P REFERENCES I MPLICIT IN THE S TATE OF THE W ORLD
A BSTRACT
Reinforcement learning (RL) agents optimize only the features speciﬁed in a reward function and are indifferent to anything left out inadvertently.
This means that we must not only specify what to do, but also the much larger space of what not to do.
It is easy to forget these preferences, since these preferences are already satisﬁed in our environment.
This motivates our key insight: when a robot is deployed in an environment that humans act in, the state of the environment is already optimized for what humans want .
We can therefore use this implicit preference information from the state to ﬁll in the blanks.
We develop an algorithm based on Maximum Causal Entropy IRL and use it to evaluate the idea in a suite of proof-of-concept environments designed to show its properties.
We ﬁnd that information from the initial state can be used to infer both side effects that should be avoided as well as preferences for how the environment should be organized.
Our code can be found at https://github.com/HumanCompatibleAI/rlsp .
I NTRODUCTION
Deep reinforcement learning (deep RL) has been shown to succeed at a wide variety of complex tasks given a correctly speciﬁed reward function.
Unfortunately, for many real-world tasks it can be challenging to specify a reward function that captures human preferences, particularly the preference for avoiding unnecessary side effects while still accomplishing the goal (Amodei et al., 2016).
As a result, there has been much recent work (Christiano et al., 2017; Fu et al., 2017; Sadigh et al., 2017) that aims to learn speciﬁcations for tasks a robot should perform.
Typically when learning about what people want and don’t want, we look to human action as evidence: what reward they specify (Hadﬁeld-Menell et al., 2017), how they perform a task (Ziebart, 2010; Fu et al., 2017), what choices they make (Christiano et al., 2017; Sadigh et al., 2017), or how they rate certain options (Daniel et al., 2014).
Here, we argue that there is an additional source of information that is potentially rather helpful, but that we have been ignoring thus far:
The key insight of this paper is that when a robot is deployed in an environment that humans have been acting in, the state of the environment is already optimized for what humans want .
For example, consider an environment in which a household robot must navigate to a goal location without breaking any vases in its path, illustrated in Figure 1.
The human operator, Alice, asks the robot to go to the purple door, forgetting to specify that it should also avoid breaking vases along the way.
However, since the robot has been deployed in a state that only contains unbroken vases, it can infer that while acting in the environment (prior to robot’s deployment), Alice was using one of the relatively few policies that do not break vases, and so must have cared about keeping vases intact.
The initial state s 0 can contain information about arbitrary preferences, including tasks that the robot should actively perform.
For example, if the robot observes a basket full of apples near an apple tree, it can reasonably infer that Alice wants to harvest apples.
However, s 0 is particularly useful for acts Go to ! is deployed 👍 👎 👍 👎
Which Alice’s reward is consistent with ?
[Figure] [Figure] inferring which side effects humans care about. Recent approaches avoid unnecessary side effects by penalizing changes from an inaction baseline (Krakovna et al., 2018; Turner, 2018).
However, this penalizes all side effects.
The inaction baseline is appealing precisely because the initial state has already been optimized for human preferences, and action is more likely to ruin s 0 than inaction.
If our robot infers preferences from s 0 , it can avoid negative side effects while allowing positive ones.
This work is about highlighting the potential of this observation, and as such makes unrealistic assumptions, such as known dynamics and hand-coded features.
Given just s 0 , these assumptions are necessary: without dynamics, it is hard to tell whether some feature of s 0 was created by humans or not.
Nonetheless, we are optimistic that these assumptions can be relaxed, so that this insight can be used to improve deep RL systems.
We suggest some approaches in our discussion.
Our contributions are threefold.
First, we identify the state of the world at initialization as a source of information about human preferences.
Second, we leverage this insight to derive an algorithm, Reward Learning by Simulating the Past (RLSP), which infers reward from initial state based on a Maximum Causal Entropy (Ziebart, 2010) model of human behavior.
Third, we demonstrate the properties and limitations of this idea on a suite of proof-of-concept environments: we use it to avoid side effects, as well as to learn implicit preferences that require active action.
In Figure 1 the robot moves to the purple door without breaking the vase, despite the lack of an explicit penalty for breaking vases.
R ELATED WORK
Preference learning.
Much recent work has learned preferences from different sources of data, such as demonstrations (Ziebart, 2010; Ramachandran and Amir, 2007; Ho and Ermon, 2016; Fu et al., 2017;
Finn et al., 2016), comparisons (Christiano et al., 2017; Sadigh et al., 2017; Wirth et al., 2017), ratings (Daniel et al., 2014), human reinforcement signals (Knox and Stone, 2009; Warnell et al., 2017; MacGlashan et al., 2017), proxy rewards (Hadﬁeld-Menell et al., 2017), etc.
