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Bayesian multitask inverse reinforcement learning
Christos Dimitrakakis, Constantin Rothkopf
TL;DR
The paper addresses inference of motivations and goals from demonstrations across tasks or experts. It formulates multitask inverse reinforcement learning with structured Bayesian priors, including a prior on policy optimality; the authors report learning from multiple experts and differentiating their goals, while noting limits to finite reward spaces for stated optimality guarantees.
Problem
The paper addresses how to infer agents’ motivations and goals from demonstrations across tasks or from experts with different preferences solving the same task.
Source: §1 · p. 1Method
The framework models multitask reward-policy relationships with structured Bayesian priors and introduces a prior on the optimality of demonstrated policies.
Source: §2.1 · p. 3, §4 · p. 6Results
The authors report that the framework supports learning from multiple experts and differentiating their goals; MRP-MH shows additional improvement over RP-MH, increasing with the number of demonstrations.
Source: §5 · pp. 9–10Takeaways & Limitations
The approach provides a statistical framework for interpreting behavioural experiments and learning from multiple teachers with different preferences.
Source: §1 · p. 1, §6 · p. 11The stated optimality-sampling theorem does not directly apply to infinite reward spaces; obtaining guarantees there is beyond the paper’s scope.
Source: §4 · p. 7
Abstract
from arXiv · showhide
We generalise the problem of inverse reinforcement learning to multiple tasks, from multiple demonstrations. Each one may represent one expert trying to solve a different task, or as different experts trying to solve the same task. Our main contribution is to formalise the problem as statistical preference elicitation, via a number of structured priors, whose form captures our biases about the relatedness of different tasks or expert policies. In doing so, we introduce a prior on policy optimality, which is more natural to specify. We show that our framework allows us not only to learn to efficiently from multiple experts but to also effectively differentiate between the goals of each. Possible applications include analysing the intrinsic motivations of subjects in behavioural experiments and learning from multiple teachers.
1 Introduction
The paper frames multitask inverse reinforcement learning as inferring agents’ motivations and goals from demonstrations across tasks or experts. It presents a principled statistical formulation and develops hierarchical and policy-optimality models.
1 Introduction
- Multitask inverse reinforcement learning infers an agent’s motivations and goals from demonstrations across tasks or experts with differing preferences.The problem also applies when one expert performs different tasks.Source: §1 · p. 1
- The authors identify behavioural experiments with humans and animals as a potential application of the framework.They describe the approach as providing principled statistical tools for interpreting such experiments.Source: §1 · p. 1
- The paper presents a principled statistical formulation, extending prior single-task work to a hierarchical population model and introducing a prior on demonstration optimality.The authors describe this as, to their knowledge, the only principled statistical formulation of multitask inverse reinforcement learning.Source: §1 · p. 1
2 The general model
The general model infers task rewards and policies from demonstrations under a shared distribution of environment dynamics. It uses hierarchical reward-policy priors, with two model variants that place the dependency between rewards and policies differently.
2 The general model
- All tasks assume environment dynamics drawn from the same distribution, which may be singular, and the environment is modeled as a controlled Markov process.If dynamics are unknown, the model can maintain a probability measure over controlled Markov processes.Source: §2 · p. 2
- The model infers each task’s reward function and policy from all demonstrations and prior beliefs, linking rewards to discounted future utility.A demonstration records sequences of states and actions; rewards are defined over state-action pairs.Source: §2 · pp. 2–3
2.1 Multitask priors on reward functions and policies
- A hierarchical model places a hyperprior on joint reward-policy priors and conditions the posterior on the demonstrations.When the joint prior is unspecified, the hierarchical Bayes approach estimates its posterior distribution given the data.Source: §2.1 · p. 3
- The two variants differ in how they model reward-policy dependence: one starts with product priors on rewards and policy parameters, while the other uses priors on policies and policy optimality.In the latter model, reward functions are distributed conditionally on policies.Source: §2.1 · pp. 3–4
3 Multitask Reward-Policy prior (MRP)
The MRP model hierarchically draws task rewards and policies, with policies conditioned on rewards and a shared environment. Its policy and reward priors are structured separately, and estimation uses Monte Carlo or alternative samplers.
3 Multitask Reward-Policy prior (MRP)
- MRP draws reward and policy distributions from a product hyperprior, then draws each task’s reward and policy conditionally, given the environment and reward.The policy distribution is conditional on the task reward.Source: §3 · p. 4
3.1 The policy prior
- The policy prior can use stationary softmax policies parameterized by inverse temperature, with a prior on that parameter given the reward.Inference can use standard Monte Carlo methods; if rewards are estimated well enough, the resulting policies may outperform demonstrators.Source: §3.1 · p. 4
3.2 Reward priors
- For finite state spaces, the reward prior can be Dirichlet, treating each sampled reward as multinomial parameters and separating rewards from the policy’s inverse-temperature parameter.This is more constrained than the earlier product-Beta prior on states or state-action pairs [20].Source: §3.2 · p. 5
3.3 Estimation
- MRP-MC samples directly from the product prior on reward and inverse temperature; Metropolis-Hastings or hybrid Gibbs alternatives are also described.Metropolis-Hastings may be more efficient in practice if a good proposal distribution is available.Source: §3.3 · p. 5
4 Multitask Policy Optimality prior (MPO)
The MPO model places a prior on policy optimality rather than specifying a parametric policy form, then combines posterior policy samples with that prior to infer reward functions. Its finite-reward procedure estimates the posterior using a loss matrix and supports a finite-sample value-estimation bound, while the theorem does not directly cover infinite reward spaces.
