Generative Adversarial Imitation Learning

Jonathan Ho
OpenAI
hoj@openai.com

Stefano Ermon
Stanford University
ermon@cs.stanford.edu

Abstract
Consider learning a policy from example expert behavior, without interaction with
the expert or access to a reinforcement signal. One approach is to recover the
expert’s cost function with inverse reinforcement learning, then extract a policy
from that cost function with reinforcement learning. This approach is indirect
and can be slow. We propose a new general framework for directly extracting a
policy from data as if it were obtained by reinforcement learning following inverse
reinforcement learning. We show that a certain instantiation of our framework
draws an analogy between imitation learning and generative adversarial networks,
from which we derive a model-free imitation learning algorithm that obtains significant performance gains over existing model-free methods in imitating complex
behaviors in large, high-dimensional environments.

1

Introduction

We are interested in a specific setting of imitation learning—the problem of learning to perform a
task from expert demonstrations—in which the learner is given only samples of trajectories from
the expert, is not allowed to query the expert for more data while training, and is not provided a
reinforcement signal of any kind. There are two main approaches suitable for this setting: behavioral
cloning [18], which learns a policy as a supervised learning problem over state-action pairs from
expert trajectories; and inverse reinforcement learning [23, 16], which finds a cost function under
which the expert is uniquely optimal.
Behavioral cloning, while appealingly simple, only tends to succeed with large amounts of data, due
to compounding error caused by covariate shift [21, 22]. Inverse reinforcement learning (IRL), on
the other hand, learns a cost function that prioritizes entire trajectories over others, so compounding
error, a problem for methods that fit single-timestep decisions, is not an issue. Accordingly, IRL has
succeeded in a wide range of problems, from predicting behaviors of taxi drivers [29] to planning
footsteps for quadruped robots [20].
Unfortunately, many IRL algorithms are extremely expensive to run, requiring reinforcement learning
in an inner loop. Scaling IRL methods to large environments has thus been the focus of much recent
work [6, 13]. Fundamentally, however, IRL learns a cost function, which explains expert behavior
but does not directly tell the learner how to act. Given that the learner’s true goal often is to take
actions imitating the expert—indeed, many IRL algorithms are evaluated on the quality of the optimal
actions of the costs they learn—why, then, must we learn a cost function, if doing so possibly incurs
significant computational expense yet fails to directly yield actions?
We desire an algorithm that tells us explicitly how to act by directly learning a policy. To develop such
an algorithm, we begin in Section 3, where we characterize the policy given by running reinforcement
learning on a cost function learned by maximum causal entropy IRL [29, 30]. Our characterization
introduces a framework for directly learning policies from data, bypassing any intermediate IRL step.
Then, we instantiate our framework in Sections 4 and 5 with a new model-free imitation learning
algorithm. We show that our resulting algorithm is intimately connected to generative adversarial
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.

networks [8], a technique from the deep learning community that has led to recent successes in
modeling distributions of natural images: our algorithm harnesses generative adversarial training to fit
distributions of states and actions defining expert behavior. We test our algorithm in Section 6, where
we find that it outperforms competing methods by a wide margin in training policies for complex,
high-dimensional physics-based control tasks over various amounts of expert data.

2

Background

Preliminaries R will denote the extended real numbers R ∪ {∞}. Section 3 will work with
finite state and action spaces S and A, but our algorithms and experiments later in the paper will
run in high-dimensional continuous environments. Π is the set of all stationary stochastic policies
that take actions in A given states in S; successor states are drawn from the dynamics model
P (s0 |s, a). We work in the γ-discounted infinite horizon setting, and we will use an expectation
with respect a policy
P∞ π ∈ Π to denote an expectation with respect to the trajectory it generates:
Eπ [c(s, a)] , E [ t=0 γ t c(st , at )], where s0 ∼ p0 , at ∼ π(·|st ), and st+1 ∼ P (·|st , at ) for t ≥ 0.
We will use Êτ to denote an empirical expectation with respect to trajectory samples τ , and we will
always use πE to refer to the expert policy.
Inverse reinforcement learning Suppose we are given an expert policy πE that we wish to rationalize with IRL. For the remainder of this paper, we will adopt and assume the existence of solutions
of maximum causal entropy IRL [29, 30], which fits a cost function from a family of functions C with
the optimization problem


