Title: e-COP : Episodic Constrained Optimization of Policies
Venue: NeurIPS 2024
Accepted: yes
Keywords: reinforcement learning

policy optimization

constrained MDPs
Abstract: In this paper, we present the e-COP algorithm, the first policy optimization algorithm for constrained Reinforcement Learning (RL) in episodic (finite horizon) settings. Such formulations are applicable when there are separate sets of optimization criteria and constraints on a system's behavior. We approach this problem by first establishing a policy difference lemma for the episodic setting, which provides the theoretical foundation for the algorithm. Then, we propose to combine a set of established and novel solution ideas to yield the e-COP  algorithm that is easy to implement and numerically stable, and provide a theoretical guarantee on optimality under certain scaling assumptions. Through extensive empirical analysis using benchmarks in the Safety Gym suite, we show that our algorithm has similar or better performance than SoTA (non-episodic) algorithms adapted for the episodic setting. The scalability of the algorithm  opens the door to its application in safety-constrained Reinforcement Learning from Human Feedback for Large Language or Diffusion Models.

e-COP : Episodic Constrained Optimization of Policies

Abstract

In this paper, we present the e-COP algorithm, the first policy optimization algorithm for constrained Reinforcement Learning (RL) in episodic (finite horizon) settings. Such formulations are applicable when there are separate sets of optimization criteria and constraints on a system's behavior. We approach this problem by first establishing a policy difference lemma for the episodic setting, which provides the theoretical foundation for the algorithm. Then, we propose to combine a set of established and novel solution ideas to yield the e-COP algorithm that is easy to implement and numerically stable, and provide a theoretical guarantee on optimality under certain scaling assumptions. Through extensive empirical analysis using benchmarks in the Safety Gym suite, we show that our algorithm has similar or better performance than SoTA (non-episodic) algorithms adapted for the episodic setting. The scalability of the algorithm opens the door to its application in safety-constrained Reinforcement Learning from Human Feedback for Large Language or Diffusion Models.

1 Introduction

RL problems may be formulated in order to satisfy multiple simultaneous objectives. These can include performance objectives that we want to maximize, and physical, operational or other objectives that we wish to constrain rather than maximize. For example, in robotics, we often want to optimize a task completion objective while obeying physical safety constraints. Similarly, in generative AI models, we want to optimize for human preferences while ensuring that the output generations remain safe (expressed perhaps as a threshold on an automatic safety score that penalizes violent or other undesirable content). Scalable policy optimization algorithms such as TRPO [31], PPO [33], etc have been central to the achievements of RL over the last decade [34, 38, 7]. In particular, these have found utility in generative models, e.g., in the training of Large Language Models (LLMs) to be aligned to human preferences through the RL with Human Feedback (RLHF) paradigm [2]. But these algorithms are designed primarily for the unconstrained infinite-horizon discounted setting: Their use for constrained problems via optimization of the Lagrangian often gives unsatisfactory constraint satisfaction results. This has prompted development of a number of constrained policy optimization algorithms for the infinite-horizon discounted setting [3, 39, 43, 41, 19, 6, 5], and for the average setting [4].

However, many RL problems are more accurately formulated as episodic, i.e., having a finite time horizon. For instance, in image diffusion models [10, 21], the denoising sequence is really a finite step trajectory optimization problem, better suited to be solved via RL algorithms for the episodic setting. When existing algorithms for infinite horizon discounted setting are used for such problem, they exhibit sub-optimal performance or fail to satisfy task-specific constraints by prioritizing short-term constraint satisfaction over episodic goals [11, 28, 17]. Furthermore, the episodic setting allows for objective functions to be time-dependent which the infinite-horizon formulations do not. Even when the objective functions are

Rahul Jain

Sahil Singla

time-invariant, there is a key difference: for non-episodic settings, a stationary policy that is optimal exists whereas for episodic settings, the optimal policy is always non-stationary and time-dependent. This necessitates development of policy optimization algorithms specifically for the episodic constrained setting. We note that such policy optimization algorithms do not exist even for the unconstrained episodic setting.

In this paper, we introduce e-COP , a policy optimization algorithm for episodic constrained RL problems. Inspired by PPO, it uses deep learning-based function approximation, a KL divergence-based proximal trust region and gradient clipping, along with several novel ideas specifically for the finite horizon episodic and constrained setting. The algorithm is based on a technical lemma (Lemma 3.1) on the difference between two policies, which leads to a new loss function for the episodic setting. We also introduce other ideas that obviate the need for matrix inversion thus improving scalability and numerical stability. The resulting algorithm improves performance over state-of-the-art baselines (after adapting them for the episodic setting). In sum, the e-COP algorithm has the following advantages: (i) Solution equivalence: We show that the solution set of our e-COP loss function is same as that of the original CMDPproblem, leading to precise cost control during training and avoidance of approximation errors (see Theorem 3.3); (ii) Stable convergence: The e-COP algorithm converges tightly to safe optimal policies without the oscillatory behavior seen in other algorithms like PDO [13] and FOCOPS [43]; (iii) Easy integration: e-COP follows the skeleton structure of PPO-style algorithms using clipping instead of steady state distribution approximation, and hence can be easily integrated into existing RL pipelines; and (iv) Empiricial performance: the e-COP algorithm demonstrates superior performance on several benchmark problems for constrained optimal control compared to several state-of-the-art baselines.

