
Thananjeyan et al.
[17], Zhu et al.
[30] present ILC algorithms which rapidly improveupon suboptimal demonstrations when system dynamics are unknown.
However, these algorithmseither require knowledge of system dynamics [28, 29] or are limited to low-dimensional statespaces [24, 17, 30] and cannot be ﬂexibly applied to visuomotor control tasks.
2.3 Reinforcement Learning from Pixels
Reinforcement learning and model-based planning from visual observations is gaining signiﬁcantrecent interest as RGB images provide an easily available observation space for robot learning [1, 31].
Recent work has proposed a number of model-free and model-based algorithms that have seen successin laboratory settings in a number of robotic tasks when learning from visual observations [32, 33,10, 34, 12, 13, 1, 35, 31].
However, two core issues that prevent application of many RL algorithmsin practice, inefﬁcient exploration and safety, are signiﬁcantly exacerbated when learning fromhigh-dimensional visual observations in which the space of possible behaviors is very large and thefeatures required to determine whether the robot is safe are not readily exposed.
There has beensigniﬁcant prior work on addressing inefﬁciencies in exploration for visuomotor control such aslatent space planning [2, 31, 35] and goal-conditioned reinforcement learning [13, 10].
However, safereinforcement learning for visuomotor tasks has received substantially less attention.
Thananjeyanet al.
[36] and Kahn et al.
[37] present reinforcement learning algorithms which estimate the likelihoodof constraint violations to avoid them [36] or reduce the robot’s velocity [37].
Unlike these algorithms,which focus on presenting methods to avoid violating user-speciﬁed constraints, LS 3 additionallyprovides consistent task completion during learning by limiting exploration to the neighborhood ofprior task successes.
This difference makes LS 3 less susceptible to the challenges of unconstrainedexploration present in standard model-free reinforcement learning algorithms.
Problem Statement
We consider an agent interacting in a ﬁnite horizon goal-conditioned Markov Decision Processes(MDP) which can be described with the tuple M =
( S , G , A , P ( ·|· , · ) , R ( · , · ) , µ, T ) .
S and A are 121 the state and action spaces, P :
S ⇥ A ⇥ S !
[0 , 1] maps a state and action to a probabilitydistribution over subsequent states, R : S ⇥
A ⇥ S !
R is the reward function, µ is the initialstate distribution ( s 0 ⇠ µ ), and T is the time horizon.
In this work, the agent is only provided with
RGB image observations s t 2 R W ⇥ H ⇥ 3 + = S , where W and H are the image width and height inpixels, respectively.
We consider iterative tasks, where the agent must reach a ﬁxed goal set G ✓ S as 126 efﬁciently as possible and the support of µ is small.
While there are a number of possible choices ofreward functions that would encourage fast convergence to G , providing shaped reward functions canbe exceedingly challenging, especially when learning from high dimensional observations.
Thus, asin Thananjeyan et al.
[17], we consider a sparse reward function that only indicates task completion:
R ( s, a, s 0 )
=
if s 0 2 G and  1 otherwise.
To incorporate constraints, we augment M with an extraconstraint indicator function C : S ! { 0 , 1 } which indicates whether a state satisﬁes user-speciﬁed 132 state-space constraints, such as avoiding known obstacles.
This is consistent with the modiﬁed CMDPformulation used in [36].
We assume that R and C can be evaluated on the current state of the system,but may be approximated using prior data for use during planning.
We make this assumption because
[Figure]
in practice we plan over predicted future states, which may not be predicted at sufﬁciently highﬁdelity to expose the necessary information to directly evaluate R and C during planning.
Given a policy ⇡ : S !
A , we deﬁne its expected total return in M as R ⇡ = E ⇡ ,µ,P [
P t R ( s t , a t )] .
Furthermore, we deﬁne P ⇡ C ( s ) as the probability of future constraint violation (within time horizon T )under policy ⇡ from state s .
The objective is to maximize the expected return R ⇡ while maintaininga constraint violation probability lower than  C .
This can be written formally as follows:
[Equation]
We assume that the agent is provided with an ofﬂine dataset D of transitions in the environment ofwhich some subset D constraint ( D are constraint violating and some subset D success ( D appear in 143 successful demonstrations from a suboptimal supervisor.
As in [36],
D constraint contains examples of 144 constraint violating behaviors (for example from prior runs of different policies or collected underhuman supervision) so that the agent can learn about states which violate user-speciﬁed constraints.
Latent Space Safe Sets (LS 3 )
We describe how LS 3 uses demonstrations and online interaction to safely learn iteratively improvingpolicies.