We suggest preference learning with a new source of data: the state of the environment when the robot is ﬁrst deployed.
It can also be seen as a variant of Maximum Causal Entropy Inverse Reinforcement
Learning (Ziebart, 2010): while inverse reinforcement learning (IRL) requires demonstrations, or at least state sequences without actions (Edwards et al., 2018; Yu et al., 2018), we learn a reward function from a single state, albeit with the simplifying assumption of known dynamics.
Frame properties.
The frame problem in AI (McCarthy and Hayes, 1981) refers to the issue that we must specify what stays the same in addition to what changes.
In formal veriﬁcation, this manifests as a requirement to explicitly specify the many quantities that the program does not change (Andreescu, 2017).
Analogously, rewards are likely to specify what to do (the task), but may forget to say what not to do (the frame properties).
One of our goals is to infer frame properties automatically.
Side effects.
An impact penalty can mitigate reward speciﬁcation problems, since it penalizes unnecessary “large” changes (Armstrong and Levinstein, 2017).
Compared to an inaction baseline of doing nothing, we could penalize a reduction in the number of reachable states (Krakovna et al., 2018) or attainable utility (Turner, 2018).
However, such approaches will penalize all irreversible effects, including ones that humans want to happen.
In contrast, by taking a preference inference approach, we can infer which effects humans care about.
Goal states as speciﬁcations.
Desired behavior in RL can be speciﬁed with an explicitly chosen goal state (Kaelbling, 1993; Schaul et al., 2015; Nair et al., 2018;
Bahdanau et al., 2018; Andrychowicz et al., 2017).
In our setting, the robot observes the initial state s 0 where it starts acting, which is not explicitly chosen by the designer, but nonetheless contains preference information.
P RELIMINARIES
A ﬁnite-horizon Markov decision process (MDP) is a tuple M =
(cid:104)S , A , T , r, T (cid:105) , where S is the set of states, A is the set of actions, T : S ×
A × S (cid:55)→ [0 , 1] is the transition probability function, r : S (cid:55)→ R is the reward function, and T ∈ Z
+ is the ﬁnite planning horizon.
We consider MDPs where the reward is linear in features, and does not depend on action: r ( s ; θ )
=
θ T f ( s ) , where θ are the parameters deﬁning the reward function and f computes features of a given state.
Inverse Reinforcement Learning (IRL).
In IRL, the aim is to infer the reward function r given an MDP without reward M\ r and expert demonstrations D
= { τ 1 , ..., τ n } , where each τ i = ( s 0 , a 0 , ..., s T , a T ) is a trajectory sampled from the expert policy acting in the MDP.
It is assumed that each τ i is feasible, so that T ( s j +
|
s j , a j ) > 0 for every j .
Maximum Causal Entropy IRL (MCEIRL).
As human demonstrations are rarely optimal, Ziebart (2010) models the expert as a Boltzmann-rational agent that maximizes total reward and causal entropy of the policy.
This leads to the policy π t ( a | s, θ ) =
exp( Q t ( s, a ; θ ) − V t ( s ; θ )) , where V t ( s ; θ ) =
ln (cid:80) a exp ( Q t ( s, a ; θ )) plays the role of a normalizing constant.
Intuitively, the expert is assumed to act close to randomly when the difference in expected total reward across the actions is small, but nearly always chooses the best action when it leads to a substantially higher expected return.
The soft Bellman backup for the state-action value function Q is the same as usual, and is given by Q t ( s, a ; θ )
{+= θ T f ( s ) + (cid:80) s (cid:48) T ( s (cid:48) | s, a ) V t +
+}
( s (cid:48) ; θ ) .
The likelihood of a trajectory τ given the reward parameters θ is:
[Equation]
MCEIRL ﬁnds the reward parameters θ ∗ that maximize the log-likelihood of the demonstrations:
[Equation]
θ ∗ gives rise to a policy whose feature expectations match those of the expert demonstrations.
R EWARD L EARNING BY S IMULATING THE P AST
We solve the problem of learning the reward function of an expert Alice given a single ﬁnal state of her trajectory; we refer to this problem as IRL from a single state .
Formally, we aim to infer Alice’s reward θ given an environment M\ r and the last state of the expert’s trajectory s 0 .
Formulation.
To adapt MCEIRL to the one state setting we modify the observation model from
Equation 1.
Since we only have a single end state s 0 of the trajectory τ 0 = ( s − T , a − T , ..., s 0 , a 0 ) , we marginalize over all of the other variables in the trajectory:
[Equation]
where p ( τ 0 | θ ) is given in Equation 1.