Our Contributions and Novelty. We introduce the first policy optimization algorithm for episodic RL problems, both with and without constraints (no constraints is a special case). While some of the policy optimization algorithms can be adapted for the constrained setting via a Lagrangian formulation, as we show, they don't work so well empirically in terms of constraint satisfaction and optimality. The algorithm is based on a policy difference (technical) lemma, which is novel. We have gotten rid of Hessian matrix inversion, a common feature of policy optimization algorithms (see, for example, PPO [33], CPO [3], P3O [41], etc.) and replaced it with a quadratic penalty term which also improves numerical stability near the edge of the constraint set boundary - a problem unique to constrained RL problems. We provide an instance-dependent hyperparameter tuning routine that generalizes to various testing scenarios. And finally, our extensive empirical results against an extensive suite of baseline algorithms (e.g., adapted PPO [33], FOCOPS [43], CPO [3], PCPO [39], and P3O [41]) show that e-COP performs the best or near-best on a range of Safety Gym [12] benchmark problems.

Related Work. Abroad view of planning and model-free RL techniques for Constrained MDPs is provided in [27] and [18]. The development of SOTA policy optimization started with the TRPO algorithm [31], which used a trust region to update the current policy, and was further improved in PPO by use of proximal ideas [33]. This led to works like CPO [3], RCPO [36], and PCPO [39] for constrained RL problems in the infinite-horizon discounted setting. ACPO [4] extended CPO to the infinite-horizon average setting. These methods typically require inversion of a computationally-expensive Fischer information matrix at each update step, thus limiting scalability. Lagrangian-based algorithms [29, 30] showed that you could incorporate constraints but constrained satisfaction remained a concern. Algorithms like PDO [13] and RCPO [36] also use Lagrangian duality but to solve risk-constrained RL problems and suffer from computational overhead. Some other notable algorithms include IPO [26], P3O [41], APPO [15], etc. that use penalty terms, and hence do not suffer from computational overhead but have other drawbacks. For example, IPO assumes feasible intermediate iterations, which cannot be fulfilled in practice, P3O requires arbritarily large penalty factors for feasibility which can lead to significant estimation errors. We note that all the above algorithms are for the infinite-horizon discounted (non-episodic) setting (except ACPO [4], which is for the average setting). We are not aware of any policy optimization algorithm for the episodic RL problem, with or without constraints.

2 Preliminaries

An episodic, or fixed horizon Markov decision process (MDP) is a tuple, M : ' p S, A, r, P, µ, H q , where S is the set of states, A is the set of actions, r : S ˆ A ˆ S Ñ R is the reward function, P : S ˆ A ˆ S Ñr 0 , 1 s is the transition probability function such that P p s 1 | s, a q is the probability of transitioning to state s 1 from state s by taking action a , µ : S Ñr 0 , 1 s is the initial state distribution, and H is the time horizon for each episode (characterized by a terminal state s H ).

A policy π : S Ñ ∆ p A q is a mapping from states to probability distributions over the actions, with π p a | s q denoting the probability of selecting action a in state s , and ∆ p A q is the probability simplex over the action space A . However, due to the temporal nature of episodic RL, the optimal policies are generally not stationary, and we index the policy at time h by π h , and denote π 1: H ' p π h q H h ' 1 . Then, the total undiscounted reward objective within an episode is defined as

where τ refers to the sample trajectory p s 1 , a 1 , s 2 , a 2 , . . . , s H q generated when following a policy sequence, i.e., a h ' π h p¨| s h q , s h ` 1 ' P p¨| s h , a h q , and s 1 ' µ .

Let R h : H p τ q denote the total reward of a trajectory τ starting from time h until episode terminal time H generated by following the policy sequence π h : H . We also define the state-value function of a state s at step h as V π h p s q : ' E τ ' π r R h : H p τ q | s h ' s s and the action-value function as Q π h p s, a q : ' E τ ' π r R h : H p τ q | s h ' s, a h ' a s . The advantage function is A π h p s, a q : ' Q π h p s, a q ´ V π h p s q . We also define P π h p s | s 1 q ' ř a P A P π h p s, a | s 1 q , where the term P π h p s, a | s 1 q is the probability of reaching p s, a q at time step h following π and starting from s 1 .

Constrained MDPs. A constrained Markov decision process (CMDP) is an MDP augmented with constraints that restrict the set of allowable policies for that MDP. Specifically, we have m cost functions, C 1 , ¨ ¨ ¨ , C m (with each function C i : S ˆ A ˆ S Ñ R mapping transition tuples to costs, similar to the reward function), and bounds d 1 , ¨ ¨ ¨ , d m . And similar to the value function for the reward objective, we define the expected total cost objective for each cost function C i (called cost value for the constraint) as

The goal then, in each episode, is to find a policy sequence π ‹ 1: H such that

is the set of feasible policies for a CMDP for some given class of policies Π . Lastly, analogous to V π h , Q π h , and A π h , we can also define quantities for the cost functions C i p¨q by replacing, and denote them by V π C i ,h , Q π C i ,h , and A π C i ,h . Proofs of theorems and statements, if not given, are available in Appendix A.

Notation. r N s denotes t 1 , . . . , N u for some N P N . π h refers to the policy at time step h within an episode. Denote π s : t : ' p π s , π s ` 1 , . . . , π t q for some s ď t with s, t P r H s . We shall only write π k,h when it is necessary to differentiate policies from different episodes but at the same time h . It then naturally follows to define π k,s : t to be the sequence π s : t in episode k . We will denote π k ' π k, 1: H , and where not needed drop the index for the episode so that π ' π k .

3 Episodic Constrained Optimization of Policies ( e-COP )

In this section, we propose a constrained policy optimization algorithm for episodic MDPs. Policy optimization algorithms for MDPs have proven remarkably effective given their ability to computationally scale up to high dimensional continuous state and action spaces [31-33]. Such algorithms have also been proposed for infinite-horizon constrained MDPs with discounted criterion [1] as well as the average criterion [4] but not for the finite horizon (or as it is often called, the episodic) setting.

We note that finite horizon is not simply a special case of infinite-horizon discounted setting since the reward/cost functions in the former can be time-varying while the latter only allows for time-invariant objectives. Furthermore, even with time-invariant objectives, the optimal policy is time-dependent, while for the latter setting there an optimal policy that is stationary exists.

