agametheoreticframeworkformodelbasedreinforcementlearning/full.md

A Game Theoretic Framework for Model Based Reinforcement Learning

Aravind Rajeswaran $^{1,2}$ Igor Mordatch $^{1}$ Vikash Kumar $^{1}$

Abstract

Designing stable and efficient algorithms for model-based reinforcement learning (MBRL) with function approximation has remained challenging despite growing interest in the field. To help expose the practical challenges in MBRL and simplify algorithm design from the lens of abstraction, we develop a new framework that casts MBRL as a game between: (1) a policy player, which attempts to maximize rewards under the learned model; (2) a model player, which attempts to fit the real-world data collected by the policy player. We show that a near-optimal policy for the environment can be obtained by finding an approximate equilibrium for aforementioned game, and we develop two families of algorithms to find the game equilibrium by drawing upon ideas from Stackelberg games. Experimental studies suggest that the proposed algorithms achieve state of the art sample efficiency, match the asymptotic performance of model-free policy gradient, and scale gracefully to high-dimensional tasks like dexterous hand manipulation. Project page: https:// sites.google.com/view/mbrl-game.

1. Introduction

We study the problem of model-based reinforcement learning (MBRL) where a world model is learned from data to aid policy search. Model-based algorithms can incorporate historical off-policy data and generic priors like knowledge of physics, making them highly sample efficient. In addition, the learned models can also be re-purposed to solve new tasks. As a result, there has been a recent surge of interest in MBRL. However, a clear algorithmic framework to understand MBRL and unify insights from recent works has been lacking. To bridge this gap, and to facilitate the

¹Google Brain, Mountain View, USA. ²University of Washington, Seattle, USA. Work performed at Google Brain.. Correspondence to: Aravind Rajeswaran aravraj@cs.washington.edu.

Proceedings of the $37^{th}$ International Conference on Machine Learning, Online, PMLR 119, 2020. Copyright 2020 by the author(s).

design of stable and efficient algorithms, we develop a new framework that casts MBRL as a two-player game.

Classical frameworks for MBRL, adaptive control (Åström & Murray, 2004), and dynamic programming (Puterman, 1994), are often confined to simple linear models or tabular representations. They also rely on building global models through ideas like persistent excitation (Narendra & Annaswamy, 1987) or tabular generative models (Kearns & Singh, 1998a). Such settings and assumptions are often limiting for modern applications. To obtain a globally accurate model, we need the ability to collect data from all parts of the state space (Agarwal et al., 2019b), which is often impossible. Furthermore, learning globally accurate models may be unnecessary, unsafe, and inefficient. For example, to make an autonomous car drive on the road, we should not require accurate models in situations where it tumbles and crashes in different ways. This motivates a class of incremental methods for MBRL that interleave policy and model learning to gradually construct and refine models in the task-relevant parts of the state space. This is in sharp contrast to a two-stage approach of first building a model of the world, and subsequently planning in it.

A unifying framework for incremental MBRL can connect insights from different approaches and help simplify the algorithm design process from the lens of abstraction. As an example, distribution or domain shift is known to be a major challenge for incremental MBRL. When improving the policy using the learned model, the policy will attempt to shift the distribution over visited states. The learned model may be inaccurate for this modified distribution, resulting in a greatly biased policy update. A variety of approaches have been developed to mitigate this issue. One class of approaches (Levine & Abbeel, 2014; Sun et al., 2018a; Kakade & Langford, 2002), inspired by trust region methods, make conservative changes to the policy to constrain the distribution between successive iterates. In sharp contrast, an alternate set of approaches do not constrain the policy updates in any way, but instead rely on data aggregation to mitigate distribution shift (Ross & Bagnell, 2012; Chua et al., 2018; Nagabandi et al., 2019). Our game-theoretic framework for MBRL reveals that these two seemingly disparate approaches are essentially dual approaches to solve the same game.

Our Contributions:

1. We develop a novel framework that casts MBRL as a game between: (a) a policy player, which maximizes rewards in the learned model; and (b) a model player, which minimizes prediction error of data collected by policy player. Theoretically, we establish that at equilibrium, the policy is near-optimal for the environment.
2. Developing learning algorithms for general continuous games is well known to be challenging. To develop stable and convergent algorithms, we setup a Stackelberg game (von Stackelberg, 1934) between the two players, which can be solved efficiently through (approximate) bi-level optimization.
3. Stackelberg games are asymmetric games where players make decisions in a pre-specified order. The leader plays first and subsequently the follower. Due to the asymmetric nature, the MBRL game can take two forms depending on choice of leader player. This gives rise to two natural families of algorithms that have complementary strengths. Together, they unify and generalize many prior MBRL algorithms.
4. Experimentally, we show that our algorithms outperform prior model-based and model-free algorithms in sample efficiency; match the asymptotic performance of model-free policy gradient algorithms; and scale gracefully to high-dimensional tasks like dexterous manipulation.