maximize min −H(π) + Eπ [c(s, a)] − EπE [c(s, a)]
(1)
c∈C

π∈Π

where H(π) , Eπ [− log π(a|s)] is the γ-discounted causal entropy [3] of the policy π. In practice,
πE will only be provided as a set of trajectories sampled by executing πE in the environment, so the
expected cost of πE in Eq. (1) is estimated using these samples. Maximum causal entropy IRL looks
for a cost function c ∈ C that assigns low cost to the expert policy and high cost to other policies,
thereby allowing the expert policy to be found via a certain reinforcement learning procedure:
RL(c) = arg min −H(π) + Eπ [c(s, a)]

(2)

π∈Π

which maps a cost function to high-entropy policies that minimize the expected cumulative cost.

3

Characterizing the induced optimal policy

To begin our search for an imitation learning algorithm that both bypasses an intermediate IRL
step and is suitable for large environments, we will study policies found by reinforcement learning
on costs learned by IRL on the largest possible set of cost functions C in Eq. (1): all functions
RS×A = {c : S × A → R}. Using expressive cost function classes, like Gaussian processes [14]
and neural networks [6], is crucial to properly explain complex expert behavior without meticulously
hand-crafted features. Here, we investigate the best IRL can do with respect to expressiveness by
examining its capabilities with C = RS×A .
Of course, with such a large C, IRL can easily overfit when provided a finite dataset. Therefore,
we will incorporate a (closed, proper) convex cost function regularizer ψ : RS×A → R into our
study. Note that convexity is a not particularly restrictive requirement: ψ must be convex as a
function defined on all of RS×A , not as a function defined on a small parameter space; indeed, the
cost regularizers of Finn et al. [6], effective for a range of robotic manipulation tasks, satisfy this
requirement. Interestingly, ψ will play a central role in our discussion and will not serve as a nuisance
in our analysis.
Let us define an IRL primitive procedure, which finds a cost function such that the expert performs
better than all other policies, with the cost regularized by ψ:


IRLψ (πE ) = arg max −ψ(c) + min −H(π) + Eπ [c(s, a)] − EπE [c(s, a)]
(3)
c∈RS×A

π∈Π

2

Let c̃ ∈ IRLψ (πE ). We are interested in a policy given by RL(c̃)—this is the policy given by
running reinforcement learning on the output of IRL. To characterize RL(c̃), let
us first define for a
P∞
policy π ∈ Π its occupancy measure ρπ : S × A → R as ρπ (s, a) = π(a|s) t=0 γ t P (st = s|π).
The occupancy measure can be interpreted as the unnormalized distribution of state-action pairs
that an agent encounters
P when navigating the environment with the policy π, and it allows us to
write Eπ [c(s, a)] = s,a ρπ (s, a)c(s, a) for any cost function c. We will also need the concept of a
convex conjugate: for a function f : RS×A → R, its convex conjugate f ∗ : RS×A → R is given by
f ∗ (x) = supy∈RS×A xT y − f (y).
Now, we are prepared to characterize RL(c̃), the policy learned by RL on the cost recovered by IRL:
Proposition 3.1. RL ◦ IRLψ (πE ) = arg minπ∈Π −H(π) + ψ ∗ (ρπ − ρπE )
(4)
The proof of Proposition 3.1 can be found in Appendix A.1. It relies on the observation that the
optimal cost function and policy form a saddle point of a certain function. IRL finds one coordinate
of this saddle point, and running RL on the output of IRL reveals the other coordinate.
Proposition 3.1 tells us that ψ-regularized inverse reinforcement learning, implicitly, seeks a policy
whose occupancy measure is close to the expert’s, as measured by ψ ∗ . Enticingly, this suggests that
various settings of ψ lead to various imitation learning algorithms that directly solve the optimization
problem given by Proposition 3.1. We explore such algorithms in Sections 4 and 5, where we show
that certain settings of ψ lead to both existing algorithms and a novel one.
The special case when ψ is a constant function is particularly illuminating, so we state and show it
directly using concepts from convex optimization.
Proposition 3.2. Suppose ρπE > 0. If ψ is a constant function, c̃ ∈ IRLψ (πE ), and π̃ ∈ RL(c̃),
then ρπ̃ = ρπE .
In other words, if there were no cost regularization at all, the recovered policy will exactly match the
expert’s occupancy measure. (The condition ρπE > 0, inherited from Ziebart et al. [30], simplifies
our discussion and in fact guarantees the existence of c̃ ∈ IRLψ (πE ). Elsewhere in the paper, as
mentioned in Section 2, we assume the IRL problem has a solution.) To show Proposition 3.2, we need
the basic result that the set of valid occupancy measures D , {ρπ : π ∈ Π} can be written as a feasible
set of affine constraints
distribution of starting states and P (s0 |s, a) is the dynamics
n [19]: if p0 (s) is the
o
P
P
0
0
model, then D = ρ : ρ ≥ 0 and
a ρ(s, a) = p0 (s) + γ
s0 ,a P (s|s , a)ρ(s , a) ∀ s ∈ S .
Furthermore, there is a one-to-one correspondence between Π and D:
Lemma 3.1 (Theorem
P 2 of Syed et al. [27]). If ρ ∈ D, then ρ is the occupancy measure for
πρ (a|s) , ρ(s, a)/ a0 ρ(s, a0 ), and πρ is the only policy whose occupancy measure is ρ.
We are therefore justified in writing πρ to denote the unique policy for an occupancy measure ρ. We
also need a lemma that lets us speak about causal entropies of occupancy measures:
P
P
Lemma 3.2. Let H̄(ρ) = − s,a ρ(s, a) log(ρ(s, a)/ a0 ρ(s, a0 )). Then, H̄ is strictly concave,
and for all π ∈ Π and ρ ∈ D, we have H(π) = H̄(ρπ ) and H̄(ρ) = H(πρ ).
The proof of this lemma is in Appendix A.1. Lemma 3.1 and Lemma 3.2 together allow us to
freely switch between policies and occupancy measures when considering functions involving causal
entropy and expected costs, as in the following lemma:
P
Lemma 3.3. If L(π, c) = −H(π) + Eπ [c(s, a)] and L̄(ρ, c) = −H̄(ρ) + s,a ρ(s, a)c(s, a), then,
for all cost functions c, L(π, c) = L̄(ρπ , c) for all policies π ∈ Π, and L̄(ρ, c) = L(πρ , c) for all
occupancy measures ρ ∈ D.
Now, we are ready to verify Proposition 3.2.
P
Proof of Proposition 3.2. Define L̄(ρ, c) = −H̄(ρ) + s,a c(s, a)(ρ(s, a) − ρE (s, a)). Given that
ψ is a constant function, we have the following, due to Lemma 3.3:
c̃ ∈ IRLψ (πE ) = arg max min −H(π) + Eπ [c(s, a)] − EπE [c(s, a)] + const.
(5)
c∈RS×A π∈Π

= arg max min −H̄(ρ) +
c∈RS×A ρ∈D

X

ρ(s, a)c(s, a) −

X
s,a

s,a

3

ρE (s, a)c(s, a) = arg max min L̄(ρ, c). (6)
c∈RS×A ρ∈D

This is the dual of the optimization problem
minimize −H̄(ρ) subject to
ρ∈D

ρ(s, a) = ρE (s, a) ∀ s ∈ S, a ∈ A

(7)