A Policy Difference Lemma for Episodic MDPs. Most policy optimization RL algorithms are based on a value or policy difference technical lemma [23]. Unfortunately, the policy difference lemmas that have been derived previously for the infinite-horizon discounted [31] and average case [4] are not applicable here and hence, we derive a new policy difference lemma for the episodic setting.

Lemma 3.1. For an episode of length H and two policies, π and π 1 , the difference in their performance assuming identical initial state distribution µ (i.e., s 1 ' µ ) is given by

The proof can be found in Appendix A.1. A key difference to note between the above and similar results for infinite-horizon settings [31, 4] is that considering stationary policies (and hence corresponding occupation measures) is not enough for the episodic setting since, in general, such a policy may be far from optimal. This explains why Lemma 3.1 looks so different (e.g., see (2) in [31], and Lemma 3.1 in [4]). Indeed, the lemma above indicates that policy updates do not have to recurse backwards from the terminal time as dynamic programming algorithms do for episodic settings, which is somewhat surprising.

A Constrained Policy Optimization Algorithm for Episodic MDPs. Iterative policy optimization algorithms achieve state of the art performance [33, 36, 39] on RL problems. Most such algorithms maximize the advantage function based on a suitable policy difference lemma, solving an unconstrained RL problem. Some additionally ensure satisfaction of infinite horizon expectation constraints [3, 4]. However, given that our policy lemma for the episodic setting (Lemma 3.1) is significantly different, we need to re-design the algorithm based on it. A first attempt is presented as Algorithm 1, where each iteration corresponds to an update with a full horizon H episode.

Algorithm 1 I terative P olicy O ptimization for C onstrained E pisodic (IPOCE) RL

1: Input: Initial policy π 0 , number of episodes K , episode horizon H .

2: for k ' 1 , 2 , . . . , K do

3: Run π k ´ 1 to collect trajectories τ .

4: Evaluate A π k ´ 1 h and A π k ´ 1 C i ,h for h P r H s from τ .

5: for t ' H,H ´ 1 , . . . , 1 do

7: end for

9: end for

The iterative constrained policy optimization algorithm introduced above uses the current iterate of the policy π k to collect a trajectory τ , and use them to evaluate A π k ´ 1 h and A π k ´ 1 C i ,h for h P r H s . At the end of the episode, we solve H optimization problems (one for each h P r H s ) that result in a new sequence of policies π . As is natural in episodic problems, we do backward iteration in time, i.e., solve the problem in step (6) at h ' H , and then go backwards towards h ' 1 .

Note that the expectation of advantage functions in equation (3) is with respect to the policy π (the optimization variable) and its corresponding time-dependent state occupation distribution ρ π h . In the infinite-horizon settings, the expectation is with respect to the steady state stationary distribution, but that does not exist in the episodic setting.

Using current policy for action selection. Algorithm 1 represents an exact principled solution tothe constrained episodic MDP, but the intractable optimization performed in (3) makes it impractical (as in the case of infinite horizon policy optimization algorithms [31, 3, 39]). We proceed to introduce a sequence of ideas that make the algorithm practical (e.g., by avoiding computationally expensive Hessian matrix inversion for use with trust-region methods [33, 14, 41]). However, getting rid of trust regions leads to large updates on policies, but PPO [33] and P3O [41] successfully overcome this problem by clipping the advantage function and adding a ReLU-like penalty term to the minimization objective. Motivated by this, we rewrite the optimization problem in (3) as follows by parameterizing the policy π k,t in episode k and time step t by θ k,t :

where ρ p θ h q ' π θ k,h π θ k ´ 1 ,h is the importance sampling ratio, λ t,i is a penalty factor for constraint C i , and π k,θ h ' π k,h ' θ k,h . Note that the ReLU-like penalty term above is different from the traditional first-order and second-order gradient approximations that are employed in trust-region methods [3, 39]. In essence, the penalty is applied when the agent breaches the associated constraint, while the objective remains consistent with standard policy optimization when all constraints are satisfied.

Introducing quadratic damping penalty. It has been noted in such iterative policy optimization algorithms that the behaviour of the learning agent when it nears the constraint threshold is quite volatile during training [3, 39, 42]. This is because the penalty term is active only when the constraints are violated which results in sharp behavior change for the agent. To alleviate this problem, we introduce an additional quadratic damping term to the objective above, which provides stable cost control to compliment the lagged Lagrangian multipliers. This has proved effective in physics-based control applications [16, 25, 20] resulting in improved convergence since the damping term provides stability, while keeping the solution set the same as for the original Problem (3) and the adapted Problem (4) (as we prove later).

For brevity, we denote the constraint term in Problem (4) as

Now introduce a slack variable x t,i ě 0 for each constraint to convert the inequality constraint ( Ψ C i ,t p¨ , ¨q ď 0 ) to equality by letting

With this notation, we restate Problem (4) as:

Now we introduce the quadratic damping term and the intermediate loss function then takes the form,

where β is the damping factor, λ t ' p λ t,i q m i ' 1 , and x t ' p x t,i q m i ' 1 . We can then construct a primal-dual solution to the max-min optimization problem. The need for a slack variable x can be obviated by setting the partial derivative of L t p¨q with respect to x equal to 0. This leads to a ReLU-like solution: x ‹ t,i ' max ` 0 , ´ Ψ C i ,t p π k ´ 1 , π k q ´ λ t,i β ˘ . The intermediate problem then takes the form as below.

