To manage the distribution shift expertise by learning-based controllers, we search a mechanism for constraining the agent to areas of excessive knowledge density all through its trajectory (left). Right here, we current an method which achieves this objective by combining options of density fashions (center) and Lyapunov features (proper).

With a view to make use of machine studying and reinforcement studying in controlling actual world methods, we should design algorithms which not solely obtain good efficiency, but in addition work together with the system in a protected and dependable method. Most prior work on safety-critical management focuses on sustaining the protection of the bodily system, e.g. avoiding falling over for legged robots, or colliding into obstacles for autonomous automobiles. Nevertheless, for learning-based controllers, there may be one other supply of security concern: as a result of machine studying fashions are solely optimized to output right predictions on the coaching knowledge, they’re susceptible to outputting misguided predictions when evaluated on out-of-distribution inputs. Thus, if an agent visits a state or takes an motion that may be very completely different from these within the coaching knowledge, a learning-enabled controller could “exploit” the inaccuracies in its discovered part and output actions which are suboptimal and even harmful.

To stop these potential “exploitations” of mannequin inaccuracies, we suggest a brand new framework to purpose concerning the security of a learning-based controller with respect to its coaching distribution. The central thought behind our work is to view the coaching knowledge distribution as a security constraint, and to attract on instruments from management concept to manage the distributional shift skilled by the agent throughout closed-loop management. Extra particularly, we’ll focus on how Lyapunov stability could be unified with density estimation to supply Lyapunov density fashions, a brand new sort of security “barrier” operate which can be utilized to synthesize controllers with ensures of maintaining the agent in areas of excessive knowledge density. Earlier than we introduce our new framework, we’ll first give an summary of current methods for guaranteeing bodily security by way of barrier operate.

In management concept, a central matter of examine is: given identified system dynamics, $s_{t+1}=f(s_t, a_t)$, and identified system constraints, $s in C$, how can we design a controller that’s assured to maintain the system inside the specified constraints? Right here, $C$ denotes the set of states which are deemed protected for the agent to go to. This downside is difficult as a result of the desired constraints have to be happy over the agent’s whole trajectory horizon ($s_t in C$ $forall 0leq t leq T$). If the controller makes use of a easy “grasping” technique of avoiding constraint violations within the subsequent time step (not taking $a_t$ for which $f(s_t, a_t) notin C$), the system should still find yourself in an “irrecoverable” state, which itself is taken into account protected, however will inevitably result in an unsafe state sooner or later whatever the agent’s future actions. With a view to keep away from visiting these “irrecoverable” states, the controller should make use of a extra “long-horizon” technique which includes predicting the agent’s whole future trajectory to keep away from security violations at any level sooner or later (keep away from $a_t$ for which all attainable ${ a_{hat{t}} }_{hat{t}=t+1}^H$ result in some $bar{t}$ the place $s_{bar{t}} notin C$ and $t<bar{t} leq T$). Nevertheless, predicting the agent’s full trajectory at each step is extraordinarily computationally intensive, and sometimes infeasible to carry out on-line throughout run-time.

Illustrative instance of a drone whose objective is to fly as straight as attainable whereas avoiding obstacles. Utilizing the “grasping” technique of avoiding security violations (left), the drone flies straight as a result of there’s no impediment within the subsequent timestep, however inevitably crashes sooner or later as a result of it might’t flip in time. In distinction, utilizing the “long-horizon” technique (proper), the drone turns early and efficiently avoids the tree, by contemplating all the future horizon way forward for its trajectory.

Management theorists sort out this problem by designing “barrier” features, $v(s)$, to constrain the controller at every step (solely permit $a_t$ which fulfill $v(f(s_t, a_t)) leq 0$). With a view to make sure the agent stays protected all through its whole trajectory, the constraint induced by barrier features ($v(f(s_t, a_t))leq 0$) prevents the agent from visiting each unsafe states and irrecoverable states which inevitably result in unsafe states sooner or later. This technique basically amortizes the computation of trying into the long run for inevitable failures when designing the protection barrier operate, which solely must be performed as soon as and could be computed offline. This fashion, at runtime, the coverage solely must make use of the grasping constraint satisfaction technique on the barrier operate $v(s)$ to be able to guarantee security for all future timesteps.