with Lagrangian L̄, for which the costs c(s, a) serve as dual variables for equality constraints. Thus,
c̃ is a dual optimum for (7). In addition, strong duality holds for (7): D is compact and convex, −H̄
is convex, and, since ρE > 0, there exists a feasible point in the relative interior of the domain D.
Moreover, Lemma 3.2 guarantees that −H̄ is in fact strictly convex, so the primal optimum can
be uniquely recovered from the dual optimum [4, Section 5.5.5] via ρ̃ = arg minρ∈D L̄(ρ, c̃) =
P
arg minρ∈D −H̄(ρ) + s,a c̃(s, a)ρ(s, a) = ρE , where the first equality indicates that ρ̃ is the
unique minimizer of L̄(·, c̃), and the third follows from the constraints in the primal problem (7). But
if π̃ ∈ RL(c̃), then Lemma 3.3 implies ρπ̃ = ρ̃ = ρE .
Let us summarize our conclusions. First, IRL is a dual of an occupancy measure matching
problem, and the recovered cost function is the dual optimum. Classic IRL algorithms that solve
reinforcement learning repeatedly in an inner loop, such as the algorithm of Ziebart et al. [29] that
runs a variant of value iteration in an inner loop, can be interpreted as a form of dual ascent, in which
one repeatedly solves the primal problem (reinforcement learning) with fixed dual values (costs).
Dual ascent is effective if solving the unconstrained primal is efficient, but in the case of IRL, it
amounts to reinforcement learning! Second, the induced optimal policy is the primal optimum.
The induced optimal policy is obtained by running RL after IRL, which is exactly the act of recovering
the primal optimum from the dual optimum; that is, optimizing the Lagrangian with the dual variables
fixed at the dual optimum values. Strong duality implies that this induced optimal policy is indeed
the primal optimum, and therefore matches occupancy measures with the expert. IRL is traditionally
defined as the act of finding a cost function such that the expert policy is uniquely optimal, but we can
alternatively view IRL as a procedure that tries to induce a policy that matches the expert’s occupancy
measure.

4

Practical occupancy measure matching

We saw in Proposition 3.2 that if ψ is constant, the resulting primal problem (7) simply matches
occupancy measures with expert at all states and actions. Such an algorithm is not practically useful.
In reality, the expert trajectory distribution will be provided only as a finite set of samples, so in large
environments, most of the expert’s occupancy measure values will be small, and exact occupancy
measure matching will force the learned policy to rarely visit these unseen state-action pairs simply
due to lack of data. Furthermore, in the cases in which we would like to use function approximation to
learn parameterized policies πθ , the resulting optimization problem of finding an appropriate θ would
have an intractably large number of constraints when the environment is large: as many constraints as
points in S × A.
Keeping in mind that we wish to eventually develop an imitation learning algorithm suitable for large
environments, we would like to relax Eq. (7) into the following form, motivated by Proposition 3.1:
minimize dψ (ρπ , ρE ) − H(π)
π

(8)

by modifying the IRL regularizer ψ so that dψ (ρπ , ρE ) , ψ ∗ (ρπ − ρE ) smoothly penalizes violations
in difference between the occupancy measures.
Entropy-regularized apprenticeship learning It turns out that with certain settings of ψ, Eq. (8)
takes on the form of regularized variants of existing apprenticeship learning algorithms, which
indeed do scale to large environments with parameterized policies [10]. For a class of cost functions
C ⊂ RS×A , an apprenticeship learning algorithm finds a policy that performs better than the expert
across C, by optimizing the objective
minimize max Eπ [c(s, a)] − EπE [c(s, a)]
π

c∈C

(9)

Classic apprenticeship learning algorithms restrict C to convex sets given by linear combinations
of basis functions f1 , . . . , fd , which give rise a feature vector f (s, a) = [f1 (s, a), . . . , fd (s, a)] for
each state-action pair. Abbeel and Ng [1] and Syed et al. [27] use, respectively,
P
P
P
Clinear = { i wi fi : kwk2 ≤ 1} and Cconvex = { i wi fi :
(10)
i wi = 1, wi ≥ 0 ∀i} .
4

