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0 | 0 | Title | Y6LzLWXlS8E_00_00_00 | 0 | 0 | Title | WNev_iSes_00_00_00 | 0 | Deconfounded Imitation Learning | Deconfounded Imitation Learning | {
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1 | 0 | Paragraph | Y6LzLWXlS8E_01_00_00 | 1 | null | null | null | null | Anonymous Author(s) | {
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3 | 0 | Abstract | Y6LzLWXlS8E_01_02_00 | 1 | 0 | Abstract | WNev_iSes_01_00_00 | 1 | Abstract | Abstract | {
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4 | 0 | Abstract | Y6LzLWXlS8E_01_03_00 | 1 | 0 | Abstract | WNev_iSes_01_01_00 | 1 | Standard imitation learning can fail when the expert demonstrators have differentsensory inputs than the imitating agent. | Standard imitation learning can fail when the expert demonstrators have different sensory inputs than the imitating agent. | {
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5 | 0 | Abstract | Y6LzLWXlS8E_01_03_01 | 1 | 0 | Abstract | WNev_iSes_01_01_01 | 1 | This partial observability gives rise tohidden confounders in the causal graph, which lead to the failure to imitate. | This is because partial observability gives rise to hidden confounders in the causal graph. | {
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6 | 0 | Abstract | Y6LzLWXlS8E_01_03_02 | 1 | 0 | Abstract | WNev_iSes_01_01_02 | 1 | Webreak down the space of confounded imitation learning problems and identify threesettings with different data requirements in which the correct imitation policy canbe identified. | We break down the space of confounded imitation learning problems and identify three settings with different data requirements in which the correct imitation policy can be identified. | {
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7 | 0 | Abstract | Y6LzLWXlS8E_01_03_03 | 1 | 0 | Abstract | WNev_iSes_01_01_03 | 1 | We then introduce an algorithm for deconfounded imitation learning,which trains an inference model jointly with a latent-conditional policy. | We then introduce an algorithm for deconfounded imitation learning, which trains an inference model jointly with a latent-conditional policy. | {
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8 | 0 | Abstract | Y6LzLWXlS8E_01_03_04 | 1 | 0 | Abstract | WNev_iSes_01_01_04 | 1 | At testtime, the agent alternates between updating its belief over the latent and actingunder the belief. | At test time, the agent alternates between updating its belief over the latent and acting under the belief. | {
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9 | 0 | Abstract | Y6LzLWXlS8E_01_03_05 | 1 | 0 | Abstract | WNev_iSes_01_01_05 | 1 | We show in theory and practice that this algorithm convergesto the correct interventional policy, solves the confounding issue, and can undercertain assumptions achieve an asymptotically optimal imitation performance | We show in theory and practice that this algorithm converges to the correct interventional policy, solves the confounding issue, and can under certain assumptions achieve an asymptotically optimal imitation performance. | {
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10 | 0 | Section | Y6LzLWXlS8E_02_00_00 | 2 | 0 | Section | WNev_iSes_02_00_00 | 2 | Introduction | Introduction | {
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11 | 0 | Paragraph | Y6LzLWXlS8E_02_00_00 | 2 | 0 | Paragraph | WNev_iSes_02_00_00 | 2 | In imitation learning (IL), an agent learns a policy directly from expert demonstrations withoutrequiring the specification of a reward function. | In imitation learning (IL), an agent learns a policy directly from expert demonstrations without requiring the specification of a reward function. | {
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12 | 0 | Paragraph | Y6LzLWXlS8E_02_00_01 | 2 | 0 | Paragraph | WNev_iSes_02_00_01 | 2 | This paradigm could be essential for solving realworld problems in autonomous driving and robotics where reward functions can be difficult to shapeand online learning may be dangerous. | This paradigm could be essential for solving realworld problems in autonomous driving and robotics where reward functions can be difficult to shape and online learning may be dangerous. | {
