1 Preamble/Disclaimer
This preamble was written by Pablo Samuel Castro
This is a paper I wrote during my postdoc with Daqing Zhang at the Institut MinesTELECON/TELECOM SudParis. We submitted it to NIPS (now called NeurIPS) 2011 and it was rejected. I finished my postdoc shortly after the rejection and joined Google as a software engineer, saying goodbye to academia for ever (at the time I couldn’t envision a future where I’d be lucky enough to rejoin the academic world). This was at a time when putting papers up on ArXiv was not a common thing (at least not for me), so the paper lay dormant in my email.
During a recent visit to Scott Niekum’s group in UT Austin I decided to dust it off, as they’re doing research on inverse reinforcement learning. One of his students, Daniel Brown^{1}^{1}1Check out some of his recent papers, including (Brown et al., 2019; Brown and Niekum, 2018). mentioned that he liked the proposition I have in this paper, and wondered how he could cite it. And that is what led me to put this up.
This paper is presented asis: except for a few minor grammatical corrections, it is unchanged from my original submission. I considered revising it and bringing the relatedwork section uptodate, but upon rereading it there are many things I feel would need to be changed, updated, or completely rewritten. In short: this is not a paper I would send out for review in its current form. ^{2}^{2}2Some of the things I would do are: Include an appendix with all the results discussed but not included; fix the colors on the GridWorld plots (they make no sense as they are right now!); run on more domains.
So why am I putting up on ArXiv then? I do feel that some of the ideas presented here are interesting enough that they could perhaps be of value to other researchers in the field (at least 1 seemed interesting enough to Daniel to be worth citing).
However, I am not currently working on this problem, I no longer have access to the code I wrote for it, nor to the GPS data I used, so it is extermely unlikely that I will work on an updatedandimproved version of this paper. If you find this idea interesting and want to make a proper paper out of it, be my guest!
2 Introduction
Markov Decision Processes (MDPs) are a popular mathematical framework for encoding a sequential decision making problem, where an agent aims to find an optimal way of acting in said environment. The optimality of the agent’s behaviour has traditionally been quantified by means of a reward/cost function, supplied as part of the MDP. Whether the optimal behaviour is computed directly or learned through experience, the reward function plays a central role in this process, enabling the agent to rank the actions according to their expected numerical return.
Many problems have been successfully encoded as MDPs and have produced successful behaviours. Nevertheless, there are many problems for which specifying a reward function can be very difficult. Consider the problem of learning to ride a bicycle: although a reward function could be specified that gives a penalty for falling down or losing one’s balance, there are many other factors which characterize a good rider. Indeed, when we are taught to ride a bicycle, our teachers describe or demonstrate the type of behaviour that will enable us to ride the bicycle properly. In these situations, it is often easier to demonstrate an optimal behaviour, and have the agent attempt to emulate it. This problem is known as imitation learning, apprenticeship learning, learning from demonstration, amongst others, and has been studied extensively during the last two decades, often for robotics applications using supervised learning methods
(Hayes and Demiris, 1994; Amit and Mataric, 2002; Pomerleau, 1989). More recently, Abbeel and Ng (2004) proposed an apprenticeship learning algorithm that guarantees the learned policy will be close to the expert’s policy.A similar approach is to construct a reward function under which the expert’s behaviour is optimal. In the Artificial Intelligence community this is known as the Inverse Reinforcement Learning (IRL) problem and was first formally studied in
(Ng and Russell, 2000). This approach has the advantage of producing a reward function which can be used afterwards (such as for Reinforcement Learning (RL) algorithms), and reward functions are usually a more compact representation of a domain than a policy.Both approaches mentioned above are based on the observed behaviours of an expert.^{3}^{3}3In most cases a single expert is considered, although (Ratliff et al., 2006) is a notable exception. In other words, the sole purpose of the demonstrations is to inform the agent of what it should do. This is somewhat in contrast to the way humans learn: we learn to behave not only by attempting to imitate experts, but also by avoiding the behaviours of nonexperts. The field of imitation learning is based on the assumption that the expert’s behaviour is optimal with respect to an underlying reward function, and as long as there are enough expert demonstrations to “cover” the environment, the extracted reward function should be sufficient to produce a similar behaviour. This may be so, but as we will prove below, the resulting reward function may be rather superficial, as it may fail to disclose certain “negative” aspects of the true reward function. Moreover, if the expert is aware of certain areas in the environment that have low reward, she may avoid them altogether, resulting in incomplete information about the environment.
