Learner-aware Teaching: Inverse Reinforcement Learning with Preferences and Constraints

06/02/2019 ∙ by Sebastian Tschiatschek, et al. ∙ Max Planck Institute for Software Systems Microsoft ETH Zurich 0

Inverse reinforcement learning (IRL) enables an agent to learn complex behavior by observing demonstrations from a (near-)optimal policy. The typical assumption is that the learner's goal is to match the teacher's demonstrated behavior. In this paper, we consider the setting where the learner has her own preferences that she additionally takes into consideration. These preferences can for example capture behavioral biases, mismatched worldviews, or physical constraints. We study two teaching approaches: learner-agnostic teaching, where the teacher provides demonstrations from an optimal policy ignoring the learner's preferences, and learner-aware teaching, where the teacher accounts for the learner's preferences. We design learner-aware teaching algorithms and show that significant performance improvements can be achieved over learner-agnostic teaching.



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1 Introduction

Inverse reinforcement learning (IRL) enables a learning agent (learner) to acquire skills from observations of a teacher’s demonstrations. The learner infers a reward function explaining the demonstrated behavior and optimizes her own behavior accordingly. IRL has been studied extensively Abbeel and Ng (2004); Ratliff et al. (2006); Ziebart (2010); Boularias et al. (2011); Osa et al. (2018) under the premise that the learner can and is willing to imitate the teacher’s behavior.

In real-world settings, however, a learner typically does not blindly follow the teacher’s demonstrations, but also has her own preferences and constraints. For instance, consider demonstrating to an auto-pilot of a self-driving car how to quickly navigate from A to B by going through a pedestrian zone. These demonstrations might conflict with the constraint of the auto-pilot to drive only on lanes in order to ensure maximum safety of human beings. Similarly, in robot-human interaction with the goal of teaching people how to cook, a teaching robot might demonstrate to a human user how to cook “roast chicken”, which could conflict with the preferences of the learner who is “vegetarian”. To give yet another example, consider a surgical training simulator which provides virtual demonstrations of expert behavior; a novice learner might not be confident enough to imitate a difficult procedure because of safety concerns. In all these examples, the learner might not be able to acquire useful skills from the teacher’s demonstrations.

In this paper, we formalize the problem of teaching a learner with preferences and constraints. First, we are interested in understanding the suboptimality of learner-agnostic teaching, i.e., ignoring the learner’s preferences. Second, we are interested in designing learner-aware teachers who account for the learner’s preferences and thus enable more efficient learning. To this end, we study a learner model with preferences and constraints in the context of the Maximum Causal Entropy (MCE) IRL framework Ziebart (2010); Ziebart et al. (2013); Zhou et al. (2018). This enables us to formulate the teaching problem as an optimization problem, and to derive and analyze algorithms for learner-aware teaching.

Our main contributions are:

  1. We formalize the problem of IRL under preference constraints (Section 2 and Section 3).

  2. We analyze the problem of optimizing demonstrations for the learner when preferences are known to the teacher, and we propose a bilevel optimization approach to the problem (Section 4).

  3. We propose strategies for adaptively teaching a learner with preferences unknown to the teacher, and we provide theoretical guarantees under natural assumptions (Section 5).

  4. We empirically show that significant performance improvements can be achieved by learner-aware teachers as compared to learner-agnostic teachers (Section 6).

2 Problem Setting

Environment. Our environment is described by a Markov decision process (MDP) . Here and denote finite sets of states and actions. describes the state transition dynamics, i.e.,

is the probability of landing in state

by taking action from state . is the discounting factor. is an initial distribution over states. is the reward function. We assume that there exists a feature map such that the reward function is linear, i.e., for some . Note that a bound of ensures that for all .

Basic definitions. A policy is a map such that

is a probability distribution over actions for every state

. We denote by the set of all such policies. The performance measure for policies we are interested in is the expected discounted reward , where the expectation is taken with respect to the distribution over trajectories induced by together with the transition probabilities and the initial state distribution . A policy is optimal for the reward function if , and we denote optimal policies by . Note that , where ,

, is the map taking a policy to its vector of

(discounted) feature expectations. We denote by the image of this map. Note that the set is convex (see (Ziebart, 2010, Theorem 2.8) and Abbeel and Ng (2004)), and also bounded due to the discounting factor . For a finite collection of trajectories obtained by executing a policy in the MDP , we denote the empirical counterpart of by .

