## 1 Introduction

While reinforcement learning (RL, Sutton1998Reinforcement) algorithms have achieved impressive results in games, for example on the Atari platform (Mnih2015Humanlevel), they are rarely applied to real-world physical systems (e.g., robots) outside of academia. The main reason is that RL algorithms provide optimal policies only in the long-term, so that intermediate policies may be unsafe, break the system, or harm their environment. This is especially true in safety-critical systems that can affect human lives. Despite this, safety in RL has remained largely an open problem (Amodei2016Concrete).

Consider, for example, a self-driving car. While it is desirable for the algorithm that drives the car to improve over time (e.g., by adapting to driver preferences and changing environments), any policy applied to the system has to guarantee safe driving. Thus, it is not possible to learn about the system through random exploratory actions, which almost certainly lead to a crash. In order to avoid this problem, the learning algorithm needs to consider its ability to safely recover from exploratory actions. In particular, we want the car to be able to recover to a safe state, for example, driving at a reasonable speed in the middle of the lane. This ability to recover is known as *asymptotic stability* in control theory (Khalil1996Nonlinear). Specifically, we care about the *region of attraction* of the closed-loop system under a policy. This is a subset of the state space that is forward invariant so that any state trajectory that starts within this set stays within it for all times and converges to a goal state eventually.

In this paper, we present a RL algorithm for continuous state-action spaces that provides these kind of high-probability safety guarantees for policies. In particular, we show how, starting from an initial, safe policy we can expand our estimate of the region of attraction by collecting data inside the safe region and adapt the policy to both increase the region of attraction and improve control performance.

### Related work

Safety is an active research topic in RL and different definitions of safety exist (Pecka2014Safe; Garcia2015Comprehensive). *Discrete*Markov decision processes (MDPs) are one class of tractable models that have been analyzed. In risk-sensitive RL, one
specifies risk-aversion in the reward (Coraluppi1999Risksensitive). For example, (Geibel2005RiskSensitive) define risk as the probability of driving the agent to a set of known, undesirable states. Similarly, robust MDPs maximize rewards when transition probabilities are uncertain (Tamar2014Scaling; Wiesemann2012Robust). Both (Moldovan2012Safe) and (Turchetta2016Safe) introduce algorithms to safely explore MDPs so that the agent never gets stuck without safe actions. All these methods require an accurate probabilistic model of the system.

In *continuous* state-action spaces, model-free policy search algorithms have been successful. These update policies without a system model by repeatedly executing the same task (Peters2006Policy). In this setting, (Achiam2017Constrained) introduces safety guarantees in terms of constraint satisfaction that hold in expectation. High-probability worst-case safety guarantees are available for methods based on Bayesian optimization (Mockus1989Bayesian) together with Gaussian process models (GP, (Rasmussen2006Gaussian)) of the cost function. The algorithms in (Schreiter2015Safe) and (Sui2015Safe) provide high-probability safety guarantees for any parameter that is evaluated on the real system. These methods are used in (Berkenkamp2016Safe) to safely optimize a parametric control policy on a quadrotor. However, resulting policies are task-specific and require the system to be reset.

In the *model-based* RL setting, research has focused on safety in terms of state constraints. In (Garcia2012Safe; Hans2008Safe), a priori known, safe global backup policies are used, while (Perkins2003Lyapunov) learns to switch between several safe policies. However, it is not clear how one may find these policies in the first place. Other approaches use model predictive control with constraints, a model-based technique where the control actions are optimized online. For example, (Sadigh2016Safe) models uncertain environmental constraints, while (Ostafew2016Robust) uses approximate uncertainty propagation of GP dynamics along trajectories. In this setting, robust feasability and constraint satisfaction can be guaranteed for a learned model with bounded errors using robust model predictive control Aswani2013Provably. The method in (Akametalu2014Reachability)

uses reachability analysis to construct safe regions in the state space. The theoretical guarantees depend on the solution to a partial differential equation, which is approximated.

Theoretical guarantees for the stability exist for the more tractable stability analysis and verification under a *fixed* control policy. In control, stability of a known system can be verified using a Lyapunov function (Bobiti2016Sampling). A similar approach is used by (Berkenkamp2016Lyapunov) for deterministic, but unknown dynamics that are modeled as a GP, which allows for provably safe learning of regions of attraction for fixed policies. Similar results are shown in (Vinogradska2016Stability) for stochastic systems that are modeled as a GP. They use Bayesian quadrature to compute provably accurate estimates of the region of attraction. These approaches do not update the policy.