It is usually assumed that an expert is “requested” to act in an enviornment, from which the behaviours can be extracted. This is not problematic in simulated or controlled environments, but may be difficult in large, realworld problems. Nevertheless, due to the growing pervasiveness of devices capable of logging a user’s contextual information (such as location, actions, etc.), we may have access to many experts and nonexperts. This paper introduces a method for Inverse Reinforcement Learning using multiple experts with varying degrees of expertise. Our approach makes use of maximum margin ordinal regression algorithms (Shashua and Levin, 2002) to produce a reward function that preserves the ranking of our experts, as well as maximizing the margin between the different ranks. We illustrate our approach on the problem of finding optimal taxi driver strategies for finding new passengers, where we make use of the driving trajectories from multiple GPSequipped taxis. As we will demonstrate, our approach is able to produce meaningful and useful results.
The paper is organized as follows. In section 3 we introduce Markov Decision Processes and discuss the problem of apprenticeship learning, along with related works. We present our proposed method in section 4, along with a proof of the “superficiality” of solutions based solely on expert’s behaviours. In section 5 we highlight the advantages of our approach over existing approaches using a simple grid, and demonstrate the efficacy of our method on a large problem involving providing passengerfinding strategies to taxi drivers in section 6. We conclude our results and discuss future avenues of research in section 7.
3 Preliminaries
3.1 Markov Decision Processes
A finite Markov Decision Process (MDP) is a 4tuple , where is a finite set of states, is a finite set of actions available at each state, ^{4}^{4}4
is the set of all probability distributions on a set
. is a probabilistic transition function, and is the reward function.The behaviour of an agent in an MDP is formalized as a policy ; let be the set of all policies. We define the value of a policy from a state as , where is a discount factor and
is a random variable representing the reward received at time step
, given that we started at state and followed policy . We may sometimes refer to as thedimensional vector containing the values for all states.
In most situations, one is interested in finding an optimal behaviour: a policy satisfying for all . If the MDP parameters are known, there are a number of exact and approximate methods for finding (Puterman, 1994). Reinforcement learning is a popular approach to learning to act optimally when neither the transition or reward functions are known a priori, but are revealed to the agent as it interacts with the environment (Sutton and Barto, 1998).
3.2 Apprenticeship Learning
In the situations mentioned above, it is implicitly assumed that the agent has access to a fully specified reward function. Although this is a reasonable assumption in many problems, there are many situations where specifying a reward function can be difficult. In these situations, it may be easier to specify or demonstrate a desired behaviour, rather than specify a reward function.
For the rest of the paper we will assume that MDPs are not equipped with a reward function; when a reward function is specified, we will denote the resulting value function under policy as . The Inverse Reinforcement Learning (IRL) problem aims to determine a reward function that “explains” the behaviour, or policy , of an expert . Specifically, it aims to find a reward function such that . In (Ng and Russell, 2000)
the authors characterize the set of reward functions with this property. They point out that many of these reward functions will be degenerate, and suggest some heuristics to overcome this degeneracy. The heuristics are based on the principle of choosing a reward function that maximizes the difference between the value of the expert’s policy and other, suboptimal, policies. In
(Ziebart et al., 2008)the authors apply the principle of maximum entropy to overcome this degeneracy.