We could invert this and sample from p ( θ | s 0 ) ; the resulting algorithm is presented in Appendix C, but is relatively noisy and slow.
We instead ﬁnd the MLE:
[Equation]
Solution.
Similarly to MCEIRL, we use a gradient ascent algorithm to solve the IRL from one state problem.
We explain the key steps here and give the full derivation in Appendix B. First, we express the gradient in terms of the gradients of trajectories:
[Equation]
This has a nice interpretation – compute the Maximum Causal Entropy gradients for each trajectory, and then take their weighted sum, where each weight is the probability of the trajectory given the evidence s 0 and current reward θ .
We derive the exact gradient for a trajectory instead of the approximate one in Ziebart (2010) in Appendix A and substitute it in to get:
[Equation]
where we have suppressed the dependence on θ for readability.
F t ( s t ) denotes the expected features when starting at s t at time t and acting until time 0 under the policy implied by θ .
Since we combine gradients from simulated past trajectories, we name our algorithm Reward Learning by Simulating the Past (RLSP).
The algorithm computes the gradient using dynamic programming, detailed in Appendix B. We can easily incorporate a prior on θ by adding the gradient of the log prior to the gradient in Equation 5.
E VALUATION
Evaluation of RLSP is non-trivial.
The inferred reward is very likely to assign state s 0 maximal reward, since it was inferred under the assumption that when Alice optimized the reward she ended up at s 0 .
If the robot then starts in state s 0 , if a no-op action is available (as it often is), the RLSP reward is likely to incentivize no-ops, which is not very interesting.
Ultimately, we hope to use RLSP to correct badly speciﬁed instructions or reward functions.
So, we created a suite of environments with a true reward R true , a speciﬁed reward R spec , Alice’s ﬁrst state s − T , and the robot’s initial state s 0 , where R spec ignores some aspect(s) of R true .
RLSP is used to infer a reward θ Alice from s 0 , which is then combined with the speciﬁed reward to get a ﬁnal reward θ ﬁnal =
θ Alice + λθ spec .
(We considered another method for combining rewards; see Appendix D for details.)
We inspect the inferred reward qualitatively and measure the expected amount of true reward obtained when planning with θ ﬁnal , as a fraction of the expected true reward from the optimal policy.
We tune the hyperparameter λ controlling the tradeoff between R spec and the human reward for all algorithms, including baselines.
We use a Gaussian prior over the reward parameters.
5. B ASELINES
Speciﬁed reward policy π spec .
We act as if the true reward is exactly the speciﬁed reward.
Policy that penalizes deviations π deviation .
This baseline minimizes change by penalizing deviations from the observed features f ( s 0 ) , giving R ﬁnal ( s )
=
θ T spec f ( s ) +
λ || f ( s ) − f ( s 0 )
|| .
Relative reachability policy π reachability .
Relative reachability (Krakovna et al., 2018) considers a change to be negative when it decreases coverage , relative to what would have happened had the agent done nothing.
Here, coverage is a measure of how easily states can be reached from the current state.
We compare against the variant of relative reachability that uses undiscounted coverage and a baseline policy where the agent takes no-op actions, as in the original paper.
Relative reachability requires known dynamics but not a handcoded featurization.
A version of relative reachability that operates in feature space instead of state space would behave similarly.
5. C OMPARISON TO BASELINES
We compare RLSP to our baselines with the assumption of known s − T , because it makes it easier to analyze RLSP’s properties.
We consider the case of unknown s − T in Section 5.3.
We summarize the results in Table 1, and show the environments and trajectories in Figure 2.
[Table]
s − T a nd sπ s p ec t r a j ec t o r i e s π R L
SP t r a j ec t o r i e s (a) ( [Figure] [Figure]
[Figure]
[Figure]
“Unseen”
Room
far
Side effects: Room with vase (Figure 2a) .
The room tests whether the robot can avoid breaking a vase as a side effect of going to the purple door.
There are features for the number of broken vases, standing on a carpet, and each door location.
Since Alice didn’t walk over the vase, RLSP infers a negative reward on broken vases, and a small positive reward on carpets (since paths to the top door usually involve carpets).
So, π RLSP successfully avoids breaking the vase.
The penalties also achieve the desired behavior: π deviation avoids breaking the vase since it would change the “number of broken vases” feature, while relative reachability avoids breaking the vase since doing so would result in all states with intact vases becoming unreachable.
Distinguishing environment effects: Toy train (Figure 2b) .
To test whether algorithms can distinguish between effects caused by the agent and effects caused by the environment, as suggested in
Krakovna et al.
