ATIVE ADVERSARIAL IMITATION LEARNING
Year: 2020

Abstract:
Generative Adversarial Imitation Learning (GAIL) is a powerful and practical approach for learning sequential decision-making policies. Different from Reinforcement Learning (RL), GAIL takes advantage of demonstration data by experts (e.g., human), and learns both the policy and reward function of the unknown environment. Despite the significant empirical progresses, the theory behind GAIL is still largely unknown. The major difficulty comes from the underlying temporal dependency of the demonstration data and the minimax computational formulation of GAIL without convex-concave structure. To bridge such a gap between theory and practice, this paper investigates the theoretical properties of GAIL. Specifically, we show: (1) For GAIL with general reward parameterization, the generalization can be guaranteed as long as the class of the reward functions is properly controlled; (2) For GAIL, where the reward is parameterized as a reproducing kernel function, GAIL can be efficiently solved by stochastic first order optimization algorithms, which attain sublinear convergence to a stationary solution. To the best of our knowledge, these are the first results on statistical and computational guarantees of imitation learning with reward/policy function approximation. Numerical experiments are provided to support our analysis.

heading: 1 INTRODUCTION
text: As various robots (Tail et al., 2018), self-driving cars (Kuefler et al., 2017), unmanned aerial vehicles (Pfeiffer et al., 2018) and other intelligent agents are applied to complex and unstructured environments, programming their behaviors/policy has become increasingly challenging. These intelligent agents need to accommodate a huge number of tasks with unique environmental demands. To address these challenges, many reinforcement learning (RL) methods have been proposed for learning sequential decision-making policies (Sutton et al., 1998; Kaelbling et al., 1996; Mnih et al., 2015). These RL methods, however, heavily rely on human expert domain knowledge to design proper reward functions. For complex tasks, which are often difficult to describe formally, these RL methods become impractical.
The Imitation Learning (IL, Argall et al. (2009); Abbeel & Ng (2004)) approach is a powerful and practical alternative to RL. Rather than having a human expert handcrafting a reward function for learning the desired policy, the imitation learning approach only requires the human expert to demonstrate the desired policy, and then the intelligent agent (a.k.a. learner) learns to match the demonstration. Most of existing imitation learning methods fall in the following two categories:
• Behavioral Cloning (BC, Pomerleau (1991)). BC treats the IL problem as supervised learning. Specifically, it learns a policy by fitting a regression model over expert demonstrations, which directly maps states to actions. Unfortunately, BC has a fundamental drawback. Recall that in supervised learning, the distribution of the training data is decoupled from the learned model, whereas in imitation learning, the agent’s policy affects what state is queried next. The mismatch between training and testing distributions, also known as covariate shift (Ross & Bagnell, 2010; Ross et al., 2011), yields significant compounding errors. Therefore, BC often suffers from poor generalization.
• Inverse Reinforcement Learning (IRL, Russell (1998); Ng et al. (2000); Finn et al. (2016); Levine & Koltun (2012)). IRL treats the IL problem as bi-level optimization. Specifically, it finds a reward
function, under which the expert policy is uniquely optimal. Though IRL does not have the error compounding issue, its computation is very inefficient. Many existing IRL methods need to solve a sequence of computationally expensive reinforcement learning problems, due to their bi-level optimization nature. Therefore, they often fail to scale to large and high dimensional environments.
More recently, Ho & Ermon (2016) propose a Generative Adversarial Imitation Learning (GAIL) method, which obtains significant performance gains over existing IL methods in imitating complex expert policies in large and high-dimensional environments. GAIL generalizes IRL by formulating the IL problem as minimax optimization, which can be solved by alternating gradient-type algorithms in a more scalable and efficient manner.
Specifically, we consider an infinite horizon Markov Decision Process (MDP), where S denotes the state space, A denotes the action space, P denotes the Markov transition kernel, r∗ denotes the reward function, and p0 denotes the distribution of the initial state. We assume that the Markov transition kernel P is fixed and there is an unknown expert policy π∗ : S → P(A), where P(A) denotes the set of distributions over the action space. As can be seen, {st}T−1t=0 essentially forms a Markov chain with the transition kernel induced by π∗ as Pπ ∗ (s, s′) = ∑ a∈A π
∗(a | s)·P (s′ | s, a). Given n demonstration trajectories from π∗ denoted by {s(i)t , a(i)t }T−1t=0 , where i = 1, ..., n, s0 ∼ p0, at ∼ π∗(· | st), and st+1 ∼ P (· | st, at), GAIL aims to learn π∗ by solving the following minimax optimization problem,
minπ maxr∈R[Eπr(s, a)− Eπ∗nr(s, a)], (1)
where Eπ[r(s, a)] = limT→∞ E[ 1T ∑T−1 t=0 r(st, at)|π] denotes the average reward under the policy π when the reward function is r, and Eπ∗n [r(s, a)] = 1 nT ∑n i=1 ∑T−1 t=0 [r(s (i) t , a (i) t )] denotes the empirical average reward over the demonstration trajectories. As shown in (1), GAIL aims to find a policy, which attains an average reward similar to that of the expert policy with respect to any reward belonging to the function classR. For large and high-dimensional imitation learning problems, we often encounter infinitely many states. To ease computation, we need to consider function approximations. Specifically, suppose that for every s ∈ S and a ∈ A, there are feature vectors ψs ∈ RdS and ψa ∈ RdA associated with a and s, respectively. Then we can approximate the policy and reward as
π(·|s) = π̃ω(ψs) and r(s, a) = r̃θ(ψs, ψa),
where π̃ and r̃ belong to certain function classes (e.g. reproducing kernel Hilbert space or deep neural networks, Ormoneit & Sen (2002); LeCun et al. (2015)) associated with parameters ω and θ, respectively. Accordingly, we can optimize (1) with respect to the parameters ω and θ by scalable alternating gradient-type algorithms.
Although GAIL has achieved significant empirical progresses, its theoretical properties are still largely unknown. There are three major difficulties when analyzing GAIL: 1). There exists temporal dependency in the demonstration trajectories/data due to their sequential nature (Howard, 1960; Puterman, 2014; Abounadi et al., 2001); 2). GAIL is formulated as a minimax optimization problem. Most of existing learning theories, however, focus on empirical risk minimization problems, and therefore are not readily applicable (Vapnik, 2013; Mohri et al., 2018; Anthony & Bartlett, 2009); 3). The minimax optimization problem in (1) does not have a convex-concave structure, and therefore existing theories in convex optimization literature cannot be applied for analyzing the alternating stochastic gradient-type algorithms (Willem, 1997; Ben-Tal & Nemirovski, 1998; Murray & Overton, 1980; Chambolle & Pock, 2011; Chen et al., 2014). Some recent results suggest to use stage-wise stochastic gradient-type algorithms (Rafique et al., 2018; Dai et al., 2017). More specifically, at every iteration, they need to solve the inner maximization problem up to a high precision, and then apply stochastic gradient update to the outer minimization problem. Such algorithms, however, are rarely used by practitioners, as they are inefficient in practice (due to the computationally intensive inner maximization).
To bridge such a gap between practice and theory, we establish the generalization properties of GAIL and the convergence properties of the alternating mini-batch stochastic gradient algorithm for solving (1). Specifically, our contributions can be summarized as follows:
•We formally define the generalization of GAIL under the “so-called”R-reward distance, and then show that the generalization of GAIL can be guaranteed under reward distance as long as the class of the reward functions is properly controlled;
•We provide sufficient conditions, under which an alternating mini-batch stochastic gradient algorithm can efficiently solve the minimax optimization in (1), and attains sublinear convergence to a stationary solution.
To the best of our knowledge, these are the first results on statistical and computational theories of imitation learning with reward/policy function approximations.
Our work is related to Syed et al. (2008); Cai et al. (2019). Syed et al. (2008) study the generalization and computational properties of apprenticeship learning. Since they assume that the state space of the underlying Markov decision process is finite, they do not consider any reward/policy function approximations; Cai et al. (2019) study the computational properties of imitation learning under a simple control setting. Their assumption on linear policy and quadratic reward is very restrictive, and does not hold for many real applications. Notation. Given a vector x = (x1, ..., xd)> ∈ Rd, we define ‖x‖22 = ∑d j=1 x 2 j . Given a function f : Rd 7→ R, we denote its `∞ norm as ‖f‖∞ = maxx |f(x)|.