An IRL learner and a teacher. We consider a learner implementing an inverse reinforcement learning (IRL) algorithm and a teacher . The teacher has access to the full MDP ; the learner knows the MDP and the parametric form of reward function but does not know the true reward parameter . The learner, upon receiving demonstrations from the teacher, outputs a policy using her algorithm. The teacher’s objective is to provide a set of demonstrations to the learner that ensures that the learner’s output policy achieves high reward .

The standard IRL algorithms are based on the idea of feature matching Abbeel and Ng (2004); Ziebart (2010); Osa et al. (2018): The learner’s algorithm finds a policy that matches the feature expectations of the received demonstrations, ensuring that where specifies a desired level of accuracy. In this standard setting, the learner’s primary goal is to imitate the teacher (via feature matching) and this makes the teaching process easy. In fact, the teacher just needs to provide a sufficiently rich pool of demonstrations obtained by executing , ensuring . This guarantees that . Furthermore, the linearity of rewards and ensures that the learner’s output policy satisfies .

(a) Environment
(b) Set of vectors
Figure 1: An illustrative example to showcase the suboptimality of teaching when the learner has preferences and constraints. Environment: Figure 0(a) shows a grid-world environment inspired by the object-world and gathering game environments Levine et al. (2010); Leibo et al. (2017); Mendez et al. (2018). Each cell represents a state, there are five actions given by “left", “up", “right", "down", “stay", the transitions are deterministic, and the starting state is the top-left cell. The agent’s goal is to collect objects in the environment: Collecting a “star" provides a reward of and a “plus" a reward of ; objects immediately appear again upon collection, and the rewards are discounted with close to 1. The optimal policy is to go to the nearest “star" and then “stay" there. Preferences: A small number of states in the environment are distractors, depicted by colored cells in Figure 0(a). We consider a learner who prefers to avoid “green" distractors: she has a hard constraint that the probability of having a “green" distractor within a neighborhood is at most . Feature expectation vectors: Figure 0(b) shows the set of feature expectation vectors . The -axis and the -axis represent the discounted feature count for collecting “star" and “plus" objects, respectively. The striped region represents policies that are feasible w.r.t. the learner’s constraint. Suboptimality of teaching: Upon receiving demonstrations from an optimal policy with feature vector , the learner under her preference constraint can best match the teacher’s demonstrations (in a sense of minimizing ) by outputting a policy with , which is clearly suboptimal w.r.t. the true rewards. Policy with feature vector represents an alternate teaching policy which would have led to higher reward for the learner.

Key challenges in teaching a learner with preference constraints. In this paper, we study a novel setting where the learner has her own preferences which she additionally takes into consideration when learning a policy using teacher’s demonstrations. We formally specify our learner model in the next section; here we highlight the key challenges that arise in teaching such a learner. Given that the learner’s primary goal is no longer just imitating the teacher via feature matching, the learner’s output policy can be suboptimal with respect to the true reward even if she had access to , i.e., the feature expectation vector of an optimal policy . Figure 1 provides an illustrative example to showcase the suboptimality of teaching when the learner has preferences and constraints. The key challenge that we address in this paper is that of designing a teaching algorithm that selects demonstrations while accounting for the learner’s preferences.

3 Learner Model

In this section we describe the learner models we consider, including different ways of defining preferences and constraints. First, we introduce some notation and definitions that will be helpful. We capture learner’s preferences via a feature map . We define as a concatenation of the two feature maps and given by and let . Similar to the map , we define , and , . Similar to , we define and as the images of the maps and . Note that for any policy , we have .

Standard (discounted) MCE-IRL. Our learner models build on the (discounted) Maximum Causal Entropy (MCE) IRL framework Ziebart et al. (2008); Ziebart (2010); Ziebart et al. (2013); Zhou et al. (2018). In the standard (discounted) MCE-IRL framework, a learning agent aims to identify a policy that matches the feature expectations of the teacher’s demonstrations while simultaneously maximizing the (discounted) causal entropy given by . More background is provided in Appendix C.