(2018), we add a toy train that moves along a predeﬁned track.
The train breaks if the agent steps on it.
We add a new feature indicating whether the train is broken and new features for each possible train location.
As before, the speciﬁed reward only has a positive weight on the purple door, while the true reward also penalizes broken trains and vases.
RLSP infers a negative reward on broken vases and broken trains, for the same reason as before.
It also infers not to put any weight on any particular train location, even though it changes frequently, because it doesn’t help explain s 0 .
As a result, π RLSP walks over a carpet, but not a vase or a train.
π deviation immediately breaks the train to keep the train location the same.
π reachability deduces that breaking the train is irreversible, and so follows the same trajectory as π RLSP .
Implicit reward: Apple collection (Figure 2d) .
This environment tests whether the algorithms can learn tasks implicit in s 0 .
There are three trees that grow apples, as well as a basket for collecting apples, and the goal is for the robot to harvest apples.
However, the speciﬁed reward is zero: the robot must infer the task from the observed state.
We have features for the number of apples in baskets, the number of apples on trees, whether the robot is carrying an apple, and each location that the agent could be in.
s 0 has two apples in the basket, while s − T has none.
π spec is arbitrary since every policy is optimal for the zero reward.
π deviation does nothing, achieving zero reward, since its reward can never be positive.
π reachability also does not harvest apples.
RLSP infers a positive reward on apples in baskets, a negative reward for apples on trees, and a small positive reward for carrying apples.
Despite the spurious weights, π RLSP harvests apples as desired.
Desirable side effect: Batteries (Figure 2c) .
This environment tests whether the algorithms can tell when a side effect is allowed.
We take the toy train environment, remove vases and carpets, and add batteries.
The robot can pick up batteries and put them into the (now unbreakable) toy train, but the batteries are never replenished.
If the train runs for 10 timesteps without a new battery, it stops operating.
There are features for the number of batteries, whether the train is operational, each train location, and each door location.
There are two batteries at s − T but only one at s 0 .
The true reward incentivizes an operational train and being at the purple door.
We consider two variants for the task reward – an “easy” case, where the task reward equals the true reward, and a “hard” case, where the task reward only rewards being at the purple door.
Unsurprisingly, π spec succeeds at the easy case, and fails on the hard case by allowing the train to run out of power.
Both π deviation and π reachability see the action of putting a battery in the train as a side effect to be penalized, and so neither can solve the hard case.
They penalize picking up the batteries, and so only solve the easy case if the penalty weight is small.
RLSP sees that one battery is gone and that the train is operational, and infers that Alice wants the train to be operational and doesn’t want batteries (since a preference against batteries and a preference for an operational train are nearly indistinguishable).
So, it solves both the easy and the hard case, with π RLSP picking up the battery, then staying at the purple door except to deliver the battery to the train.
“Unseen” side effect: Room with far away vase (Figure 2e) .
This environment demonstrates a limitation of our algorithm: it cannot identify side effects that Alice would never have triggered.
In this room, the vase is nowhere close to the shortest path from the Alice’s original position to her goal, but is on the path to the robot’s goal.
Since our baselines don’t care about the trajectory the human takes, they all perform as before: π spec walks over the vase, while π deviation and π reachability both avoid it.
Our method infers a near zero weight on the broken vase feature, since it is not present on any reasonable trajectory to the goal, and so breaks it when moving to the goal.
Note that this only applies when Alice is known to be at the bottom left corner at s − T : if we have a uniform prior over s − T (considered in Section 5.3) then we do consider trajectories where vases are broken.
5. C OMPARISON BETWEEN KNOWING s − T VS . A DISTRIBUTION OVER s − T
So far, we have considered the setting where the robot knows s − T , since it is easier to analyze what happens.
However, typically we will not know s − T , and will instead have some prior over s − T .
Here, we compare RLSP in two settings: perfect knowledge of s − T (as in Section 5.2), and a uniform distribution over all states.
Side effects: Room with vase (Figure 2a) and toy train (Figure 2b) .
In both room with vase and toy train, RLSP learns a smaller negative reward on broken vases when using a uniform prior.
This is because RLSP considers many more feasible trajectories when using a uniform prior, many of which do not give Alice a chance to break the vase, as in Room with far away vase in Section 5.2.
In room with vase, the small positive reward on carpets changes to a near-zero negative reward on carpets.
With known s − T , RLSP overﬁts to the few consistent trajectories, which usually go over carpets, whereas with a uniform prior it considers many more trajectories that often don’t go over carpets, and so it correctly infers a near-zero weight.