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13 | 0 | Paragraph | Y6LzLWXlS8E_02_00_02 | 2 | 0 | Paragraph | WNev_iSes_02_00_02 | 2 | However, standard IL requires that the conditions under whichthe agent operates exactly match those encountered by the expert. | However, standard IL requires that the conditions under which the agent operates exactly match those encountered by the expert. | {
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14 | 0 | Paragraph | Y6LzLWXlS8E_02_00_03 | 2 | 0 | Paragraph | WNev_iSes_02_00_03 | 2 | In particular, they assume thatthere are no latent confounders —variables that affect the expert behavior, but that are not observed bythe agent. | In particular, they assume that there are no latent confounders —variables that affect the expert behavior, but that are not observed by the agent. | {
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15 | 0 | Paragraph | Y6LzLWXlS8E_02_00_04 | 2 | 0 | Paragraph | WNev_iSes_02_00_04 | 2 | This assumption is often unrealistic. | This assumption is often unrealistic. | {
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16 | 0 | Paragraph | Y6LzLWXlS8E_02_00_05 | 2 | 0 | Paragraph | WNev_iSes_02_00_05 | 2 | Consider a human driver who is aware of the weatherforecast and lowers its speed in icy conditions, even if those are not visible from observations. | Consider a human driver who is aware of the weather forecast and lowers its speed in icy conditions, even if those are not visible from observations. | {
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17 | 0 | Paragraph | Y6LzLWXlS8E_02_00_06 | 2 | 0 | Paragraph | WNev_iSes_02_00_06 | 2 | Animitator agent without access to the weather forecast will not be able to adapt to such conditions. | An imitator agent without access to the weather forecast will not be able to adapt to such conditions. | {
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18 | 0 | Paragraph | Y6LzLWXlS8E_02_01_00 | 2 | 0 | Paragraph | WNev_iSes_02_01_00 | 2 | In such a situation, an imitating agent may take their own past actions as evidence for the values ofthe confounder. | In such a situation, an imitating agent may take their own past actions as evidence for the values of the confounder. | {
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19 | 0 | Paragraph | Y6LzLWXlS8E_02_01_01 | 2 | 0 | Paragraph | WNev_iSes_02_01_01 | 2 | A self-driving car, for instance, could conclude that it is driving fast, thus there can beno ice on the road. | A self-driving car, for instance, could conclude that it is driving fast, thus there can be no ice on the road. | {
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20 | 0 | Paragraph | Y6LzLWXlS8E_02_01_02 | 2 | 0 | Paragraph | WNev_iSes_02_01_02 | 2 | This issue of causal delusion was first pointed out in Ortega and Braun [2010a,b]and studied in more depth by Ortega et al. | This issue of causal delusion was first pointed out in Ortega and Braun [2010a,b] and studied in more depth by Ortega et al. | {
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21 | 0 | Paragraph | Y6LzLWXlS8E_02_01_03 | 2 | 0 | Paragraph | WNev_iSes_02_01_03 | 2 | The authors analyze the causal structure of thisproblem and argue that an imitator needs to learn a policy that corresponds to a certain interventionaldistribution. | The authors analyze the causal structure of this problem and argue that an imitator needs to learn a policy that corresponds to a certain interventional distribution. | {
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22 | 0 | Paragraph | Y6LzLWXlS8E_02_01_04 | 2 | 0 | Paragraph | WNev_iSes_02_01_04 | 2 | They then show that the classic DAgger algorithm [Ross et al., 2011], which requiresquerying experts at each time step, solves this problem and converges to the interventional policy. | They then show that the classic DAgger algorithm [Ross et al., 2011], which requires querying experts at each time step, solves this problem and converges to the interventional policy. | {