heading: 2 GENERALIZATION OF GAIL
text: To analyze the generalization properties of GAIL, we first assume that we can access an infinite number of the expert’s demonstration trajectories (underlying population), and that the reward function is chosen optimally within some large class of functions. This allows us to remove the maximum operation from (1), which leads to an interpretation of how and in what sense the resulting policy is close to the true expert policy. Before we proceed, we first introduce some preliminaries. Definition 1 (Stationary Distribution). Note that any policy π induces a Markov chain on S × A. The transition kernel is given by
Pπ(s ′, a′ | s, a) = π(a′ | s′) · P (s′ | s, a), ∀(s, a), (s′, a′) ∈ S ×A.
When such a Markov chain is aperiodic and recurrent, we denote its stationary distribution as ρπ .
Note that a policy π is uniquely determined by its stationary distribution ρπ in the sense that π(a | s) = ρπ(s, a)/ ∑ a∈A ρπ(s, a).
Then we can write the expected average reward of r(s, a) under the policy π as Eπ[r(s, a)] = limT→∞ E [ 1 T ∑T−1 t=0 r(st, at) ∣∣π] = Eρπ[r(s, a)] = ∑(s,a)∈S×A ρπ(s, a) · r(s, a). We further define theR-distance between two policies π and π′ as follows. Definition 2. Let R denote a class of symmetric reward functions from S × A to R, i.e., if r ∈ R, then −r ∈ R. Given two policy π′ and π, theR-distance for GAIL is defined as
dR(π, π ′) = supr∈R[Eπr(s, a)− Eπ′r(s, a)].
The R-distance over policies for Markov decision processes is essentially an Integral Probability Metric (IPM) over stationary distributions (Müller, 1997). For different choices ofR, we have variousR-distances. For example, we can chooseR as the class of all 1-Lipschitz continuous functions, which yields that dR(π, π′) is the Wasserstein distance between ρπ and ρπ′ (Vallender, 1974). For computational convenience, GAIL and its variants usually choose R as a class of functions from some reproducing kernel Hilbert space, or a class of neural network functions. Definition 3. Given n demonstration trajectories from time 0 to T − 1 obtained by an expert policy π∗ denoted by (s(i)t , a (i) t ) T−1 t=0 , where i = 1, ..., n, a policy π̂ learned by GAIL generalizes under the R-distance dR(·, ·) with generalization error , if with high probability, we have |dR(π∗n, π̂)− dR(π∗, π̂)| ≤ , where dR(π∗n, π̂) is the empiricalR-distance between π∗ and π̂ defined as dR(π ∗ n, π̂) = supr∈R[Eπ∗nr(s, a)− Eπ̂r(s, a)] with Eπ∗n [r(s, a)] = 1nT ∑n i=1 ∑T−1 t=0 [r(s (i) t , a (i) t )].
The generalization of GAIL implies that theR-distance between the expert policy π∗ and the learned policy π̂ is close to the empirical R-distance between them. Our analysis aims to prove the former distance to be small, whereas the latter one is what we attempts to minimize in practice.
We then introduce the assumptions on the underlying Markov decision process and expert policy.
Assumption 1. Under the expert policy π∗, (st, at)T−1t=0 forms a stationary and exponentially βmixing Markov chain, i.e.,
β(k) = supn EB∈σn0 supA∈σ∞n+k |P(A|B)− P(A)| ≤ β0 exp(−β1k α),
where β0, β1, α are positive constants, and σ j i is the σ-algebra generated by (st, at) j t=i for i ≤ j.
Moreover, for every s ∈ S and a ∈ A, there are feature vectors ψs ∈ RdS and ψa ∈ RdA associated with a and s, respectively, and ψs and ψa are uniformly bounded, where
‖ψs‖2 ≤ 1 and ‖ψa‖2 ≤ 1, ∀s ∈ S and ∀a ∈ A.
Assumption 1 requires the underlying MDP to be ergodic (Levin & Peres, 2017), which is a commonly studied assumption in exiting reinforcement learning literature on maximizing the expected average reward (Strehl & Littman, 2005; Li et al., 2011; Brafman & Tennenholtz, 2002; Kearns & Singh, 2002). The feature vectors associated with a and s allow us to apply function approximations to parameterize the reward and policy functions. Accordingly, we write the reward function as r(s, a) = r̃(ψs, ψa), which is assumed to be bounded. Assumption 2. The reward function class is uniformly bounded, i.e., ‖r‖∞ ≤ Br for any r ∈ R.
Now we proceed with our main result on generalization properties of GAIL. We useN (R, , ‖ · ‖∞) to denote the covering number of the function classR under the `∞ distance ‖ · ‖∞. Theorem 1 (Main Result). Suppose Assumptions 1-2 hold, and the policy learned by GAIL satisfies
dR(π ∗ n, π̂)− infπ dR(π∗n, π) < ,
where the infimum is taken over all possible learned policies. Then with probability at least 1 − δ over the joint distribution of {(a(i)t , s(i)t )T−1t=0 }ni=1, we have
dR(π ∗, π̂)− inf
π dR(π
∗, π) ≤ O (
Br√ nT/ζ
√ logN ( R, √
ζ nT , ‖ · ‖∞ ) +Br
√ log(1/δ)
nT/ζ
) + ,
where ζ = (β−11 log β0T δ ) 1 α .
Theorem 1 implies that the policy π̂ learned by GAIL generalizes as long as the complexity of the function class R is well controlled. To the best of our knowledge, this is the first result on the generalization of imitation learning with function approximations. As the proof of Theorem 1 is involved, we only present a sketch due to space limit. More details are provided in Appendix A.1.
Proof Sketch. Our analysis relies on characterizing the concentration property of the empirical average reward under the expert policy. For notational simplicity, we define
φ = Eπ∗r(s, a)− 1nT ∑n i=1 ∑T−1 t=0 r(s (i) t , a (i) t ).
The key challenge comes from the fact that (s(i)t , a (i) t )’s are dependent. To handle such a dependency, we adopt the independent block technique from Yu (1994). Specifically, we partition every trajectory into disjoint blocks (where the block size is of the order O((log(T ) + log(1/δ))1/α), and construct two separable trajectories: One contains all blocks with odd indices (denoted by Bodd), and the other contains all those with even indices (denoted by Beven). We define
φ1 = Eπ∗r(s, a)− 2nT ∑n i=1 ∑ (s
(i) t ,a (i) t )∈Bodd
r(s (i) t , a (i) t ),
and analogously for φ2 with (s (i) t , a (i) t ) ∈ Beven. Then we have
P(supr∈R φ ≥ ε) ≤ P(supr∈R φ12 + supr∈R φ2 2 ≥ ε) ≤ P(supr∈R φ1 ≥ ε) + P(supr∈R φ2 ≥ ε).
We consider a block-wise independent counterpart of φ1 denoted by φ̃1, where each block is sampled independently from the same Markov chain as φ1, i.e., φ̃1 has independent blocks of samples from the same exponentially β-mixing Markov chain . Accordingly, we denote φ̃1 as
φ̃1 = Eπ∗r(s, a)− 2nT ∑n i=1 ∑ (s
(i) t ,a (i) t )∈B̃odd
r(s (i) t , a (i) t ),
where B̃odd denotes i.i.d. blocks of samples. Now we bound the difference between φ1 and φ̃1 by P(sup r∈R φ1 − E[sup r∈R φ̃1] ≥ ε− E[sup r∈R φ̃1]) ≤ P(sup r∈R φ̃1 − E[sup r∈R φ̃1] ≥ ε− E[sup r∈R φ̃1])
+ CβT/(log(T ) + log(1/δ))1/α,
where C is a constant, and β is the mixing coefficient, and P(supr∈R φ̃1 − E[supr∈R φ̃1] ≥ ε − E[supr∈R φ̃1]) can be bounded using the empirical process technique for independent random variables. The details of the above inequality can be found in Corollary 3 in Appendix A.1, where the proof technique is adapted from Lemma 1 in Mohri & Rostamizadeh (2009). Let φ̃2 be defined analogously as φ̃1. With a similar argument further applied to φ2 and φ̃2, we obtain
P(sup r∈R φ ≥ ε) ≤ 2P(sup r∈R φ̃1 − E[sup r∈R φ̃1] ≥ ε− E[sup r∈R φ̃1]) + 2CβT/(log(T ) + log(1/δ)) 1/α.
The rest of our analysis follows the PAC-learning framework using Rademacher complexity and is omitted (Mohri et al., 2018). We complete the proof sketch.
Example 1: Reproducing Kernel Reward Function. One popular option to parameterize the reward by functions is the reproducing kernel Hilbert space (RKHS, Kim & Park (2018); Li et al. (2018)). There have been several implementations of RKHS, and we consider the feature mapping approach. Specifically, we consider g : RdS × RdA → Rq , and the reward can be written as
r(s, a) = r̃θ(ψs, ψa) = θ >g(ψs, ψa),
where θ ∈ Rq . We require g to be Lipschitz continuous with respect to (ψa, ψs). Assumption 3. The feature mapping g satisfies g(0, 0) = 0, and there exists a constant ρg such that for any ψa, ψ′a, ψs and ψ ′ s, we have
‖g(ψs, ψa)− g(ψ′s, ψ′a)‖22 ≤ ρg √ ‖ψs − ψ′s‖22 + ‖ψa − ψ′a‖22.
Assumption 3 is mild and satisfied by popular feature mappings, e.g., random Fourier feature mapping1 (Rahimi & Recht, 2008; Bach, 2017). The next corollary presents the generalization bound of GAIL using feature mapping. Corollary 1. Suppose ‖θ‖2 ≤ Bθ. For large enough n and T , with probability at least 1 − δ over the joint distribution of {(a(i)t , s(i)t )T−1t=0 }ni=1, we have
dR(π ∗, π̂)− inf
π dR(π
∗, π) ≤ O (
ρgBθ√ nT/ζ
√ q log ( ρgBθ √ nT/ζ ) + ρgBθ √ log(1/δ)
nT/ζ
) + .
Corollary 1 indicates that with respect to a class of properly normalized reproducing kernel reward functions, GAIL generalizes in terms of theR-distance. Example 2: Neural Network Reward Function. Another popular option to parameterize the reward function is to use neural networks. Specifically, let σ(v) = [max{v1, 0}, ...,max{vd, 0}]> denote the ReLU activation for v ∈ Rd. We consider a D-layer feedforward neural network with ReLU activation as follows,
r(s, a) = r̃W(ψs, ψa) = W > Dσ(WD−1σ(...σ(W1[ψ > a , ψ > s ] >))),
whereW = {Wk | Wk ∈ Rdk−1×dk , k = 1, ..., D − 1, WD ∈ RdD−1} and d0 = dA + dS . The next corollary presents the generalization bound of GAIL using neural networks.
1More precisely, Assumption 3 actually holds with overwhelming probability over the distribution of the random mapping.
Corollary 2. Suppose ‖Wi‖2 ≤ 1, where i = 1, ..., D. For large enough n and T , with probability at least 1− δ over the joint distribution of {(a(i)t , s(i)t )T−1t=0 }ni=1, we have
dR(π ∗, π̂)− inf
π dR(π
∗, π) ≤ O (
1√ nT/ζ
√ d2D log ( D √ dnT/ζ ) + √ log(1/δ)
nT/ζ
) + .
Corollary 2 indicates that with respect to a class of properly normalized neural network reward functions, GAIL generalizes in terms of theR-distance. Remark 1 (The Tradeoff between Generalization and Representation of GAIL). As can be seen from Definition 2, the R-distances are essentially differentiating two policies. For the Wassersteintype distance, i.e., R contains all 1-Lipschitz continuous functions, if dR(π, π′) is small, it is safe to conclude that two policies π and π′ are nearly the same almost everywhere. However, when we choose R to be the reproducing kernel Hilbert space or the class of neural networks with relatively small complexity, dR(π, π′) can be small even if π and π′ are not very close. Therefore, we need to choose a sufficiently diverse class of reward functions to ensure that we recover the expert policy.
As Theorem 1 suggests, however, that we need to control the complexity of the function class R to guarantee the generalization. This implies that when parameterizing the reward function, we need to carefully choose the function class to attain the optimal tradeoff between generalization and representation of GAIL.

