Keeping Learning-Based Control Safe by Regulating Distributional Shift

Keeping Learning-Based Control Safe by Regulating Distributional Shift


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To regulate 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). Here, we current an strategy which achieves this purpose by combining options of density fashions (center) and Lyapunov features (proper).

In order to utilize machine studying and reinforcement studying in controlling actual world methods, we should design algorithms which not solely obtain good efficiency, but additionally work together with the system in a protected and dependable method. Most prior work on safety-critical management focuses on sustaining the security of the bodily system, e.g. avoiding falling over for legged robots, or colliding into obstacles for autonomous autos. However, for learning-based controllers, there’s one other supply of security concern: as a result of machine studying fashions are solely optimized to output appropriate predictions on the coaching knowledge, they’re liable 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 realized part and output actions which might be suboptimal and even harmful.

To stop these potential “exploitations” of mannequin inaccuracies, we suggest a brand new framework to motive in regards to the security of a learning-based controller with respect to its coaching distribution. The central concept 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. More particularly, we’ll focus on how Lyapunov stability will be unified with density estimation to supply Lyapunov density fashions, a brand new sort of security “barrier” perform which can be utilized to synthesize controllers with ensures of maintaining the agent in areas of excessive knowledge density. Before we introduce our new framework, we are going to first give an outline of present strategies for guaranteeing bodily security through barrier perform.

In management concept, a central matter of research 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? Here, $C$ denotes the set of states which might be deemed protected for the agent to go to. This downside is difficult as a result of the required constraints must be glad 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 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. In order to keep away from visiting these “irrecoverable” states, the controller should make use of a extra “long-horizon” technique which entails 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 potential ${ a_{hat{t}} }_{hat{t}=t+1}^H$ result in some $bar{t}$ the place $s_{bar{t}} notin C$ and $t<bar{t} leq T$). However, predicting the agent’s full trajectory at each step is extraordinarily computationally intensive, and infrequently infeasible to carry out on-line throughout run-time.

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Illustrative instance of a drone whose purpose is to fly as straight as potential whereas avoiding obstacles. Using 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 could possibly’t flip in time. In distinction, utilizing the “long-horizon” technique (proper), the drone turns early and efficiently avoids the tree, by contemplating your entire future horizon way forward for its trajectory.

Control theorists sort out this problem by designing “barrier” features, $v(s)$, to constrain the controller at every step (solely enable $a_t$ which fulfill $v(f(s_t, a_t)) leq 0$). In order 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 primarily amortizes the computation of wanting into the longer term for inevitable failures when designing the security barrier perform, which solely must be executed as soon as and will be computed offline. This manner, at runtime, the coverage solely must make use of the grasping constraint satisfaction technique on the barrier perform $v(s)$ as a way to guarantee security for all future timesteps.

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

Here, we used the notion of a “barrier” perform as an umbrella time period to explain plenty of completely different sorts of features whose functionalities are to constrain the controller as a way to make long-horizon ensures. Some particular examples embrace management Lyapunov features for guaranteeing stability, management barrier features for guaranteeing basic security constraints, and the worth perform in Hamilton-Jacobi reachability for guaranteeing basic security constraints beneath exterior disturbances. More 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 troublesome to design. However, prior works in each conventional and learning-based barrier features are primarily centered on making ensures of bodily security. In the subsequent part, we are going to 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 resulting from distribution shift, many learning-based management algorithms constrain or regularize the controller to stop the agent from taking low-likelihood actions or visiting low probability states, for example in offline RL, model-based RL, and imitation studying. However, 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 is just not 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 close to the information distribution. Based on this statement, we suggest a brand new sort of barrier perform: the Lyapunov density mannequin (LDM), which merges the dynamics-aware side of a Lyapunov perform with the data-aware side of a density mannequin (it’s in reality a generalization of each kinds of perform). Analogous to how Lyapunov features retains the system from changing 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 damaging log densities, the place the values of $G(s, a)$ characterize one of the best knowledge density the agent is ready to keep above all through its trajectory. It will 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 damaging 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 enable 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 change into “recoverable” with respect to their up to date worth. As a end result, 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.


Example of information distributions (center) and their related LDMs (proper) for a 2D linear system (left). LDMs will be considered 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 $-log P(s, a)$. Under 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 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 rising the present the worth of $G(s, a)$, we’re “correcting” the LDM such that its constraints wouldn’t embrace “irrecoverable” states. Here, one LDM operator replace captures the impact of wanting into the longer term for one timestep. If we repeatedly apply the LDM operator on $G(s, a)$ till convergence, the ultimate LDM can be freed from “irrecoverable” states within the agent’s whole future trajectory.

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


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

So far, now we have solely mentioned the properties of a “excellent” LDM, which will 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: regardless that the position of the LDM is to stop distribution shift, the LDM itself can even undergo from the damaging results of distribution shift, which degrades its effectiveness for stopping distribution shift. To perceive 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 depends upon the dimensions of the errors within the studying process. Empirically, we approximate the LDM utilizing deep neural networks, and present that utilizing a realized LDM to constrain the learning-based controller nonetheless supplies efficiency enhancements in comparison with utilizing single-step density fashions on a number of domains.


Evaluation of our methodology (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.

Currently, one of many greatest 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 centered on making ensures about how methods evolve. However, 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 concept behind our work is to as an alternative view the coaching knowledge distribution as a security constraint. This permits 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 relies on the paper “Lyapunov Density Models: Constraining Distribution Shift in Learning-Based Control”, 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 precious suggestions on this weblog submit.

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() To regulate 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). Here, we current an strategy which achieves this purpose by combining options of density fashions (center) and Lyapunov features (proper). In order to utilize machine studying…