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Title: Disagreement-Regularized Imitation Learning

Paper content:
We present a simple and effective algorithm designed to address the covariate shift problem in imitation learning.
It operates by training an ensemble of policies on the expert demonstration data, and using the variance of their predictions as a cost which is minimized with RL together with a supervised behavioral cloning cost.
Unlike adversarial imitation methods, it uses a fixed reward function which is easy to optimize.
We prove a regret bound for the algorithm in the tabular setting which is linear in the time horizon multiplied by a coefficient which we show to be low for certain problems in which behavioral cloning fails.
We evaluate our algorithm empirically across multiple pixel-based Atari environments and continuous control tasks, and show that it matches or significantly outperforms behavioral cloning and generative adversarial imitation learning.
Training artificial agents to perform complex tasks is essential for many applications in robotics, video games and dialogue.
If success on the task can be accurately described using a reward or cost function, reinforcement learning (RL) methods offer an approach to learning policies which has been shown to be successful in a wide variety of applications (Mnih et al., 2015; Hessel et al., 2018) However, in other cases the desired behavior may only be roughly specified and it is unclear how to design a reward function to characterize it.
For example, training a video game agent to adopt more human-like behavior using RL would require designing a reward function which characterizes behaviors as more or less human-like, which is difficult.
Imitation learning (IL) offers an elegant approach whereby agents are trained to mimic the demonstrations of an expert rather than optimizing a reward function.
Its simplest form consists of training a policy to predict the expert's actions from states in the demonstration data using supervised learning.
While appealingly simple, this approach suffers from the fact that the distribution over states observed at execution time can differ from the distribution observed during training.
Minor errors which initially produce small deviations from the expert trajectories become magnified as the policy encounters states further and further from its training distribution.
This phenomenon, initially noted in the early work of (Pomerleau, 1989) , was formalized in the work of (Ross & Bagnell, 2010) who proved a quadratic O( T 2 ) bound on the regret and showed that this bound is tight.
The subsequent work of (Ross et al., 2011) showed that if the policy is allowed to further interact with the environment and make queries to the expert policy, it is possible to obtain a linear bound on the regret.
However, the ability to query an expert can often be a strong assumption.
In this work, we propose a new and simple algorithm called DRIL (Disagreement-Regularized Imitation Learning) to address the covariate shift problem in imitation learning, in the setting where the agent is allowed to interact with its environment.
Importantly, the algorithm does not require any additional interaction with the expert.
It operates by training an ensemble of policies on the demonstration data, and using the disagreement in their predictions as a cost which is optimized through RL together with a supervised behavioral cloning cost.
The motivation is that the policies in the ensemble will tend to agree on the set of states covered by the expert, leading to low cost, but are more likely to disagree on states not covered by the expert, leading to high cost.
The RL cost thus pushes the agent back towards the distribution of the expert, while the supervised cost ensures that it mimics the expert within the expert's distribution.
Our theoretical results show that, subject to realizability and optimization oracle assumptions, our algorithm obtains a O( κ T ) regret bound for tabular MDPs, where κ is a measure which quantifies a tradeoff between the concentration of the demonstration data and the diversity of the ensemble outside the demonstration data.
We evaluate DRIL empirically across multiple pixel-based Atari environments and continuous control tasks, and show that it matches or significantly outperforms behavioral cloning and generative adversarial imitation learning, often recovering expert performance with only a few trajectories.
Addressing covariate shift has been a long-standing challenge in imitation learning.
In this work, we have proposed a new method to address this problem by penalizing the disagreement between an ensemble of different policies sampled from the posterior.
Importantly, our method requires no additional labeling by an expert.
Our experimental results demonstrate that DRIL can often match expert performance while using only a small number of trajectories across a wide array of tasks, ranging from tabular MDPs to pixel-based Atari games and continuous control tasks.
On the theoretical side, we have shown that our algorithm can provably obtain a low regret bound for tabular problems in which the κ parameter is low.
There are multiple directions for future work.
On the theoretical side, extending our analysis to continuous state spaces and characterizing the κ parameter on a larger array of problems would help to better understand the settings where our method can expect to do well.
Empirically, there are many other settings in structured prediction (Daumé et al., 2009 ) where covariate shift is an issue and where our method could be applied.
For example, in dialogue and language modeling it is common for generated text to become progressively less coherent as errors push the model off the manifold it was trained on.
Our method could potentially be used to fine-tune language or translation models (Cho et al., 2014; Welleck et al., 2019) after training by applying our uncertainty-based cost function to the generated text.
A PROOFS
Proof.
We will first show that for any π ∈ Π and U ⊆ S, we have
. We can rewrite this as:
We begin by bounding the first term:
We next bound the second term:
Now observe we can decompose the RL cost as follows:
Putting these together, we get the following:
Here we have used the fact that β(U) ≤ 1 since 0 ≤ π(a|s) ≤ 1 and α(U) ≥ s∈U
Taking the minimum over subsets U ⊆ S, we get J exp (π) ≤ κJ alg (π).
Proof.
Plugging the optimal policy into J alg , we get:
We will first bound Term 1:
We will next bound Term 2:
The last step follows from our optimization oracle assumption:
Combining the bounds on the two terms, we get J alg (π ) ≤ 2 .
Since π ∈ Π, the result follows.
Theorem 1.
Letπ be the result of minimizing J alg using our optimization oracle, and assume that
Proof.
By our optimization oracle and Lemma 2, we have
Combining with Lemma 1, we get:
Applying Theorem 1 from (Ross et al., 2011) , we get J(π) ≤ J(π ) + 3uκ T .

assistant: text: Method for addressing covariate shift in imitation learning using ensemble uncertainty