Proposition 3.2. The inner optimization problem in (5) with respect to x is a convex quadratic program with non-negative constraints, which can be solved to yield the following intermediate problem:

The proof can be found in Appendix A.1. One can see that the cost penalty is active when Ψ C i ,t p π k ´ 1 , π k q ě ´ λ t,i β rather than when Ψ C i ,t p π k ´ 1 , π k q ě 0 . This allows the agent to act in a constrained manner even before the constraint is violated. Further, as we show next, the introduction of the damping factor and the RELU-like penalty does not change the solution of the problem (under some suitable assumptions):

Theorem 3.3. Let π (3) ‹ be a solution to Problem (3) , and let ` π (6) ‹ , λ (6) ‹ ˘ be a solution to Problem (6) . Then, for sufficiently large β ą ¯ β and λ t,i ą ¯ λ @ i , π (3) ‹ is a solution to Problem (6) , and π (6) ‹ is a solution to Problem (3) .

We refer the reader to Appendix A.1 for the proof. This theorem implies that we can search for the optimal feasible policies of the CMDP Problem (1) by iteratively solving Problem (6). Next, we make some further modifications to Problem (6) that give us our final tractable algorithm.

Removing Lagrange multiplier dependency. Problem (6) requires a primal-dual algorithm that will iteratively solve over the policies and the dual variable λ . But from the Lagrangian, we can actually take a derivative with respect to λ , and then solve for it, which yields the following update rule for it:

This update rule simplifies the optimization problem and updates the Lagrange multipliers in the k th episode based on the constraint violation in the p k ´ 1 q th episode.

Clipping the advantage functions. Solving the optimization problem presented in equation (6) is intractable since we do not know ρ π beforehand. Hence, we replace ρ π by the empirical distribution observed with the policy of the previous episode, π k ´ 1 , i.e., ρ π k,h « ρ π k ´ 1 ,h @ h . Similar to [33] for PPO, we also use clipped surrogate objective functions for both the reward and cost advantage functions. Thus, the final problem combining equation (4) and equation (6) can be constructed as follows.

If we let

then, the final loss function r L t p¨q of the final problem is:

Usually for experiments, Gaussian policies with means and variances predicted from neural networks are used [31, 3, 33, 39]. We employ the same approach and since we work in the finite horizon setting, the reward and constraint advantage functions can easily be calculated from any trajectory τ ' π . The surrogate problem in equation (8) then includes the pessimistic bounds on Problem (6), which is unclipped.

Adaptive parameter selection. The value of β is required to be larger than the unknown ¯ β according to Theorem 3.3, but we also know that too large a β decays the performance (as seen in harmonic oscillator kinetic energy formulations [25, 20, 40]). To manage this tradeoff, we provide an instance-dependent adaptive way to adjust the damping factor as a hyperparameter. In each episode k , we update the damping parameter whenever a secondary constraint cost value denoted by C p π k q is larger than some threshold c k . Using Proposition 3.2, we provide the following definitions.

Algorithm 2 E pisodic C onstrained O ptimization of P olicies ( e-COP )

1: Input: Initial policy θ 0 : ' π 0 : ' π θ 0 , critic networks V ϕ 0 and V ψ 0 C i @ i , penalty factor β , number of episodes K , episode horizon H , learning rate α , penalty update factor p . 2: for k ' 1 , 2 , . . . , K do 3: Collect a set of trajectories D k ´ 1 with policy π k ´ 1 and update the critic network. 4: Get updated λ p k q using equation (7). 5: for t ' H,H ´ 1 , . . . , 1 do 6: Update the policy θ k,t Ð θ k,t ` 1 ´ α ∇ θ r L t p θ, λ p k q , β q using equation (8). 7: end for 8: if C p θ k q ě c k then 9: β ' min p β max , p β q 10: end if 11: end for

(a) Humanoid

(b) Circle

(c) Reach

(d) Grid

(e) Bottleneck

(f) Navigation

Hence, we increase β by a constant factor p ą 1 after every episode if C p π k q ě c k until a stopping condition is fulfilled, typically a constant β max . This leads to constraint-satisfying iterations that are more stable, and we show that it enables a fixed β to generalize well across various tasks. The initial β can simply be selected by a quantified line-search to obtain a feasible β ą ¯ β [3, 4].

We note that the final loss function in equation (8) is differentiable almost everywhere, so we could easily solve it via any first-order optimizer [24]. The final practical algorithm, e-COP is given in Algorithm 2.

4 Experiments

We conducted extensive experimental evaluation on the relative empirical performance of the e-COP algorithm to arrive at the following conclusions: (i) The e-COP algorithm performs better or nearly as well as all baseline algorithms for infinite-horizon discounted safe RL tasks in maximizing episodic return while satisfying given constraints. (ii) It is more robust to stochastic and complex environments [30], even where previous methods struggle. (iii) It has stable behavior and more accurate cost control as compared to other baselines near the constraint threshold due to the damping term.

Environments. For a comprehensive empirical evaluation, we selected eight scenarios from wellknown safe RL benchmark environments - Safe MuJoCo [43] and Safety Gym [30], as well as MuJoCo environments. These include: Humanoid , PointCircle , AntCircle , PointReach , AntReach , Grid , Bottleneck , and Navigation . See Figure 1 for an overview of the tasks and scenarios. Note that Navigation is a multi-constraint task and for the Reach environment, we set the reward as a function of the Euclidean distance between agent's position and goal position. In addition, we make it impossible for the agent to reach the goal before the end of the episode. For more information see Appendix A.2.1.

Baselines. We compare our e-COP algorithm with the following baseline algorithms: CPO [3], PCPO [39], FOCOPS [43], PPO with Lagrangian relaxation [33, 35], and penalty-based P3O [41]. Since the above state-of-the-art baseline algorithms are already well understood to outperform other algorithms such as PDO [13], IPO [26], and CPPO-PID [35] in prior benchmarking studies, we do not compare against them. Moreover, since PPO does not originally incorporate constraints, for fair comparison, we introduce constraints using a Lagrangian relaxation (called PPO-L). In addition, for each algorithm, we report its performance with the discount factor that achieves the best performance. See Appendix A.3.1 for more details.

