Generative Adversarial Imitation Learning with Neural Networks: Global Optimality and Convergence Rate

We consider a Markov decision process (MDP) $({\mathcal{S}},\mathcal{A},r,P,\rho,\gamma)$. Here ${\mathcal{S}}\subseteq\mathbb{R}^{d_{1}}$ is the state space, $\mathcal{A}\subseteq\mathbb{R}^{d_{2}}$ is the action space, which is assumed to be finite throughout this paper, $r:{\mathcal{S}}\times\mathcal{A}\rightarrow\mathbb{R}$ is the reward function, $P:{\mathcal{S}}\times\mathcal{A}\rightarrow{\mathcal{P}}({\mathcal{S}})$ is the transition kernel, $\rho\in{\mathcal{P}}({\mathcal{S}})$ is the initial state distribution, and $\gamma\in(0,1)$ is the discount factor. Without loss of generality, we assume that ${\mathcal{S}}\times\mathcal{A}$ is compact and that $\|(s,a)\|_{2}\leq 1$ for any $(s,a)\in{\mathcal{S}}\times\mathcal{A}\subseteq\mathbb{R}^{d}$, where $d=d_{1}+d_{2}$. An agent following a policy $\pi:{\mathcal{S}}\rightarrow{\mathcal{P}}(\mathcal{A})$ interacts with the environment in the following manner. At the state $s_{t}\in{\mathcal{S}}$, the agent takes the action $a_{t}\in\mathcal{A}$ with probability $\pi(a_{t}\,|\,s_{t})$ and receives the reward $r_{t}=r(s_{t},a_{t})$. The environment then transits into the next state $s_{t+1}$ with probability $P(s_{t+1}\,|\,s_{t},a_{t})$. Given a policy $\pi$ and a reward function $r(s,a)$, we define the state-action value function $Q^{\pi}_{r}:{\mathcal{S}}\times\mathcal{A}\rightarrow\mathbb{R}$ as follows,

Here the expectation $\mathbb{E}_{\pi}$ is taken with respect to $a_{t}\sim\pi(\cdot\,|\,s_{t})$ and $s_{t+1}\sim P(\cdot\,|\,s_{t},a_{t})$. Correspondingly, we define the state value function $V^{\pi}_{r}:{\mathcal{S}}\rightarrow\mathbb{R}$ and the advantage function $A^{\pi}_{r}:{\mathcal{S}}\times\mathcal{A}\rightarrow\mathbb{R}$ as follows,

The goal of RL is to maximize the following expected cumulative reward,

The policy $\pi$ induces a state visitation measure $d_{\pi}\in{\mathcal{P}}({\mathcal{S}})$ and a state-action visitation measure $\nu_{\pi}\in{\mathcal{P}}({\mathcal{S}}\times\mathcal{A})$, which take the forms of

It then holds that $J(\pi;r)=\mathbb{E}_{(s,a)\sim\nu_{\pi}}[r(s,a)]$. Meanwhile, we assume that the policy $\pi$ induces a state stationary distribution $\varrho_{\pi}$ over ${\mathcal{S}}$, which satisfies that

We denote by $\rho_{\pi}(s,a)=\varrho(s)\cdot\pi(a\,|\,s)$ the state-action stationary distribution over ${\mathcal{S}}\times\mathcal{A}$.

The goal of imitation learning (IL) is to learn a policy that outperforms the expert policy $\pi_{\text{{\tiny E}}}$ based on the trajectory $\{(s_{i}^{\text{{\tiny E}}},a_{i}^{\text{{\tiny E}}})\}_{i\in[T_{\text{{\tiny E}}}]}$ of $\pi_{\text{{\tiny E}}}$. We denote by $\nu_{\text{{\tiny E}}}=\nu_{\pi_{\text{{\tiny E}}}}$ and $d_{\text{{\tiny E}}}=d_{\pi_{\text{{\tiny E}}}}$ the state-action and state visitation measures induced by the expert policy, respectively, and assume that the expert trajectory $\{(s_{i},a_{i})\}_{i\in[T_{\text{{\tiny E}}}]}$ is drawn from $\nu_{{\text{{\tiny E}}}}$. To this end, we parameterize the policy and the reward function by $\pi_{\theta}$ for $\theta\in{\mathcal{X}}_{\Pi}$ and $r_{\beta}(s,a)$ for $\beta\in{\mathcal{X}}_{R}$, respectively, and solve the following minimax optimization problem known as GAIL (Ho and Ermon, 2016),

Here $J(\pi;r)$ is the expected cumulative reward defined in (2.2), $\psi:{\mathcal{X}}_{R}\rightarrow\mathbb{R}_{+}$ is the regularizer, and $\lambda\geq 0$ is the regularization parameter. Given a reward function class $\mathcal{R}$, we define the $\mathcal{R}$-distance between two policies $\pi_{1}$ and $\pi_{2}$ as follows,