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23 | 0 | Paragraph | Y6LzLWXlS8E_02_02_00 | 2 | 0 | Paragraph | WNev_iSes_02_02_00 | 2 | In this paper, we present a solution to a confounded IL problem, where both the expert policy andthe environment dynamics are Markovian. | In this paper, we present a solution to a confounded IL problem, where both the expert policy and the environment dynamics are Markovian. | {
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24 | 0 | Paragraph | Y6LzLWXlS8E_02_02_01 | 2 | 0 | Paragraph | WNev_iSes_02_02_01 | 2 | The solution does not require querying experts. | The solution does not require querying experts. | {
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25 | 0 | Paragraph | Y6LzLWXlS8E_02_02_02 | 2 | 0 | Paragraph | WNev_iSes_02_02_02 | 2 | We firstpresent a characterization of confounded IL problems depending on properties of the environmentand expert policy (section 3). | We first present a characterization of confounded IL problems depending on properties of the environment and expert policy (Section 3). | {
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26 | 0 | Paragraph | Y6LzLWXlS8E_02_02_03 | 2 | 0 | Paragraph | WNev_iSes_02_02_03 | 2 | We then show theoretically that an imitating agent can learn behaviorsthat approach optimality when the above Markov assumptions and a recurrence property hold. | We then show theoretically that an imitating agent can learn behaviors that approach optimality when the above Markov assumptions and a recurrence property hold. | {
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27 | 0 | Paragraph | Y6LzLWXlS8E_02_03_00 | 2 | 1 | Paragraph | WNev_iSes_02_03_00 | 2 | We then introduce a practical algorithm for deconfounded imitation learning that does not require expert queries (section 4). | We then introduce a practical algorithm for deconfounded imitation learning that does not require expert queries (Section 4). | {
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28 | 0 | Paragraph | Y6LzLWXlS8E_02_03_01 | 2 | 1 | Paragraph | WNev_iSes_02_03_01 | 2 | An agent jointly learns an inference network for the valueof latent variables that explain the environment dynamics as well as a latent-conditional policy. | An agent jointly learns an inference network for the value of latent variables that explain the environment dynamics as well as a latent-conditional policy. | {
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29 | 0 | Paragraph | Y6LzLWXlS8E_02_04_00 | 2 | 1 | Paragraph | WNev_iSes_02_03_02 | 2 | At test time, the agent iteratively samples latents from its belief, acts in the environ- ment, and updates the belief based on the environment dynamics. | At test time, the agent iteratively samples latents from its belief, acts in the environment, and updates the belief based on the environment dynamics. | {
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30 | 0 | Paragraph | Y6LzLWXlS8E_02_04_01 | 2 | 1 | Paragraph | WNev_iSes_02_03_03 | 2 | An imitator steering a selfdriving car, for instance, would learn how to infer the weather condition from the dynamics ofthe car on the road. | An imitator steering a self-driving car, for instance, would learn how to infer the weather condition from the dynamics of the car on the road. | {
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31 | 0 | Paragraph | Y6LzLWXlS8E_02_04_02 | 2 | 1 | Paragraph | WNev_iSes_02_03_04 | 2 | This inference model can be applied both to its own online experienceas well as to expert trajectories, allowing it to imitate the behavior adequate for the weather. | This inference model can be applied both to its own online experience as well as to expert trajectories, allowing it to imitate the behavior adequate for the weather. | {
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32 | 1 | Figure | Y6LzLWXlS8E_02_05_00 | 2 | 1 | Figure | WNev_iSes_02_04_00 | 2 | [Figure] | [Figure] | {
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33 | 1 | Paragraph | Y6LzLWXlS8E_02_06_00 | 2 | 1 | Paragraph | WNev_iSes_02_05_00 | 2 | Finally, our deconfounded imitation learning algorithm isdemonstrated in a multi-armed bandit problem. | Finally, our deconfounded imitation learning algorithm is demonstrated in a multi-armed bandit problem. | {