Clinear leads to feature expectation matching [1], which minimizes `2 distance between expected
feature vectors: maxc∈Clinear Eπ [c(s, a)]−EπE [c(s, a)] = kEπ [f (s, a)]−EπE [f (s, a)]k2 . Meanwhile,
Cconvex leads to MWAL [26] and LPAL [27], which minimize worst-case excess cost among the
individual basis functions, as maxc∈Cconvex Eπ [c(s, a)] − EπE [c(s, a)] = maxi∈{1,...,d} Eπ [fi (s, a)] −
EπE [fi (s, a)].
We now show how Eq. (9) is a special case of Eq. (8) with a certain setting of ψ. With the indicator
function δC : RS×A → R, defined by δC (c) = 0 if c ∈ C and +∞ otherwise, we can write the
apprenticeship learning objective (9) as
X
max Eπ [c(s, a)]−EπE [c(s, a)] = max −δC (c) +
(ρπ (s, a)−ρπE(s, a))c(s, a) = δC∗ (ρπ −ρπE)
c∈RS×A

c∈C

s,a

Therefore, we see that entropy-regularized apprenticeship learning
minimize −H(π) + max Eπ [c(s, a)] − EπE [c(s, a)]
π

c∈C

(11)

is equivalent to performing RL following IRL with cost regularizer ψ = δC , which forces the implicit
IRL procedure to recover a cost function lying in C. Note that we can scale the policy’s entropy
regularization strength in Eq. (11) by scaling C by a constant α as {αc : c ∈ C}, recovering the
original apprenticeship objective (9) by taking α → ∞.
Cons of apprenticeship learning It is known that apprenticeship learning algorithms generally do
not recover expert-like policies if C is too restrictive [27, Section 1]—which is often the case for the
linear subspaces used by feature expectation matching, MWAL, and LPAL, unless the basis functions
f1 , . . . , fd are very carefully designed. Intuitively, unless the true expert cost function (assuming it
exists) lies in C, there is no guarantee that if π performs better than πE on all of C, then π equals πE .
With the aforementioned insight based on Proposition 3.1 that apprenticeship learning is equivalent
to RL following IRL, we can understand exactly why apprenticeship learning may fail to imitate: it
forces πE to be encoded as an element of C. If C does not include a cost function that explains expert
behavior well, then attempting to recover a policy from such an encoding will not succeed.
Pros of apprenticeship learning While restrictive cost classes C may not lead to exact imitation,
apprenticeship learning with such C can scale to large state and action spaces with policy function
approximation. Ho et al. [10] rely on the following policy gradient formula for the apprenticeship
objective (9) for a parameterized policy πθ :
∇θ max Eπθ [c(s, a)] − EπE [c(s, a)] = ∇θ Eπθ [c∗ (s, a)] = Eπθ [∇θ log πθ (a|s)Qc∗ (s, a)]
c∈C
(12)
where c∗ = arg max Eπθ [c(s, a)] − EπE [c(s, a)], Qc∗ (s̄, ā) = Eπθ [c∗ (s̄, ā) | s0 = s̄, a0 = ā]
c∈C

Observing that Eq. (12) is the policy gradient for a reinforcement learning objective with cost c∗ , Ho
et al. propose an algorithm that alternates between two steps:
1. Sample trajectories of the current policy πθi by simulating in the environment, and fit a
cost function c∗i , as defined in Eq. (12). For the cost classes Clinear and Cconvex (10), this cost
fitting amounts to evaluating simple analytical expressions [10].
2. Form a gradient estimate with Eq. (12) with c∗i and the sampled trajectories, and take a trust
region policy optimization (TRPO) [24] step to produce πθi+1 .
This algorithm relies crucially on the TRPO policy step, which is a natural gradient step constrained
to ensure that πθi+1 does not stray too far πθi , as measured by KL divergence between the two
policies averaged over the states in the sampled trajectories. This carefully constructed step scheme
ensures that the algorithm does not diverge due to high noise in estimating the gradient (12). We refer
the reader to Schulman et al. [24] for more details on TRPO.
With the TRPO step scheme, Ho et al. were able train large neural network policies for apprenticeship learning with linear cost function classes (10) in environments with hundreds of observation
dimensions. Their use of these linear cost function classes, however, limits their approach to settings
in which expert behavior is well-described by such classes. We will draw upon their algorithm to
develop an imitation learning method that both scales to large environments and imitates arbitrarily
complex expert behavior. To do so, we first turn to proposing a new regularizer ψ that wields more
expressive power than the regularizers corresponding to Clinear and Cconvex (10).
5

