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Title: SQIL: Imitation Learning via Reinforcement Learning with Sparse Rewards

Paper content:
Learning to imitate expert behavior from demonstrations can be challenging, especially in environments with high-dimensional, continuous observations and unknown dynamics.
Supervised learning methods based on behavioral cloning (BC) suffer from distribution shift: because the agent greedily imitates demonstrated actions, it can drift away from demonstrated states due to error accumulation.
Recent methods based on reinforcement learning (RL), such as inverse RL and generative adversarial imitation learning (GAIL), overcome this issue by training an RL agent to match the demonstrations over a long horizon.
Since the true reward function for the task is unknown, these methods learn a reward function from the demonstrations, often using complex and brittle approximation techniques that involve adversarial training.
We propose a simple alternative that still uses RL, but does not require learning a reward function.
The key idea is to provide the agent with an incentive to match the demonstrations over a long horizon, by encouraging it to return to demonstrated states upon encountering new, out-of-distribution states.
We accomplish this by giving the agent a constant reward of r=+1 for matching the demonstrated action in a demonstrated state, and a constant reward of r=0 for all other behavior.
Our method, which we call soft Q imitation learning (SQIL), can be implemented with a handful of minor modifications to any standard Q-learning or off-policy actor-critic algorithm.
Theoretically, we show that SQIL can be interpreted as a regularized variant of BC that uses a sparsity prior to encourage long-horizon imitation.
Empirically, we show that SQIL outperforms BC and achieves competitive results compared to GAIL, on a variety of image-based and low-dimensional tasks in Box2D, Atari, and MuJoCo.
This paper is a proof of concept that illustrates how a simple imitation method based on RL with constant rewards can be as effective as more complex methods that use learned rewards.
Many sequential decision-making problems can be tackled by imitation learning: an expert demonstrates near-optimal behavior to an agent, and the agent attempts to replicate that behavior in novel situations (Argall et al., 2009 ).
This paper considers the problem of training an agent to imitate an expert, given expert action demonstrations and the ability to interact with the environment.
The agent does not observe a reward signal or query the expert, and does not know the state transition dynamics.
Standard approaches based on behavioral cloning (BC) use supervised learning to greedily imitate demonstrated actions, without reasoning about the consequences of actions (Pomerleau, 1991) .
As a result, compounding errors cause the agent to drift away from the demonstrated states (Ross et al., 2011) .
The problem with BC is that, when the agent drifts and encounters out-of-distribution states, the agent does not know how to return to the demonstrated states.
Recent methods based on inverse reinforcement learning (IRL) overcome this issue by training an RL agent not only to imitate demonstrated actions, but also to visit demonstrated states (Ng et al., 2000; Wulfmeier et al., 2015; Finn et al., 2016b; Fu et al., 2017) .
This is also the core idea behind generative adversarial imitation learning (GAIL) (Ho & Ermon, 2016) , which implements IRL using generative adversarial networks (Goodfellow et al., 2014; Finn et al., 2016a) .
Since the true reward function for the task is unknown, these methods construct a reward signal from the demonstrations through adversarial training, making them difficult to implement and use in practice (Kurach et al., 2018) .
The main idea in this paper is that the effectiveness of adversarial imitation methods can be achieved by a much simpler approach that does not require adversarial training, or indeed learning a reward function at all.
Intuitively, adversarial methods encourage long-horizon imitation by providing the agent with (1) an incentive to imitate the demonstrated actions in demonstrated states, and (2) an incentive to take actions that lead it back to demonstrated states when it encounters new, out-ofdistribution states.
One of the reasons why adversarial methods outperform greedy methods, such as BC, is that greedy methods only do (1), while adversarial methods do both (1) and (2).
Our approach is intended to do both (1) and (2) without adversarial training, by using constant rewards instead of learned rewards.
The key idea is that, instead of using a learned reward function to provide a reward signal to the agent, we can simply give the agent a constant reward of r = +1 for matching the demonstrated action in a demonstrated state, and a constant reward of r = 0 for all other behavior.