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34 | 1 | Paragraph | Y6LzLWXlS8E_02_06_01 | 2 | 1 | Paragraph | WNev_iSes_02_05_01 | 2 | We showthat the agent quickly adapts to the unobserved propertiesof the environment and then behaves optimally (section 5). | We show that the agent quickly adapts to the unobserved properties of the environment and then behaves optimally (Section 5). | {
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35 | 1 | Section | Y6LzLWXlS8E_03_00_00 | 3 | 1 | Section | WNev_iSes_03_00_00 | 3 | Imitation learning and latent confounders | Imitation learning and latent confounders | {
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36 | 1 | Paragraph | Y6LzLWXlS8E_03_00_00 | 3 | 1 | Paragraph | WNev_iSes_03_00_00 | 3 | We begin by introducing the problem of confounded imitation learning. | We begin by introducing the problem of confounded imitation learning. | {
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37 | 1 | Paragraph | Y6LzLWXlS8E_03_00_01 | 3 | 1 | Paragraph | WNev_iSes_03_00_01 | 3 | Following Ortega et al. | Following Ortega et al. | {
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38 | 1 | Paragraph | Y6LzLWXlS8E_03_00_02 | 3 | 1 | Paragraph | WNev_iSes_03_00_02 | 3 | [2021], we discusshow behavioral cloning fails in the presence of latent confounders. | [2021], we discuss how behavioral cloning fails in the presence of latent confounders. | {
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39 | 1 | Paragraph | Y6LzLWXlS8E_03_00_03 | 3 | 1 | Paragraph | WNev_iSes_03_00_03 | 3 | We then define the interventional policy, whichsolves the problem of confounded imitation learning. | We then define the interventional policy, which solves the problem of confounded imitation learning. | {
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40 | 1 | Section | Y6LzLWXlS8E_04_00_00 | 4 | 1 | Section | WNev_iSes_04_00_00 | 4 | 2.1 Imitation learning | 2.1 Imitation learning | {
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41 | 1 | Paragraph | Y6LzLWXlS8E_04_00_00 | 4 | 1 | Paragraph | WNev_iSes_04_00_00 | 4 | Imitation learning learns a policy from a dataset of expert demonstrations via supervised learning. | Imitation learning learns a policy from a dataset of expert demonstrations via supervised learning. | {
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42 | 1 | Paragraph | Y6LzLWXlS8E_04_01_00 | 4 | 1 | Paragraph | WNev_iSes_04_00_01 | 4 | The expert is a policy that acts in a (reward-free) | The expert is a policy that acts in a (reward-free) | {
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43 | 1 | Paragraph | Y6LzLWXlS8E_04_01_01 | 4 | 1 | Paragraph | WNev_iSes_04_00_02 | 4 | Markov decision process (MDP) defined by a tuple M = ( S , A , P ( s ′ | s, a ) , P ( s 0 )) , where S is the set of states, A is the set of actions, P ( s ′ | s, a ) isthe transition probability, and P ( s 0 ) is a distribution over initial states. | Markov decision process (MDP) defined by a tuple M = ( S , A , P ( s ′ | s, a ) , P ( s 0 )) , where S is the set of states, A is the set of actions, P ( s ′ | s, a ) is the transition probability, and P ( s 0 ) is a distribution over initial states. | {
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44 | 1 | Paragraph | Y6LzLWXlS8E_04_02_01 | 4 | 1 | Paragraph | WNev_iSes_04_00_03 | 4 | The expert’s interaction withthe environment produces a trajectory τ = | The expert’s interaction with the environment produces a trajectory τ = | {
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45 | 1 | Paragraph | Y6LzLWXlS8E_04_02_02 | 4 | 1 | Paragraph | WNev_iSes_04_00_04 | 4 | ( s 0 , a 0 , . . | ( s 0 , a 0 , . . | {
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47 | 1 | Paragraph | Y6LzLWXlS8E_04_02_04 | 4 | 1 | Paragraph | WNev_iSes_04_00_06 | 4 | The expert may maximize theexpectation over some reward function, but this is not necessary (and some tasks cannot be expressedthrough Markov rewards Abel et al. | The expert may maximize the expectation over some reward function, but this is not necessary (and some tasks cannot be expressed through Markov rewards Abel et al. | {