heading: 3 COMPUTATION OF GAIL
text: To investigate the computational properties of GAIL, we parameterize the reward by functions belonging to some reproducing kernel Hilbert space. The implementation is based on feature mapping, as mentioned in the previous section. The policy can be parameterized by functions belonging to some reproducing kernel Hilbert space or some class of deep neural networks with parameter ω. Specifically, we denote π(a|s) = π̃ω(ψs), where π̃ω(ψs) is the parametrized policy mapping from RdS to a simplex in RdA with |A| = d. For computational convenience, we consider solving a slightly modified minimax optimization problem:
minω max‖θ‖2≤κ Eπ̃ω [r̃θ(s, a)]− Eπ∗ r̃θ[(ψs, ψa)]− λH(π̃ω)− µ 2 ‖θ‖ 2 2 , (2)
where r̃θ(s, a) = θ>g(ψs, ψa), H(π̃ω) is some regularizer for the policy (e.g., causal entropy regularizer, Ho & Ermon (2016)), and λ > 0 and µ > 0 are tuning parameters. Compared with (1), the additional regularizers in (2) can improve the optimization landscape, and help mitigate computational instability in practice.
We apply the alternating mini-batch stochastic gradient algorithm to (2). Specifically, we denote the objective function in (2) as F (ω, θ) for notational simplicity. At the (t+ 1)-th iteration, we take
θ(t+1) = Πκ ( θ(t) + ηθqθ ∑ j∈M(t)θ ∇θfj(ω(t), θ(t)) ) and (3)
ω(t+1) = ω(t) − ηωqω ∑ j∈M(t)ω ∇ω f̃j(ω(t), θ(t+1)), (4)
where ηθ and ηω are learning rates, the projection Πκ(v) = 1(‖v‖2 ≤ κ) · v + 1(‖v‖2 > κ) · κ · v/ ‖v‖2 , ∇fj’s and ∇f̃j’s are independent stochastic approximations of ∇F (Sutton et al., 2000), and M(t)θ , M (t) ω are mini-batches with sizes qθ and qω , respectively. Before we proceed with the convergence analysis, we impose the follow assumptions on the problem.
Assumption 4. There are two positive constants Mω and Mθ such that for any ω and ‖θ‖2 ≤ κ,
Unbiased : E∇fj(ω, θ) = E∇f̃j(ω, θ) = ∇F (ω, θ), Bounded : E‖∇ω f̃j(ω, θ)−∇ωF (ω, θ)‖22 ≤Mω and E‖∇θfj(ω, θ)−∇θF (ω, θ)‖22 ≤Mθ.
Assumption 4 requires the stochastic gradient to be unbiased with a bounded variance, which is a common assumption in existing optimization literature (Nemirovski et al., 2009; Ghadimi & Lan, 2013; Duchi et al., 2011; Bottou, 2010).
Assumption 5. (i) For any ω, there exists some constant χ > 0 and υ ∈ (0, 1) such that ‖(Pπ̃ω )tρ0 − ρπ̃ω‖TV ≤ χυt, where Pπ̃ω (s ′, a′ | s, a) = π̃ω(a′|s′)P (s′ | s, a) is the transition kernel induced by π̃ω , ρ0 is the initial distribution of (s0, a0), and ρπ̃ω is the stationary distribution induced by π̃ω . (ii) There exist constants Sπ̃, Bω, Lρ, LQ > 0 such that for any ω, ω′, we have ‖∇ω log(π̃ω(a|s))−∇ω log(π̃ω′(a|s))‖2 ≤ Sπ̃ ‖ω − ω′‖2 , ‖∇ω log π̃ω(a|s)‖2 ≤ Bω, ‖ρπ̃ω − ρπ̃′ω‖TV ≤ Lρ‖ω − ω′‖2, ‖Qπ̃ω −Qπ̃ω′‖∞ ≤ LQ ‖ω − ω′‖2 , where Qπ̃ω (s, a) = ∑∞ t=0 E [r̃(st, at)− Eπ̃ω [r̃] | s0 = s, a0 = a, π̃ω] is the action-value function. (iii) There exist constants BH and SH > 0 such that for any ω, ω′, we have
H(π̃ω) ≤ BH , and ‖∇ωH(π̃ω)−∇ωH(π̃ω′)‖2 ≤ SH ‖ω − ω′‖2 .
Note that (i) of Assumption 5 requires the Markov Chain to be geometrically mixing. (ii) and (iii) state some commonly used regularity conditions for policies (Sutton et al., 2000; Pirotta et al., 2015).
We then define L-stationary points of F . Specifically, we say that (ω∗, θ∗) is a stationary point of F , if and only if, for any fixed α > 0,
∇ωF (ω∗, θ∗) = 0 and θ∗ −Πκ(θ∗ + α∇θF (ω∗, θ∗)) = 0. The L-stationarity is a generalization of the stationary point for unconstrained optimization, and is a necessary condition for optimality. Accordingly, we take α = 1 and measure the sub-stationarity of the algorithm at the iteration N by
JN = min1≤t≤N E‖θ(t) −Πκ(θ(t) +∇θF (ω(t), θ(t)))‖22 + E‖∇ωF (ω(t), θ(t+1))‖22. We then state the global convergence of the alternating mini-batch stochastic gradient algorithm. Theorem 2. Suppose Assumptions 1-5 hold. We choose step sizes ηθ, ηω satisfying
ηω ≤ min {
Lω Sω(8Lω + 2) , 1 2Lω
} , ηθ ≤ min { 1
150µ ,
7Lω + 1
150S2ω ,
1
100(2µ+ Sω)
} ,
and meanwhile ηω/ηθ ≤ µ/(30Lω + 5), where Lω = 2 √
2(Sπ̃ + 2BωLρ)κρgχ/(1− υ) +BωLQ, and Sω = 2 √ 2qκρgχBω/(1 − υ). Given any > 0, we choose batch sizes qθ = Õ(1/ ) and
qω = Õ(1/ ). Then we need at most
N = η(C0 + 4 √ 2ρgκ+ µκ 2 + 2λBH) −1
iterations such that JN ≤ , where C0 depends on the initialization, and η depends on ηω and ηθ.
Here Õ hides linear or quardic dependence on some constants in Assumptions 1-5. Theorem 2 shows that though the minimax optimization problem in (2) does not have a convex-concave structure, the alternating mini-batch stochastic gradient algorithm still guarantees to converge to a stationary point. We are not aware of any similar results for GAIL in existing literature.
Proof Sketch. We prove the convergence by showing∑N i=1 E ∥∥θ(t) −Πκ(θ(t) +∇θF (ω(t), θ(t)))∥∥22 + E∥∥∇ωF (ω(t), θ(t+1))∥∥22 ≤ C +N /2, (5) where C is a constant and N /2 is the accumulation of noise in stochastic approximations of ∇F . Then we straightforwardly have NJN ≤ C + N /2. Dividing both sizes by N , we can derive the desired result. The main difficulty of showing (5) comes from the fact that the outer minimization problem is nonconvex and we cannot solve the inner maximization problem exactly. To overcome this difficulty, we construct a monotonically decreasing potential function:
E(t) = EF (ω(t), θ(t)) + s ( (1 + 2ηωLω)/2 · E ∥∥ω(t) − ω(t−1)∥∥2 2
+ (ηω/2ηθ − µηω/4 + 3ηωηθµ2/2) · E ∥∥θ(t+1) − θ(t)∥∥2
2 + µηω/8 · E ∥∥θ(t) − θ(t−1)∥∥2 2 ) ,
for a constant s to be chosen later. Denote ξ(t)θ and ξ (t) ω as the i.i.d. noise of the stochastic gradients. The following lemma characterizes the decrement of the potential function at each iteration.
Lemma 1. With the step sizes ηθ and ηω chosen as in Theorem 2, we have
E(t+1) − E(t) ≤− k1E‖ω(t+1) − ω(t)‖22 − k2E‖ω(t) − ω(t−1)‖22 − k3E‖θ(t+2) − θ(t+1)‖22 − k4E‖θ(t+1) − θ(t)‖22 − k5E‖θ(t) − θ(t−1)‖22 + ν(E ∥∥ξ(t)ω ‖22 + E‖ξ(t)θ ‖22), where ν is a constant depending on F , ηθ, and ηω . Moreover, we have constants k1, k2, k3, k4, k5 > 0 for s = 8/(η2ω(58Lω + 9)) .
Let k = 1/min{k1, k4} and φ = max{1, 1/η2θ , 1/η2ω}. We obtain∑N t=1 E ∥∥θ(t) −Πκ(θ(t) +∇θF (ω(t), θ(t)))∥∥22 + E∥∥∇ωF (ω(t), θ(t+1))∥∥22 (i) ≤φ∑Nt=1 E[‖θ(t+1) − θ(t)‖22 + ‖ω(t+1) − ω(t)‖22] (ii)≤ kφ(E(1) − E(N)) + kφNνE[‖ξ(t)ω ‖22 + ‖ξ(t)θ ‖22], where (i) follows from plugging in the update (3) as well as the contraction property of projection, and (ii) follows from Lemma 1. Choosing qθ = 4kφνMθ/ and qω = 4kφνMω/ , we obtain∑N t=1 E
∥∥θ(t) −Πκ(θ(t) +∇θF (ω(t), θ(t)))∥∥22 + E∥∥∇ωF (ω(t), θ(t+1))∥∥22 ≤ kφ(E(1) − E(N)) + N 2 . We have E(N) ≥ EF (ω(N), θ(N)) by the construction of E(N). It is easy to verify that F is lower bounded (Lemma 10 in Appendix B). Eventually, we complete the proof by substituting the lower bound and choosing N = kφ(2E(1) + 4 √ 2ρgκ+ µκ 2 + 2λBH) −1.

heading: 4 EXPERIMENT
text: To verify our theory in Section 3, we conduct experiments in three reinforcement learning tasks: Acrobot, MountainCar, and Hopper. For each task, we first train an expert policy using the proximal policy optimization (PPO) algorithm in (Schulman et al., 2017) for 500 iterations, and then use the expert policy to generate the demonstration data. The demonstration data for every task contains 500 trajectories, each of which is a series of state action pairs throughout one episode in the environment. When training GAIL, we randomly select a mini-batch of trajectories, which contain at least 8192 state action pairs. We use PPO to update the policy parameters. This avoids the instability of the policy gradient algorithm, and improves the reproducibility of our experiments.
We use the same neural network architecture for all the environments. For policy, we use a fully connected neural network with two hidden layers of 128 neurons in each layer and tanh activation. For reward, we use a fully connected ReLU neural network with two hidden layers of 1024 and 512 neurons, respectively. To implement the kernel reward, we fix the first two layers of the neural network after random initialization and only update the third layer, i.e., the first two layers mimic the random feature mapping. We choose κ = 1 and µ = 0.3. When updating the neural network reward, we use weight normalization in each layer (Salimans & Kingma, 2016).
When updating the kernel reward at each iteration, we choose to take the stochastic gradient ascent step for either once (i.e., alternating update in Section 3) or 10 times. When updating the neural network reward at each iteration, we choose to take the stochastic gradient ascent step for only once. We tune step size parameters for updating the policy and reward, and summarize the numerical results of the step sizes attaining the maximal average episode reward in Figure 1.
As can be seen, using multiple stochastic gradient ascent steps for updating the reward at each iteration yields similar performance as that of one step. We present the convergence analysis of using multiple stochastic gradient ascent steps for updating the reward in Appendix C. Moreover, we observe that parameterizing the reward by neural networks slightly outperform that of the kernel reward. However, its training process tends to be unstable and takes longer time to converge.

heading: 5 DISCUSSIONS
text: Our proposed theories of GAIL are closely related to Generative Adversarial Networks (Goodfellow et al., 2014; Arjovsky et al., 2017): (1) The generalization of GANs is defined based on the integral probabilistic metric (IPM) between the synthetic distribution obtained by the generator network and the distribution of the real data (Arora et al., 2017). As the real data in GANs are considered as independent realizations of the underlying distribution, the generalization of GANs can be analyzed using commonly used empirical process techniques for i.i.d. random variables. GAIL, however, involves dependent demonstration data from experts, and therefore the analysis is more involved. (2) Our computational theory of GAIL can be applied to MMD-GAN and its variants, where the IPM is induced by some reproducing kernel Hilbert space (Li et al., 2017; Bińkowski et al., 2018; Arbel et al., 2018). The alternating mini-batch stochastic gradient algorithm attains a similar sublinear rate of convergence to a stationary solution.
Moreover, our computational theory of GAIL only considers the policy gradient update when learning the policy (Sutton et al., 2000). Extending to other types of updates such as natural policy gradient (Kakade, 2002), proximal policy gradient (Schulman et al., 2017) and trust region policy optimization (Schulman et al., 2015) is a challenging, but important future direction.