When $\mathcal{R}$ is the class of $1$-Lipschitz functions, $\mathbb{D}_{\mathcal{R}}(\pi_{1},\pi_{2})$ is the Wasserstein-$1$ metric between the state-action visitation measures induced by $\pi_{1}$ and $\pi_{2}$. However, $\mathbb{D}_{\mathcal{R}}(\pi_{1},\pi_{2})$ is not a metric in general. When $\mathbb{D}_{\mathcal{R}}(\pi_{1},\pi_{2})\leq 0$, the policy $\pi_{2}$ outperforms the policy $\pi_{1}$ for any reward function $r\in\mathcal{R}$. Such a notion of $\mathcal{R}$-distance is used in Chen et al. (2020). We denote by $\mathcal{R}_{\beta}=\{r_{\beta}(s,a)\,|\,\beta\in{\mathcal{X}}_{R}\}$ the reward function class parameterized with $\beta$. Hence, the optimization problem in (2.4) minimizes the $\mathcal{R}_{\beta}$-distance $\mathbb{D}_{\mathcal{R}_{\beta}}(\pi_{\text{{\tiny E}}},\pi_{\theta})$ (up to the regularizer $\lambda\cdot\psi(\beta)$), which searches for a policy $\bar{\pi}$ that (approximately) outperforms the expert policy given any reward function $r_{\beta}\in\mathcal{R}_{\beta}$.

We define a two-layer neural network with rectified linear units (ReLU) as follows,

Here $m\in\mathbb{N}_{+}$ is the width of the neural network, $b=(b_{1},\ldots,b_{m})^{\top}\in\mathbb{R}^{m}$ and $W=([W]_{1}^{\top},\ldots,[W]_{m}^{\top})^{\top}\in\mathbb{R}^{md}$ are the parameters, and $\phi_{W,b}(s,a)=([\phi_{W,b}(s,a)]^{\top}_{1},\ldots,[\phi_{W,b}(s,a)]^{\top}_{m})^{\top}\in\mathbb{R}^{md}$ is called the feature vector in the sequel, where

It then holds that $u_{W,b}(s,a)=W^{\top}\phi_{W,b}(s,a)$. Note that the feature vector $\phi_{W,b}(s,a)$ depends on the parameters $W$ and $b$. We consider the following random initialization,

Throughout the training process, we keep the parameter $b$ fixed while updating $W$. For notational simplicity, we write $u_{W,b}(s,a)$ as $u_{W}(s,a)$ and $\phi_{W,b}(s,a)$ as $\phi_{W}(s,a)$ in the sequel. We denote by $\mathbb{E}_{\text{init}}$ the expectation taken with respect to the random initialization in (3.3). For an absolute constant $B>0$, we define the parameter domain as

which is the ball centered at $W_{0}$ with the domain radius $B$.

In the sequel, we consider the following energy-based policy,

where $\tau\geq 0$ is the inverse temperature parameter and $u_{\theta}(s,a)$ is the energy function parameterized by the neural network defined in (3.1) with $W=\theta$. In the sequel, we call $\theta$ the policy parameter. Meanwhile, we parameterize the reward function $r_{\beta}(s,a)$ as follows,

where $u_{\beta}(s,a)$ is the neural network defined in (3.1) with $W=\beta$ and $\gamma$ is the discount factor. Here we use the scaling parameter $(1-\gamma)^{-1}$ to ensure that for any policy $\pi$ the state-action value function $Q^{\pi}_{r_{\beta}}(s,a)$ defined in (2.1) is well approximated by the neural network defined in (3.1). In the sequel, we call $\beta$ the reward parameter and define the reward function class as

where $S_{B_{\beta}}$ is the parameter domain of $\beta$ defined in (3.4) with domain radius $B_{\beta}$. To facilitate algorithm design, we establish the following proposition, which calculates the explicit expressions of the gradients $\nabla L(\theta,\beta)$ and the Fisher information $\mathcal{I}(\theta)$. Recall that the Fisher information is defined as

Note that the expression of the policy gradient $\nabla_{\theta}L(\theta,\beta)$ in (3.10) of Proposition 3.1 involves the state-action value function $Q^{\pi_{\theta}}_{r_{\beta}}(s,a)$. To this end, we estimate the state-action value function $Q_{r}^{\pi}(s,a)$ by $\widehat{Q}_{\omega}(s,a)$, which is parameterized as follows,

Here $u_{\omega}(s,a)$ is the neural network defined in (3.1) with $W=\omega$. In the sequel, we call $\omega$ the value parameter.

We employ an actor-critic scheme with alternating updates of the policy and the reward function, which is presented in Algorithm 1. Specifically, we update the policy parameter $\theta$ via natural policy gradient and update the reward parameter $\beta$ via gradient ascent in the actor step, while we estimate the state-action value function $Q^{\pi}_{r}(s,a)$ via neural temporal difference (TD) (Cai et al., 2019c) in the critic step.