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48 | 1 | Paragraph | Y6LzLWXlS8E_04_02_05 | 4 | 1 | Paragraph | WNev_iSes_04_01_00 | 4 | In the simplest form of imitation learning, a behavioralcloning policy π η ( a | s ) parametrized by η is learned by minimizing the loss − (cid:80) s,a ∈D log π η ( a | s ) ,where D is the dataset of state-action pairs collected by the expert’s policy. | In the simplest form of imitation learning, a behavioral cloning policy π η ( a | s ) parametrized by η is learned by minimizing the loss − (cid:80) s,a ∈D log π η ( a | s ) , where D is the dataset of state-action pairs collected by the expert’s policy. | {
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49 | 1 | Section | Y6LzLWXlS8E_05_00_00 | 5 | 1 | Section | WNev_iSes_05_00_00 | 5 | 2.2 Confounded imitation learning | 2.2 Confounded imitation learning | {
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50 | 1 | Paragraph | Y6LzLWXlS8E_05_00_00 | 5 | 1 | Paragraph | WNev_iSes_05_00_00 | 5 | We now extend the imitation learning setup to allow for some variables θ ∈ Θ that are observed bythe expert, but not the imitator. | We now extend the imitation learning setup to allow for some variables θ ∈ Θ that are observed by the expert, but not the imitator. | {
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51 | 1 | Paragraph | Y6LzLWXlS8E_05_00_01 | 5 | 1 | Paragraph | WNev_iSes_05_00_01 | 5 | We define a family of Markov Decision processes as a latent space Θ ,a distribution P ( θ ) , and for each θ ∈ Θ , a reward-free MDP M θ = ( S , A , P ( s ′ | s, a, θ ) , P ( s 0 | θ )) . | We define a family of Markov Decision processes as a latent space Θ , a distribution P ( θ ) , and for each θ ∈ Θ , a reward-free MDP M θ = ( S , A , P ( s ′ | s, a, θ ) , P ( s 0 | θ )) . | {
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52 | 1 | Paragraph | Y6LzLWXlS8E_05_01_00 | 5 | 1 | Paragraph | WNev_iSes_05_01_00 | 5 | We assume there exists an expert policy π exp ( a | s, θ ) for each MDP. | We assume there exists an expert policy π exp ( a | s, θ ) for each MDP. | {
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53 | 1 | Paragraph | Y6LzLWXlS8E_05_01_01 | 5 | 1 | Paragraph | WNev_iSes_05_01_01 | 5 | When it interacts with theenvironment, it generates the following distribution over trajectories τ : | When it interacts with the environment, it generates the following distribution over trajectories τ : | {
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54 | 1 | Equation | Y6LzLWXlS8E_05_02_00 | 5 | 1 | Equation | WNev_iSes_05_02_00 | 5 | [Equation] | [Equation] | {
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55 | 1 | Paragraph | Y6LzLWXlS8E_05_03_00 | 5 | 1 | Paragraph | WNev_iSes_05_03_00 | 5 | The imitator does not observe the latent θ . | The imitator does not observe the latent θ . | {
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56 | 1 | Paragraph | Y6LzLWXlS8E_05_03_01 | 5 | 1 | Paragraph | WNev_iSes_05_03_01 | 5 | It may thus need to implicitly infer it from the pasttransitions, so we take it to be a non-Markovian policy π η ( a t | s 1 , a 1 , . . . | It may thus need to implicitly infer it from the past transitions, so we take it to be a non-Markovian policy π η ( a t | s 1 , a 1 , . . . | {
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57 | 1 | Paragraph | Y6LzLWXlS8E_05_03_02 | 5 | 1 | Paragraph | WNev_iSes_05_03_02 | 5 | , s t ) , parameterized by η . | , s t ) , parameterized by η . | {
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58 | 1 | Paragraph | Y6LzLWXlS8E_05_04_00 | 5 | 1 | Paragraph | WNev_iSes_05_04_00 | 5 | The imitator generates the following distribution over trajectories: | The imitator generates the following distribution over trajectories: | {
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59 | 1 | Equation | Y6LzLWXlS8E_05_05_00 | 5 | 1 | Equation | WNev_iSes_05_05_00 | 5 | [Equation] | [Equation] | {