heading: A PROOFS IN SECTION 2

heading: A.1 PROOF OF THEOREM 1
text: We first consider n = 1. For notational simplicity, we denote xt = (st, at) and τ = {xt}T−1t=0 . Then the generalization gap is bounded by
dR(π ∗, π̂)− inf
π dR(π
∗, π) = dR(π ∗, π̂)− dR(π∗n, π̂)
+ dR(π ∗ n, π̂)− inf
π dR(π
∗ n, π)
+ inf π dR(π
∗ n, π)− inf
π dR(π
∗, π)
≤ 2 (
sup r∈R
Eπ∗r(s, a)− 1
T T−1∑ t=0 r(st, at)
) + .
Denote Φ(τ) = supr∈R Eπ∗r(s, a)− 1T T−1∑ t=0 r(st, at) = supr∈R Eπ∗r(x)− 1T T−1∑ t=0 r(xt). We utilize the independent block technique first proposed in Yu (1994) to show the concentration of Φ(τ). Specifically, we partition τ into 2m blocks of equal size b. We denote two alternating sequences as
τ0 = (X1, X2, · · · , Xm) Xi = (x(2i−1)b+1, · · · , x(2i−1)b+b) τ1 = (X (1) 1 , X (1) 2 , · · · , X(1)m ) Xi = (x2ib+1, · · · , x2ib+b),
We now define a new sequence
τ̃0 = (X̃1, X̃2, · · · , X̃m), (6)
where X̃’s are i.i.d blocks of size b, and each block X̃i follows the same distribution as Xi. We define rb : X → R as rb(X) = 1b ∑b k=1 r(xk). Note that rb is essentially the average reward on a block. Accordingly, we denoteRb as the set of all rb’s induced by r ∈ R. Before we proceed, we need to introduce a lemma which characterizes the relationship between the expectations of a bounded measurable function with respect to τ0 and τ̃0. Lemma 2. (Yu, 1994) Suppose h is a measurable function bounded by M > 0 defined over the blocks Xi, then the following inequality holds:
|Eτ0(h)− Eτ̃0(h)| ≤ (m− 1)Mβ(b), where Eτ0 denotes the expectation with repect to τ0, and Eτ̃0 denotes the expectation with respect to τ̃0. Corollary 3. Applying Lemma 2, we have
Pτ (Φ(τ) > ) ≤ 2Pτ̃0 (Φ(τ̃0)− Eτ̃0 [Φ(τ̃0)] > − Eτ̃0 [Φ(τ̃0)]) + 2(m− 1)β(b). (7)
Proof. Consider P(Φ(τ) > ),we have
Pτ (Φ(τ) > ) = Pτ (sup r∈R
Eπ∗r(x)− 1
T T∑ t=0 r(xt) > )
≤ Pτ  supr∈R Eπ∗r(x)− 2 T ∑ t∈τ0 r(xt)
2 +
supr∈R Eπ∗r(x)− 2T ∑ t∈τ1 r(xt)
2 >  (8)
= Pτ (Φ(τ0) + Φ(τ1) > 2 ) ≤ Pτ0(Φ(τ0) > ) + Pτ1(Φ(τ1) > ) = 2Pτ0(Φ(τ0) > ) = 2Pτ0 (Φ(τ0)− Eτ̃0 [Φ(τ̃0)] > − Eτ̃0 [Φ(τ̃0)]) ,
where the first inequality (8) follows from the convexity of supremum.
Applying Lemma 2 and setting h = 1 {(Φ(τ̃0)− Eτ̃0 [Φ(τ̃0)] > − Eτ̃0 [Φ(τ̃0)]}, we obtain Pτ0(Φ(τ0)− Eτ̃0 [Φ(τ̃0)] > − Eτ̃0 [Φ(τ̃0)])
≤ Pτ̃0((Φ(τ̃0)− Eτ̃0 [Φ(τ̃0)] > − Eτ̃0 [Φ(τ̃0)]) + 2(m− 1)β(b).
Now since τ̃0 consists of independent blocks, we can apply McDiarmid’s inequality to rb by viewing X̃i’s as i.i.d samples. We rewrite Φ(τ̃0) as
Φ(τ̃0) = sup rb∈Rb
Eπ∗rb(X̃i)− 1
m m∑ i=1 rb(X̃i).
Given samples X̃1, · · · , X̃i, · · · , X̃m and X̃1, · · · , X̃ ′i, · · · , X̃m , we have∣∣∣Φ(X̃1, · · · , X̃i, · · · , X̃m)− Φ(X̃1, · · · , X̃ ′i, · · · , X̃m)∣∣∣ ≤ 2m ∣∣∣rb(X̃i)∣∣∣ ≤ 2Brm . Then by McDiarmid’s inequality, we have
Pτ̃0 (Φ(τ̃0)− Eτ̃0 [Φ(τ̃0)] > − Eτ̃0 [Φ(τ̃0)]) ≤ exp (−m( − Eτ̃0 [Φ(τ̃0)])2
2B2r
) . (9)
Now combining (7) and (9), we obtain Pτ (Φ(τ) > ) ≤ 2 exp (−m( − Eτ̃0 [Φ(τ̃0)])2
2B2r
) + 2(m− 1)β(b). (10)
By the argument of symmetrization, we have Eτ̃0 [Φ(τ̃0)] ≤ 2Eτ̃0,σ [ 1
m sup rb∈Rb m∑ i=1 σirb(X̃i)
] , (11)
where σi’s are i.i.d. Rademacher random variables. Now we relate the Rademacher complexity (11) to its counterpart taking i.i.d samples. Specifically, we denote x(t)j as the j-th point of the t-th block. Denote τ̃ j0 as the collection of the j-th sample from each independent block X̃i for i = 1, . . . ,m. Plugging in the definition of rb, we have
2Eτ̃0,σ  1 m sup r∈R m∑ t=1 σt 1 b b∑ j=1 r(x (t) j )  ≤ 2Eτ̃0,σ 1 b b∑ j=1 1 m sup r∈R m∑ t=1 σtr(x (t) j )  ≤ 2 b b∑ j=1 Eτ̃0,σ [ 1 m sup r∈R m∑ t=1 σtr(x (t) j ) ]
= 2
b b∑ j=1 Eτ̃j0 ,σ
[ 1
m sup r∈R m∑ t=1 σtr(x (t) j )
]
= 2Eτ̃10 ,σ
[ 1
m sup r∈R m∑ t=1 σtr(x (t) 1 )
] . (12)
Setting the right-hand side of (10) to be δ2 and substituting (12), we obtain, with probability at least 1− δ2 , for all r ∈ R,
Φ(τ) ≤ 2Eτ̃10 ,σ [ 1
m sup r∈R m∑ t=1 σtr(x (t) 1 )
] + 2Br √ log 4δ′
2m , (13)
where δ′ = δ − 4(m− 1)β(b).
Then we denote
Rademacher complexity for X̃i’s: RD̃m = Eτ̃10 ,σ
[ 1
m sup r∈R m∑ t=1 σir(x (t) 1 )
] ,
Empirical Rademacher complexity for X̃i’s: R̂τ̃10 = Eσ
[ 1
m sup r∈R m∑ t=1 σir(x̃ (t) 1 )
] ,
Empirical Rademacher complexity for Xi’s: R̂m = Eσ
[ 1
m sup r∈R m∑ t=1 σir(x (t) 1 )
] .
Applying Lemma 2 to the indicator function 1{RD̃m − R̂m > }, we obtain
P(RD̃m − R̂m > ) ≤ P(RD̃m − R̂τ̃10 > ) + (m− 1)β(2b− 1) ≤ P(R D̃ m − R̂τ̃10 > ) + (m− 1)β(b).
It is straightforward to verify that given x(1)1 , . . . , x (i) 1 , . . . , x (m) 1 and x (1) 1 , . . . , x ′(i) 1 , . . . , x (m) 1 , the Rademacher complexity satisfies ∣∣∣R̂′τ̃10 − R̂τ̃10 ∣∣∣ ≤ 2Brm . Then by applying McDiarmid’s Inequality again, we obtain
P ( RD̃m − R̂m > ) ≤ exp (−m 2 2B2r ) + (m− 1)β(b).
Thus with probability at least 1− δ2 , we have
RD̃m − R̂m ≤ 2Br
√ log 1δ/2−(m−1)β(b)
2m ≤ 2Br
√ log 4δ′
2m . (14)
Combining (13) and (14), we have with probability 1− δ,
Φ(τ) ≤ 2R̂m + 6Br
√ log 4δ′
2m . (15)
We apply the Dudley’s entropy integral to bound R̂m. Specifically, we have
R̂m ≤ 4α√ m + 12 m ∫ √mBr α √ logN (R, , ‖ · ‖∞)d
≤ 4α√ m + 12Br√ m
√ logN (R, α, ‖ · ‖∞). (16)
It suffices to pick α = 1√ m . By combining (6), (15), and (16), we have with probability at least 1− δ,
dR(π ∗, π̂)− inf
π dR(π ∗, π) ≤ 16 m + 48Br√ m
√ logN (R, 1/√m, ‖ · ‖∞) + 12Br √ log 4δ′
2m + ,
where δ′ = δ − 4(m− 1)β(b) and 2bm = T . Substituting m = T/2b, we have
dR(π ∗, π̂)− inf
π dR(π ∗, π) ≤ 32b T + 48Br√ T/2b
√ logN (R, 1/ √ T/2b, ‖ · ‖∞) + 12Br √ log 4δ′
T/b + .
(17)
Now we instantiate a choice of b and m for expotentially β-mixing sequences, where the mixing coefficient β(b) ≤ β0 exp(−β1bα) for constants β0, β1, α > 0. We set δ′ > δ− 4(m− 1)β(b) = δ2 .
By a simple calculation, it is enough to choose b = ( log(4β0T/δ)β1 ) 1/α. Substituting such a b into (17). We have with probability at least 1− δ:
dR(π ∗, π̂)− inf
π dR(π
∗, π) ≤ O  Br√ T/ζ √ logN (R, √ ζ T , ‖ · ‖∞) +Br √ log(1/δ) T/ζ + , (18)
where ζ = (β−11 log β0T δ ) 1 α .
When n > 1, we concatenate n trajectories to form a sequence of length nT , and such a sequence is still exponentially β-mixing. Applying the same technique, we partition the whole sequence into 2nm blocks of equal size b. Then with probability at least 1− δ, we have
dR(π ∗, π̂)− inf
π dR(π
∗, π) ≤ O  Br√ nT/ζ √ logN (R, √ ζ nT , ‖ · ‖∞) +Br √ log(1/δ) nT/ζ + .

heading: A.2 PROOF OF COROLLARY 1
text: The reward function can be bounded by
|r(s, a)| = |θ>g(ψs, ψa)| ≤ ‖θ‖2‖g(ψs, ψa)‖2 ≤ √ 2Bθρg,
where the first inequality comes from Cauchy-Schwartz inequality.
To compute the covering number, we exploit the Lipschitz continuity of r(s, a) with respect to parameter θ. Specifically, for two different parameters θ and θ ′ , we have
‖r(s, a)− r′(s, a)‖∞ = ∥∥(θ − θ′)>g(ψs, ψa)∥∥∞
(i) ≤ ‖θ − θ′‖2 sup (s,a)∈S×A
‖g(ψs, ψa)‖2 (ii)
≤ √
2‖θ − θ′‖2ρg sup (s,a)∈S×A
√ ‖ψs‖22 + ‖ψa‖ 2 2 (iii) ≤ √ 2ρg ‖θ − θ′‖2 ,
where (i) comes from Cauchy-Schwartz inequality, (ii) comes from the Lipschitz continuity of g, and (iii) comes from the boundedness of ψs and ψa.
Denote Θ = {θ ∈ Rq : ‖θ‖2 ≤ Bθ}. By the standard argument of the volume ratio, we have N (Θ, , ‖ · ‖2) ≤ ( 1 + 2Bθ )q .
Accordingly, we have
N ( R, √ 2(β−11 log(4β0T/δ)) 1 χ
T , ‖ · ‖∞
) ≤ N ( Θ,
1√ 2ρg
√ 2(β−11 log(4β0T/δ)) 1 χ
T , ‖ · ‖2
)
≤ ( 1 + 2 √
2ρgBθ
√ T
2(β−11 log(4β0T/δ)) 1 χ
)q . (19)
Plugging (19) into (18), we have
dR(π ∗, π̂)− inf
π dR(π
∗, π)
= O  ρgBθ√ T/ζ √√√√q log(ρgBθ √ T ζ ) + ρgBθ √ log(1/δ) T/ζ + hold, with probability at least 1− δ.

heading: A.3 PROOF OF COROLLARY 2
text: We investigate the Lipschitz continuity of r with respect to the weight matrices W1, · · · ,WD. Specifically, given two different sets of matrices W1, · · · ,WD and W ′1, · · · ,W ′D, we have ‖r(s, a)− r′(s, a)‖∞ ≤ ∥∥W>Dσ(WD−1σ(...σ(W1[ψ>a , ψ>s ]>)...))− (W ′D)>σ(W ′D−1σ(...σ(W ′1[ψ>a , ψ>s ]>)...))∥∥2 ≤ ∥∥W>Dσ(WD−1σ(...σ(W1[ψ>a , ψ>s ]>)...))− (W ′D)>σ(WD−1σ(...σ(W1[ψ>a , ψ>s ]>)...))∥∥2 + ∥∥(W ′D)>σ(WD−1σ(...σ(W1[ψ>a , ψ>s ]>)...))− (W ′D)>σ(W ′D−1σ(...σ(W ′1[ψ>a , ψ>s ]>)...))∥∥2
≤‖WD −W ′D‖2 ∥∥σ(WD−1σ(...σ(W1[ψ>a , ψ>s ]>)...))∥∥2
+ ‖W ′D‖2 ∥∥σ(WD−1σ(...σ(W1[ψ>a , ψ>s ]>)...))− σ(W ′D−1σ(...σ(W ′1[ψ>a , ψ>s ]>)...))∥∥2 .
Note that we have∥∥σ(WD−1σ(...σ(W1[ψ>a , ψ>s ]>)...))∥∥2 (i)≤ ∥∥WD−1σ(...σ(W1[ψ>a , ψ>s ]>)...)∥∥2 ≤ ‖WD−1‖2
∥∥σ(...σ(W1[ψ>a , ψ>s ]>)...)∥∥2 (ii)≤ ∥∥[ψ>a , ψ>s ]>∥∥2 (iii)≤ √2, where (i) comes from the definition of the ReLU activation, (ii) comes from ‖Wi‖2 ≤ 1 and recursion, and (iii) comes from the boundedness of ψs and ψa. Accordingly, we have
‖r(s, a)− r′(s, a)‖∞ ≤ √ 2 ‖WD −W ′D‖2 + ‖W ′D‖2 ∥∥σ(WD−1σ(...)− σ(W ′D−1σ(...)∥∥2
(i) ≤ √ 2 ‖WD −W ′D‖2 + ∥∥WD−1σ(...)−W ′D−1σ(...)∥∥2
(ii) ≤ D∑ i=1 √ 2 ‖Wi −W ′i‖2 ,
where (i) comes from the Lipschitz continuity of the ReLU activation, and (ii) comes from the recursion. We then derive the covering number ofR by the Cartesian product of the matrix covering of W1, ...,WD:
N (R, , ‖ · ‖∞) ≤ D∏ i=1 N ( Wi, D √ 2 , ‖ · ‖2 ) ≤ ( 1 + √ 2D √ d )d2D , (20)
where the second inequality comes from the standard argument of the volume ratio. Plugging (20) into (18), we have
dR(π ∗, π̂)− inf
π dR(π
∗, π)
= O  1√ T/ζ √√√√d2D log(D√dT ζ ) + √ log(1/δ) T/ζ + hold, with probability at least 1− δ.

heading: B PROOF OF THEOREM 2
text: The proof is arranged as follows. We first prove the bounded ness of Q fucntion and characterize the Lipschitz properties of the gradients of F with respect to ω and θ, respectively, in Section B.2. Then we provide some important lemmas in Section B.3. Using these lemmas, we prove Lemma 1 in Section B.4. We prove Theorem 2 in Section B.5. For notational simplicity, we denote 〈·, ·〉 as the vector inner product throughout the rest of our analysis.

heading: B.1 BOUNDEDNESS OF Q FUNCTION
text: Lemma 3. For any ω, we have ∥∥∥Qπ̃ω∥∥∥ ∞ ≤ BQ, where BQ = 2 √ 2κρgχ 1−υ .

heading: Proof.
text: Qπ̃ω (s, a) = ∞∑ t=0 E[r̃θ(st, at)− Eπ̃ω r̃θ | s0 = s, a0 = a, π̃ω]
= ∞∑ t=0 [ ∫ S×A r̃θ(s, a)ρ0(s, a)(Pπω ) td(s, a)− ∫ S×A r̃θ(s, a)ρπ̃ω (s, a)d(s, a) ]
= ∞∑ t=0 ∫ S×A r̃θ(s, a)[ρ0(s, a)(Pπω ) t − ρπ̃ω (s, a)]d(s, a)
≤ ∞∑ t=0 2 ‖r̃θ‖∞ ∥∥ρ0(Pπω )t − ρπ̃ω∥∥TV ≤ 2 √
2κρg ∞∑ t=0 χυt = 2 √ 2κρgχ 1− υ ,
where the first inequality comes from the definition of Total Variance distance of probability measures and the second inequality results from (i) of Assumption 5.