5

Generative adversarial imitation learning

As discussed in Section 4, the constant regularizer leads to an imitation learning algorithm that exactly
matches occupancy measures, but is intractable in large environments. The indicator regularizers
for the linear cost function classes (10), on the other hand, lead to algorithms incapable of exactly
matching occupancy measures without careful tuning, but are tractable in large environments. We
propose the following new cost regularizer that combines the best of both worlds, as we will show in
the coming sections:


EπE [g(c(s, a))] if c < 0
−x − log(1 − ex ) if x < 0
ψGA (c) ,
where g(x) =
(13)
+∞
otherwise
+∞
otherwise
This regularizer places low penalty on cost functions c that assign an amount of negative cost to
expert state-action pairs; if c, however, assigns large costs (close to zero, which is the upper bound
for costs feasible for ψGA ) to the expert, then ψGA will heavily penalize c. An interesting property of
ψGA is that it is an average over expert data, and therefore can adjust to arbitrary expert datasets. The
indicator regularizers δC , used by the linear apprenticeship learning algorithms described in Section 4,
are always fixed, and cannot adapt to data as ψGA can. Perhaps the most important difference between
ψGA and δC , however, is that δC forces costs to lie in a small subspace spanned by finitely many basis
functions, whereas ψGA allows for any cost function, as long as it is negative everywhere.
Our choice of ψGA is motivated by the following fact, shown in the appendix (Corollary A.1.1):
∗
ψGA
(ρπ − ρπE ) =
sup
Eπ [log(D(s, a))] + EπE [log(1 − D(s, a))]
(14)
D∈(0,1)S×A

where the supremum ranges over discriminative classifiers D : S × A → (0, 1). Equation (14) is proportional to the optimal negative log loss of the binary classification problem of distinguishing between
state-action pairs of π and πE . It turns out that this optimal loss is, up to a constant shift and scaling,
the Jensen-Shannon divergence DJS (ρ̄π , ρ̄πE ) , DKL (ρ̄π k(ρ̄π + ρ̄E )/2) + DKL (ρ̄E k(ρ̄π + ρ̄E )/2),
which is a squared metric between the normalized occupancy distributions ρ̄π = (1 − γ)ρπ and
ρ̄πE = (1 − γ)ρπE [8, 17]. Treating the causal entropy H as a policy regularizer controlled by λ ≥ 0
and dropping the 1 − γ occupancy measure normalization for clarity, we obtain a new imitation
learning algorithm:
∗
minimize ψGA
(ρπ − ρπE ) − λH(π) = DJS (ρπ , ρπE ) − λH(π),
(15)
π

which finds a policy whose occupancy measure minimizes Jensen-Shannon divergence to the expert’s.
Equation (15) minimizes a true metric between occupancy measures, so, unlike linear apprenticeship
learning algorithms, it can imitate expert policies exactly.
Algorithm Equation (15) draws a connection between imitation learning and generative adversarial
networks [8], which train a generative model G by having it confuse a discriminative classifier
D. The job of D is to distinguish between the distribution of data generated by G and the true
data distribution. When D cannot distinguish data generated by G from the true data, then G has
successfully matched the true data. In our setting, the learner’s occupancy measure ρπ is analogous
to the data distribution generated by G, and the expert’s occupancy measure ρπE is analogous to the
true data distribution.
We now present a practical imitation learning algorithm, called generative adversarial imitation
learning or GAIL (Algorithm 1), designed to work in large environments. GAIL solves Eq. (15) by
finding a saddle point (π, D) of the expression
Eπ [log(D(s, a))] + EπE [log(1 − D(s, a))] − λH(π)
(16)
with both π and D represented using function approximators: GAIL fits a parameterized policy
πθ , with weights θ, and a discriminator network Dw : S × A → (0, 1), with weights w. GAIL
alternates between an Adam [11] gradient step on w to increase Eq. (16) with respect to D, and a
TRPO step on θ to decrease Eq. (16) with respect to π (we derive an estimator for the causal entropy
gradient ∇θ H(πθ ) in Appendix A.2). The TRPO step serves the same purpose as it does with the
apprenticeship learning algorithm of Ho et al. [10]: it prevents the policy from changing too much
due to noise in the policy gradient. The discriminator network can be interpreted as a local cost
function providing learning signal to the policy—specifically, taking a policy step that decreases
expected cost with respect to the cost function c(s, a) = log D(s, a) will move toward expert-like
regions of state-action space, as classified by the discriminator.
6