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60 | 1 | Paragraph | Y6LzLWXlS8E_05_06_00 | 5 | 1 | Paragraph | WNev_iSes_05_06_00 | 5 | The Bayesian networks associated to these distributions are shown in figure 1. | The Bayesian networks associated to these distributions are shown in Figure 1. | {
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61 | 1 | Paragraph | Y6LzLWXlS8E_05_07_00 | 5 | 1 | Paragraph | WNev_iSes_05_07_00 | 5 | The goal of imitation learning in this setting is to learn imitator parameters η such that when theimitator is executed in the environment, the imitator agrees with the expert’s decisions, meaning wewish to maximise | The goal of imitation learning in this setting is to learn imitator parameters η such that when the imitator is executed in the environment, the imitator agrees with the expert’s decisions, meaning we wish to maximise | {
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62 | 1 | Equation | Y6LzLWXlS8E_05_08_00 | 5 | 2 | Equation | WNev_iSes_05_08_00 | 5 | [Equation] | [Equation] | {
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63 | 1 | Paragraph | Y6LzLWXlS8E_05_09_00 | 5 | 2 | Paragraph | WNev_iSes_05_09_00 | 5 | If the expert solves some task (e. g. maximizes some reward function), this amounts to solving thesame task. | If the expert solves some task (e. g. maximizes some reward function), this amounts to solving the same task. | {
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64 | 2 | Section | Y6LzLWXlS8E_06_00_00 | 6 | 2 | Section | WNev_iSes_06_00_00 | 6 | 2.3 Naive behavioral cloning | 2.3 Naive behavioral cloning | {
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65 | 2 | Paragraph | Y6LzLWXlS8E_06_00_00 | 6 | 2 | Paragraph | WNev_iSes_06_00_00 | 6 | If we have access to a data set of expert demonstrations, one can learn an imitator via behavioralcloning on the expert’s demonstrations. | If we have access to a data set of expert demonstrations, one can learn an imitator via behavioral cloning on the expert’s demonstrations. | {
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66 | 2 | Paragraph | Y6LzLWXlS8E_06_00_01 | 6 | 2 | Paragraph | WNev_iSes_06_00_01 | 6 | At optimality, this learns the conditional policy : | At optimality, this learns the conditional policy : | {
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67 | 2 | Equation | Y6LzLWXlS8E_06_01_00 | 6 | 2 | Equation | WNev_iSes_06_01_00 | 6 | [Equation] | [Equation] | {
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68 | 2 | Paragraph | Y6LzLWXlS8E_06_02_00 | 6 | 2 | Paragraph | WNev_iSes_06_02_00 | 6 | Following Ortega et al. | Following Ortega et al. | {
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69 | 2 | Paragraph | Y6LzLWXlS8E_06_02_01 | 6 | 2 | Paragraph | WNev_iSes_06_02_01 | 6 | [2021], consider the following example of a confounded multi-armed banditwith | [2021], consider the following example of a confounded multi-armed bandit with A = | {
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70 | 2 | Paragraph | Y6LzLWXlS8E_06_02_02 | 6 | 2 | Paragraph | WNev_iSes_06_02_02 | 6 | A = Θ = { 1 , . . | Θ = { 1 , . . | {
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71 | 2 | Paragraph | Y6LzLWXlS8E_06_02_03 | 6 | 2 | Paragraph | WNev_iSes_06_02_03 | 6 | , 5 } and S = { 0 , 1 } : | , 5 } and S = { 0 , 1 } : | {
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72 | 2 | Equation | Y6LzLWXlS8E_06_03_00 | 6 | 2 | Equation | WNev_iSes_06_03_00 | 6 | [Equation] | [Equation] [Equation] | {
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73 | 2 | Paragraph | Y6LzLWXlS8E_06_04_00 | 6 | 2 | Paragraph | WNev_iSes_06_05_00 | 6 | The expert knows which bandit arm is special (and labeled by θ ) and pulls it with high probability,while the imitating agent does not have access to this information. | The expert knows which bandit arm is special (and labeled by θ ) and pulls it with high probability, while the imitating agent does not have access to this information. | {