### Our contributions

We introduce a novel algorithm that can safely optimize policies in continuous state-action spaces while providing high-probability safety guarantees in terms of stability. Moreover, we show that it is possible to exploit the regularity properties of the system in order to *safely learn* about the dynamics and thus improve the policy and increase the estimated safe region of attraction without ever leaving it. Specifically, starting from a policy that is known to stabilize the system locally, we gather data at informative, safe points and improve the policy safely based on the improved model of the system and prove that any exploration algorithm that gathers data at these points reaches a natural notion of *full exploration*. We show how the theoretical results transfer to a *practical algorithm* with safety guarantees and apply it to a simulated inverted pendulum stabilization task.

## 2 Background and Assumptions

We consider a deterministic, discrete-time dynamic system

(1) |

with states and control actions and a discrete time index . The true dynamics consist of two parts: is a known, prior model that can be obtained from first principles, while represents a priori unknown model errors. While the model errors are unknown, we can obtain noisy measurements of by driving the system to the state and taking action . We want this system to behave in a certain way, e.g., the car driving on the road. To this end, we need to specify a control policy that, given the current state, determines the appropriate control action that drives the system to some goal state, which we set as the origin without loss of generality (Khalil1996Nonlinear). We encode the performance requirements of how to drive the system to the origin through a positive cost that is associated with states and actions and has . The policy aims to minimize the cumulative, discounted costs for each starting state.

The goal is to safely learn about the dynamics from measurements and adapt the policy for performance, without encountering system failures. Specifically, we define the safety constraint on the state divergence that occurs when leaving the region of attraction. This means that adapting the policy is not allowed to decrease the region of attraction and exploratory actions to learn about the dynamics are not allowed to drive the system outside the region of attraction. The region of attraction is not known a priori, but is implicitly defined through the system dynamics and the choice of policy. Thus, the policy not only defines performance as in typical RL, but also determines safety and where we can obtain measurements.

### Model assumptions

In general, this kind of safe learning is impossible without further assumptions. For example, in a discontinuous system even a slight change in the control policy can lead to drastically different behavior. Moreover, to expand the safe set we need to generalize learned knowledge about the dynamics to (potentially unsafe) states that we have not visited. To this end, we restrict ourselves to the general and practically relevant class of models that are Lipschitz continuous. This is a typical assumption in the control community Khalil1996Nonlinear. Additionally, to ensure that the closed-loop system remains Lipschitz continuous when the control policy is applied, we restrict policies to the rich class of -Lipschitz continuous functions , which also contains certain types of neural networks (Szegedy2014Intriguing).

###### Assumption 1 (continuity).

The dynamics and in creftype 1 are - and Lipschitz continuous with respect to the 1-norm. The considered control policies lie in a set of functions that are -Lipschitz continuous with respect to the 1-norm.

To enable safe learning, we require a reliable statistical model. While we commit to GPs for the exploration analysis, for safety any suitable, well-calibrated model is applicable.

###### Assumption 2 (well-calibrated model).

Let and denote the posterior mean and covariance matrix functions of the statistical model of the dynamics creftype 1 conditioned on noisy measurements. With , there exists a such that with probability at least it holds for all , , and that

This assumption ensures that we can build confidence intervals on the dynamics that, when scaled by an appropriate constant

, cover the true function with high probability. We introduce a specific statistical model that fulfills both assumptions under certain regularity assumptions in Sec. 3.### Lyapunov function

To satisfy the specified safety constraints for safe learning, we require a tool to determine whether individual states and actions are safe. In control theory, this safety is defined through the region of attraction, which can be computed for a fixed policy using Lyapunov functions (Khalil1996Nonlinear). Lyapunov functions are continuously differentiable functions with and for all . The key idea behind using Lyapunov functions to show stability of the system creftype 1 is similar to that of gradient descent on strictly quasiconvex functions: if one can show that, given a policy , applying the dynamics on the state maps it to strictly smaller values on the Lyapunov function (‘going downhill’), then the state eventually converges to the equilibrium point at the origin (minimum). In particular, the assumptions in Theorem 1 below imply that is strictly quasiconvex within the region of attraction if the dynamics are Lipschitz continuous. As a result, the one step decrease property for all states within a level set guarantees eventual convergence to the origin.

###### Theorem 1 ((Khalil1996Nonlinear)).

Let be a Lyapunov function, Lipschitz continuous dynamics, and a policy. If for all within the level set , , then is a region of attraction, so that implies for all and .

It is convenient to characterize the region of attraction through a level set of the Lyapunov function, since it replaces the challenging test for convergence with a one-step decrease condition on the Lyapunov function. For the theoretical analysis in this paper, we assume that a Lyapunov function is given to determine the region of attraction. For ease of notation, we also assume for all , which ensures that level sets are connected if . Since Lyapunov functions are continuously differentiable, they are -Lipschitz continuous over the compact set .