heading: B.2 LIPSCHITZ PROPERTIES OF THE GRADIENTS
text: Lemma 4. Suppose that Assumption 1, 3 and 5 hold. For any ω, ω′, θ and θ′, we have
‖∇ωF (ω, θ)−∇ωF (ω′, θ)‖2 ≤ Lω ‖ω − ω′‖2 , ‖∇θF (ω, θ)−∇θF (ω, θ′)‖2 ≤ µ ‖θ − θ′‖2 ,
where Lω = 2 √ 2(Sπ̃+2BωLρ)κρgχ
1−υ +BωLQ.
Proof. By the Policy Gradient Theorem (Sutton et al., 2000), we have
∇ωF (ω, θ) = Eπ̃ω∇ log(π̃ω(a | s))Qπ̃ω (s, a). Therefore,
‖∇ωF (ω, θ)−∇ωF (ω′, θ)‖2 = ∥∥∥Eπ̃ω∇ log(π̃ω(a | s))Qπ̃ω (s, a)− Eπ̃ω′∇ log(π̃ω′(a | s))Qπ̃ω′ (s, a)∥∥∥
2 ≤ ∥∥∥Eπ̃ω∇ log(π̃ω(a | s))Qπ̃ω (s, a)− Eπ̃ω∇ log(π̃ω′(a | s))Qπ̃ω′ (s, a)∥∥∥
2 + ∥∥∥Eπ̃ω∇ log(π̃ω′(a | s))Qπ̃ω′ (s, a)− Eπ̃ω′∇ log(π̃ω′(a | s))Qπ̃ω′ (s, a)∥∥∥
2 ≤ (Sπ̃BQ +BωLQ) ‖ω − ω′‖2 + 2BωBQ ∥∥ρπ̃ω − ρπ̃ω′∥∥TV
≤ (Sπ̃BQ +BωLQ + 2BωBQLρ) ‖ω − ω′‖2 , (21) where the second and the third inequality results from (ii) of Assumption 5. Plugging BQ = 2 √ 2κρgχ 1−υ into (21) yields the desired result.
Similarly, we have
‖∇θF (ω, θ)−∇θF (ω, θ′)‖2 ≤ ‖−µ(θ − θ′)‖2 ≤ µ ‖θ − θ′‖2 .
We then characterize the Lipschitz continuity of∇θF with respect to ω. Lemma 5. Suppose Assumptions 1, 3 and 5 hold. For any ω, ω′ and θ, we have
‖∇θF (ω, θ)−∇θF (ω′, θ)‖2 ≤ Sω‖ω − ω′‖2,
where Sω = 2 √ 2qκρgχBω 1−υ .
Proof. We have ∇θF (ω, θ) = −µθ −∇θ [ Eρπ̃ω [ θ>g(ψst , ψat) ] − Eρ∗ [ θ>g(ψst , ψat) ]] = −µθ − [ Eρπ̃ω [ g(ψst , ψat) ] − Eρ∗ [ g(ψst , ψat) ]] . Therefore, we have ∥∥∥∇θF (ω, θ)−∇θF (ω′, θ)∥∥∥ 2
= ∥∥∥Eρπ̃ω [g(ψst , ψat)]− Eρπ̃ω′ [g(ψst , ψat)]∥∥∥2 ≤ √q max 1≤j≤q
∣∣∣Eρπ̃ω g(ψst , ψat)j − Eρπ̃ω′ g(ψst , ψat)j∣∣∣. (22)

heading: Suppose j∗ = argmax1≤j≤q
text: ∣∣∣Eρπ̃ω g(ψst , ψat)j − Eρπ̃ω′ g(ψst , ψat)j∣∣∣, by Mean Value Theorem, there exists vector ω̃, which is some interpolation between vectors ω and ω′, such that
Eρπ̃ω g(ψst , ψat)j − Eρπ̃ω′ g(ψst , ψat)j = 〈∇ωEρπ̃ω̃ g(ψst , ψat)j , ω − ω ′〉. (23)
By Policy Gradient Theorem, we have∥∥∥∇ωEρπ̃ω̃ g(ψst , ψat)j∥∥∥2 = ∥∥∥Eρπ̃ω̃∇ log π̃ω̃(a | s)Qπ̃ω̃g (s, a)∥∥∥2 ≤ sup
(s,a)∈S×A |∇ log π̃ω̃(a | s)||Qπ̃ω̃g (s, a)|
≤ BQBω, (24) where Qπ̃ω̃g (s, a) = ∑∞ t=0 E[g(st, at)j∗ −Eπ̃ω̃gj∗ | s0 = s, a0 = a, π̃ω̃]. Combining (22), (23), (24) and using Cauchy-Schwartz Inequality, we prove the lemma.