We motivate this approach theoretically, by showing that it implements a regularized variant of BC that learns long-horizon imitation by
(a) imposing a sparsity prior on the reward function implied by the imitation policy, and
(b) incorporating information about the state transition dynamics into the imitation policy.
Intuitively, our method accomplishes
(a) by training the agent using an extremely sparse reward function -+1 for demonstrations, 0 everywhere else -and accomplishes
(b) by training the agent with RL instead of supervised learning.
We instantiate our approach with soft Q-learning (Haarnoja et al., 2017) by initializing the agent's experience replay buffer with expert demonstrations, setting the rewards to a constant r = +1 in the demonstration experiences, and setting rewards to a constant r = 0 in all of the new experiences the agent collects while interacting with the environment.
Since soft Q-learning is an off-policy algorithm, the agent does not necessarily have to visit the demonstrated states in order to experience positive rewards.
Instead, the agent replays the demonstrations that were initially added to its buffer.
Thus, our method can be applied in environments with stochastic dynamics and continuous states, where the demonstrated states are not necessarily reachable by the agent.
We call this method soft Q imitation learning (SQIL).
The main contribution of this paper is SQIL: a simple and general imitation learning algorithm that is effective in MDPs with high-dimensional, continuous observations and unknown dynamics.
We run experiments in four image-based environments -Car Racing, Pong, Breakout, and Space Invadersand three low-dimensional environments -Humanoid, HalfCheetah, and Lunar Lander -to compare SQIL to two prior methods: BC and GAIL.
We find that SQIL outperforms BC and achieves competitive results compared to GAIL.
Our experiments illustrate two key benefits of SQIL: (1) that it can overcome the state distribution shift problem of BC without adversarial training or learning a reward function, which makes it easier to use, e.g., with images, and (2) that it is simple to implement using existing Q-learning or off-policy actor-critic algorithms.
Related work.
Concurrently with SQIL, two other imitation learning algorithms that use constant rewards instead of a learned reward function were developed (Sasaki et al., 2019; Wang et al., 2019) .
We see our paper as contributing additional evidence to support this core idea, rather than proposing a competing method.
First, SQIL is derived from sparsity-regularized BC, while the prior methods are derived from an alternative formulation of the IRL objective (Sasaki et al., 2019) and from support estimation methods (Wang et al., 2019) , showing that different theoretical approaches independently lead to using RL with constant rewards as an alternative to adversarial training -a sign that this idea may be a promising direction for future work.
Second, SQIL is shown to outperform BC and GAIL in domains that were not evaluated in Sasaki et al. (2019) or Wang et al. (2019) -in particular, tasks with image observations and significant shift in the state distribution between the demonstrations and the training environment.
Summary.
We contribute the SQIL algorithm: a general method for learning to imitate an expert given action demonstrations and access to the environment.
Simulation experiments on tasks with high-dimensional, continuous observations and unknown dynamics show that our method outperforms BC and achieves competitive results compared to GAIL, while being simple to implement on top of existing off-policy RL algorithms.
Limitations and future work.
We have not yet proven that SQIL matches the expert's state occupancy measure in the limit of infinite demonstrations.
One direction for future work would be to rigorously show whether or not SQIL has this property.
Another direction would be to extend SQIL to recover not just the expert's policy, but also their reward function; e.g., by using a parameterized reward function to model rewards in the soft Bellman error terms, instead of using constant rewards.
This could provide a simpler alternative to existing adversarial IRL algorithms.
(s, a) ) .
Splitting up the squared soft Bellman error terms for D demo and D samp in Equation 8,
Setting γ 1 turns the inner sum in the first term into a telescoping sum:
Since s T is assumed to be absorbing, V (s T ) is zero.
Thus,
In our experiments, we have that all the demonstration rollouts start at the same initial state s 0 .

assistant: text: A simple and effective alternative to adversarial imitation learning: initialize experience replay buffer with demonstrations, set their reward to +1, set reward for all other data to 0, run Q-learning or soft actor-critic to train.