In toy train, the negative reward on broken trains becomes slightly more negative, while other features remain approximately the same.
This may be because when Alice starts out closer to the toy train, she has more of an opportunity to break it, compared to the known s − T case.
Implicit preference: Apple collection (Figure 2d) .
Here, a uniform prior leads to a smaller positive weight on the number of apples in baskets compared to the case with known s − T .
Intuitively, this is because RLSP is considering cases where s − T already has one or two apples in the basket, which implies that Alice has collected fewer apples and so must have been less interested in them.
States where the basket starts with three or more apples are inconsistent with the observed s 0 and so are not considered.
Following the inferred reward still leads to good apple harvesting behavior.
Desirable side effects: Batteries (Figure 2c) .
With the uniform prior, we see the same behavior as in Apple collection, where RLSP with a uniform prior learns a slightly smaller negative reward on the batteries, since it considers states s − T where the battery was already gone.
In addition, due to the particular setup the battery must have been given to the train two timesteps prior, which means that in any state where the train started with very little charge, it was allowed to die even though a battery could have been provided before, leading to a near-zero positive weight on the train losing charge.
Despite this, RLSP successfully delivers the battery to the train in both easy and hard cases.
“Unseen” side effect: Room with far away vase (Figure 2e) .
With a uniform prior, we “see” the side effect: if Alice started at the purple door, then the shortest trajectory to the black door would break a vase.
As a result, π RLSP successfully avoids the vase (whereas it previously did not).
Here, uncertainty over the initial state s − T can counterintuitively improve the results, because it increases the diversity of trajectories considered, which prevents RLSP from “overﬁtting” to the few trajectories consistent with a known s − T and s 0 .
Overall, RLSP is quite robust to the use of a uniform prior over s − T , suggesting that we do not need to be particularly careful in the design of that prior.
5. R OBUSTNESS TO THE CHOICE OF A LICE ’ S PLANNING HORIZON
We investigate how RLSP performs when assuming the wrong value of Alice’s planning horizon T .
We vary the value of T assumed by RLSP, and report the true return achieved by π RLSP obtained using the inferred reward and a ﬁxed horizon for the robot to act.
For this experiment, we used a uniform prior over s − T , since with known s − T , RLSP often detects that the given s − T and s 0 are incompatible (when T is misspeciﬁed).
The results are presented in Figure 3. [Figure]
The performance worsens when RLSP assumes that Alice had a smaller planning horizon than she actually had.
Intuitively, if we assume that Alice has only taken one or two actions ever, then even if we knew the actions they could have been in service of many goals, and so we end up quite uncertain about Alice’s reward is.
When the assumed T is larger than the true horizon, RLSP correctly infers things the robot should not do.
Knowing that the vase was not broken for longer than T timesteps is more evidence to suspect that Alice cared about not breaking the vase.
However, overestimated T leads to worse performance at inferring implicit preferences, as in the Apples environment.
If we assume Alice has only collected two apples in 100 timesteps, she must not have cared about them much, since she could have collected many more.
The batteries environment is unusual – assuming that Alice has been acting for 100 timesteps, the only explanation for the observed s 0 is that Alice waited until the 98th timestep to put the battery into the train.
This is not particularly consistent with any reward function, and performance degrades.
Overall, T is an important parameter and needs to be set appropriately.
However, even when T is misspeciﬁed, performance degrades gracefully to what would have happened if we optimized θ spec by itself, so RLSP does not hurt.
In addition, if T is larger than it should be, then RLSP still tends to accurately infer parts of the reward that specify what not to do.
L IMITATIONS AND FUTURE WORK
Summary.
Our key insight is that when a robot is deployed, the state that it observes has already been optimized to satisfy human preferences.
This explains our preference for a policy that generally avoids side effects.
We formalized this by assuming that Alice has been acting in the environment prior to the robot’s deployment.
We developed an algorithm, RLSP, that computes a MAP estimate of Alice’s reward function.
The robot then acts according to a tradeoff between Alice’s reward function and the speciﬁed reward function.
Our evaluation showed that information from the initial state can be used to successfully infer side effects to avoid as well as tasks to complete, though there are cases in which we cannot infer the relevant preferences.
While we believe this is an important step forward, there is still much work to be done to make this accurate and practical.
Realistic environments.
The primary avenue for future work is to scale to realistic environments, where we cannot enumerate states, we don’t know dynamics, and the reward function may be nonlinear.
This could be done by adapting existing IRL algorithms (Fu et al., 2017; Ho and Ermon, 2016; Finn et al., 2016).
Unknown dynamics is particularly challenging, since we cannot learn dynamics from a single state observation.