heading: B.3 SOME IMPORTANT LEMMAS FOR PROVING LEMMA 1
text: We denote
ξ (t) θ = ∇θF (ω(t), θ(t))−
1
qθ ∑ j∈M(t)θ ∇θfj(ω(t), θ(t))
and ξ(t)ω = ∇ωF (ω(t), θ(t+1))− 1
qω ∑ j∈M(t)ω ∇ωfj(ω(t), θ(t+1))
as the i.i.d. noise of the stochastic gradient, respectively. Throughout the rest of the analysis, the expectation E is taken with respect to all the noise in each iteration of the alternating mini-batch stochastic gradient descent algorithm. The next lemma characterizes the progress at the (t + 1)-th iteration. For notational simplicity, we define vector function
G(π) = Eρπg(ψs, ψa) (25) Lemma 6. At the (t+ 1)-th iteration, we have
EF (ω(t+1), θ(t+1))− EF (ω(t), θ(t)) ≤ ( Lω − 1
ηω
) E‖ω(t+1) − ω(t)‖22 + Sω 2 · E ∥∥∥ω(t) − ω(t−1)∥∥∥2 2
+ ( 1
2ηθ + Sω 2
+ µ ) E ∥∥∥θ(t+1) − θ(t)∥∥∥2
2 + (
1 2ηθ + µ 2 )E ∥∥∥θ(t) − θ(t−1)∥∥∥2 2
+ ηωE ∥∥∥ξ(t)ω ∥∥∥2
2 +
1 2µ E ∥∥∥ξ(t−1)θ ∥∥∥2 2 .
Proof. We have
EF (ω(t+1), θ(t+1))− EF (ω(t), θ(t)) = EF (ω(t+1), θ(t+1))− EF (ω(t), θ(t+1)) + EF (ω(t), θ(t+1))− EF (ω(t), θ(t)).
By the mean value theorem, we have
〈∇ωF (ω̃(t), θ(t+1)), ω(t+1) − ω(t)〉 = F (ω(t+1), θ(t+1))− F (ω(t), θ(t+1)),
where ω̃(t) is some interpolation between ω(t+1) and ω(t). Then we have
EF (ω(t+1), θ(t+1))− EF (ω(t), θ(t+1)) = E〈∇ωF (ω̃(t), θ(t+1))−∇ωF (ω(t), θ(t+1)), ω(t+1) − ω(t)〉
+ E〈∇ωF (ω(t), θ(t+1)), ω(t+1) − ω(t)〉. (26) By Cauchy- Swartz inequality, we have
E〈∇ωF (ω̃(t), θ(t+1)) −∇ωF (ω(t), θ(t+1)), ω(t+1) − ω(t)〉 ≤ E‖∇ωF (ω̃(t), θ(t+1))−∇ωF (ω(t), θ(t+1))‖2‖ω(t+1) − ω(t)‖2 ≤ LωE‖ω(t+1) − ω(t)‖22,
where the last inequality comes from Lemma 5. Morever, (4) implies
ω(t+1) − ω(t) = −ηω(∇ωF (ω(t), θ(t+1)) + ξ(t)ω ). Therefore, we have
E〈∇ωF (ω(t), θ(t+1)), ω(t+1) − ω(t)〉 =− 1
ηω E‖ω(t+1) − ω(t)‖22 − E〈ξ(t)ω , ω(t+1) − ω(t)〉
=− E〈ξ(t)ω ,−ηω(∇ωF (ω(t), θ(t+1)) + ξ(t)ω )〉
− 1 ηω E‖ω(t+1) − ω(t)‖22
=− 1 ηω
E‖ω(t+1) − ω(t)‖22 + ηωE ∥∥∥ξ(t)ω ∥∥∥2
2 .
Thus, we have
EF (ω(t+1), θ(t+1))− EF (ω(t), θ(t+1)) ≤ (Lω − 1
ηω )E‖ω(t+1) − ω(t)‖22 + ηωE ∥∥∥ξ(t)ω ∥∥∥2 2 . (27)
By (3), the increment of F (ω, θ) takes the form
F (ω(t), θ(t+1))− F (ω(t), θ(t)) = 〈 G(π̃ω(t))−G(π∗), θ(t+1) − θ(t) 〉 − µ
2 (‖θ(t+1)‖22 − ‖θ(t)‖22) ≤ 〈 G(π̃ω(t))−G(π∗)− µθ(t), θ(t+1) − θ(t) 〉 . (28)
For notational simplicity, we define (t+1) = θ(t+1) − ( θ(t) + ηθ ( ∇θF (ω(t), θ(t)) + ξ(t)θ )) . (29)
Note that we have
∇θF (ω(t), θ(t)) = G(π̃ω(t))−G(π∗)− µθ(t). (30) Plugging (29) and (30) into (28), we obtain
F (ω(t), θ(t+1))− F (ω(t), θ(t)) ≤ 〈 θ(t+1) − θ(t) − (t+1)
ηθ − ξ(t)θ , θ(t+1) − θ(t)
〉 .
Since θ belongs to the convex set {θ | ‖θ‖2 ≤ κ}, we have
〈 (t), θ(t+1) − θ(t)〉 ≥ 0.
Then we obtain F (ω(t), θ(t+1))− F (ω(t), θ(t))
≤ 1 ηθ 〈θ(t+1) − θ(t) + (t) − (t+1), θ(t+1) − θ(t)〉 − 〈ξ(t)θ , θ(t+1) − θ(t)〉
= 1 ηθ 〈 (t) − (t+1), θ(t+1) − θ(t)〉+ 1 ηθ ∥∥∥θ(t+1) − θ(t)∥∥∥2 2 − 〈ξ(t)θ , θ(t+1) − θ(t)〉.
(31)
By the definition of (t) in (29), we have (t+1) = θ(t+1) − ( θ(t) + ηθ ( ∇θF (ω(t), θ(t)) + ξ(t)θ )) (32)
and
(t) = θ(t) − ( θ(t−1) + ηθ ( ∇θF (ω(t−1), θ(t−1)) + ξ(t−1)θ )) . (33)
Subtracting (33) from (32),we obtain (t) − (t+1) = (θ(t) − θ(t+1))− (θ(t−1) − θ(t))− ηθ ( ∇θF (ω(t−1), θ(t−1))−∇θF (ω(t), θ(t)) ) − ηθ(ξ(t−1)θ − ξ (t) θ ). (34)
Plugging (34) into the first term on the right hand side of (31), we obtain 1
ηθ 〈 (t) − (t+1), θ(t+1) − θ(t)〉
= 1
ηθ 〈θ(t) − θ(t−1), θ(t+1) − θ(t)〉︸ ︷︷ ︸
(A)
+ 〈∇θF (ω(t), θ(t))−∇θF (ω(t−1), θ(t−1)), θ(t+1) − θ(t)︸ ︷︷ ︸ (B) 〉
− 1 ηθ ∥∥∥θ(t+1) − θ(t)∥∥∥2 2 + 〈ξ(t)θ − ξ (t−1) θ , θ (t+1) − θ(t)〉. (35)
For term (A), we apply the Cauchy-Schwarz inequality to obtain an upper bound as follows. 1
ηθ 〈θ(t) − θ(t−1), θ(t+1) − θ(t)〉 ≤ 1 ηθ ∥∥∥θ(t) − θ(t−1)∥∥∥ 2 · ∥∥∥θ(t+1) − θ(t)∥∥∥ 2
≤ 1 2ηθ ∥∥∥θ(t) − θ(t−1)∥∥∥2 2 + 1 2ηθ ∥∥∥θ(t+1) − θ(t)∥∥∥2 2 . (36)
To derive the upper bound of (B), we apply Lemma 5 to obtain
〈∇θF (ω(t), θ(t))−∇θF (ω(t−1), θ(t−1)), θ(t+1) − θ(t)〉 = 〈∇θF (ω(t), θ(t))−∇θF (ω(t−1), θ(t)), θ(t+1) − θ(t)〉
+ 〈∇θF (ω(t−1), θ(t))−∇θF (ω(t−1), θ(t−1)), θ(t+1) − θ(t)〉 ≤ Sω ∥∥∥ω(t) − ω(t−1)∥∥∥ 2 · ∥∥∥θ(t+1) − θ(t)∥∥∥ 2 + µ · ∥∥∥θ(t) − θ(t−1)∥∥∥ 2 · ∥∥∥θ(t+1) − θ(t)∥∥∥ 2 ≤ Sω 2 ∥∥∥ω(t) − ω(t−1)∥∥∥2 2 + Sω 2 · ∥∥∥θ(t+1) − θ(t)∥∥∥2 2
+ µ 2 · ∥∥∥θ(t) − θ(t−1)∥∥∥2 2 + µ 2 · ∥∥∥θ(t+1) − θ(t)∥∥∥2 2 . (37)
Plugging (36) and (37) into (35), we obtain 1
ηθ 〈 (t) − (t+1), θ(t+1) − θ(t)〉 ≤
( − 1
2ηθ + µ 2 + Sω 2 )∥∥∥θ(t+1) − θ(t)∥∥∥2 2
+
( 1
2ηθ + µ 2 )∥∥∥θ(t) − θ(t−1)∥∥∥2 2
+ Sω 2 ∥∥∥ω(t) − ω(t−1)∥∥∥2 2 + 〈ξ(t)θ − ξ (t−1) θ , θ (t+1) − θ(t)〉. (38)
Further plugging (38) into (31), we obtain
F (ω(t), θ(t+1))− F (ω(t), θ(t)) ≤ ( 1
2ηθ + Sω 2 + µ 2 )∥∥∥θ(t+1) − θ(t)∥∥∥2 2 + ( 1 2ηθ + µ 2 )∥∥∥θ(t) − θ(t−1)∥∥∥2 2
+ Sω 2 ∥∥∥ω(t) − ω(t−1)∥∥∥2 2 − 〈ξ(t−1)θ , θ(t+1) − θ(t)〉
≤ ( 1
2ηθ + Sω 2 + µ )∥∥∥θ(t+1) − θ(t)∥∥∥2 2 + ( 1 2ηθ + µ 2 )∥∥∥θ(t) − θ(t−1)∥∥∥2 2
+ Sω 2 ∥∥∥ω(t) − ω(t−1)∥∥∥2 2 +
∥∥∥ξ(t−1)θ ∥∥∥2 2
2µ . (39)
Finally, taking expectation of (39) with respect to the noise and together with (27) , we prove the final result.
EF (ω(t+1), θ(t+1))− EF (ω(t), θ(t)) ≤ ( Lω − 1
ηω
) E‖ω(t+1) − ω(t)‖22 + Sω 2 E ∥∥∥ω(t) − ω(t−1)∥∥∥2 2
+
( 1
2ηθ + Sω 2 + µ
) E ∥∥∥θ(t+1) − θ(t)∥∥∥2
2
+
( 1
2ηθ + µ 2
) E ∥∥∥θ(t) − θ(t−1)∥∥∥2
2 + ηωE ∥∥∥ξ(t)ω ∥∥∥2 2 + 1 2µ E ∥∥∥ξ(t−1)θ ∥∥∥2 2 .
We then characterize the update of ω.
Lemma 7. The update of ω satisfies
E 〈 ω(t+1) − ω(t) − (ω(t) − ω(t−1)), ω(t+1) − ω(t) 〉 ≤− ηω
ηθ · E 〈 (θ(t+2) − θ(t+1))− (θ(t+1) − θ(t))− ( (t+2) − (t+1)), θ(t+1) − θ(t) 〉
− µηω 2 · E ∥∥∥θ(t+1) − θ(t)∥∥∥2 2 + ηω(5Lω + 1) 2 · E ∥∥∥ω(t+1) − ω(t)∥∥∥2 2 + ηωLω 2 · ∥∥∥ω(t) − ω(t−1)∥∥∥2 2 + ( ηω 2µ + η2ω)E ∥∥∥ξ(t)ω ∥∥∥2 2 + ηω 2 E ∥∥∥ξ(t−1)ω ∥∥∥2 2 .
Proof. By the update policy, we have
E 〈 ω(t+1) − ω(t) − (ω(t) − ω(t−1)), ω(t+1) − ω(t) 〉 =− ηωE 〈 ∇ωF (ω(t), θ(t+1)) + ξ(t)ω −∇ωF (ω(t−1), θ(t))− ξ(t−1)ω , ω(t+1) − ω(t)
〉 =− ηωE 〈 ∇ωF (ω(t), θ(t+1))−∇ωF (ω(t), θ(t)), ω(t+1) − ω(t)
〉 − ηωE 〈 ∇ωF (ω(t), θ(t))−∇ωF (ω(t−1), θ(t)), ω(t+1) − ω(t)
〉 − ηωE 〈 ξ(t)ω ,−ηω(∇ωF (ω(t), θ(t+1)) + ξ(t)ω ) 〉 + ηωE 〈 ξ(t−1)ω , ω (t+1) − ω(t) 〉
≤−ηωE 〈 ∇ωF (ω(t), θ(t+1))−∇ωF (ω(t), θ(t)), ω(t+1) − ω(t) 〉 ︸ ︷︷ ︸
(C) −ηωE 〈 ∇ωF (ω(t), θ(t))−∇ωF (ω(t−1), θ(t)), ω(t+1) − ω(t) 〉 ︸ ︷︷ ︸
(D) + η2ωE ∥∥∥ξ(t)ω ∥∥∥2 2 + ηω 2 E ∥∥∥ξ(t−1)ω ∥∥∥2 2 + ηω 2 E ∥∥∥ω(t+1) − ω(t)∥∥∥2 2 . (40)
Then for (C), by the definition of objective function, we have
(C) =− ηωE 〈 ∇ω ∑ j G(π̃ω(t))j(θ (t+1) − θ(t))j , ω(t+1) − ω(t) 〉
=− ηωE ∑ j 〈 ∇ωG(π̃ω(t))j −∇ωG(π̃ω̃(t)j )j +∇ωG(π̃ω̃(t)j )j), ω (t+1) − ω(t) 〉 ·
(θ(t+1) − θ(t))j =− ηωE ∑ j (θ(t+1) − θ(t))j 〈 ∇ωG(π̃ω(t))j −∇ωG(π̃ω̃(t)j )j), ω (t+1) − ω(t) 〉
− ηωE ∑ j (θ(t+1) − θ(t))j 〈 ∇ωG(π̃ω̃(t)j )j), ω (t+1) − ω(t) 〉 , (41)
where for every j, π̃ ω̃
(t) j
is some interpolation between vectors π̃ω(t) and π̃ω(t+1) such that
〈 ∇ωG(π̃ω̃(t)j )j , ω (t+1) − ω(t) 〉 = G(π̃ω(t+1))j −G(π̃ω(t))j . (42)
For the first term in the right hand side of (41), by Lemma 4, we have
−ηωE ∑ j (θ(t+1) − θ(t))j 〈 ∇ωG(π̃ω(t))j −∇ωG(π̃ω̃(t)j )j), ω (t+1) − ω(t) 〉
≤2ηωLωE ∥∥∥ω(t+1) − ω(t)∥∥∥2
2 . (43)
For the second term in the right hand side of (41), by (42) we have
− ηωE ∑ j (θ(t+1) − θ(t))j 〈 ∇ωG(π̃ω̃(t)j )j , ω (t+1) − ω(t) 〉
=− ηωE 〈 G(π̃ω(t+1))−G(π̃ω(t)), θ(t+1) − θ(t) 〉 .
Then by the defintion of objective function, we have − ηωE ∑ j (θ(t+1) − θ(t))j 〈 ∇ωG(π̃ω̃(t)j )j , ω (t+1) − ω(t) 〉
=− ηωE 〈 1 ηθ ( θ(t+2) − θ(t+1) − (t+2) ) − ξ(t+1)θ + µθ(t+1)
− 1 ηθ
( θ(t+1) − θ(t) − (t+1) ) + ξ
(t) θ − µθ(t), θ(t+1) − θ(t) 〉 =− ηω
ηθ E 〈 (θ(t+2) − θ(t+1))− (θ(t+1) − θ(t))− ( (t+2) − (t+1)), θ(t+1) − θ(t) 〉
+ ηωE〈ξ(t+1)θ − ξ (t) θ , θ (t+1) − θ(t)〉 − µηωE ∥∥∥θ(t+1) − θ(t)∥∥∥2
2
≤− ηω ηθ
E 〈 (θ(t+2) − θ(t+1))− (θ(t+1) − θ(t))− ( (t+2) − (t+1)), θ(t+1) − θ(t) 〉
− µηω 2
E ∥∥∥θ(t+1) − θ(t)∥∥∥2
2 + ηω 2µ
E ∥∥∥ξ(t)θ ∥∥∥2
2 . (44)
For (D), applying Cauchy-Schwartz inequality we have
−ηωE〈∇ωF (ω(t), θ(t))−∇ωF (ω(t−1), θ(t)), ω(t+1) − ω(t)〉 ≤ηωLωE ∥∥∥ω(t+1) − ω(t)∥∥∥ 2 ∥∥∥ω(t) − ω(t−1)∥∥∥ 2
≤ηωLω 2
E ∥∥∥ω(t+1) − ω(t)∥∥∥2
2 + ηωLω 2 E ∥∥∥ω(t) − ω(t−1)∥∥∥2 2 . (45)
Finally, combining (40)-(45), we prove Lemma 7.
For notational simplicity, we define
δ(t+2) = (θ(t+2) − θ(t+1))− (θ(t+1) − θ(t)). (46) Lemma 8. The first term on the right hand side of Lemma (7) satisfies
− ηω ηθ
E 〈 (θ(t+2) − θ(t+1))− (θ(t+1) − θ(t))− ( (t+2) − (t+1)), θ(t+1) − θ(t) 〉
(47)
≤− ηω 2ηθ
E ∥∥∥θ(t+2) − θ(t+1)∥∥∥2
2 +
( 3µ2ηωηθ
2 + ηω 2ηθ
) E ∥∥∥θ(t+1) − θ(t)∥∥∥2
2
+ 3ηωηθS
2 ω 2 · E ∥∥∥ω(t+1) − ω(t)∥∥∥2 2 + 3 2 ηωηθ ( E ∥∥∥ξ(t+1)θ ∥∥∥2 2 + E ∥∥∥ξ(t)θ ∥∥∥2 2 ) . (48)
Proof. Plugging (46) into (47), we obtain
− ηω ηθ
E 〈 (θ(t+2) − θ(t+1))− (θ(t+1) − θ(t))− ( (t+2) − (t+1)), θ(t+1) − θ(t) 〉
= ηω ηθ
E 〈 δ(t+2) − ( (t+2) − (t+1)), δ(t+2) − (θ(t+2) − θ(t+1)) 〉 = ηω ηθ E 〈 δ(t+2) − ( (t+2) − (t+1)), δ(t+2)) 〉 − ηω ηθ E 〈 δ(t+2), θ(t+2) − θ(t+1)
〉 + ηω ηθ E 〈 ( (t+2) − (t+1)), θ(t+2) − θ(t+1) 〉 . (49)
By applying the equality
〈u, v〉 = 1 2 (‖u‖22 + ‖v‖ 2 2 − ‖u− v‖ 2 2)
to the first two terms on the right hand side of (49), we obtain
− ηω ηθ
E 〈 (θ(t+2) − θ(t+1))− (θ(t+1) − θ(t))− ( (t+2) − (t+1)), θ(t+1) − θ(t) 〉
= ηω 2ηθ
E (∥∥∥δ(t+2) − ( (t+2) − (t+1))∥∥∥2 2 + ∥∥∥δ(t+2)∥∥∥2 2 − ∥∥∥ (t+2) − (t+1)∥∥∥2 2 ) − ηω
2ηθ E (∥∥∥δ(t+2)∥∥∥2 2 + ∥∥∥θ(t+2) − θ(t+1)∥∥∥2 2 − ∥∥∥θ(t+1) − θ(t)∥∥∥2 2 ) + ηω ηθ E〈 (t+2) − (t+1), θ(t+2) − θ(t+1)〉. (50)
Recall that
θ(t+2) = Πκ ( θ(t+1) + ηθ ( ∇θF (ω(t+1)θ(t+1)) + ξ(t+1)θ )) ,
and
θ(t+1) = Πκ ( θ(t) + ηθ ( ∇θF (ω(t), θ(t)) + ξ(t)θ )) .
Following from the convexity of {θ| ‖θ‖2 ≤ κ}, we have
〈 (t+2), θ(t+2) − θ(t+1)〉 ≤ 0 and 〈 (t+1), θ(t+2) − θ(t+1)〉 ≥ 0. Thus, the last term on the right side of (50) is negative. By rearranging the terms in (50), we obtain
− ηω ηθ
E 〈 (θ(t+2) − θ(t+1))− (θ(t+1) − θ(t))− ( (t+2) − (t+1)), θ(t+1) − θ(t) 〉
≤ ηω 2ηθ · E ∥∥∥δ(t+2) − ( (t+2) − (t+1))∥∥∥2 2 − ηω 2ηθ E ∥∥∥θ(t+2) − θ(t+1)∥∥∥2 2 + ηω 2ηθ E ∥∥∥θ(t+1) − θ(t)∥∥∥2 2 .
(51)
By definition of δ(t+2) in (46), we have
δ(t+2) − ( (t+2) − (t+1)) = (θ(t+2) − θ(t+1) − (t+2))− (θ(t+1) − θ(t) − (t+1)) = ηθ[G(π̃ω(t+1))−G(π̃ω(t))− µ(θ(t+1) − θ(t)) + ξ(t+1)θ − ξ (t) θ ].
Using the Cauchy-Schwarz inequality, we obtain ηω 2ηθ · E ∥∥∥δ(t+2) − ( (t+2) − (t+1))∥∥∥2 2
≤ 3ηωηθ 2 · ( E ‖G(π̃ω(t+1))−G(π̃ω(t))‖22 + E ∥∥∥µ(θ(t+1) − θ(t))∥∥∥2 2 + E ∥∥∥ξ(t+1)θ − ξ(t)θ ∥∥∥2 2 ) ≤ 3ηωηθS 2 ω 2 · E ∥∥∥ω(t+1) − ω(t)∥∥∥2 2 + 3 2 µ2ηωηθ · E ∥∥∥θ(t+1) − θ(t)∥∥∥2 2 + 3ηωηθ 2 · E ∥∥∥ξ(t+1)θ − ξ(t)θ ∥∥∥2 2 .
(52)
Plugging (52) into (51) yields (48), which concludes the proof of Lemma 8.
Lemma 9. For the update of ω, we have
1 2 · E ∥∥∥ω(t+1) − ω(t)∥∥∥2 2 − 1 2 · E ∥∥∥ω(t) − ω(t−1)∥∥∥2 2
≤ ( ηω(5Lω + 1)
2 +
3ηωηθS 2 ω
2
) E ∥∥∥ω(t+1) − ω(t)∥∥∥2
2 + ηωLω 2 · E ∥∥∥ω(t) − ω(t−1)∥∥∥2 2
− ηω 2ηθ · E ∥∥∥θ(t+2) − θ(t+1)∥∥∥2 2 + ( 3 2 ηωηθµ 2 + ηω 2ηθ − µηω 2 ) E ∥∥∥θ(t+1) − θ(t)∥∥∥2 2
+ (η2ω + ηω 2µ
)E ∥∥∥ξ(t)ω ∥∥∥2 2 + ηω 2 · E ∥∥∥ξ(t−1)ω ∥∥∥2 2 + 3 2 ηωηθ · ( E ∥∥∥ξ(t+1)θ ∥∥∥2 2 + E ∥∥∥ξ(t)θ ∥∥∥2 2 ) .
Proof. Combing Lemma 7 and Lemma 8, we obtain
E〈ω(t+1) − ω(t) − (ω(t) − ω(t−1)), ω(t+1) − ω(t)〉
≤ − ηω 2ηθ
E ∥∥∥θ(t+2) − θ(t+1)∥∥∥2
2 +
( 3
2 ηωηθµ 2 + ηω 2ηθ
) E ∥∥∥θ(t+1) − θ(t)∥∥∥2
2
+ 3ηωηθS
2 ω 2 E ∥∥∥ω(t+1) − ω(t)∥∥∥2 2 − µηω 2 E ∥∥∥θ(t+1) − θ(t)∥∥∥2 2
+ 2ηωLωE ∥∥∥ω(t+1) − ω(t)∥∥∥2
2 + ηω(Lω + 1) 2 E ∥∥∥ω(t+1) − ω(t)∥∥∥2 2 + ηωLω 2 E ∥∥∥ω(t) − ω(t−1)∥∥∥2 2
+ (η2ω + ηω 2µ
)E ∥∥∥ξ(t)ω ∥∥∥2 2 + ηω 2 E ∥∥∥ξ(t−1)ω ∥∥∥2 2 + 3 2 ηωηθ · (E ∥∥∥ξ(t+1)θ ∥∥∥2 2 + E ∥∥∥ξ(t)θ ∥∥∥2 2 ). (53)
Note that we have
E〈ω(t+1) − ω(t) − (ω(t) − ω(t−1)), ω(t+1) − ω(t)〉
= 1 2 E ∥∥∥ω(t+1) − ω(t)∥∥∥2 2 − 1 2 E ∥∥∥ω(t) − ω(t−1)∥∥∥2 2 + 1 2 E ∥∥∥(ω(t+1) − ω(t))− (ω(t) − ω(t−1))∥∥∥2 2
≥ 1 2 E ∥∥∥ω(t+1) − ω(t)∥∥∥2 2 − 1 2 E ∥∥∥ω(t) − ω(t−1)∥∥∥2 2 . (54)
Combining (53) and (54), we conclude the proof of Lemma 9.
Lemma 10. Suppose Assumption 1, 3 and 5 hold. F (ω(t), θ(t)) is lower bounded throughout all iterations.
Proof. By definition of F in (2), we have
F (ω(t), θ(t)) ≥ Eπ̃ ω(t) r̃θ(t)(s, a)− Eπ∗ r̃θ(t)(ψs, ψa)− µ 2 ‖θ‖22 − λBH
≥ − ( 2 √ 2ρgκ+ µ
2 κ2 + λBH
) . (55)