Algorithm 1 Generative adversarial imitation learning
1: Input: Expert trajectories τE ∼ πE , initial policy and discriminator parameters θ0 , w0
2: for i = 0, 1, 2, . . . do
3:
Sample trajectories τi ∼ πθi
4:
Update the discriminator parameters from wi to wi+1 with the gradient

Êτi [∇w log(Dw (s, a))] + ÊτE [∇w log(1 − Dw (s, a))]
5:

(17)

Take a policy step from θi to θi+1 , using the TRPO rule with cost function log(Dwi+1 (s, a)).
Specifically, take a KL-constrained natural gradient step with
Êτi [∇θ log πθ (a|s)Q(s, a)] − λ∇θ H(πθ ),
where Q(s̄, ā) = Êτi [log(Dwi+1 (s, a)) | s0 = s̄, a0 = ā]

(18)

6: end for

6

Experiments

We evaluated GAIL against baselines on 9 physics-based control tasks, ranging from low-dimensional
control tasks from the classic RL literature—the cartpole [2], acrobot [7], and mountain car [15]—to
difficult high-dimensional tasks such as a 3D humanoid locomotion, solved only recently by modelfree reinforcement learning [25, 24]. All environments, other than the classic control tasks, were
simulated with MuJoCo [28]. See Appendix B for a complete description of all the tasks.
Each task comes with a true cost function, defined in the OpenAI Gym [5]. We first generated expert
behavior for these tasks by running TRPO [24] on these true cost functions to create expert policies.
Then, to evaluate imitation performance with respect to sample complexity of expert data, we sampled
datasets of varying trajectory counts from the expert policies. The trajectories constituting each
dataset each consisted of about 50 state-action pairs. We tested GAIL against three baselines:
1. Behavioral cloning: a given dataset of state-action pairs is split into 70% training data and
30% validation data. The policy is trained with supervised learning, using Adam [11] with
minibatches of 128 examples, until validation error stops decreasing.
2. Feature expectation matching (FEM): the algorithm of Ho et al. [10] using the cost function
class Clinear (10) of Abbeel and Ng [1]
3. Game-theoretic apprenticeship learning (GTAL): the algorithm of Ho et al. [10] using the
cost function class Cconvex (10) of Syed and Schapire [26]
We used all algorithms to train policies of the same neural network architecture for all tasks: two
hidden layers of 100 units each, with tanh nonlinearities in between. The discriminator networks
for GAIL also used the same architecture. All networks were always initialized randomly at the
start of each trial. For each task, we gave FEM, GTAL, and GAIL exactly the same amount of
environment interaction for training. We ran all algorithms 5-7 times over different random seeds in
all environments except Humanoid, due to time restrictions.
Figure 1 depicts the results, and Appendix B provides exact performance numbers and details of our
experiment pipeline, including expert data sampling and algorithm hyperparameters. We found that on
the classic control tasks (cartpole, acrobot, and mountain car), behavioral cloning generally suffered
in expert data efficiency compared to FEM and GTAL, which for the most part were able produce
policies with near-expert performance with a wide range of dataset sizes, albeit with large variance
over different random initializations of the policy. On these tasks, GAIL consistently produced
policies performing better than behavioral cloning, FEM, and GTAL. However, behavioral cloning
performed excellently on the Reacher task, on which it was more sample efficient than GAIL. We
were able to slightly improve GAIL’s performance on Reacher using causal entropy regularization—in
the 4-trajectory setting, the improvement from λ = 0 to λ = 10−3 was statistically significant over
training reruns, according to a one-sided Wilcoxon rank-sum test with p = .05. We used no causal
entropy regularization for all other tasks.
7