Including preference constraints. The standard framework can be readily extended to include learner’s preferences in the form of constraints on the preference features . Clearly, the learner’s preferences can render exact matching of the teacher’s demonstrations infeasible and hence we relax this condition. To this end, we consider the following generic learner model:


Here, are convex functions representing preference constraints. We denote the parameters and variables in vector notation as , , and . The coefficients and are parameters and quantify the importance of matching the teacher’s demonstrations and satisfying the learner’s preferences. Next, we discuss two special instances of this generic learner model.

3.1 Learner Model with Hard Preferences Constraints

It is instructive to study a special case of the above-mentioned generic learner model with , and a limiting case with and . Intuitively, the preferences take the form of hard constraints, i.e., the learner’s output policy must satisfy . Additionally, while satisfying these hard constraints, the learner minimizes the norm distance to the teacher’s demonstration. We formally describe the learner’s behavior below.

First, we define the learner’s constraint set as . Similar to , we define and . Also, note that and are projections of the set to the subspaces and respectively. Then, the learner’s behavior can be approximated as:

  1. Learner can match: When , the learner outputs a policy s.t. .

  2. Learner cannot match: Otherwise, the learner outputs a policy such that is given by the norm projection of the vector onto the set .

Figure 1 provides an illustration of the behavior of this learner model. We will design learner-aware teaching algorithms for this learner model in Section 4.1 and Section 5.

3.2 Learner Model with Soft Preference Constraints

Another interesting learner model that we study in this paper arises from the generic learner when we consider number of box-type linear constraints with . We consider norm penalty on violation, and for simplicity we consider . In this case, the learner’s model is given by


The solution to the above problem corresponds to a softmax policy with a reward function where is parametrized by . The optimal parameters can be computed efficiently and the corresponding softmax policy is then obtained by Soft-Value-Iteration procedure (see (Ziebart, 2010, Algorithm. 9.1), Zhou et al. (2018)). Details are provided in Appendix D. We will design learner-aware teaching algorithms for this learner in Section 4.2.

4 Learner-aware Teaching under Known Constraints

In this section, we analyze the setting when the teacher has full knowledge of the learner’s constraints.

4.1 A Learner-aware Teacher for Hard Preferences: Aware-CMDP

Here, we design a learner-aware teaching algorithm when considering the learner from Section 3.1. Given that the teacher has full knowledge of the learner’s preferences, it can compute an optimal teaching policy by maximizing the reward over policies that satisfy the learner’s preference constraints, i.e., the teacher solves a constrained-MDP problem (see De (1960); Altman (1999)) given by

We refer to an optimal solution of this problem as and the corresponding teacher as Aware-CMDP. We can make the following observation formalizing the value of learner-aware teaching:

Theorem 1.

For simplicity, assume that the teacher can provide an exact feature expectation of a policy instead of providing demonstrations to the learner. Then, the value of learner-aware teaching is

When the set

is defined via a set of linear constraints, the above problem can be formulated as a linear program and solved exactly; details are provided in Appendix 


4.2 A Learner-aware Teacher for Soft Preferences via Bi-level Optimization: Aware-BiL

For the learner models in Section 3, the optimal learner-aware teaching problem can be naturally formalized as the following bi-level optimization problem:


where stands for the IRL problem solved by the learner given demonstrations from and can include preferences of the learner (see Eq. 1 in Section 3).

There are many possibilities for solving this bi-level optimization problem—see for example Sinha et al. (2018) for an overview. In this paper we adopted a single-level reduction approach to simplify the above bi-level optimization problem as this results in particularly intuitive optimiziation problems for the teacher. The basic idea of single-level reduction is to replace the lower-level problem, i.e., , by the optimality conditions for that problem given by the Karush-Kuhn-Tucker conditions Boyd and Vandenberghe (2004); Sinha et al. (2018). For the learner model outlined in Section 3.2, these reductions take the following form (see Appendix F in the supplementary material for details):


where corresponds to a softmax policy with a reward function for . Thus, finding optimal demonstrations means optimization over softmax teaching policies while respecting the learner’s preferences. To actually solve the above optimization problem and find good teaching policies, we use an approach inspired by the Frank-Wolfe algorithm Jaggi (2013) detailed in Appendix F. We refer to a teacher implementing this approach as Aware-BiL.