heading: B.4 PROOF OF LEMMA 1
text: Recall that we construct a potential function that dacays monotonically along the solution path, which takes the form
E(t+1) = EF (ω(t+1), θ(t+1)) + s · (1 + 2ηωLω 2 E ∥∥∥ω(t+1) − ω(t)∥∥∥2 2
+ ( ηω 2ηθ − µηω 4 + 3 2 ηωηθµ 2 ) E ∥∥∥θ(t+2) − θ(t+1)∥∥∥2 2 + µηω 8 E ∥∥∥θ(t+1) − θ(t)∥∥∥2 2 ) . (56)
for some constant s > 0. We define five constants k1, k2, k3, k4, k5 as
k1 = 1
2ηω − s · (ηω(7Lω + 1) 2 + 3ηωηθS 2 ω 2 ) , k2 = s · ηωLω 2 − Sω 2 ,
k3 = s · ( ηωµ
4 − 3 2 ηωηθµ 2
) , k5 = s · µηω 8 − ( 1 2ηθ + µ 2 ) ,
k4 = s · µηω 8 − ( 1 2ηθ + Sω + 2µ 2 ) .
Here, we restate Lemma 1 and then prove it.
Lemma 1. We choose step sizes ηθ, ηω satisfying ηω ≤ min {
Lω Sω(8Lω + 2) , 1 2Lω
} , ηθ ≤ min { 1
150µ ,
7Lω + 1
150S2ω ,
1
100(2µ+ Sω)
} ,
and meanwhile ηω/ηθ ≤ µ/(30Lω + 5), where Lω = √ 2κρgSπ|A|+ λSH . Then we have
E(t+1) − E(t) ≤− k1E ∥∥∥ω(t+1) − ω(t)∥∥∥2
2 − k2E ∥∥∥ω(t) − ω(t−1)∥∥∥2 2 − k3E ∥∥∥θ(t+2) − θ(t+1)∥∥∥2 2
− k4E ∥∥∥θ(t+1) − θ(t)∥∥∥2
2 − k5E ∥∥∥θ(t) − θ(t−1)∥∥∥2 2
+ ηωE ∥∥∥ξ(t)ω ∥∥∥2
2 +
1 2 E ∥∥∥ξ(t−1)θ ∥∥∥2 2
+ s · ( η2ωE ∥∥∥ξ(t)ω ∥∥∥2 2 + ηω 2 E ∥∥∥ξ(t−1)ω ∥∥∥2 2 + 3ηωηθ 2 · (E ∥∥∥ξ(t+1)θ ∥∥∥2 2 + E ∥∥∥ξ(t)θ ∥∥∥2 2 ) ) . (57)
Moreover, we have k1, k2, k3, k4, k5 > 0 for
s = 8
η2ω(58Lω + 9) .
Proof. For notational simplicity, we define
K(t+1) = 1 + 2ηωLω 2 E ∥∥∥ω(t+1) − ω(t)∥∥∥2 2
+ ( ηω 2ηθ − µηω 4 +
3ηωηθµ 2
2 ) · E ∥∥∥θ(t+2) − θ(t+1)∥∥∥2 2 + µηω 8 E ∥∥∥θ(t+1) − θ(t)∥∥∥2 2 . (58)
By rearranging the inequality in Lemma (9), we obtain K(t+1) −K(t) ≤ (ηω(7Lω + 1)
2 +
3ηωηθS 2 ω
2
) E ∥∥∥ω(t+1) − ω(t)∥∥∥2
2
− ηωLω 2
E ∥∥∥ω(t) − ω(t−1)∥∥∥2 2 − (µηω 4 − 3µ 2ηωηθ 2 ) E ∥∥∥θ(t+2) − θ(t+1)∥∥∥2 2
− µηω 8
E ∥∥∥θ(t+1) − θ(t)∥∥∥2
2 − µηω 8 E ∥∥∥θ(t) − θ(t−1)∥∥∥2 2
+ (η2ω + ηω 2µ
)E ∥∥∥ξ(t)ω ∥∥∥2 2 + ηω 2 E ∥∥∥ξ(t−1)ω ∥∥∥2 2 + 3ηωηθ 2 · (E ∥∥∥ξ(t+1)θ ∥∥∥2 2 + E ∥∥∥ξ(t)θ ∥∥∥2 2 ). (59)
By definition of P (t) in (56), we have E(t) = F (ω(t), θ(t)) + s ·K(t)
for some constant s > 0. Combining (59) and Lemma 6, since ηω < 12Lω , we obtain E(t+1) − E(t) ≤ − ( 1
2ηω − s · (ηω(7Lω + 1) 2 + 3ηωηθS 2 ω 2 )) ∥∥∥ω(t+1) − ω(t)∥∥∥2 2
− ( s · ηωLω
2 − Sω 2 )∥∥∥ω(t) − ω(t−1)∥∥∥2 2 − s · (µηω 4 − 3µ 2ηωηθ 2 )∥∥∥θ(t+2) − θ(t+1)∥∥∥2 2
− ( s · µηω 8 − ( 1 2ηθ + 2µ+ Sω 2 )) ∥∥∥θ(t+1) − θ(t)∥∥∥2 2
− ( s · µηω 8 − ( 1 2ηθ + µ 2 )) ∥∥∥θ(t) − θ(t−1)∥∥∥2 2
+ ηωE ∥∥∥ξ(t)ω ∥∥∥2
2 +
1 2µ E ∥∥∥ξ(t−1)θ ∥∥∥2 2
+ s · (
(η2ω + ηω 2µ
)E ∥∥∥ξ(t)ω ∥∥∥2 2 + ηω 2 E ∥∥∥ξ(t−1)ω ∥∥∥2 2 + 3ηωηθ 2 · (E ∥∥∥ξ(t+1)θ ∥∥∥2 2 + E ∥∥∥ξ(t)θ ∥∥∥2 2 ) .
(60)
Since ηθ < 1150µ , we have k3 > 0. Now we choose a proper constant s such that k1, k2, k4, k5 are positive. Note that they are positive if and only if
1/(2ηω) > s · ((ηω(7Lω + 1)/2 + 3ηωηθS
2 ω
2 )), (61)
s · ηωLω/2 > Sω/2, (62) s · (µηω/8) > 1/(2ηθ) + (2µ+ Sω)/2, (63) s · (µηω)/8 > 1/(2ηθ) + µ/2. (64)
Rearranging the terms in (61), (62), (63), (64) , we obtain Sω ηωLω < s < 1/(2ηω) ηω(7Lω + 1 + 3ηθS2ω)/2 , (65)
1/(2ηθ) + (Sω + 2µ)/2
µηω/8 < s <
1/(2ηω)
ηω(7Lω + 1 + 3ηθS2ω)/2 . (66)
Since ηθ ≤ 7Lω + 1
150S2ω and ηθ <
1
100(2µ+ Sω) ,
by rearranging the terms in (65) and (66) and taking the leading terms, we obtain
ηω < Lω
Sω(8Lω + 2) and ηω ηθ < µ 30Lω + 5 .
Therefore, we have ηω ηθ < µ 30Lω + 5 , ηω < ηω and ηθ < ηθ, where
ηω = min
{ Lω
Sω(8Lω + 2) ,
1
2Lω
} ,
ηθ = min
{ 1
150µ ,
7Lω + 1
150S2ω ,
1
100(2µ+ Sω)
} .