Evaluation Details and Protocol. For the Circle task, we use a a point-mass with S Ď R 9 , A Ď R 2 and for the Reach task, an ant robot with S Ď R 16 , A Ď R 8 . The Grid task has S Ď R 56 , A Ď R 4 . We use two hidden layer neural networks to represent Gaussian policies for the tasks. For Circle and Reach, size is (32,32) for both layers, and for Grid and Navigation the layer sizes are (16,16) and (25,25). We set the step size δ to 10 ´ 4 , and for each task, we conduct 5 independent runs of K ' 500 episodes each of horizon H ' 200 . Since there are two objectives (rewards in the objective and costs in the constraints), we show the plots which maximize the reward objective while satisfying the cost constraint.

4.1 Performance Analysis

Table 1 lists the numerical performance of all tested algorithms in seven single constraint scenarios, and one multiple constraint scenario. We find that overall, the e-COP algorithm in most cases outperforms (green) all other baseline algorithms in finding the optimal policy while satisfying the constraints, and in other cases comes a close second (light green).

From Figure 2, we can see how the e-COP algorithm is able to improve the reward objective over the baselines while having approximate constraint satisfaction. We also see that updates of e-COP are faster and smoother than other baselines due to the added damping penalty, which ensures smoother

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convergence with only a few constraint-violating behaviors during training. In particular, e-COP is the only algorithm that best learns almost-constraint-satisfying maximum reward policies across all tasks: in simple Humanoid and Circle environments, e-COP is able to almost exactly track the cost constraint values to within the given threshold. However, for the high dimensional Grid environment we have more constraint violations due to complexity of the policy behavior, leading to higher variance in episodic rewards as compared to other environments. Regardless, overall in these environments, e-COP still outperforms all other baselines with the least episodic constraint violation. For the multiple constraint Navigation environment, see Figure 3.

4.2 Secondary Evaluation

In this section, we take a deeper dive into the empirical performance of e-COP . We discuss its dependence on various factors, and try to verify its merits.

Generalizability. From the discussion above, it's clear that e-COP demonstrates accurate safety satisfaction across tasks of varying difficulty levels. From Figure 4, we further see that e-COP satisfies the

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constraints in all cases and precisely converges to the specified cost limit. Furthermore, the fluctuation observed in the baseline Lagrangian-based algorithms is shown not to be tied to a specific cost limit.

We also conducted a set of experiments wherein we study how e-COP effectively adapts to different cost thresholds. For this, we use the hyperparameters of a pre-trained e-COP agent, which is trained with a particular cost threshold in an environment, for learning on different cost thresholds within the same environment. Figure 6 in Appendix A.3.4 illustrates the training curves of these pre-trained agents, and we see that while e-COP can generalize well across different cost thresholds, other baseline algorithms may require further tuning to accommodate different constraint thresholds.

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Sensitivity. The effectiveness and performance of e-COP would not be justified if it was not robust to the damping hyperparameter β , which varies across tasks depending on the values of C p¨q and c h . Since this damping penalty enables e-COP to have stable continuous cost control, we update it adaptively as described in Algorithm 2. As seen in Table 2, damping penalty indeed stabilizes the training process and helps in converging to an optimal safe policy.

5 Conclusion

In this paper, we have introduced an easy to implement, scalable policy optimization algorithm for episodic RL problems with constraints due to safety or other considerations. It is based on a policy difference lemma for the episodic setting, which surprisingly has quite a different form than the ones for infinitehorizon discounted or average settings. This provides the theoretical foundation for the algorithm, which is designed by incorporating several time-tested, practical as well as novel ideas. Policy optimization algorithms for Constrained MDPs tend to be numerical unstable and non-scalable due to the need for inverting the Fisher information matrix. We sidestep both of these issues by introducing a quadratic damping penalty term that works remarkably well. The algorithm development is well supported by theory, as well as with extensive empirical analysis on a range of Safety Gym and Safe MuJoco benchmark environments against a suite of baseline algorithms adapted from their non-episodic roots.

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A Appendix

A.1 Proofs

Lemma A.1. For an episode of length H and two policies, π and π 1 , the difference in their performance assuming identical initial state distribution µ (i.e., s 1 ' µ ) is given by

Proof. Let us consider s 1 ' µ and categorize the value function difference between the two policies. Also define P π h p s | s 1 q ' ř a P A P π h p s, a | s 1 q , where the term P π h p s, a | s 1 q is the probability of reaching p s, a q at time step h following π and starting from s 1 .

where a 1 ' π 1 p¨| s 1 q , s 2 ' P p¨| s 1 , π 1 p s 1 qq and s 1 ' µ , the initial state distribution.

Now recursively apply the same procedure to the term V π h p s h q ´ V π 1 h p s h q @ h P t 2 , . . . , H u to obtain the following:

Now we know that J p π q ' E s ' µ r V π 1 p s qs , this means that we combine this with the above to obtain the final result.

Proposition A.2. The inner optimization problem in (5) with respect to x is a convex quadratic program with non-negative constraints, which can be solved to yield the following intermediate problem:

Proof. As in Equation (4), we have the following equivalent problem:

Letting Ψ C i ,t p π k ´ 1 , π k q : ' ř H h ' t E s ' ρ π k,h a ' π k ´ 1 ,h ' ρ p θ h q A π k ´ 1 C i ,h p s, a q ‰ `p J C i p π k ´ 1 q ´ d i q , and introducing slack variables x t,i ě 0 and defining w t,i p π k q : ' Ψ C i ,t p π k ´ 1 , π k q ` x t,i ' 0 , we get the quadratic damped problem same as Equation (5) below.