5 Learner-Aware Teaching Under Unknown Constraints

In this section, we consider the more realistic and challenging setting in which the teacher does not know the learner ’s constraint set . Without feedback from , can generally not do better than the agnostic teacher who simply ignores any constraints. We therefore assume that and interact in rounds as described by Algorithm 1. The two versions of the algorithm we describe in Sections 5.1 and 5.2 are obtained by specifying how adapts the teaching policy in each round.

1:Initial teaching policy (e.g., optimal policy ignoring any constraints)
2:for round  do
3:     Teacher provides demonstrations with feature vector using policy
4:     Learner upon receiving computes a policy with feature vector
5:     Teacher observes learner’s feature vector and adapts the teaching policy
Algorithm 1 Teacher-learner interaction in the adaptive teaching setting

In this section, we assume that is as described in Section 3.1: Given demonstrations , finds a policy such that matches the -projection of onto . For the sake of simplifying the presentation and the analysis, we also assume that and can observe the exact feature expectations of their respective policies, e.g., if is sampled from .

5.1 An Adaptive Learner-aware Teacher Using Volume Search: AdAware-Vol

In our first adaptive teaching algorithm AdAware-Vol,

maintains an estimate

of the learner’s constraint set, which in each round gets updated by intersecting the current version with a certain affine halfspace, thus reducing the volume of . The new teaching policy is then any policy which is optimal under the constraint that . The interaction ends as soon as for a threshold . Details on the algorithm are provided in Appendix B.1.

Theorem 2.

Upon termination of AdAware-Vol, ’s output policy satisfies for any policy which is optimal under ’s constraints. For the special case that is a polytope defined by linear inequalities, the algorithm terminates in iterations.

5.2 An Adaptive Learner-aware Teacher Using Line Search: AdAware-Lin

In our second adaptive teaching algorithm, AdAware-Lin, adapts the teaching policy by performing a binary search on a line segment of the form to find a vector that is the vector of feature expectations of a policy; here are fixed constants. If that is not successful, the teacher finds a teaching policy with . The following theorem analyzes the convergence of ’s performance to under the assumption that ’s search succeeds in every round. Further details and the proof of the theorem are provided in Appendix B.2.

Theorem 3.

Fix some and assume that there exists a constant such that, as long as , the teacher can find a teaching policy satisfying for some . Then the learner’s performance increases monotonically in each round of AdAware-Lin, i.e., . Moreover, after at most teaching steps, the learner’s performance satisfies . Here we abbreviate .

6 Experimental Evaluation

In this section we evaluate our teaching algorithms for different types of learners on the environment introduced in Figure 1. The environment we consider here has three reward objects, i.e., a “star" object with reward of , a “plus" object with reward of and a “dot" object with reward of such that . Two objects of each type are placed randomly on the grid. Furthermore, there are two types of distractors: (i) two “green" distractors are randomly placed at a distance of 0-cell and 1-cell to the “star" objects; (ii) two “yellow" distractors are randomly placed at a distance of 1-cell and 2-cells to the “plus" objects, see Figure 1(a). We have a total of 6 preference features with as follows: the first three features in are binary-indicators whether there is a “green" distractor at a distance of 0-cell, 1-cell, and 2-cells; similarly the next three features are binary-indicators for the “yellow" distractor. We use a discount factor of . Upon collecting an object, there is a probability of transiting to a terminal state.

We consider the learner with soft constraints from Section 3.2 for , and . We have a total of different learners depending on the preference features used by them out of total preference features discussed above, see Figure 1(a), e.g., L1 learner has no preference features, L2 learner has first two preference features (i.e., ), L3 learner has first four preference features, L4 learner has first five preference features, and L5 learner has all six preference features. The first row in Figure 2 shows the considered object-worlds and indicates the preference of the learners to avoid certain regions by the gray area.