In general, it is not easy to find suitable Lyapunov functions. However, for physical models, like the prior model in creftype 1, the energy of the system (e.g., kinetic and potential for mechanical systems) is a good candidate Lyapunov function. Moreover, it has recently been shown that it is possible to compute suitable Lyapunov functions (Li2016Computation; Giesl2015Review). In our experiments, we exploit the fact that value functions in RL are Lyapunov functions if the costs are strictly positive away from the origin. This follows directly from the definition of the value function, where . Thus, we can obtain Lyapunov candidates as a by-product of approximate dynamic programming.

### Initial safe policy

Lastly, we need to ensure that there exists a safe starting point for the learning process. Thus, we assume that we have an initial policy that renders the origin of the system in creftype 1 asymptotically stable within some small set of states . For example, this policy may be designed using the prior model in creftype 1, since most models are locally accurate but deteriorate in quality as state magnitude increases. This policy is explicitly not safe to use throughout the state space .

## 3 Theory

In this section, we use these assumptions for safe reinforcement learning. We start by computing the region of attraction for a fixed policy under the statistical model. Next, we optimize the policy in order to expand the region of attraction. Lastly, we show that it is possible to safely learn about the dynamics and, under additional assumptions about the model and the system’s reachability properties, that this approach expands the estimated region of attraction safely. We consider an idealized algorithm that is amenable to analysis, which we convert to a practical variant in LABEL:sec:experiments. See Fig. 1 for an illustrative run of the algorithm and examples of the sets defined below.

### Region of attraction

We start by computing the region of attraction for a fixed policy. This is an extension of the method in (Berkenkamp2016Lyapunov) to discrete-time systems. We want to use the Lyapunov decrease condition in Theorem 1 to guarantee safety for the statistical model of the dynamics. However, the posterior uncertainty in the statistical model of the dynamics means that one step predictions about are uncertain too. We account for this by constructing high-probability confidence intervals on : From Assumption 2 together with the Lipschitz property of , we know that is contained in with probability at least . For our exploration analysis, we need to ensure that safe state-actions cannot become unsafe; that is, an initial set of safe set remains safe (defined later). To this end, we intersect the confidence intervals: , where the set is initialized to when and otherwise. Note that is contained in with the same probability as in Assumption 2. The upper and lower bounds on are defined as and .

Given these high-probability confidence intervals, the system is stable according to Theorem 1 if for all . However, it is intractable to verify this condition directly on the continuous domain without additional, restrictive assumptions about the model. Instead, we consider a discretization of the state space into cells, so that holds for all . Here, denotes the point in with the smallest distance to . Given this discretization, we bound the decrease variation on the Lyapunov function for states in and use the Lipschitz continuity to generalize to the continuous state space .

###### Theorem 2.

Under Assumptions 2 and 1 with , let be a discretization of such that for all . If, for all with , , and for some it holds that then holds for all with probability at least and is a region of attraction for creftype 1 under policy .

The proof is given in LABEL:sec:proofs_stability. Theorem 2 states that, given confidence intervals on the statistical model of the dynamics, it is sufficient to check the stricter decrease condition in Theorem 2 on the discretized domain to guarantee the requirements for the region of attraction in the continuous domain in Theorem 1. The bound in Theorem 2 becomes tight as the discretization constant and go to zero. Thus, the discretization constant trades off computation costs for accuracy, while approaches as we obtain more measurement data and the posterior model uncertainty about the dynamics, decreases. The confidence intervals on and the corresponding estimated region of attraction (red line) can be seen in the bottom half of Fig. 1.

### Policy optimization

So far, we have focused on estimating the region of attraction for a fixed policy. Safety is a property of states under a fixed policy. This means that the policy directly determines which states are safe. Specifically, to form a region of attraction all states in the discretizaton within a level set of the Lyapunov function need to fulfill the decrease condition in Theorem 2 that depends on the policy choice. The set of all state-action pairs that fulfill this decrease condition is given by

(2) |

see Fig. 1(c) (top, red shaded). In order to estimate the region of attraction based on this set, we need to commit to a policy. Specifically, we want to pick the policy that leads to the largest possible region of attraction according to Theorem 2. This requires that for each discrete state in the corresponding state-action pair under the policy must be in the set . Thus, we optimize the policy according to

(3) |

The region of attraction that corresponds to the optimized policy according to creftype 3 is given by , see Fig. 1(b). It is the largest level set of the Lyapunov function for which all state-action pairs that correspond to discrete states within are contained in . This means that these state-action pairs fulfill the requirements of Theorem 2 and is a region of attraction of the true system under policy . The following theorem is thus a direct consequence of Theorem 2 and creftype 3.

###### Theorem 3.

Let be the true region of attraction of creftype 1 under the policy . For any , we have with probability at least that for all .

Thus, when we optimize the policy subject to the constraint in creftype 3 the estimated region of attraction is always an inner approximation of the true region of attraction. However, solving the optimization problem in creftype 3 is intractable in general. We approximate the policy update step in LABEL:sec:experiments.