4 Multitask Policy Optimality prior (MPO)
- The MPO model places a prior on policy optimality, then combines it with a posterior over policies to obtain a posterior distribution over reward functions.The prior assigns probability to ε-optimality; the policy posterior is calculated from the demonstrations.Source: §4 · p. 6
- The policy-optimality prior is presented as more natural to specify than a parametric policy form such as softmax.The method can estimate the integral using policy samples and the prior over ε-optimality.Source: §4 · pp. 6–7
- For finite reward sets, the procedure samples policies from their demonstration-conditioned posterior and evaluates each policy–reward pair to form a loss matrix.The matrix supports estimating the reward posterior; its finite number of distinct loss values makes the relevant function piecewise constant in ε.Source: §4 · pp. 6–7
- Theorem 1 bounds samples needed for small value-estimation loss, with minor modifications for single- and multitask cases when the reward set is finite.The stated theorem does not directly apply to infinite reward sets; deriving guarantees there is beyond the paper’s scope.Source: §4 · p. 7
5 Experiments
The experiments compare multitask models and samplers with imitators and other baselines across chain and random MDP tasks. They report an additional multitask improvement on random tasks, increasing gains over imitation as task count rises, and higher MPO sample efficiency when demonstrations are longer.
5 Experiments
- The sampler study compares MTPP sampling strategies and MRP versus MPO Monte Carlo estimates using expected loss as total samples increase.The experiment averages results over 103 runs.Source: Figure 2 · p. 8, §5 · p. 8
- With more demonstration data, MPO is much more sample-efficient because it samples directly from ξ(π | D), whereas MRP-MC is very inefficient in that setting.The single-task comparison also reports MPO-MC consistently better than the imitator, while MRP-MC converges slowly.Source: §5 · pp. 8–9, Figure 3 · p. 9
- As task count increases, MRP-MC’s performance gain over an imitator increases with 10 total demonstrations.The chain-task comparison covers 1, 2, 5, and 10 tasks.Source: §5 · p. 8, Figure 3 · p. 9
- For the five-state chain task, the data lets the imitator perform rather well, while MPO-MC remains consistently better and MRP-MC converges slowly.The experiment uses rewards r1 = 0.2, r2 = 0, r3 = 1.Source: §5 · p. 8, Figure 3 · p. 9
- MRP-MH shows clear additional improvement over flat MH on random MDP tasks, and its gain increases with the number of demonstrations.RP-MH degrades slowly under its assumption that all demonstrations share a common reward function.Source: §5 · p. 10, Figure 4 · p. 10
6 Related work and discussion
The discussion positions the approach alongside prior work in inverse reinforcement learning, multitask learning, and preference estimation. It frames the contribution as a Bayesian multitask extension with a policy-optimality prior, while noting that unknown MDPs make optimal-policy calculation harder.
6 Related work and discussion
- The approach generalises utility estimation to dynamic inverse reinforcement learning, multiple-intention learning to full Bayesian estimation, and single-task IRL [20] to multiple tasks.Related work also includes Bayesian multitask reinforcement learning with priors on MDPs or value functions, and multi-agent IRL addresses joint policies.Source: §6 · pp. 10–11
- The paper presents its formulation as a principled statistical approach to learning from multiple teachers with different preferences and inferring a distribution over task reward functions.It also introduces a more natural policy-optimality prior with computational complexity bounds.Source: §6 · p. 11
- When the MDP is unknown, calculating an optimal policy is generally much harder; the authors identify near-optimal memoryless policies as applicable to this setting.Non-parametric priors for the policy-optimality model are left for future work.Source: §6 · p. 11
A Auxillary results and proofs
The appendix derives a uniform empirical approximation bound from Hoeffding’s inequality, then applies it to bound expected loss using the value function’s total-variation bound.
A Auxillary results and proofs
- The auxiliary lemma concerns independent random variables bounded within intervals, with their expectations and a positive threshold specified.The supplied passage gives the lemma’s conditions but not its inequality expression.Source: A Auxillary results and proofs · p. 11
- The corollary bounds the uniform deviation between an expectation f and its sample-based estimate f_n with probability at least 1 − δ.It defines f as the expectation of g(x, y) under P and f_n from a sample drawn from P.Source: A Auxillary results and proofs · p. 11
- The proof fixes x, applies Hoeffding’s inequality to h_x(y) = g(x, y), and uses its expectation f(x) to establish the corollary’s bound.The passage identifies the empirical mean of h_x over sampled y values before applying the inequality.Source: A Auxillary results and proofs · p. 11
- Theorem 1’s proof applies the corollary with c = 2(1 − γ)^2, using the value function’s total variation bound of 1/(1 − γ) to bound expected loss.The resulting bound is not specified in the supplied passage.Source: A Auxillary results and proofs · p. 12