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74 | 2 | Paragraph | Y6LzLWXlS8E_06_05_00 | 6 | 2 | Paragraph | WNev_iSes_06_06_00 | 6 | If we roll out the naive behavioral cloning policy in this environment, shown in Figure 2, we see thecausal delusion at work. | If we roll out the naive behavioral cloning policy in this environment, shown in Figure 2, we see the causal delusion at work. | {
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75 | 2 | Paragraph | Y6LzLWXlS8E_06_05_01 | 6 | 2 | Paragraph | WNev_iSes_06_06_01 | 6 | At time t , the latent that is inferred by p cond takes past actions as evidencefor the latent variable. | At time t , the latent that is inferred by p cond takes past actions as evidence for the latent variable. | {
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76 | 2 | Paragraph | Y6LzLWXlS8E_06_05_02 | 6 | 2 | Paragraph | WNev_iSes_06_06_02 | 6 | This makes sense on the expert demonstrations, as the expert is cognizantof the latent variable. | This makes sense on the expert demonstrations, as the expert is cognizant of the latent variable. | {
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77 | 2 | Paragraph | Y6LzLWXlS8E_06_05_03 | 6 | 2 | Paragraph | WNev_iSes_06_06_03 | 6 | However, during an imitator roll-out, the past actions are not evidence of thelatent, as the imitator is blind to it. | However, during an imitator roll-out, the past actions are not evidence of the latent, as the imitator is blind to it. | {
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78 | 2 | Paragraph | Y6LzLWXlS8E_06_05_04 | 6 | 2 | Paragraph | WNev_iSes_06_06_04 | 6 | Concretely, the imitator will take its first action uniformly andlater tends to repeat that action, as it mistakenly takes the first action to be evidence for the latent. | Concretely, the imitator will take its first action uniformly and later tends to repeat that action, as it mistakenly takes the first action to be evidence for the latent. | {
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79 | 2 | Section | Y6LzLWXlS8E_07_00_00 | 7 | 2 | Section | WNev_iSes_07_00_00 | 7 | 2.4 Interventional policy | 2.4 Interventional policy | {
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80 | 2 | Paragraph | Y6LzLWXlS8E_07_00_00 | 7 | 2 | Paragraph | WNev_iSes_07_00_00 | 7 | A solution to this issue is to only take as evidence the data that was actually informed by the latent,which are the transitions. | A solution to this issue is to only take as evidence the data that was actually informed by the latent, which are the transitions. | {
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81 | 2 | Paragraph | Y6LzLWXlS8E_07_00_01 | 7 | 2 | Paragraph | WNev_iSes_07_00_01 | 7 | This defines the following imitator policy: | This defines the following imitator policy: | {
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82 | 2 | Equation | Y6LzLWXlS8E_07_01_00 | 7 | 2 | Equation | WNev_iSes_07_01_00 | 7 | [Equation] | [Equation] | {
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83 | 2 | Paragraph | Y6LzLWXlS8E_07_02_00 | 7 | 2 | Paragraph | WNev_iSes_07_02_00 | 7 | In a causal framework, that corresponds to treating the choice of past actions as interventions. | In a causal framework, that corresponds to treating the choice of past actions as interventions. | {
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84 | 2 | Paragraph | Y6LzLWXlS8E_07_02_01 | 7 | 2 | Paragraph | WNev_iSes_07_02_01 | 7 | Inthe notation of the do-calculus [Pearl, 2009], this equals p ( a t | s 1 , do( a 1 ) , s 2 , do( a 2 ) , . . . | In the notation of the do-calculus [Pearl, 2009], this equals p ( a t | s 1 , do( a 1 ) , s 2 , do( a 2 ) , . . . | {
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85 | 2 | Paragraph | Y6LzLWXlS8E_07_02_02 | 7 | 2 | Paragraph | WNev_iSes_07_02_02 | 7 | , s t ) . | , s t ) . | {
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86 | 2 | Paragraph | Y6LzLWXlS8E_07_02_03 | 7 | 2 | Paragraph | WNev_iSes_07_02_03 | 7 | Thepolicy in equation (5) is therefore known as interventional policy [Ortega et al., 2021]. | The policy in Equation (5) is therefore known as interventional policy [Ortega et al., 2021]. | {
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87 | 2 | Section | Y6LzLWXlS8E_08_00_00 | 8 | null | null | null | null | Deconfounding imitation learning | {