This maximum margin principle lies at the heart of the algorithms proposed in (Abbeel and Ng, 2004) and (Ratliff et al., 2006). Abbeel and Ng (2004) assume there is a feature vectore and that the reward function can be expressed as a linear combination of these features: , where is a weight vector. Under this assumption, it can be shown that
where is an initial state distribution, is a random variable representing the state at time step when starting at and following policy , and are the feature expectations of policy
. The authors assume access to an estimate of the expert’s feature expectations,
. Their algorithm begins with a random policy , and at each iteration chooses a weight vector that maximizes the margin between and for all ; is then set to be an optimal policy for the MDP augmented with the reward function and the process is repeated untilgoes below a prespecified threshold. As the authors point out, at each iteration we are finding the maximum margin hyperplane separating
from the set of, for which a Support Vector Machine (SVM) or Quadratic Programming (QP) solver can be used.
The maximum margin approach was further studied in (Ratliff et al., 2006)
, where the authors use a predefined loss function that quantifies the “closeness” of the computed policy to the expert’s policy. The chosen loss function directly impacts the performance of the algorithm; indeed, the resulting margin will scale with the loss function. Rather than starting from a single expert and computing new policies at each iteration as in
(Abbeel and Ng, 2004), the authors assume access to a set of experts (each potentially defined in a separate MDP). Their algorithm seeks to find a weight vector such that the optimal policy resulting from each input MDP augmented with the reward function is “close” to the supplied expert’s behaviour.4 Rank IRL
As mentioned in the introduction, there are many situations where we cannot simply request an expert to interact with an environment in order to observe their behaviour. However, with the surge in ubiquitous devices such as GPS and mobile phones, we may have access to many experts. Although we may combine to obtain an “average” expert, we can capitalize on the difference amongst the various experts, and in particular, nonexperts. The following result demonstrates that the reward functions produced by an IRL or apprenticeship learning solver may be quite superficial.
Proposition 1.
There exists an MDP with true reward function , expert policy , approximate reward function , and nonexpert policies and such that
Proof.
Consider the following MDP with deterministic transitions. Each transition is labelled by the action name, and the true reward (if any) received upon entering a state is indicated in the state’s node.
Here is some arbitrary constant. Obviously , and it is easy to see that , , and yield the desired result. ∎
Our method will make use of feature expectations, as policies are usually difficult to specify precisely. Let us assume that we have access to the expected features of a number of demonstrators: , and to a ranking function that classifies each expected feature into one of ranks. Let us also define the set . We wish to find a weight vector such that for all if and only if . For all we would also like to maximize , where
In other words, should maximize the difference between ranks.
Ranking the instances properly is an instance of the ordinal regression problem, and applying the maximum margin principle yields the following (primal) quadratic program (QP):
(1)  
subject to  
This is the sumofmargins strategy introduced in (Shashua and Levin, 2002). Traditional SVM solvers maximize a single margin between two classes of points (a notion which is generalized to ordinal regression in (Shashua and Levin, 2002) as the fixedmargin strategy). In our current situation we wish to maximally distinguish between ranks, which corresponds to maximizing margins. Rather than searching for a single separating hyperplane, we are searching for parallel hyperplanes with normal vector (see Figure 1). Note that one of the constraints in QP Equation 1 is that the norm of is at most one. In (Shashua and Levin, 2002) they demonstrate that using this convex constraint is equivalent to using the nonconvex constraint . Hence, the margin between ranks and is given by . The performance of our method depends directly on the quality of the supplied ranking. The slack variables and allow for some instances to be misclassified or be on the wrong side of or , respectively. The constant trades off between the sum of the margins and margin errors (see (Shashua and Levin, 2002) for a lengthier discussion of the meaning of ). We set for all of our experiments below. In the conclusion we suggest some simple mechanisms for dealing with unreliable rankings.
5 Illustration
As mentioned above, our method is motivated by the fact that agents can learn how not to act from suboptimal demonstrations. 1 demonstrates that the policy/behaviour of an expert can obfuscate certain aspects of the true underlying reward function. We illustrate this point further by comparing our method against the apprenticeship learning algorithm of Abbeel and Ng (2004) on a simple grid domain. Consider a 16x16 room with reward function as given in the left panel of Figure 2
and optimal policy in the right panel. As can be seen, even though there is a clear advantage to being in the even rows as opposed to the odd rows, this advantage is not evident in the optimal policy. This optimal policy was used to generate the expert’s expected features for both algorithms.