Like the Lagrangian method, we can alternately update π, λ , and x to find the optimal triplet. Consider updating π and x by minimizing L t p π, λ , x , β q at any iteration:

The inner optimization problem with respect to x is a convex quadratic programming problem with non-negative constraints,

The optimal solution is x ‹ t,i ' max ! 0 , ´ λ t,i β ´ Ψ C i ,t p π k ´ 1 , π k q ) . Then,

Substituting back into Equation (9), we get

Hence, we finally get

Lemma A.3. Consider two problems, Problem (P) and Problem (Q) below. For sufficiently large λ i ą ¯ λ @ i and β ą ¯ β for some finite ¯ λ and finite ¯ β , the optimal solution set of Problem (Q) (equivalent version of Problem (6) ) is identical to the optimal solution set of Problem (P) . Problem (P) :

Problem (Q) :

, where x ` : ' max p 0 , x q , and

Proof. This proof uses some ideas given in [41] for Part 1 below.

Part 1 - Solution of Problem (P) is solution of Problem (Q).

Suppose ¯ θ t is the optimum of the constrained Problem (P) augmented with the quadratic penalty. Let ¯ λ t be the corresponding Lagrange multiplier vector for its dual problem, and ¯ β be the additive quadratic penalty coefficient. Then for λ t,i ě || ¯ λ || 8 @ i and β ě || ¯ β || 8 , ¯ θ is also a minimizer of its ReLU-penalized optimization Problem (Q) as below. Let Ω p θ t q : ' ř H h ' t E s ' ρ π k,h a ' π k ´ 1 ,h ' ´ ρ p θ h q A π k ´ 1 h p s, a q ‰ . Then it follows that:

By assumption, ¯ θ t is a Karush-Kuhn-Tucker point in the constrained Problem (P), at which KKT conditions are satisfied with the Lagrange multiplier vector ¯ λ and ¯ β . We then have:

, where the first line holds because ¯ θ t minimizes the Lagrange function, and the second line is derived from the complementary slackness. Thus, we conclude that for the objective function of Problem (Q), call it L Q p θ t q , we have L Q p θ t q ě L Q p ¯ θ t q for all θ t P Θ , which means ¯ θ t is a minimizer of the quadratic damped optimization Problem (Q).

Part 2 - Solution of Problem (Q) is solution of Problem (P).

Let r θ t be an optimal point of the quadratic damped Problem (Q), with ¯ θ t and ¯ λ being the same as defined above. Then, if r θ t is in the set of feasible solutions S feasible ' t θ | Ψ C i ,t p π k ´ 1 , π k , θ q ď 0 @ i u , we have:

The inequality above indicates r θ t is also optimal in the constrained Problem (P). Now, if r θ is not feasible, we have:

, which is a contradiction to the assumption that r θ t is a minimizer of the penalized optimization Problem (Q). Thus, r θ t can only be the feasible optimal solution for Problem (P).

Lemma A.4. Consider two problems, Problem (P') and Problem (R) . For sufficiently large β ą ¯ β for some finite ¯ β , the feasible optimal solution set of Problem (R) (equivalent version of Problem (3) ) is identical to the solution set of Problem (P') .

Problem (P') :

Problem (R) :

Proof. Recall that we are using parameterized policies, hence we overload notation as θ ' π frequently. For brevity, denote Ω t p π q : ' ř H h ' t E s ' ρ π k,h a ' π k ´ 1 ,h ' ρ p θ h q A π k ´ 1 h p s, a q ‰ . We will also go back and forth between the equivalent problems of Problem (P') and Problem (6) of the main paper in Section 3.

Part 1. Solution of Problem (R) is solution of Problem (P').

Suppose that π ‹ is the optimal feasible policy for the primal Problem (R), which is a Lagrangian version of Problem (3). Consider the corresponding Langrangian dual parameter λ ‹ of π ‹ , which satisfies the KKT conditon,

and the second-order sufficient condition that for all non-zero vectors u that satisfy u T ∇ π w t,i p π ‹ k q ' 0 , we have

Compare Equation (R) and Equation (P'), we have,

, where we use w t,i p π ‹ k q : ' Ψ C i ,t p π k ´ 1 , π ‹ q ` x ‹ t,i ' 0 with the feasible policy π ‹ . Moreover,

To prove that p π ‹ k,t , λ ‹ q is a strict minimum solution to L P 1 t p π k , λ , x , β q , we only need to prove the following is true for sufficiently large β ,

If the above is not true, then for any large β , there exists u t such that } u t } ' 1 and satisfies

Therefore, t u h u is a bounded sequence and there must be a limit point, denoted by 8 u . Then

The above contradicts Equation (A), so the conclusion. Hence, π ‹ k,t is also the optimal feasible policy for the primal-dual Problem (P').

Part 2. Solution of Problem (P') is solution of Problem (R).

This part is straightforward since it is a standard result. Please see Chapter 2 and Chapter 9 of [9], and Chapter 2 and Chapter 4 of [8] for the proof. For completeness, we provide the result below.

Suppose π ‹ k,t in the feasible optimal solution set of the primal-dual Problem (P'). Let λ ‹ be the corresponding dual parameter of π ‹ k,t . Consider Problem (6), which is an equivalent version of Problem (P'). For any feasible π k , we have

Now we have two cases:

where the last step uses β ą 0 , Ψ C i ,t p π k ´ 1 , π k q ă 0 , and λ ‹ t,i β ` Ψ C i ,t p π k ´ 1 , π k q ą 0 .

Case 2. When λ ‹ t,i β ` Ψ C i ,t p π k ´ 1 , π ‹ k q ď 0 , we have

Now, combining both cases above, we have L t p π k , λ ‹ , β q ď ´ Ω t p π k,t q . On the other hand, L t p π ‹ k , λ ‹ , β q ' L t p π ‹ k , λ ‹ , x ‹ , β q ' ´ Ω t p π ‹ k,t q . Thus, the combining all of the above we get

Theorem 3.3. Let π (3) ‹ be a solution to Problem (3) , and let ` π (6) ‹ , λ (6) ‹ ˘ be a solution to Problem (6) . Then, for sufficiently large β ą ¯ β and λ t,i ą ¯ λ @ i , π (3) ‹ is a solution to Problem (6) , and π (6) ‹ is a solution to Problem (3) .