Section 4.1 describes how we learn a low-dimensional latent representation of image observations to facilitate efﬁcient model-based planning.
To enable this planning, we learn a probabilisticforward dynamics model as in [26] in the learned latent space and models to estimate whether planswill likely complete the task (Section 4.2) and to estimate future rewards and constraint violations(Section 4.3) from predicted trajectories.
In Section 4.4, we discuss how these components aresynthesized in LS 3 .
Dataset D is expanded using online rollouts of LS 3 and used to update all latentspace models (Sections 4.2 and 4.3) after every K rollouts.
See Algorithm 1 and the supplement forfurther details on training procedures and data collection.
Require: ofﬂine dataset D , number of updates
4.1 Learning a Latent Space for Planning
Learning compressed representations of images has been a popular approach in vision based controlto facilitate efﬁcient algorithms for planning and control which can reason about lower dimensional inputs [2, 35, 6, 38, 39, 31].
To learn such a representation, we train a  -variational autoencoder [40]on states in D to map states to a probability distribution over a d -dimensional latent space Z .
Theresulting encoder network f enc ( z | s ) is then used to sample latent vectors z t ⇠ f enc ( z t | s t ) to train aforward dynamics model, value function, reward estimator, constraint classiﬁer, safe set, and combinethese elements to deﬁne a policy for model-based planning.
Motivated by Laskin et al.
[41], duringtraining we augment inputs to the encoder with random cropping, which we found to be helpful inlearning representations that are useful for planning.
For all environments we use a latent dimensionof d = 32 , as in [2] and found that varying d did not signiﬁcantly affect performance.
4.2 Latent Safe Sets for Model-Based Control
LS 3 learns a binary classiﬁer for latent states to learn a latent space safe set that represents statesfrom which the agent has high conﬁdence in task completion based on prior experience.
Because theagent can reach the goal from these states, they are safe: the agent can avoid constraint violations bysimply completing the task as before.
While classical algorithms use known dynamics to constructsafe sets, we approximate this set using successful trajectories from prior iterations.
At each iterationj , the algorithm collects K trajectories in the environment.
We then deﬁne the sampled safe set atiteration j , S j , as the set of states from which the agent has successfully navigated to G in iterations0 through j of training, where demonstrations trajectories are those collected at iteration 0 .
Werefer to the dataset collecting all these states as D success .
This discrete set is difﬁcult to plan to withcontinuous-valued state distributions so we leverage data from D success (data in the sampled safe set),data from D \ D success (data outside the sampled safe set), and the learned encoder from Section 4.1 tolearn a continuous relaxation of this set in latent space (the latent safe set).
We train a neural networkwith a binary cross-entropy loss to learn a binary classiﬁer f S ( · ) that predicts the probability of astate s t with encoding z t being in S j .
To mitigate the negative bias that appears when trajectoriesthat start in safe regions fail, we utilize the intuition that if a state s t +1 2 S j then it is likely thats t is also safe.
To do this, rather than just predict S j , we train f S with a recursive objective topredict max( S j ,  S f S ( s t +1 )) .
The relaxed latent safe set is parameterized by the superlevel sets off S , where the level  S is adaptively set during execution:
S j Z = { z t | f S ( · )( z t )   S } .
4.3 Reward and Constraint Estimation
In this work, we deﬁne rewards based on whether the agent has reached a state s 2 G , but we needrewards that are deﬁned on predictions from the dynamics, which may not correspond to valid realimages.
To address this, we train a classiﬁer f G : Z ! { 0 , 1 } to map the encoding of a state to 190 whether the state is contained in
G using terminal states in D success (which are known to be in G ) and 191 other states in
D .
However, in the temporally-extended, sparse reward tasks we consider, rewardprediction alone is insufﬁcient because rewards only indicate whether the agent is in the goal set, andthus provide no signal on task progress unless the agent can plan to the goal set.
To address this, asin prior MPC-literature [17, 16, 14, 8], we train a recursively-deﬁned value function (details in thesupplement).
Similar to the reward function, we use the encoder (Section 4.1) to train a classiﬁerf C : Z !
[0 , 1] with data of constraint violating states from D constraint and the constraint satisfying 197 states in
D \ D constraint to map the encoding of a state to the probability of constraint violation.
4.4 Model-Based
Planning with LS[-
-]
LS 3 aims to maximize total rewards attained in the environment while limiting constraint violationprobability within some threshold  C (equation 1).
We optimize an approximation of this objectiveover an H -step receding horizon with model-predictive control.