While acting in the environment, we would have to learn a dynamics model or an inverse dynamics model that can be used to simulate the past, and update the learned preferences as our model improves over time.
Alternatively, if we use unsupervised skill learning (Achiam et al., 2018; Eysenbach et al., 2018; Nair et al., 2018) or exploration (Burda et al., 2018), or learn a goal-conditioned policy (Schaul et al., 2015; Andrychowicz et al., 2017), we could compare the explored states with the observed s 0 .
Hyperparameter choice.
While our evaluation showed that RLSP is reasonably robust to the choice of planning horizon T and prior over s − T , this may be speciﬁc to our gridworlds.
In the real world, we often make long term hierarchical plans, and if we don’t observe the entire plan (corresponding to a choice of T that is too small) it seems possible that we infer bad rewards, especially if we have an uninformative prior over s − T .
We do not know whether this will be a problem, and if so how bad it will be, and hope to investigate it in future work with more realistic environments.
Conﬂicts between θ spec and θ Alice .
RLSP allows us to infer θ Alice from s 0 , which we must somehow combine with θ spec to produce a reward θ ﬁnal for the robot to optimize.
θ Alice will usually prefer the status quo of keeping the state similar to s 0 , while θ spec will probably incentivize some change to the state, leading to conﬂict.
We traded off between the two by optimizing their sum, but this is not very principled and future work could improve upon this.
For example, θ Alice could be decomposed into θ Alice,task , which says which task Alice is performing (“go to the black door”), and θ frame , which consists of the frame conditions (“don’t break vases”).
The robot then optimizes θ frame + λθ spec .
This requires some way of performing the decomposition. We could model the human as pursuing multiple different subgoals, or the environment as being created by multiple humans with different goals.
θ frame would be shared, while θ task would vary, allowing us to distinguish between them.
However, combination may not be the answer – instead, perhaps the robot ought to use the inferred reward to inform Alice of any conﬂicts and actively query her for more information.
Learning tasks to perform.
The apples and batteries environments demonstrate that RLSP can learn preferences that require the robot to actively perform a task.
It is not clear that this is desirable, since the robot may perform an inferred task instead of the task Alice explicitly sets for it.
Preferences that are not a result of human optimization.
While the initial state is optimized for human preferences, this may not be a result of human optimization, as assumed in this paper.
For example, we prefer that the atmosphere contain oxygen for us to breathe.
The atmosphere meets this preference in spite of human action, and so RLSP would not infer this preference.
While this is of limited relevance for household robots, it may become important for more capable AI systems.
A CKNOWLEDGMENTS
We thank the researchers at the Center for Human Compatible AI for valuable feedback.
This work was supported by the Open Philanthropy Project and National Science Foundation Graduate Research
Fellowship Grant No. DGE 1752814.
A
Here, we derive an exact gradient for the maximum causal entropy distribution introduced in Ziebart (2010), as the existing approximation is insufﬁcient for our purposes.
Given a trajectory τ T = s 0 a 0 . . .
s T a T , we seek the gradient ∇ θ ln p ( τ T ) .
We assume that the expert has been acting according to the maximum causal entropy IRL model given in Section 3 (where we have dropped θ from the notation for clarity):
[Equation]
In the following, unless otherwise speciﬁed, all expectations over states and actions use the probability distribution over trajectories from the above model, starting from the state and action just prior.
For example, E s
[Equation]
In addition, for all probability distributions over states and actions, we drop the dependence on θ for readability, so the probability of reaching state s T is written as p ( s T ) instead of p ( s T | θ ) .
First, we compute the gradient of V t ( s ) .
We have ∇ θ V T +1 ( s ) = 0 , and for 0 ≤ t ≤ T :
[Equation]
Unrolling the recursion
[Equation]
we get that the gradient is the expected feature counts under the policy implied by θ from s t onwards, which we could prove using induction.
Deﬁne:
[Equation]
Then we have:
[Equation]
We can now calculate the gradient we actually care about:
[Equation]
only π t depends on θ
[Equation]
The last term of the summation is f ( s T ) + E s
[Equation]
[Equation]
∇ θ V T +
[Equation]
s [Equation] T +1 ) (cid:3) − ∇ θ V T ( s T ) , which simpliﬁes to f ( s T ) + 0 − F T ( s T ) =
f ( s T ) − f ( s T ) = 0 , so we can drop it.
Thus, our gradient is:
[Equation]
This is the gradient we will use in Appendix B, but a little more manipulation allows us to compare with the gradient in Ziebart (2010).