Learner ()
L1 L2 L3 L4 L5
Teacher Agnostic
Table 1: Learners’ average rewards after teaching. L1, , L5 correspond to learners with preferences similar to those in Figure 2. Results are averaged over 10 random object-worlds, standard error
(a) Environments and learners’ preferences for 5 different learners L1, , L5
(b) Learners’ rewards inferred from learner-agnostic teacher’s (Agnostic) demonstrations
(c) Learners’ rewards inferred from learner-aware teacher’s (Aware-BiL) demonstrations
Figure 2: Teaching in object-world environments under full knowledge of the learner’s preferences. Green and yellow cells indicate distractors associated with either “star" or “plus" objects, respectively. Learner’s preferences to avoid cells are indicated in gray. Learner model from Section 3.2 with , is considered for these experiments. The learner-aware teacher enable the learner to infer reward functions that are compatible with the learner’s preferences and achieve higher average rewards. In Figure 1(b) and Figure 1(c), blue color represents positive reward, red color represents negative reward, and the magnitude of the reward is indicated by color intensity.

6.1 Teaching under known constraints

Our first set of results are presented in Figure 2. The second and third row show the reward function inferred by the learner for demonstrations provided by a learner-agnostic teacher (Agnostic) and the bi-level learner-aware teacher (Aware-BiL), respectively. We observe that Agnostic fails to teach the learner about objects’ positive rewards in cases where the learners’ preferences conflict with the position of the most rewarding objects (second row). In contrast, Aware-BiL always successfully teaches the learners about rewarding objects that are compatible with the learners’ preferences (third row).

We also compare Agnostic and Aware-BiL in terms of reward achieved by the learner after teaching for object worlds of size in Table 1. The numbers show the average reward over 10 randomly generated object-worlds. We observe, that a learner can learn better policies from a teacher that knows about the learner’s preferences and takes them into account.

6.2 Teaching under unknown constraints

In this section we evaluate the teaching algorithms from Section 5. We consider the learner model from Section 3.1 that uses -projection to match reward feature expectations as studied in Section 5. Here, we study the learner who considers first two preference features (i.e., ): this corresponds to an object-world similar to that in Figure 2 (learner L2 with second grid-world). For modeling the hard constraints, we consider box-type linear constraints with for these two preference features (also see Eq. 2).

In this context it is instructive to investigate how quickly these adaptive teaching strategies converge to the performance of a teacher who would have full knowledge about the learner. Results comparing the adaptive teaching strategies (AdAware-Vol and AdAware-Lin) are shown in Figure 3. We can observe that both teaching strategies converge to the best possible performance under full knowledge about the learner (Aware-CMDP).

We also provide results showing the performance achieved by the adaptive teaching strategies on object-worlds of varying sizes, see Figure 3. Note that the performance of AdAware-Vol decreases slightly when teaching for more rounds, i.e., comparing the results after 3 teaching rounds and at the end of the teaching process. This is because of approximations when learner is computing the policy via projection, which in turn leads to errors on the teacher side when approximating . In contrast, AdAware-Lin performance always increases when teaching for more rounds.

(a) Reward over teaching rounds
Aware-CMDP 7.51 7.25 7.18
Agnostic 3.22 3.18 3.17
Conserv 1.36 1.61 1.56
AdAware-Vol () 7.39 7.13 6.92
AdAware-Vol (end) 6.9 6.66 6.37
AdAware-Lin () 6.19 6.03 6.25
AdAware-Lin (end) 7.37 7.27 7.06
(b) Varying grid-size
Figure 3: Performance of adaptive teaching strategies AdAware-Vol and AdAware-Lin. (left) Figure 2(a) shows the reward for learner’s policy over number of teaching interactions. The horizontal lines indicate the performance of learner’s policy for the learner-aware teacher with full knowledge of the learner’s constraints Aware-CMDP, the learner-agnostic teacher Agnostic who ignores any constraints, and a conservative teacher Conserv who considers all constraints. Our adaptive teaching strategies AdAware-Vol and AdAware-Lin significantly outperform baselines (Agnostic and Conserv) and quickly converge towards the optimal performance of Aware-CMDP. The dotted lines AdAware-Vol:T and AdAware-Lin:T show the rewards corresponding to teacher’s policy at a round and are shown to highlight the very different behavior of two adaptive teaching strategies. (right) Table 2(b) shows results for varying grid-size of the environment. Results are reported at round and at the “end" round when algorithm reaches it’s stopping criterion. Results are reported as average over 5 runs, where each run corresponds to a random environment.