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88 | 2 | Figure | Y6LzLWXlS8E_08_00_00 | 8 | 2 | Figure | WNev_iSes_07_03_00 | 7 | [Figure] | [Figure] | {
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90 | 2 | Paragraph | Y6LzLWXlS8E_08_01_00 | 8 | 3 | Paragraph | WNev_iSes_08_00_00 | 8 | We now present our theoretical results on how imitation learning can be deconfounded. | We now present our theoretical results on how imitation learning can be deconfounded. | {
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91 | 2 | Paragraph | Y6LzLWXlS8E_08_01_01 | 8 | 3 | Paragraph | WNev_iSes_08_00_01 | 8 | We first showthat the interventional policy is optimal in some sense, before analyzing in which settings it can belearned. | We first show that the interventional policy is optimal in some sense, before analyzing in which settings it can be learned. | {
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92 | 2 | Section | Y6LzLWXlS8E_09_00_00 | 9 | 3 | Section | WNev_iSes_09_00_00 | 9 | 3.1 Optimality of the interventional policy | 3.1 Optimality of the interventional policy | {
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93 | 2 | Figure | Y6LzLWXlS8E_09_00_00 | 9 | 3 | Paragraph | WNev_iSes_09_00_00 | 9 | [Figure] Under some reasonable assumptions, the interventional policy approaches the expert’s policy, as weprove in the appendix 2. [Figure] | Under some reasonable assumptions, the interventional policy approaches the expert’s policy, as we prove in the Appendix B. | {
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94 | 3 | Paragraph | Y6LzLWXlS8E_09_03_00 | 9 | 3 | Paragraph | WNev_iSes_09_01_00 | 9 | Theorem 3.1 (Informal) . | Theorem 3.1 (Informal) . | {
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95 | 3 | Paragraph | Y6LzLWXlS8E_09_03_01 | 9 | 3 | Paragraph | WNev_iSes_09_01_01 | 9 | If the interventional inference p int ( θ | τ <t ) approaches the true latent of theenvironment as t → ∞ on the rollouts of π int , and if the expert maximises some reward that is fixedacross all environments, then as t → ∞ , the imitator policy π int ( a t | s ) approaches the expert policy. | If the interventional inference p int ( θ | τ <t ) approaches the true latent of the environment as t → ∞ on the rollouts of π int , and if the expert maximises some reward that is fixed across all environments, then as t → ∞ , the imitator policy π int ( a t | s ) approaches the expert policy. | {
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96 | 3 | Paragraph | Y6LzLWXlS8E_09_04_00 | 9 | 3 | Paragraph | WNev_iSes_09_02_00 | 9 | Proof. | Proof. | {
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97 | 3 | Paragraph | Y6LzLWXlS8E_09_04_01 | 9 | 3 | Paragraph | WNev_iSes_09_02_01 | 9 | See lemma 2.1 in the appendix. | See Lemma 2.1 in the appendix. | {
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98 | 3 | Paragraph | Y6LzLWXlS8E_09_05_00 | 9 | 3 | Paragraph | WNev_iSes_09_03_00 | 9 | The requirement here means that the transition dynamics must be informative about the latent —we consider latent confounders that manifest in the dynamics, not those that affect only the agentbehavior. | The requirement here means that the transition dynamics must be informative about the latent — we consider latent confounders that manifest in the dynamics, not those that affect only the agent behavior. | {
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99 | 3 | Paragraph | Y6LzLWXlS8E_09_05_01 | 9 | 3 | Paragraph | WNev_iSes_09_03_01 | 9 | In this case, the interventional policy thus presents a solution to the confounding problem. | In this case, the interventional policy thus presents a solution to the confounding problem. | {
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100 | 3 | Paragraph | Y6LzLWXlS8E_09_06_00 | 9 | 3 | Paragraph | WNev_iSes_09_04_00 | 9 | In the rest of this paper we focus on the question if and how it can be learned from data. | In the rest of this paper we focus on the question if and how it can be learned from data. | {
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