Acrobot

Mountain Car

HalfCheetah

1.0

1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0.0

0.0
1

4

7

10

1.0
0.8

0.0

0.0
1

Hopper

4

7

10

Reacher
1.0

1

4

Walker

7

10

4

Ant

11

18

25

Humanoid

1.0

Performance (scaled)

Performance (scaled)

Cartpole
1.0

0.5

0.0

−0.5

1.0

1.0

0.8

0.8

0.5

0.8

4

0.6

0.6

0.0

0.6

Number of trajectories in dataset

0.4

0.4

−0.5

0.4

Expert

GAIL (¸ = 0)

0.2

0.2

0.2

Random

GAIL (¸ = 10 ¡3 )

0.0

0.0

0.0

Behavioral cloning

GAIL (¸ = 10 ¡2 )

4

11

18

25

1.0

−1.0
4

11

18

25

4

11

18

25

80

160

11

18

240

Number of trajectories in dataset
Expert
Random

Behavioral cloning
FEM

GTAL
GAIL (ours)

(a)

(b)

Figure 1: (a) Performance of learned policies. The y-axis is negative cost, scaled so that the expert
achieves 1 and a random policy achieves 0. (b) Causal entropy regularization λ on Reacher. Except
for Humanoid, shading indicates standard deviation over 5-7 reruns.

On the other MuJoCo environments, GAIL almost always achieved at least 70% of expert performance
for all dataset sizes we tested and reached it exactly with the larger datasets, with very little variance
among random seeds. The baseline algorithms generally could not reach expert performance even
with the largest datasets. FEM and GTAL performed poorly for Ant, producing policies consistently
worse than a policy that chooses actions uniformly at random. Behavioral cloning was able to reach
satisfactory performance with enough data on HalfCheetah, Hopper, Walker, and Ant, but was unable
to achieve more than 60% for Humanoid, on which GAIL achieved exact expert performance for all
tested dataset sizes.

7

Discussion and outlook

As we demonstrated, GAIL is generally quite sample efficient in terms of expert data. However, it is
not particularly sample efficient in terms of environment interaction during training. The number
of such samples required to estimate the imitation objective gradient (18) was comparable to the
number needed for TRPO to train the expert policies from reinforcement signals. We believe that we
could significantly improve learning speed for GAIL by initializing policy parameters with behavioral
cloning, which requires no environment interaction at all.
Fundamentally, our method is model free, so it will generally need more environment interaction than
model-based methods. Guided cost learning [6], for instance, builds upon guided policy search [12]
and inherits its sample efficiency, but also inherits its requirement that the model is well-approximated
by iteratively fitted time-varying linear dynamics. Interestingly, both GAIL and guided cost learning
alternate between policy optimization steps and cost fitting (which we called discriminator fitting),
even though the two algorithms are derived completely differently.
Our approach builds upon a vast line of work on IRL [29, 1, 27, 26], and hence, just like IRL,
our approach does not interact with the expert during training. Our method explores randomly
to determine which actions bring a policy’s occupancy measure closer to the expert’s, whereas
methods that do interact with the expert, like DAgger [22], can simply ask the expert for such actions.
Ultimately, we believe that a method that combines well-chosen environment models with expert
interaction will win in terms of sample complexity of both expert data and environment interaction.
Acknowledgments
We thank Jayesh K. Gupta, John Schulman, and the anonymous reviewers for assistance, advice,
and critique. This work was supported by the SAIL-Toyota Center for AI Research and by a NSF
Graduate Research Fellowship (grant no. DGE-114747).
8

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