Precisely, LS 3 solves the followingoptimization problem to generate an action to execute at timestep t :
[Equation]
In this problem, the expectations and probabilities are taken with respect to the learned, probabilisticdynamics model f dyn ( z t +1 | z t , a t ) .
The optimization problem is solved approximately using the crossentropy method (CEM) [42] which is a popular optimizer in model-based RL [43, 17, 16, 44, 36].
The objective function is the expected sum of future rewards if the agent executes a t : t + H  1 andthen subsequently executes ⇡ (equation 2).
First, the current state s t is encoded to z t (equation 3).
Then, for a candidate sequence of actions a t : t + H  1 , an H -step latent trajectory { z t +1 , . . .
, z t + H } is 209 sampled from the learned dynamics f dyn (equation 4).
LS 3 constrains exploration using two chanceconstraints: (1) the terminal latent state in the plan must fall in the safe set (equation 5) and (2) alllatent states in the trajectory must satisfy user-speciﬁed state-space constraints (equation 6).
Z Cis the set of all latent states such that the corresponding observation is constraint violating.
Theoptimizer estimates constraint satisfaction probabilities for a candidate action sequence by simulatingit repeatedly over f dyn .
The ﬁrst chance constraint ensures the agent maintains the ability to return tosafe states where it knows how to do the task within H steps if necessary.
Because the agent replans ateach timestep, the agent need not return to the safe set: during training, the safe set expands, enablingfurther exploration.
In practice, we set  S for the safe set classiﬁer f S adaptively as described inthe supplement.
The second chance constraint encourages constraint violation probability of nomore than  C .
After solving the optimization problem, the agent executes the ﬁrst action in the plan:⇡ ( z t ) =
a t where a t is the ﬁrst element of a ⇤ t : t + H  1 , observes a new state, and replans.
Experiments
We evaluate LS 3 on 3 robotic control tasks in simulation and a physical cable routing task on theda Vinci Research Kit (dVRK) [45].
Safe RL is of particular interest for surgical robots such asthe dVRK due to its delicate structure, motivating safety, and relatively imprecise controls [17, 46],motivating closed-loop control.
We study whether LS 3 can learn more safely and efﬁciently thanalgorithms that do not structure exploration based on prior task successes.
5.1 Comparisons
We evaluate LS 3 in comparison to prior algorithms that behavior clone suboptimal demonstrationsbefore exploring online ( SACfD ) [47] or leverage ofﬂine reinforcement learning to learn a policyusing all ofﬂine data before updating the policy online ( AWAC ) [48].
For both of these comparisonswe enforce constraints via a tuned reward penalty of  for constraint violations as in [49].
We alsoimplement a version of SACfD with a learned recovery policy ( SACfD+RRL ) using the Recovery
RL algorithm [36] to use prior constraint violating data to try to avoid constraint violating states.
Wethen compare LS 3 to an ablated version without the safe set constraint (just binary classiﬁcation (BC))in equation 5 ( LS 3 (  Safe Set) ) to evaluate if the safe set promotes consistent task completion andstable learning.
Finally, we compare LS 3 to a an ablated version of the safe set classiﬁer (Section 4.2)without a recursive objective, where the classiﬁer is just trained to predict S j ( LS 3 (BC SS) ).
Seethe supplement for details on hyperparameters and ofﬂine data used for LS 3 and prior algorithms.
5.2 Evaluation Metrics
For each algorithm on each domain, we aggregate statistics over random seeds (10 for simulationexperiments, 3 for the physical experiment), reporting the mean and standard error across the seeds.
We present learning curves that show the total sum reward for each training trajectory to study howefﬁciently LS 3 and the comparisons learn each task.
Because all tasks use the sparse task completionbased rewards deﬁned in Section 3, the total reward for a trajectory is the time to reach the goalset, where more negative rewards correspond to slower convergence to G .
Thus, for a task with taskhorizon T , a total reward greater than  T implies successful task completion.
The state is frozenin place upon constraint violation until the task horizon elapses.
We also report task success andconstraint satisfaction rates for LS 3 and comparisons during learning to study (1) the degree to whichtask completion inﬂuences sample efﬁciency and (2) how safely different algorithms explore.
LS 3collects K = 10 trajectories in between training phases on simulated tasks and K = 5 in betweentraining phases for physical tasks, while the SACfD and AWAC comparisons update their parametersafter each timestep.
This presents a metric in terms of the amount of data collected across algorithms.
5.3 Domains
In simulation, we evaluate LS 3 on 3 vision-based continuous control domains that are illustratedin Figure 3.