Proof. We prove this result as a two-step process.

First, we show that the solution sets of the below problems are identical. See Lemma A.3 for the proof. Problem (P).

Problem (Q).

Second, we show that the solution sets of the below problems are identical. See Lemma A.4 for the proof. Problem (P') :

(a) Humanoid

(b) Circle

(c) Reach

(d) Grid

(e) Bottleneck

(f) Navigation

Problem (R) :

Now, it follows from equivalency that the optimal solution of Problem (Q) and Problem (R), and hence Problem (6) and Problem (3), is the same.

A.2 Experiments Revisited

Below we detail the experimental attributes that we used in benchmarking. See Figure 5 for the environment details. All our experiments are run in the omnisafe module [22].

A.2.1 Environment Details

Comprehensively, our experiments consist of eight tasks ranging from more superficial (Run and Circle tasks) to relatively more stochastic and sophisticated (Bottleneck and Grid tasks), each training different robots. They come from three well-known safe RL benchmark environments, Safe MuJoCo, Bullet-SafetyGym, and Safety-Gym. For agents maneuvering on a two-dimensional plane, the cost is calculated as C p s, a q ' b v 2 x ` v 2 y . For agents moving along a straight line, the cost is calculated as C p s, a q ' | v x | , where v x and v y are the velocities of the agent in the x and y directions.

Circle This environment is inspired by [3]. Reward is maximized by moving along a circle of radius d :

but the safety region x lim is smaller than the radius d : C ' 1 r x ą x lim s .

Navigation This environment is inspired by [30]. Reward is maximized by getting close to the destination R ' Dist p target, s t ´ 1 q ´ Dist p target, s t q , but it yields a cost of +1 when the agent hits the hazard or the pillar. The two different types of cost functions are returned separately and have different thresholds. In out setting, d 1 ' 25 for the hazard constraint and d 2 ' 20 for the pillar constraint.

Since the main goal in MuJoCo is to train the robot to locomote on the plane, we call it the "Run" task in our article. Our chosen two robots are the relatively complex types in MuJoCo: Ant and Humanoid. OpenAI Gym is open source at https://github.com/openai/gym, and has a documentation

at https://www.gymlibrary.ml/. Bullet Safety Gym . The implementation of the Circle task comes from Bullet-safety-Gym (Gronauer 2022), which Stooke, Achiam, and Abbeel (2020) first proposed. The reward is dense and increases by the agent's velocity and the proximity to the boundary of the circle. Costs are received when the agent leaves the safety zone defined by the two yellow boundaries. The environment is open source at https://github.com/SvenGronauer/Bullet-Safety-Gym.

Safety Gym . The remaining two tasks, Goal and Button, are from Safety-Gym (Ray, Achiam, and Amodei 2019). Compared to Run and Circle tasks, they are more stochastic and sophisticated in that agents are challenged to maximize the return while satisfying the constraints.

The environment is open source at https://github.com/openai/safety-gym, and readers can see OpenAI's blog at https://openai.com/blog/safety-gym/ for more details.

A.3 Agents

For single-constraint scenarios, Point agent is a 2D mass point( A Ď R 2 ) and Ant is an quadruped robot( A Ď R 8 ). For the multi-constraint scenario which is modified from OpenAI SafetyGym [30], S Ď R 28 ` 16 ¨ m where m is the number of pseudo-radar (one for each type of obstacles and we set two different types of obstacles in the Navigation task) and A Ď R 2 for a mass point or a wheeled car.

A.3.1 Experimental Details

To be fair in comparison, the proposed e-COP algorithm and FOCOPS [43] are implemented with same rules and tricks on the code-base of [30].

A.3.2 Hyperparameters

Table 3 shows the hyperparameters of baseline algorithms.

A.3.3 Runtime Environment

All experiments were implemented in Pytorch 1.7 .0 with CUDA 11.0 and conducted on an Ubuntu 20.04.2 LTS with 8 CPU cores (AMD Ryzen Threadripper PRO 3975WX 8-Coresz), 127G memory and 2 GPU cards (NVIDIA GeForce RTX 4060 Ti Cards).

A.3.4 Robustness to Cost Thresholds

We conducted a set of experiments wherein we study how e-COP effectively adapts to different cost thresholds. For this, we use a pre-trained e-COP agent, which is trained with a particular cost threshold in