7 Related Work

Our work is closely related to algorithmic machine teaching Goldman and Kearns (1995); Zhu et al. (2018), whose general goal is to design teaching algorithms that optimize the data that is provided to a learning algorithm. Algorithmic teaching provides a rigorous formalism for a number of real-world applications such as personalized education and intelligent tutoring systems Patil et al. (2014); Zhu (2015); Rafferty et al. (2016); Hunziker et al. (2018), social robotics Cakmak and Thomaz (2014), and human-in-the-loop systems Singla et al. (2013, 2014).

Most works in machine teaching so far focus on supervised learning tasks and assume that the learning algorithm is fully known to the teacher, see e.g. 

Zhu (2013); Singla et al. (2014); Liu and Zhu (2016); Mac Aodha et al. (2018). In the IRL setting, few works study how to provide maximally informative demonstrations to the learner, e.g., Cakmak and Lopes (2012); Brown and Niekum (2019). In contrast to our work, their teacher fully knows the learner model and provides the demonstrations without any adaptation to the learner. The question of how a teacher should adaptively react to a learner has been addressed by Liu et al. (2018); Chen et al. (2018); Melo et al. (2018); Yeo et al. (2019), but only in the supervised setting. In recent work, Kamalaruban et al. (2019) studies interactive teaching algorithms for an IRL learner, however, they consider a sequential learner, and there is no notion of learner’s preferences and constraints in their setting.

Within the area of IRL, there is a line of work on active learning approaches

Cohn et al. (2011); Brown et al. (2018); Amin et al. (2017); Cui and Niekum (2018), which is related to our work in the sense that they consider the question of how to optimize demonstrations for a given learner. In contrast to us, they take the perspective of the learner who actively influences the demonstrations she receives. A few papers have addressed aspects of the problem that arises in IRL when the learner does not have full access to the reward features, e.g., Levine et al. (2010) and Haug et al. (2018).

Our work is also loosely related to multi-agent reinforcement learning. Dimitrakakis et al. (2017) studies the interaction between agents with misaligned models with a focus on the question of how to jointly optimize a policy. Also Hadfield-Menell et al. (2016) study the cooperation between agents which do not perfectly understand each other.

8 Conclusions

In the context of inverse reinforcement learning, we investigated the important problem of interacting with learners that have preferences and constraints that prevent them from closely approximating the teacher’s demonstrations. We demonstrated the suboptimality of learner-agnostic teaching and proposed algorithms for learner-aware teaching strategies for known and unknown preferences of the learner. In future work, we will evaluate our approach in machine-human and human-machine tasks and extend our approach to other learner models.


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Appendix A List of Appendices

In this section we provide a brief description of the content provided in the appendices of the paper.

  • Appendix B provides additional details on the adaptive teaching strategies (Section 5).

  • Appendix C provides background on the (discounted) MCE-IRL problem (Section 3).

  • Appendix D provides additional details on the (discounted) MCE-IRL problem with preferences (Section 3.2).

  • Appendix E provides the LP formulation for the teacher Aware-CMDP (Section 4.1).

  • Appendix F provides additional details on the bi-level optimization approach for the teacher Aware-BiL (Section 4.2).

Appendix B Details for Learner-Aware Teaching under Unknown Constraints (Section 5)

In this appendix, we provide more details on the adaptive teaching algorithms AdAware-Vol and AdAware-Lin described in Sections 5.1 and 5.2. Recall that both teaching algorithms are obtained from Algorithm 1 by defining the way in which the teacher adapts the teaching policy based on the learner ’s feature expectations in past rounds.

b.1 Details for AdAware-Vol (Section 5.1)

Estimation of the learner’s constraint set.