The blue area denotes the of states allowed by the barrier operate constraint, $ v(s) leq 0$. Utilizing a “long-horizon” barrier operate, the drone solely must greedily be sure that the barrier operate constraint $v(s) leq 0$ is happy for the subsequent state, to be able to keep away from security violations for all future timesteps.

Right here, we used the notion of a “barrier” operate as an umbrella time period to explain quite a few completely different sorts of features whose functionalities are to constrain the controller to be able to make long-horizon ensures. Some particular examples embrace management Lyapunov features for guaranteeing stability, management barrier features for guaranteeing normal security constraints, and the worth operate in Hamilton-Jacobi reachability for guaranteeing normal security constraints beneath exterior disturbances. Extra lately, there has additionally been some work on studying barrier features, for settings the place the system is unknown or the place barrier features are tough to design. Nevertheless, prior works in each conventional and learning-based barrier features are primarily targeted on making ensures of bodily security. Within the subsequent part, we’ll focus on how we are able to lengthen these concepts to manage the distribution shift skilled by the agent when utilizing a learning-based controller.

To stop mannequin exploitation on account of distribution shift, many learning-based management algorithms constrain or regularize the controller to forestall the agent from taking low-likelihood actions or visiting low probability states, as an example in offline RL, model-based RL, and imitation studying. Nevertheless, most of those strategies solely constrain the controller with a single-step estimate of the information distribution, akin to the “grasping” technique of maintaining an autonomous drone protected by stopping actions which causes it to crash within the subsequent timestep. As we noticed within the illustrative figures above, this technique shouldn’t be sufficient to ensure that the drone is not going to crash (or go out-of-distribution) in one other future timestep.

How can we design a controller for which the agent is assured to remain in-distribution for its whole trajectory? Recall that barrier features can be utilized to ensure constraint satisfaction for all future timesteps, which is precisely the sort of assure we hope to make as regards to the information distribution. Based mostly on this statement, we suggest a brand new sort of barrier operate: the Lyapunov density mannequin (LDM), which merges the dynamics-aware side of a Lyapunov operate with the data-aware side of a density mannequin (it’s in actual fact a generalization of each kinds of operate). Analogous to how Lyapunov features retains the system from turning into bodily unsafe, our Lyapunov density mannequin retains the system from going out-of-distribution.

An LDM ($G(s, a)$) maps state and motion pairs to detrimental log densities, the place the values of $G(s, a)$ signify one of the best knowledge density the agent is ready to keep above all through its trajectory. It may be intuitively regarded as a “dynamics-aware, long-horizon” transformation on a single-step density mannequin ($E(s, a)$), the place $E(s, a)$ approximates the detrimental log probability of the information distribution. Since a single-step density mannequin constraint ($E(s, a) leq -log(c)$ the place $c$ is a cutoff density) would possibly nonetheless permit the agent to go to “irrecoverable” states which inevitably causes the agent to go out-of-distribution, the LDM transformation will increase the worth of these “irrecoverable” states till they turn out to be “recoverable” with respect to their up to date worth. In consequence, the LDM constraint ($G(s, a) leq -log(c)$) restricts the agent to a smaller set of states and actions which excludes the “irrecoverable” states, thereby guaranteeing the agent is ready to stay in excessive data-density areas all through its whole trajectory.

Instance of knowledge distributions (center) and their related LDMs (proper) for a 2D linear system (left). LDMs could be seen as “dynamics-aware, long-horizon” transformations on density fashions.