We ran the algorithm from Abbeel and Ng (2004) against our method (henceforth referred to as RankIRL). We used a “lossless” feature vector, that is, one feature for every grid cell. Thus, the returned by the algorithms would exactly correspond to the reward function. We used four types of policies to generate three types of expected features for our method:

First rank (expert): The optimal policy

Second rank: A policy that avoids odd rows and goes right only on even rows, then goes down once in the last column

Third rank: The policy is similar to the second, but even rows are avoided and odd rows are followed towards the right

Fourth rank: A policy that avoids even rows and follows odd rows to the left
We assume a uniform distribution over all possible cells for both algorithms. In
Figure 3 we compare the reward functions of the two algorithms, where it is evident that our method produces a reward function that prefers even rows over odd rows (most evident in the states on the righthand side). The algorithm of Abbeel and Ng (2004) displayed much higher variability than our method due to the random policy used in the first iteration. As is pointed out by the authors, although their method is guaranteed to produce expected features close to those of the expert’s, it is unfortunate that the resulting vector is unreliable. We also computed the ratio of the performance of the policies from both algorithms against the performance of the true optimal policy, and our method consistently had an advantage of at least 20% over the algorithm of Abbeel and Ng (2004). This behaviour was consistent as the number of sampled trajectories used to generate the expected features were varied. A visual inspection of the resulting policies also reveals that our method explicitly avoids odd rows, while the policy produced by the algorithm of Abbeel and Ng (2004) is oblivious to the difference between even and odd states. We have not included these figures for lack of space. Finally, we also compared our algorithm against the MMP algorithm of Ratliff et al. (2006) and observed similar results. This is not surprising, since their method is still based on emulating one or more experts.6 Application: Passenger finding routes for taxi drivers
We have access to a large dataset consisting of 5000 taxi drivers’ GPS data collected over a year in Hangzhou, China. Amongst other things, each GPS entry gives us the taxi’s unique ID, its location (in latitude/longitude), speed and state (occuppied/vacant). This data is logged approximately once per minute. We constructed a digital representation of the road network of Hangzhou based on the accumulated taxi trajectories, and mapped each GPS entry to a road segment. Mapping the analogue trajectories onto a discrete domain essentially removed noise and small variations in the trajectories.
An important problem for taxi drivers is determining the best strategy to find a new passenger when unoccupied. Existing approaches include using a naive Bayesian classifier to predict the number of vacant taxis in different areas (Phithakkitnukoon et al., 2010), using handfilled surveys and simple statistical analyses (Takayama et al., 2011), using means clustering coupled with temporal analysis (Hayes and Demiris, 1994), and using norm SVM to determine whether a taxi driver should “hunt” or wait (Li et al., 2011). The granularity of these approaches is quite coarse, as most rely on splitting the city into equalsized grids. It would be more useful to provide passenger finding assistance to taxi drivers on a roadbyroad basis.
To test our method on this problem, we decided to focus on the weekdays of a single month. Each taxi driver was ranked according the proportion of time during the day they spent unoccupied (lower is better), since taxis with a low proportion probably have a good passengerfinding strategy. The size of the state space for our MDP was defined to be twice the number of road segments, resulting in two states for each road segment, one for each orientation.^{5}^{5}5The fact that certain roads may be unidirectional is not a problem, as we should not have any trajectories going in the wrong direction! The road segment and orientation is quite important for this problem, as traffic conditions and speed limits can vary greatly amongst road segments, and even within the same segment, depending on orientation. Thus, we decided to once again use a lossless feature vector, with one feature for every possible state.