We evaluate LS 3 and comparisons on a constrained visual navigation task (Pointmass
Navigation) where the agent navigates from a ﬁxed start state to a ﬁxed goal set while avoiding alarge central obstacle.
We study this domain to gain intuition and visualize the learned value function,goal/constraint indicators, and safe set in Figure 2.
We then study a constrained image-based reachingtask (Reacher) based on [50], where the objective is to navigate the end effector of a 2-link planarrobotic arm to a yellow goal position without the end-effector entering a red stay out zone.
We thenstudy a challenging sequential image-based robotic pushing domain (Sequential Pushing), in whichthe objective is to push each of the 3 blocks forward on the table without pushing them to eitherside and causing them to fall out of the workspace.
Finally, we evaluate LS 3 with an image-basedphysical experiment on the da Vinci Research Kit (dVRK) [51] (Figure 3), where the objective is toguide the endpoint of a cable to a goal region without letting the cable or end effector collide with anobstacle.
The Pointmass Navigation and Reaching domains have a task horizon of T = 100 whilethe Sequential Pushing domain and physical experiment have task horizons of T = 150 and T = 50respectively.
See the supplement for more details on all domains.
5.4 Simulation Results
We ﬁnd that LS 3 is able to learn more stably and efﬁciently than all comparisons across all simulateddomains while converging to similar performance within 250 trajectories collected online (Figure 4).
LS 3 is able to consistently complete the task during learning, while the comparisons, which do notlearn a safe set to structure exploration based on prior successes, exhibit much less stable learning.
Additionally, in Table 1 and Table 2, we report the task success rate and constraint violation rate ofall algorithms during training.
We ﬁnd that LS 3 achieves a signiﬁcantly higher task success rate thancomparisons on all tasks.
We also ﬁnd that LS 3 violates constraints less often than comparisons on the
Reacher task, but violates constraints more often than SACfD and SACfD+RRL on the other domains.
This is because SACfD and SACfD+RRL spend much less time in the neighborhood of constraintviolating states during training due to their lower task success rates.
Because they do not efﬁcientlylearn to perform the tasks, they do not violate constraints as often.
We ﬁnd that the AWAC comparisonachieves very low task performance.
While AWAC is designed for ofﬂine reinforcement learning, tothe best of our knowledge, it has not been previously evaluated on long-horizon, image-based tasksas in this paper, which we hypothesize are very challenging for it.
[Table] [Table]
We ﬁnd LS 3 has a lower success rate when the safe set constraint is removed (LS 3 (  Safe Set)) as expected.
The safe set is particularly important in the sequential pushing task, and LS 3 (  Safe Set)has a much lower task completion rate than LS 3 .
LS 3 without the recursive classiﬁcation objectivefrom Section 4.2 (LS 3 (BC SS)) has similar performance to LS 3 on the navigation environment,but learns substantially more slowly on the Reacher environment and performs signiﬁcantly worsethan LS 3 on the more challenging Pushing environment as the learned safe set is unable to exploittemporal structure to distinguish safe states from unsafe states.
See the supplement for details onexperimental parameters and ofﬂine data used for LS 3 and comparisons and ablations studying theeffect of the planning horizon and threshold used to deﬁne the safe set.
5.5 Physical Results
In physical experiments, we compare LS 3 to SACfD and SACfD+RRL (Figure 5) on the physical cablerouting task illustrated in Figure 3.
We ﬁnd LS 3 quickly outperforms the suboptimal demonstrationswhile succeeding at the task signiﬁcantly more often than both comparisons, which are unable to learnthe task and also violate constraints more than LS 3 .
We hypothesize that the difﬁculty of reasoningabout cable collisions and deformation from images makes it challenging for prior algorithms tomake sufﬁcient task progress as they do not use prior successes to structure exploration.
See thesupplement for details on experimental parameters and ofﬂine data used for LS 3 and comparisons.
Discussion and Future Work
We present LS 3 , a scalable algorithm for safe and efﬁcient policy learning for visuomotor tasks.
LS 3 structures exploration by learning a safe set in a learned latent space, which captures the set ofstates from which the agent is conﬁdent in task completion.
LS 3 then ensures that the agent can planback to states in the safe set, encouraging consistent task completion during learning.
Experimentssuggest that LS 3 can safely and efﬁciently learn 4 visuomotor control tasks, including a challengingsequential pushing task in simulation and a cable routing task on a physical robot.
In future work, weare excited to explore further physical evaluation of LS 3 on safety critical visuomotor control tasksand applications to systems with dynamic constraints on velocity or acceleration.