We reintroduce the terms that we cancelled above:
[Equation]
Ziebart (2010) states that the gradient is given by the expert policy feature expectations minus the learned policy feature expectations, and in practice uses the feature expectations from demonstrations to approximate the expert policy feature expectations.
Assuming we have N trajectories { τ i } , the gradient would be (cid:16) 1 N (cid:80) i (cid:80) Tt =
f ( s t,i ) (cid:17) − E s
[ F 0 ( s 0 )] .
The ﬁrst term matches our ﬁrst term exactly.
Our second term matches the second term in the limit of sufﬁciently many trajectories, so that the starting states s 0 follow the distribution p ( s 0 ) .
Our third term converges to zero with sufﬁciently many trajectories, since any s t , a t pair in a demonstration will be present sufﬁciently often that the empirical counts of s t +1 will match the expected proportions prescribed by T ( · | s t , a t ) .
In a deterministic environment, we have T ( s (cid:48) t +1 | s t , a t )
=
1[ s (cid:48) t +1 =
s t +1 ] since only one transition is possible.
Thus, the third term is zero and even for one trajectory the gradient reduces to (cid:16)(cid:80)
Tt =0 f
( s t ) (cid:17) − F 0 ( s 0 ) .
This differs from the gradient in Ziebart (2010) only in that it computes feature expectations from the observed starting state s 0 instead of the MDP distribution over initial states p ( s 0 ) .
In a stochastic environment, the third term need not be zero, and corrects for the “bias” in the observed states s t +1 .
Intuitively, when the expert chose action a t , she did not know which next state s (cid:48) t + would arise, but the ﬁrst term of our gradient upweights the particular next state s t +1 that we observed.
The third term downweights the future value of the observed state and upweights the future value of all other states, all in proportion to their prior probability T ( s (cid:48)
t +1 | s t , a t ) .
B
This section provides a derivation of the gradient ∇ θ ln p ( s 0 ) , which is needed to solve argmax θ ln p ( s 0 ) with gradient ascent.
We provide the results ﬁrst as a quick reference:
[Equation]
Base cases: ﬁrst, p ( s − T ) is given, second, G − T ( s − T ) = 0 , and third, F 0 ( s 0 ) =
f ( s 0 ) .
For the derivation, we start by expressing the gradient in terms of gradients of trajectories, so that we can use the result from Appendix A. Note that, by inspecting the ﬁnal form of the gradient in
Appendix A, we can see that ∇ θ p ( τ − T :0 ) is independent of a 0 .
Then, we have:
[Equation]
This has a nice interpretation – compute the gradient for each trajectory and take the weighted sum
[Equation]
where each weight is the probability of the trajectory given the evidence s 0 and current reward θ .
We can rewrite the gradient in Equation 6 as ∇ θ ln p ( τ T ) = (cid:80) T − [Equation] =
g ( s t , a t ) , where
[Equation]
We can now substitute this to get:
[Equation]
Note that we can compute p ( s t ) since we are given the distribution p ( s − T ) and we can use the recursive rule p ( s t +1 )
= (cid:80) s t ,a
[Equation]
p ( [Equation]
[Equation]
| s t )
[Equation]
( s t
[Equation]
1 | s t , a t ) .
In order to compute g ( s t , a t ) we need to compute F t ( s t ) , which has base case F 0 ( s 0 ) =
f ( s 0 ) and recursive rule:
[Equation]
For the remaining part of the gradient, deﬁne G t such that ∇ θ
[Equation]
) p
[Equation]
s 0 ) :
[Equation]
) (cid:35) .
We now derive a recursive relation for G :
[Equation]
= [Equation] = (cid:88) [Equation]

[Equation]
s
[Equation]
− T : t
[Equation]
[Equation]
s t )
[Equation]
[Equation]
t
[Equation]
T
[Equation]
(cid:33)  = (cid:88) s t
[Equation]
a t
[Equation]
(cid:18)
[Equation]
a t
[Equation]
t
[Equation]
(cid:19) .
For the base case, note that
G − T + [Equation] s − T + [Equation]
=
[Equation]
a
[Equation] T [Equation] Comparing this to the recursive rule, for the base case we can set G − T ( s − T )
= 0 .
C
Instead of estimating the MLE (or MAP if we have a prior) using RLSP, we could approximate the entire posterior distribution. One standard way to address the computational challenges involved with the continuous and high-dimensional nature of θ is to use MCMC sampling to sample from p ( θ | s 0 ) ∝ p ( s 0 | θ ) p ( θ ) . The resulting algorithm resembles Bayesian IRL (Ramachandran and
Amir, 2007) and is presented in Algorithm 1.