Our road network has 2203 road segments, which means we are searching for a weight vector . Solving a QP in this high dimensional setting was not possible in our machines^{6}^{6}6This was in 2011, so we took advantage of the “Cartesian” nature of our domain and split the city into disjoint areas. The areas were obtained dynamically by splitting the city according to most frequently traversed road intersections. The details of this decomposition method will be ommitted as it is outside the scope of this paper. The decomposition we used is displayed in Figure 4, and all of the trajectories were split according this city decomposition. We computed a weight vector for each disjoint cluster and combined them afterwards. This reduced the dimensionality of our feature vectors from 2203 to no more that 130 dimensions and enabled us to compute the solutions in parallel.
We chose 10 taxis for each rank (expert, midlevel, and low) based on the scores mentioned above and computed the expected features from the GPS trajectories. We display the average expected features for the highest ranked taxis in the left panel of Figure 5. We used the computed weight vector as a reward function and computed the value function for the road network encoded as an MDP; the resulting value function is displayed in the right panel of Figure 5. It is interesting to note that our algorithm recognizes certain important passenger pickup areas (such as the airport) even though the experts do not have very high expected features there. The areas that have highest value are in accordance with places that have intuitively higher taxi demand, such as bus stations and hospitals. Our method is also able to recognize roads with poor passenger pickup records, such as in the mountainous area, where people do not generally wait for taxis. Although it seems somewhat surprising that the downtown area is given a relatively low value, this is in accordance to the results found in (Liu et al., 1994), where they uncovered the fact that the most successful taxi drivers do not always search for passengers in areas with highest demand (such as in the downtown area), as these areas are often prone to high levels of traffic. Using this value function we can provide taxi drivers with passengerfinding strategies on a much finer granularity than existing methods.
7 Conclusion
We have presented a novel method for apprenticeship learning, based on the principle that agents should not only learn what to do, but should also learn what not to do by observing nonexperts’ behaviours. The illustration on the 16x16 grid clearly demonstrates that learning solely from experts (as in most apprenticeship algorithms) can lead to certain important aspects of the true underlying reward function to remain “hidden” from the algorithm. Furthermore, our approach is inherently unambiguous as it will always produce the same weight vector, given a fixed set of expected features, thereby overcoming solution degeneracy. This reduced variability in solutions is in stark contrast to the method in (Abbeel and Ng, 2004), whose resulting solutions are greatly dependent on the starting policy. An additional advantage of our approach over closely related methods (such as (Abbeel and Ng, 2004) and (Ratliff et al., 2006)) is that we are not required to solve any MDPs at each iteration. Rather, our method produces a weight vector based solely on the expected features of the demonstrators.
Although our results for passengerfinding strategies are very promising, we are convinced we can do much better if we consider different timeslots in isolation. In this paper we have examined the behaviour of the different drivers throughout the weekdays, disregarding important differences between different time slots (such as between rush hour and late night shifts).
We were fortunate to have a scoring function that produced a reliable ranking of the taxi drivers for our problem. For other problems, however, it may be difficult to properly rank the different demonstrators. The slack variables and provide a way to determine improperly ranked demonstrators. One could take either an incremental or a pruning approach. In the incremental approach one begins with a small set of demonstrators and incrementally adds new ones (thereby providing more information about the underlying reward function to the algorithm) as long as the new demonstrator is not coupled with a positive or value. In the pruning approach, one could remove (or swap) the expert with the highest or value.
We have also used our algorithm to find optimal navigating routes through our road network. Although there exist many algorithms for finding optimal routes to a destination, these usually rely on a preexisting reward function. Noting that taxi drivers usually know the road network of a city better than regular drivers, we would like to learn from their navigational behaviours. Finding a good ranking function has proved somewhat daunting, as we often find certain demonstrators improperly ranked. We have used the pruning approach mentioned above to remove these outliers and this has greatly improved the output of our algorithm, producing very promising preliminary results. A difficulty with this type of problem is that an expert’s expected features will probably not cover the whole road network, resulting in many areas with poor navigational suggestions. By learning from the behaviour of nonexperts, our method is able to cover a much larger area of the city.
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