heading: B.5 PROOF OF THEOREM 2
text: Let k = 1/min{k1, k4}, and φ = max{1, 1/η2ω, 1/η2θ}. Then we have
NJN ≤ N∑ t=1 φ · E( ∥∥∥ω(t+1) − ω(t)∥∥∥2 2 + ∥∥∥θ(t+1) − θ(t)∥∥∥2 2 )
≤ φk ( N∑ t=1 (E(t) − E(t+1)) + N∑ t=1 ( (ηω + sη 2 ω + sηω 2µ )E ∥∥∥ξ(t)ω ∥∥∥2 2 + 1 2µ · E ∥∥∥ξ(t−1)θ ∥∥∥2 2
+ s · (ηω/2 · E ∥∥∥ξ(t−1)ω ∥∥∥2
2 + 3ηωηθ/2 · (E ∥∥∥ξ(t+1)θ ∥∥∥2 2 + E ∥∥∥ξ(t)θ ∥∥∥2 2 ) ))
≤ φk ( (E(1) − E(N)) + N∑ t=1 ( 2 max { ηω + sη 2 ω + sηω 2µ , sηω 2 } E ∥∥∥ξ(t)ω ∥∥∥2 2
+ 3 max
{ 1
2µ , 3ηωηθ 2
} E ∥∥∥ξ(t)θ ∥∥∥2
2
)) . (67)
Now set
ν = max { 2 max { ηω + sη 2 ω +
sηω 2µ , sηω 2
} , 3 max { 1
2µ , 3ηωηθ 2 }} and divide both sides of (67) by N , we have
JN ≤ kφ(E(1) − E(N))
N + kφν ( MG qω + Mθ qθ ) . (68)
By definition of E in (56) and Lemma 10, we have E(N) ≥ F (ω(N), θ(N)) ≥ − ( 2 √ 2ρgκ+ µ
2 κ2 + λBH
) > −∞.
Now for any given > 0, we take
qθ = 4kφνMθ , qω = 4kφνMω and N = kφ 2E(1) + 4
√ 2ρgκ+ µκ 2 + 2λBH
,
and obtain JN ≤ .

heading: C EMPIRICAL MAXIMIZER CASE
text: Notice that the object function (2) is quadratic in θ, and thus we are able to get the empirical maximizer by simple calculation instead of performing stochastic gradient ascent at each iteration. This allows us to avoid using large batches while achieving the same sample complexity Õ( 1 2 ).
Given a fixed ω(t), by definition of G in (25), we can get the optimal θ∗(ω(t)) in population form:
θ∗(ω(t)) = 1
µ [G(πω(t))−G(π∗)]. (69)
At the (t+ 1)-th iteration, we can achieve the closed form of the empirical maximizer by sampling one trajectory randomly and then apply stochastic gradient descent to the outer minimization. More specifically, the updating rule is defined as
θ̃(t+1)(ω(t)) = 1
µ [G̃(πω(t))− G̃(π∗)], (70)
ω(t+1) = ω(t) − ηω∇ω f̃t(ω(t), θ̃(t+1)), (71) where G̃ is the empirical version of G obtained by sampling, and ∇ω f̃ is the stochastic approximation of ∇ωF at the (t+ 1)-the iteration. We define the stationary point as follow. Definition 4. We call ω∗ an stationary point if ∇ωF (ω∗, θ∗(ω∗)) = 0.
Before we proceed with the convergence analysis, we impose the following assumptions on the problem.
Assumption 6. There is some constantMG > 0 s.t. for any ω, θ̃ and π, the following two conditions hold.
Unbiased : E∇ω f̃t(ω, θ̃) = ∇ωF (ω, θ̃),EG̃(π) = EG(π). Gradient bounded : E ∥∥∥∇ω f̃t(ω, θ)∥∥∥2
2 ≤MG.
Assumption 6 requires the stochastic gradient to be unbiased with bounded second moment.
Corollary 4. Under Assumption 1, 3 and 5, there exists BF = 12ρ2g µ +λBH such that for any ω, we have |F (ω, θ∗(ω))| < BF .
Proof. By Equation (69),we have ‖θ∗(ω)‖2 ≤ 2 √ 2ρg µ . Plugging this into F (ω, θ
∗(ω)), we have for any ω,
|F (ω, θ∗(ω))| = |Eπ̃ω 〈θ∗(ω), g(ψst , ψat)〉 − Eπ∗ 〈θ∗(ω), g(ψst , ψat)〉 − λH(π̃ω(t))− µ
2 ‖θ∗(ω)‖22 |
≤ 2 ‖θ∗(ω)‖2 ·maxs,a ‖g(ψs, ψa)‖2 + λH(π̃ω(t)) + µ 2 ‖θ∗(ω)‖22
≤ 12ρ 2 g
µ + λMH .
For notational simplicity, We define ξ(t)ω = ∇θF (ω(t), θ̃(t+1)) − ∇θf̃t(ω(t), θ̃(t+1)) as the i.i.d stochastic noise and σ(t)θ = θ
∗(ω(t))− θ̃(t) as the i.i.d error of empirical maximizer. We denote E as the expectation over ξ(t)ω (t ≥ 1) and σ(t)θ (t ≥ 1). We measure the sub-stationarity of the algotithm at the iteration N by
IN = min 1≤t≤N
E ∥∥∥∇ωF (ω(t), θ∗(ω(t)))∥∥∥2
2 .
Then we state the global convergence of the above mentioned optimization method.
Theorem 3. Suppose Assumptions 1, 3, 5, 6 hold. Given any > 0, we take ηω = √
2(2Lω+S2ω/µ)MG , then we need at most
N = Õ
( (ρ2g/µ+ λBH)(Lω + S 2 ω/µ)MG
2 ) iterations to have IN < .
Proof. By employing the first inequality in Lemma 4, we have
F (ω(t+1), θ∗(ω(t)))− F (ω(t), θ∗(ω(t)))− 〈∇ωF (ω(t), θ∗(ω(t))), ω(t+1) − ω(t)〉
≤Lω 2 ∥∥∥ω(t+1) − ω(t)∥∥∥2 2 . (72)
Note that
E〈∇ωF (ω(t), θ∗(ω(t))), ω(t+1) − ω(t)〉 = E〈∇ωF (ω(t), θ∗(ω(t))),−ηω(∇ωF (ω(t), θ̃(t)) + ξ(t)ω )〉 = E〈∇ωF (ω(t), θ∗(ω(t))),−ηω(∇ωF (ω(t), θ∗(ω(t))− σiθ) + ξ(t)ω )〉 (i) = E〈∇ωF (ω(t), θ∗(ω(t))),−ηω(∇ωF (ω(t), θ∗(ω(t))) + ξ(t)ω )〉 (ii) = E〈∇ωF (ω(t), θ∗(ω(t))),−ηω∇ωF (ω(t), θ∗(ω(t)))〉
= −ηωE ∥∥∥∇ωF (ω(t), θ∗(ω(t)))∥∥∥2
2 , (73)
where (i) comes from the unbiased property of θ̃(t) and the fact that ∇ωF (ω, θ) is linear in θ, and (ii) comes from the unbiased property of f̃j(ω, θ̃). Now taking the expectation on both sides of (72) and plugging (73) in, we obtain
EF (ω(t+1), θ∗(ω(t)))− F (ω(t), θ∗(ω(t))) + ηωE ∥∥∥∇ωF (ω(t), , θ∗(ω(t))∥∥∥2
2 ≤ Lω 2 η2ωMG. (74)
Dividing both sides by ηω and rearranging the terms in (74), we get E ∥∥∥∇ωF (ω(t), θ∗(ω(t)))∥∥∥2
2 ≤EF (ω (t), θ∗(ω(t)))− F (ω(t+1), θ∗(ω(t))) ηω + Lω 2 ηωMG
≤EF (ω (t), θ∗(ω(t)))− F (ω(t+1), θ∗(ω(t+1)))
ηω + Lω 2 ηωMG
+ EF (ω(t+1), θ∗(ω(t+1)))− F (ω(t+1), θ∗(ω(t)))
ηω . (75)
Now consider F (ω(t+1), θ∗(ω(t+1)))− F (ω(t+1), θ∗(ω(t))), we have F (ω(t+1), θ∗(ω(t+1)))− F (ω(t+1), θ∗(ω(t)))
= 〈G(πω(t+1))−G(π∗), θ∗(ω(t+1))〉 − 〈G(πω(t+1))−G(π∗), θ∗(ω(t))〉
− µ 2
( ∥∥∥θ∗(ω(t+1))∥∥∥2 2 − ∥∥∥θ∗(ω(t))∥∥∥2 2 )
= 〈µθ∗(ω(t+1)), θ∗(ω(t+1))− θ∗(ω(t))〉 − µ
2 〈θ∗(ω(t+1)) + θ∗(ω(t)), θ∗(ω(t+1))− θ∗(ω(t))〉
= µ
2 ∥∥∥θ∗(ω(t+1))− θ∗(ω(t)∥∥∥2 2
= µ
2 ∥∥∥∥ 1µ (G(πω(t+1))−G(πω(t))) ∥∥∥∥2 2
≤ S 2 ω
2µ ∥∥∥ω(t+1) − ω(t)∥∥∥2 2 . (76)
Taking expectation on both sides of (76) with respect to the noise introduced by SGD, we have
EF (ω(t+1), θ∗(ω(t+1)))− F (ω(t+1), θ∗(ω(t))) ≤ S 2 ω
2µ η2ωMG.
Summing the equation(75) up, we have
N∑ t=1 E ∥∥∥∇ωF (ω(t), θ∗(ω(t)))∥∥∥2 2
≤ 1 ηω N∑ i=1 EF (ω(t), θ∗(ω(t)))− F (ω(t+1), θ∗(ω(t))) + Lω 2 NηωMG
≤ 1 ηω N∑ i=1 EF (ω(t), θ∗(ω(t)))− F (ω(t+1), θ∗(ω(t+1)))
+ 1
ηω N∑ i=1 EF (ω(t+1), θ∗(ω(t+1)))− F (ω(t+1), θ∗(ω(t))) + Lω 2 NηωMG
≤ 1 ηω N∑ i=1 EF (ω(t), θ∗(ω(t)))− F (ω(t+1), θ∗(ω(t+1)))
+ 1
ηω N∑ i=1 S2ω 2µ η2ωMG + Lω 2 NηωMG.
Dividing both sides of the above equation by N , we get
min 1≤t≤N
E ∥∥∥∇ωF (ω(t), θ∗(ω(t)))∥∥∥2
2 ≤|F (ω (1), θ∗(ω(1)))− EF (ω(N+1), θ∗(ω(N+1)))| Nηω
+ ( Lω 2 + S2ω 2µ )ηωMG.
Since |F (ω(1), θ∗(ω(1))) − EF (ω(N+1), θ∗(ω(N+1)))| ≤ 2BF . Take ηω = √
2BF (Lω/2+S2ω/2)MGN , then we have
min 1≤t≤N
E ∥∥∥∇ωF (ω(t), θ∗(ω(t)))∥∥∥2 2 ≤ 4 √ BF (Lω + S2ω/µ)MG N ,
where BF = 12ρ2g µ + λMH . This implies that when ηω =
√ 2(2Lω+S2ω/µ)MG , we need at most
N = Õ
( (ρ2g/µ+ λMH)(Lω + S 2 ω/µ)MG
2 ) such that IN < .

Review summaries:
This paper presents theoretical analysis of the generalization properties of GAIL, as well as the local convergence of the traditional minibatch SGD applied to its min-max optimization problem (without assuming convex-concave structure of the game).  Specifically, the authors first prove a generalization bound for GAIL that characterizes how well optimizing the empirical GAIL objective minimizes the true population version (in terms of an "R-distance" they use the characterize the expected distributions induced by the underlying policies for a given reward class).  Second, they prove a convergence result for minibatch SGD applied to the min-max game, showing that the method will converge to a stationary point regardless of any convex-concave structure.