In AdAware-Vol, maintains an estimate of ’s constraint set, starting with . After observing the feature expectations of the policy found in round , updates this estimate as follows:


The set on the right hand side of (5) with which gets intersected is a halfspace containing . This is due to the fact that is convex by assumption, and to our assumption that ’s learning algorithm is such that it outputs a policy whose feature expectations match the -projection of to . Inductively, it follows that for all .

In practice, we implement a slightly modified version of the update step in which we intersect with a halfspace that is shifted in the direction of by a small amount, i.e., we use

with a step size parameter . This helps make the algorithm more robust to noise in the learner’s feature expectations. In our experiments, we used .

Update of the teaching policy.

After updating the estimate of the learner’s constraint set to , solves a constrained MDP in order to find

Given that is cut out by linear equations, solving the constrained MDP reduces to solving an LP, as described in Appendix E.

Termination of the interaction.

The algorithm terminates as soon as the stopping criterion is satisfied. Note that implies that

for any . Therefore, after termination we have

for any policy which is optimal under ’s constraints, which is the first statement of Theorem 2.

The second statement of Theorem 2 follows from the fact that if is a convex polytope cut out by linear inequalities, the number of faces, which is in , is an upper bound on the number of iterations of the algorithm, because one face is “eliminated” in each round.

b.2 Details for AdAware-Lin (Section 5.2)

1:, , , , .
3:while  do
6:     if  then
8:     else
10:if  then
Algorithm 2 LineSearch
Figure 4:
LineSearch is the algorithm that uses in order to find a teaching policy provided that the feature expectations of ’s current policy are . Figure 4 illustrates the two cases may occur: For the right , LineSearch returns a policy whose feature expectations satisfy such that . For the left , LineSearch returns a policy whose feature expectations satisfy .

In AdAware-Lin, updates the teaching policy based on ’s feature expectations from the previous round. To do so, uses LineSearch (Algorithm 2) to perform a binary search on the line segment


in order to find a vector that is realizable as the vector of feature expectations of a policy. If the intersection of the line segment (6) with is non-empty, it is of the form for some due to the convexity of . In that case, LineSearch returns a policy with feature expectations

where is the maximal such that . If the intersection is empty, LineSearch returns a policy with feature expectations

Figure 4 illustrates the two cases that may occur.

b.2.1 Proof of Theorem 3

In this section, we provide the proof of Theorem 3, which gives a guarantee on the improvement of ’s performance in each round of the AdAware-Lin algorithm. The assumption we make here is that, in every teaching round, LineSearch returns a teaching policy such that for some , where is a fixed constant. It is easy to see that this assumption, together with our assumption on ’s algorithm and the convexity of , imply that the change in learner performance

is non-negative in every teaching round. The following proposition, which will be needed in the proof of Theorem 3, strengthens this statement:

Proposition 1.

Let be the maximally achievable learner performance. Assume that, in teaching round , can find a teaching policy whose feature expectations satisfy for some . Then


where .

Proof of Proposition 1.

Consider the plane spanned by and and denote by the unique point in with the properties that

  1. ,

  2. lies on the same side of the line through and as , and

  3. and span a right triangle with at the right-angled corner.

Note that must lie inside this triangle, i.e., on the red line segment in Figure 5: Otherwise there would a point on the line segment connecting and , and hence in by convexity, which is closer to than , contradicting the fact that is closest to among all points in . Denote by the line passing through and .

Figure 5: Illustration of the proof of Proposition 1: The smaller the performance increase , the better the upper bound on the gap .

The facts that is convex and that imply that

must lie on one side of the hyperplane

Therefore, we can upper bound in terms of the slope of the line which arises by intersecting that hyperplane with :


Note that the slope is upper bounded by the slope of . We have , where is the length of the red line segment in Figure 5, and by Pythagoras’s theorem. Using that, we obtain


The claimed estimate (7) follows by plugging this upper bound for into (8) and rearranging. ∎

Proof of Theorem 3.
Proof of Theorem 3.

The fact that , which is equivalent to , follows immediately from Proposition 1.

We now prove the claimed rate of convergence.

First, using Proposition 1, we note that the assumption that implies that


Using that, we can conclude that


Indeed, if , it follows from (10) that we must have