While this algorithm is less efﬁcient and noisier than RLSP, it gives us an estimate of the full posterior distribution. In our experiments, we collapsed the full distribution into a point estimate by taking the mean. Initial experiments showed that the algorithm was slower and noisier than the gradientbased RLSP, so we did not test it further. However, in future work we could better leverage the full distribution, for example to create risk-averse policies, to identify features that are uncertain, or to identify features that are certain but conﬂict with the speciﬁed reward, after which we could actively query Alice for more information.
Algorithm Require: MDP M , prior p ( θ ) , step size δ 1: θ ← random sample ( p ( θ )) 2: π, V = soft value iteration ( M , θ ) 3: p ← p ( s 0 | θ ) p ( θ ) 4: repeat 5: θ (cid:48) ← random sample ( N ( θ, δ )) 6: π (cid:48) , V (cid:48)
=
soft value iteration ( M , θ (cid:48) )
(cid:46)
The value function is initialized with V .
7: p (cid:48) ← p ( s 0 | θ (cid:48) )
p ( θ (cid:48) ) 8: if random sample ( Unif (0 , 1)) ≤ min(1 , p (cid:48) p )
then 9: θ ← θ (cid:48) ; V ← V (cid:48) 10: end if 11: append θ to the list of samples 12: until have generated the desired number of samples
Comparison of the methods for combining θ spec and θ H         ) U DF WL RQ  R I  P D [   WHPSHUDWXUH  $GGLWLYHEDWWHULHV$GGLWLYHURRP$GGLWLYHWUDLQ%D\HVLDQEDWWHULHV%D\HVLDQURRP%D\HVLDQWUDLQ      6WDQGDUGGHYLDWLRQ  
 WHPSHUDWXUH          WHPSHUDWXUH 
D C OMBINING THE SPECIFIED REWARD WITH THE INFERRED REWARD
In Section 5, we evaluated RLSP by combining the reward it infers with a speciﬁed reward to get a ﬁnal reward θ ﬁnal =
θ Alice + λθ spec .
As discussed in Section 6, the problem of combining θ Alice and θ spec is difﬁcult, since the two rewards incentivize different behaviors and will conﬂict.
The Additive method above is a simple way of trading off between the two.
Both RLSP and the sampling algorithm of Appendix C can incorporate a prior over θ .
Another way to combine the two rewards is to condition the prior on θ spec before running the algorithms.
In particular, we could replace our prior P ( θ Alice ) with a new prior P ( θ Alice | θ spec ) , such as a Gaussian distribution centered at θ spec .
When we use this prior, the reward returned by RLSP can be used as the ﬁnal reward θ ﬁnal .
It might seem like this is a principled Bayesian method that allows us to combine the two rewards.
However, the conﬂict between the two reward functions still exists.
In this formulation, it arises in the new prior P ( θ Alice | θ spec ) .
Modeling this as a Gaussian centered at θ spec suggests that before knowing s 0 , it seems likely that θ Alice is very similar to θ spec .
However, this is not true – Alice is probably providing the reward θ spec to the robot so that it causes some change to the state that she has optimized, and so it will be predictably different from θ spec .
On the other hand, we do need to put high probability on θ spec , since otherwise θ ﬁnal will not incentivize any of the behaviors that θ spec did.
Nonetheless, this is another simple heuristic for how we might combine the two rewards, that manages the tradeoff between θ spec and θ Alice .
We compared the Additive and Bayesian methods by evaluating their robustness.
We vary the parameter that controls the tradeoff and report the true reward obtained by π RLSP , as a fraction of the expected true reward under the optimal policy.
For the Bayesian method, we vary the standard deviation σ of the Gaussian prior over θ Alice that is centered at θ spec .
For the
Additive method, the natural choice would be to vary λ ; however, in order to make the results more comparable, we instead set λ = 1 and vary the standard deviation of the Gaussian prior used while inferring θ Alice , which is centered at zero instead of at θ spec .
A larger standard deviation allows θ Alice to become larger in magnitude (since it is penalized less for deviating from the mean of zero reward), which effectively corresponds to a smaller λ .
While we typically create π RLSP using value iteration, this leads to deterministic policies with very sharp changes in behavior that make it hard to see differences between methods, and so we also show results with soft value iteration, which creates stochastic policies that vary more continuously.
As demonstrated in Figure 4, our experiments show that overall the two methods perform very similarly, with some evidence that the Additive method is slightly more robust.
The Additive method also has the beneﬁt that it can be applied in situations where the inferred reward and speciﬁed reward are over different feature spaces, by creating the ﬁnal reward R ﬁnal ( s )
=
θ Alice T f Alice ( s ) + λR spec ( s ) .