an environment, and test its performance on different cost thresholds within the same environment. Figure 6 illustrates the training curves of these pre-trained agents, and we see that while e-COP can generalize well across different cost thresholds, other baseline algorithms may require further tuning to accommodate different constraint thresholds.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors propose a policy optimization algorithm for constrained Reinforcement Learning (RL) problem in a finite horizon setting. First, the authors develop a policy difference lemma for the finite horizon MDP problem. Following that, the authors combine a set of ideas to propose the e-COP algorithm. The authors show that the proposed algorithm is numerically much stable compared to the state-of-the-art solutions. Optimality of the proposed scheme is established under certain assumptions. Extensive simulations have demonstrated that the performance of the algorithm is better than existing methods.\n\nSTRENGTHS:\nThe paper is well-written and easy to follow. The results and claims presented in the paper appears to be correct.\n\nWEAKNESSES:\nThe idea of using a quadratic penalty term to improve the stability near the edge of the constraint set boundary is a purely heuristic based approach. The idea of approximating $\\rho_{\\pi}$ by the empirical distribution from the policy of the previous episode does not appear to be a good approximation. The idea of allowing the agent to act in a constrained manner even before the constraint is violated may lead to suppression in the original objective function.\n\nQUESTIONS:\n1. The idea of using a quadratic penalty term to improve the stability near the edge of the constraint set boundary is a purely heuristic based approach. Is there any strong theoretical argument supporting such a choice?\n\n2. In the related work section, the authors have discussed about drawbacks of using penalty terms in existing methods. Are these drawbacks applicable to the scheme proposed in the paper?\n\n3. It is not clear how the time-dependent state occupation distribution is computed. The idea of approximating $\\rho_{\\pi}$ by the empirical distribution from the policy of the previous episode does not appear to be a good approximation. Although such approach may work well in infinite horizon problems where the optimality of stationary policy holds, it may not work well in the finite horizon setting.\n\n4. The approach of using the second order penalty is not very novel and is widely used in practice for practical RL algorithms. What is the novelty of the proposed approach?\n\n5. The idea of allowing the agent to act in a constrained manner even before the constraint is violated may lead to suppression in the original objective function. In other words, when the constraint is not too tight, the proposed scheme may unnecessarily impose additional level of constraint on the system.\n\n6. How to ensure that we have sufficiently large $\\beta$ and $\\lambda_{t,i}$ which are well above the given lower bounds? The adaptive parameter selection process is just a heuristic.\n\n7. The symbol in the last term of equation (7) is not clearly defined.\n\n8. The algorithm proposed in the paper is an amalgamation of many ideas. Many of these ideas involve approximation and many are just driven from intuition without any formal guarantee of optimality. It is hard to see how this fusion work together in theory (e.g., satisfying the conditions given in Theorem 3.3).\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a new algorithm for finite-horizon constrained RL problems. The algorithm is based on three ideas: PPO-like updates for the finite-horizon setting, P3O-like treatment of constraints with adaptive penalty coefficient, and quadratic damping penalty. The proposed algorithm outperformed previous algorithms in some benchmark tasks.\n\nSTRENGTHS:\nA strong empirical performance of the proposed algorithm. The introduction of quadratic damping penalty to safe RL seems to be a new idea. The paper is easy to follow except some typos.\n\nWEAKNESSES:\nPerformance difference lemma for the finite-horizon setting is already known. See Kakade's thesis. Since PPO-like update is based on the performance difference lemma and PPO's idea, I do not see it is novel. P3O-like treatment of constraints with adaptive penalty coefficient is also not novel.\n\nThere are some parts I do not really understand. In particular I wonder if the proposed algorithm really does what Algorithm 2 states. Concretely Algorithm 2 suggests to save policy parameters for each time step, which sounds to be unusual. (Also I guess Line 6 of the algorithm has a type: $\\pi_{k, t} \\gets \\pi_{k, t+1}$ should be $\\pi_{k, t} \\gets \\pi_{k-1, t}$.)\n\nIf the algorithm is implemented as above, it is unfair to compare to previous algorithms, because previous algorithms use a much fewer number of parameters. If the algorithm is not implemented as above, there is a serious issue in presentation.\n\nEq 8 seems to contain many typos. For example $\\min_{\\pi_{k, t}} L(\\theta, \\lambda, \\beta)$ does not make sense since the loss do not have $\\pi_{k, t}$. (Sorry for the sloppy notation, but I believe the authors can understand what I mean.) Also $\\pi_{k, t}^\\star = \\min_{\\pi_{k, t}} L(\\theta, \\lambda, \\beta)$ does not make sense since $\\min_{\\pi_{k, t}} L(\\theta, \\lambda, \\beta)$ is a scalar.\n\nQUESTIONS:\n- Does the proposed algorithm what Algorithm 2 states? Algorithm 2 suggests to save policy parameters for each time step, which sounds to be unusual.\n- I looked into the code, but it seems it will not run since `_episodic_reward_loss` and `_get_surr_cadv` methods  are undefined. I checked if the Github repo of omnisafe, but I cannot find those functions. Is the code intended only for reference? (Indeed the code is provided as a text file, not python file.)\n- I currently think the main contribution of the paper is the introduction of quadratic penalty dumping to safe RL. Therefore I would like to see results of ablation studies to prove the importance of quadratic penalty dumping. Is there any result? (Or am I missing?)\n\nLIMITATIONS:\nIf the proposed algorithm really does what Algorithm 2 states, there is an issue of huge memory requirement as the horizon gets larger. I think it should be discussed. Otherwise I do not see any potential negative societal impact.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

comments: SUMMARY:
The authors introduce a policy optimization algorithm for episodic constrained RL problems, e-COP. In general, Lagrangian formulation is used for constrained optimization problems, however, the constraints are not always satisfied in real world applications. The solution approach avoids Hessian matrix inversion. The algorithm approach uses deep learning-based function approximation, a KL divergence-based proximal trust region and gradient clipping popular in improving the generalization and avoiding vanishing gradients in proximal policy optimization algorithms. The authors use SafeGym to benchmark their algorithm against the state-of-the-art algorithms such as PPO, FOCOPS, CPO, PCPO, and P3O.

STRENGTHS:
The e-COP algorithm in most cases outperforms all other baseline algorithms but fails on Humanoid, AndReach, and Grid where it provides the second best optimal results. The authors used an extensive set of baselines -- few using Lagrangian approximations. 

The idea is original. In general, the paper is well-written and clear. There are few grammatical errors and typos.

WEAKNESSES:
The authors do not talk about the limitations of the algorithms. There is a mention on the complexity of the Grid environment resulting the e-COP not to perform as best. It will be good for the completion of the paper to have a short section on limitations.

QUESTIONS:
I would like to know whether any hyper-parameter tuning has been done for the baseline algorithms? It is possible that the Lagrangian formulation may perform better after tuning.

LIMITATIONS:
The authors do not adequately address limitations.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-13