How precisely does this “dynamics-aware, long-horizon” transformation work? Given an information distribution $P(s, a)$ and dynamical system $s_{t+1} = f(s_t, a_t)$, we outline the next because the LDM operator: $mathcal{T}G(s, a) = max{-log P(s, a), min_{a’} G(f(s, a), a’)}$. Suppose we initialize $G(s, a)$ to be $-log P(s, a)$. Beneath one iteration of the LDM operator, the worth of a state motion pair, $G(s, a)$, can both stay at $-log P(s, a)$ or improve in worth, relying on whether or not the worth at one of the best state motion pair within the subsequent timestep, $min_{a’} G(f(s, a), a’)$, is bigger than $-log P(s, a)$. Intuitively, if the worth at one of the best subsequent state motion pair is bigger than the present $G(s, a)$ worth, which means that the agent is unable to stay on the present density stage no matter its future actions, making the present state “irrecoverable” with respect to the present density stage. By growing the present the worth of $G(s, a)$, we’re “correcting” the LDM such that its constraints wouldn’t embrace “irrecoverable” states. Right here, one LDM operator replace captures the impact of trying into the long run for one timestep. If we repeatedly apply the LDM operator on $G(s, a)$ till convergence, the ultimate LDM will likely be freed from “irrecoverable” states within the agent’s whole future trajectory.

To make use of an LDM in management, we are able to practice an LDM and learning-based controller on the identical coaching dataset and constrain the controller’s motion outputs with an LDM constraint ($G(s, a)) leq -log(c)$). As a result of the LDM constraint prevents each states with low density and “irrecoverable” states, the learning-based controller will have the ability to keep away from out-of-distribution inputs all through the agent’s whole trajectory. Moreover, by selecting the cutoff density of the LDM constraint, $c$, the consumer is ready to management the tradeoff between defending in opposition to mannequin error vs. flexibility for performing the specified process.

Instance analysis of ours and baseline strategies on a hopper management process for various values of constraint thresholds (x- axis). On the proper, we present instance trajectories from when the brink is simply too low (hopper falling over on account of extreme mannequin exploitation), good (hopper efficiently hopping in direction of goal location), or too excessive (hopper standing nonetheless on account of over conservatism).

Up to now, we have now solely mentioned the properties of a “excellent” LDM, which could be discovered if we had oracle entry to the information distribution and dynamical system. In observe, although, we approximate the LDM utilizing solely knowledge samples from the system. This causes an issue to come up: though the function of the LDM is to forestall distribution shift, the LDM itself also can undergo from the detrimental results of distribution shift, which degrades its effectiveness for stopping distribution shift. To grasp the diploma to which the degradation occurs, we analyze this downside from each a theoretical and empirical perspective. Theoretically, we present even when there are errors within the LDM studying process, an LDM constrained controller continues to be capable of keep ensures of maintaining the agent in-distribution. Albeit, this assure is a bit weaker than the unique assure supplied by an ideal LDM, the place the quantity of degradation will depend on the size of the errors within the studying process. Empirically, we approximate the LDM utilizing deep neural networks, and present that utilizing a discovered LDM to constrain the learning-based controller nonetheless offers efficiency enhancements in comparison with utilizing single-step density fashions on a number of domains.

Analysis of our technique (LDM) in comparison with constraining a learning-based controller with a density mannequin, the variance over an ensemble of fashions, and no constraint in any respect on a number of domains together with hopper, lunar lander, and glucose management.

At the moment, one of many largest challenges in deploying learning-based controllers on actual world methods is their potential brittleness to out-of-distribution inputs, and lack of ensures on efficiency. Conveniently, there exists a big physique of labor in management concept targeted on making ensures about how methods evolve. Nevertheless, these works often deal with making ensures with respect to bodily security necessities, and assume entry to an correct dynamics mannequin of the system in addition to bodily security constraints. The central thought behind our work is to as a substitute view the coaching knowledge distribution as a security constraint. This enables us to make use of those concepts in controls in our design of learning-based management algorithms, thereby inheriting each the scalability of machine studying and the rigorous ensures of management concept.

This submit is predicated on the paper “Lyapunov Density Fashions: Constraining Distribution Shift in Studying-Based mostly Management”, introduced at ICML 2022. You
discover extra particulars in our paper and on our web site. We thank Sergey Levine, Claire Tomlin, Dibya Ghosh, Jason Choi, Colin Li, and Homer Walke for his or her helpful suggestions on this weblog submit.