To approximately obtain the compatible function approximation (Sutton et al., 2000; Wang et al., 2019), we share the random initialization among the policy $\pi_{\theta}$, the reward function $r_{\beta}(s,a)$, and the state-action value function $\widehat{Q}_{\omega}(s,a)$. In other words, we set $\theta_{0}=\beta_{0}=\omega(0)=W_{0}$ in our algorithm, where $W_{0}$ is the random initialization in (3.3). The output of GAIL is the mixed policy $\bar{\pi}$ (Altman, 1999). Specifically, the mixed policy $\bar{\pi}$ of $\pi_{0},\ldots,\pi_{T-1}$ is executed by randomly selecting a policy $\pi_{k}$ for $k\in[0:T-1]$ with equal probability before time $t=0$ and exclusively following $\pi_{k}$ thereafter. It then holds for any reward function $r(s,a)$ that

We impose the following assumptions on the stationary distributions $\varrho_{k}\in{\mathcal{P}}({\mathcal{S}}),\rho_{k}\in{\mathcal{P}}({\mathcal{S}}\times\mathcal{A})$ and the visitation measures $d_{k}\in{\mathcal{P}}({\mathcal{S}}),\nu_{k}\in{\mathcal{P}}({\mathcal{S}}\times\mathcal{A})$.

Assumption (a) (a) holds when the probability density functions of $\rho_{k}$ and $\nu_{k}$ are uniformly upper bounded across $k$. Assumption (b) (b) states that the concentrability coefficients are uniformly upper bounded across $k$, which is commonly used in the analysis of RL (Szepesvári and Munos, 2005; Munos and Szepesvári, 2008; Antos et al., 2008; Farahmand et al., 2010; Scherrer et al., 2015; Farahmand et al., 2016; Lazaric et al., 2016).

For notational simplicity, we write $u_{0}(s,a)=u_{W_{0}}(s,a)$ and $\phi_{0}(s,a)=\phi_{W_{0}}(s,a)$, where $u_{W_{0}}(s,a)$ is the neural network defined in (3.1) with $W=W_{0}$, $\phi_{W_{0}}(s,a)$ is the feature vector defined in (3.2) with $W=W_{0}$, and $W_{0}$ is the random initialization in (3.3). We impose the following assumptions on the neural network $u_{0}(s,a)$ and the transition kernel $P$.

Assumption 4.2 (b) states that the MDP belongs to (a variant of) the class of linear MDPs (Yang and Wang, 2019a, b; Jin et al., 2019; Cai et al., 2019b). However, our class of transition kernels is infinite-dimensional, and thus, captures a rich class of MDPs. To understand Assumption 4.2 (a), recall that we initialize the neural network with $[W_{0}]_{l}\sim N(0,I_{d}/d)$ and $b_{l}\sim{\text{Unif}}(\{-1,1\})$ for any $l\in[m]$. Thus, the neural network $u_{0}(s,a)$ defined in (3.1) with $W=W_{0}$ converges to a Gaussian process indexed by $(s,a)\in{\mathcal{S}}\times\mathcal{A}$ as $m$ goes to infinity. Following from the facts that the maximum of a Gaussian process over a compact index set is sub-Gaussian (van de Geer and Muro, 2014) and that ${\mathcal{S}}\times\mathcal{A}$ is compact, it is reasonable to assume that $\sup_{(s,a)\in{\mathcal{S}}\times\mathcal{A}}|u_{0}(s,a)|$ is sub-Gaussian, which further implies the existence of the absolute constants $M_{0}$ and $v$ in (4.1) of Assumption 4.2 (a). Moreover, if we assume that $m$ is even and initialize the parameters $W_{0},b$ as follows,

we have that $u_{0}(s,a)=0$ for any $(s,a)\in{\mathcal{S}}\times\mathcal{A}$, which allows us to set $M_{0}=0$ and $v=+\infty$ in Assumption (a) (a). Also, it holds that $0=u_{0}(s,a)\in\mathcal{R}_{\beta}$, which implies that $\mathbb{D}_{\mathcal{R}_{\beta}}(\pi_{1},\pi_{2})\geq 0$ for any $\pi_{1}$ and $\pi_{2}$. The proof of our results with the random initialization in (4.2) is identical.

Finally, we impose the following assumption on the regularizer $\psi(\beta)$ and the variances of the estimators $\widehat{\mathcal{I}}(\theta)$, $\widehat{\nabla}_{\theta}L(\theta,\beta)$, and $\widehat{\nabla}_{\beta}L(\theta,\beta)$ defined in (3.16), (3.17), and (3.18), respectively.

Assumption 4.3 (a) holds when $\widehat{Q}_{\omega_{k}}(s_{i},a_{i})\cdot\iota_{\theta_{k}}(s_{i},a_{i})$, $\iota_{\theta_{k}}(s_{i},a_{i})\iota_{\theta_{k}}(s_{i},a_{i})^{\top}$, and $\phi_{\beta_{k}}(s_{i},a_{i})$ have uniformly upper bounded variances across $i\in[m]$ and $k$, and the Markov chain that generates $\{(s_{i},a_{i})\}_{i\in[N]}$ mixes sufficiently fast (Zhang et al., 2019a). Similar assumptions are also used in the analysis of policy optimization (Xu et al., 2019a, b). Also, Assumption 4.3 (b) holds for most commonly used regularizers (Ho and Ermon, 2016).