### Collecting measurements

Given these stability guarantees, it is natural to ask how one might obtain data points in order to improve the model of and thus efficiently increase the region of attraction. This question is difficult to answer in general, since it depends on the property of the statistical model. In particular, for general statistical models it is often not clear whether the confidence intervals contract sufficiently quickly. In the following, we make additional assumptions about the model and reachability within in order to provide exploration guarantees. These assumptions allow us to highlight fundamental requirements for safe data acquisition and that safe exploration is possible.

We assume that the unknown model errors have bounded norm in a reproducing kernel Hilbert space (RKHS, Scholkopf2002Learning) corresponding to a differentiable kernel , . These are a class of well-behaved functions of the form defined through representer points and weights that decay sufficiently fast with . This assumption ensures that satisfies the Lipschitz property in Assumption 1, see Berkenkamp2016Lyapunov. Moreover, with we can use GP models for the dynamics that fulfill Assumption 2 if the state if fully observable and the measurement noise is -sub-Gaussian (e.g., bounded in ), see Chowdhury2017Kernelized. Here is the information capacity. It corresponds to the amount of mutual information that can be obtained about from measurements, a measure of the size of the function class encoded by the model. The information capacity has a sublinear dependence on for common kernels and upper bounds can be computed efficiently Srinivas2012Gaussian. More details about this model are given in LABEL:sec:gaussian_process_theory.

In order to quantify the exploration properties of our algorithm, we consider a discrete action space . We define exploration as the number of state-action pairs in that we can safely learn about without leaving the true region of attraction. Note that despite this discretization, the policy takes values on the continuous domain. Moreover, instead of using the confidence intervals directly as in creftype 3, we consider an algorithm that uses the Lipschitz constants to slowly expand the safe set. We use this in our analysis to quantify the ability to generalize beyond the current safe set. In practice, nearby states are sufficiently correlated under the model to enable generalization using creftype 2.

Suppose we are given a set of state-action pairs about which we can learn safely. Specifically, this means that we have a policy such that, for any state-action pair in , if we apply action in state and then apply actions according to the policy, the state converges to the origin. Such a set can be constructed using the initial policy from Sec. 2 as . Starting from this set, we want to update the policy to expand the region of attraction according to Theorem 2. To this end, we use the confidence intervals on for states inside to determine state-action pairs that fulfill the decrease condition. We thus redefine for the exploration analysis to

(4) |

This formulation is equivalent to creftype 2, except that it uses the Lipschitz constant to generalize safety. Given , we can again find a region of attraction by committing to a policy according to creftype 3. In order to expand this region of attraction effectively we need to decrease the posterior model uncertainty about the dynamics of the GP by collecting measurements. However, to ensure safety as outlined in Sec. 2, we are not only restricted to states within , but also need to ensure that the state after taking an action is safe; that is, the dynamics map the state back into the region of attraction . We again use the Lipschitz constant in order to determine this set,

(5) |

The set contains state-action pairs that we can safely evaluate under the current policy without leaving the region of attraction, see Fig. 1 (top, white shaded).

What remains is to define a strategy for collecting data points within to effectively decrease model uncertainty. We specifically focus on the high-level requirements for any exploration scheme without committing to a specific method. In practice, any (model-based) exploration strategy that aims to decrease model uncertainty by driving the system to specific states may be used. Safety can be ensured by picking actions according to whenever the exploration strategy reaches the boundary of the safe region ; that is, when . This way, we can use as a backup policy for exploration.

The high-level goal of the exploration strategy is to shrink the confidence intervals at state-action pairs in order to expand the safe region. Specifically, the exploration strategy should aim to visit state-action pairs in at which we are the most uncertain about the dynamics; that is, where the confidence interval is the largest:

(6) |

As we keep collecting data points according to creftype 6, we decrease the uncertainty about the dynamics for different actions throughout the region of attraction and adapt the policy, until eventually we have gathered enough information in order to expand it. While creftype 6 implicitly assumes that any state within can be reached by the exploration policy, it achieves the high-level goal of any exploration algorithm that aims to reduce model uncertainty. In practice, any safe exploration scheme is limited by unreachable parts of the state space.

We compare the active learning scheme in

creftype 6 to an oracle baseline that starts from the same initial safe set and knows up to accuracy within the safe set. The oracle also uses knowledge about the Lipschitz constants and the optimal policy in at each iteration. We denote the set that this baseline manages to determine as safe with and provide a detailed definition in LABEL:sec:app_baseline.###### Theorem 4.

Assume -sub-Gaussian measurement noise and that the model error in creftype 1 has RKHS norm smaller than . Under the assumptions of Theorem 2, with , and with measurements collected according to creftype 6, let be the smallest positive integer so that where . Let be the true region of attraction of creftype 1 under a policy . For any , and , the following holds jointly with probability at least for all :

. .

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