2. Background and Notations

We treat the environment as an infinite horizon MDP characterized by: $M = {\mathcal{S},\mathcal{A},\mathcal{R},P,\gamma ,\rho }$ . Per usual notation, $\mathcal{S}\subseteq \mathbb{R}^n$ and $\mathcal{A}\subseteq \mathbb{R}^{m}$ represent the continuous state and action spaces. The transition dynamics is described by $s^\prime \sim P(\cdot |s,a)$ . $\mathcal{R}:S\to [0,R_{\max}]$ , $\gamma \in [0,1)$ , and $\rho$ represents the reward, discount, and initial state distribution respectively. Policy is a mapping from states to a probability distribution over actions, i.e. $\pi :S\rightarrow P(\mathcal{A})$ , and in practice we typically consider parameterized policies. The goal is to optimize the objective:

$$\begin{array}{cccc}& \underset{\mathit{\pi }}{\max }J(\mathit{\pi },\mathit{M}):={\mathbb{E}}_{\mathit{M},\mathit{\pi }}\left[\sum _{t=0}^{\mathrm{\infty }}{\gamma }^t\mathcal{R}\left(s_t\right)\right]& & \text{(1)}\end{array}\backslash max\; \_\; \{\backslash boldsymbol\; \{\backslash pi\}\}\; J\; (\backslash boldsymbol\; \{\backslash pi\},\; \backslash boldsymbol\; \{M\})\; :=\; \backslash mathbb\; \{E\}\; \_\; \{\backslash boldsymbol\; \{M\},\; \backslash boldsymbol\; \{\backslash pi\}\}\; \backslash left[\; \backslash sum\_\; \{t\; =\; 0\}\; ^\; \{\backslash infty\}\; \backslash gamma^\; \{t\}\; \backslash mathcal\; \{R\}\; \backslash left(s\; \_\; \{t\}\backslash right)\; \backslash right]\; \backslash tag\; \{1\}$$

Model-free methods solve this optimization by directly estimating a gradient using collected samples or through value functions. Model-based methods, in contrast, construct an explicit world model to aid policy optimization.

2.1. Model-Based Reinforcement Learning

We represent the world model with another tuple: $\widehat{M} = {S, \mathcal{A}, \mathcal{R}, \widehat{P}, \gamma, \rho}$ . The model has the same state-action space, reward function, discount, and initial state

distribution. We parameterize the transition dynamics of the model $\widehat{P}$ (as a neural network) and learn the parameters so that it approximates the environment transition dynamics, i.e. $\widehat{P} \approx P$ . For simplicity, we assume that the reward function and initial state distribution are known. This is a benign assumption for many applications in control, robotics, and operations research. If required, these quantities can also be learned from data, and are typically easier to learn than $\widehat{P}$ . Enormous quantities of experience can be cheaply generated by simulating the model, without interacting with the world, and can be used for policy optimization. Thus, model-based methods tend to be sample efficient.

Idealized Global Model Setting To motivate challenges in MBRL, we first consider the idealized setting of an approximate global model. This corresponds to the case where $\widehat{M}$ is sufficiently expressive and approximates $M$ everywhere. Lemma 1 relates the performance of a policy in the model and environment.

Lemma 1. (Simulation Lemma) Suppose $\widehat{M}$ is such that $D_{TV}\left(P(\cdot |s,a),\widehat{P} (\cdot |s,a)\right)\leq \epsilon_M\forall (s,a)$ . Then, for any policy $\pi$ , we have

$$\begin{array}{cccc}& \mid J(\pi ,M)-J(\pi ,\widehat{M})\mid \le O\left(\frac{{ϵ}_M}{(1-\gamma {)}^2}\right)\mathrm{\forall }\pi \mathrm{.}& & \text{(2)}\end{array}\backslash left|\; J\; (\backslash pi\; ,\; M)\; -\; J\; (\backslash pi\; ,\; \backslash widehat\; \{M\})\; \backslash right|\; \backslash leq\; O\; \backslash left(\backslash frac\; \{\backslash epsilon\_\; \{M\}\}\{(1\; -\; \backslash gamma)\; ^\; \{2\}\}\backslash right)\; \backslash quad\; \backslash forall\; \backslash pi\; .\; \backslash tag\; \{2\}$$

The proof is provided in the appendix. Since Lemma 1 provides a uniform bound applicable to all policies, we can expect good performance in the environment by optimizing the policy in the model, i.e. $\max_{\pi} J(\pi, \widehat{M})$ .

Beyond global models A global modeling approach as above is often impractical. To obtain a globally accurate model, we need the ability to collect data from all parts of the state space (Agarwal et al., 2019b; Pathak et al., 2017), which can be difficult. More importantly, learning globally accurate models may be unnecessary, unsafe, and inefficient. For example, to make a robot walk, we should not require accurate models in situations where it falls and crashes in different ways. This motivates the need for incremental MBRL, where models are gradually constructed and refined in the task-relevant parts of the state space. To formalize this intuition, we consider the below notion of model quality.

Definition 1. (Model approximation loss) Given $\widehat{M}$ and distribution $\mu(s, a)$ , the model approximation loss is

$$\begin{array}{cccc}& \mathrm{\ell }(\widehat{\mathit{M}},\mu )={\mathbb{E}}_{(s,a)\sim \mu }\left[D_{KL}(P(\cdot \mathrm{\mid }s,a),\widehat{P}(\cdot \mathrm{\mid }s,a))\right]\mathrm{.}& & \text{(3)}\end{array}\backslash ell\; (\backslash widehat\; \{\backslash boldsymbol\; \{M\}\},\; \backslash mu)\; =\; \backslash mathbb\; \{E\}\; \_\; \{(s,\; a)\; \backslash sim\; \backslash mu\}\; \backslash left[\; D\; \_\; \{K\; L\}\; \backslash left(P\; (\backslash cdot\; |\; s,\; a),\; \backslash widehat\; \{P\}\; (\backslash cdot\; |\; s,\; a)\backslash right)\; \backslash right].\; \backslash tag\; \{3\}$$

We use $D_{KL}$ to refer to the KL divergence which can be optimized using samples from $M$ , and is closely related to $D_{TV}$ through Pinsker's inequality. In the case of isotropic Gaussian distributions, as typically considered in continuous control applications, $D_{KL}$ reduces to the familiar $\ell_2$ loss. Importantly, the loss is intimately tied to the sampling distribution $\mu$ . In general, models that are accurate in some parts

of the state space need not generalize/transfer to other parts. As a result, a more conservative policy learning procedure is required, in contrast to the global model case.

3. Model Based RL as a Two Player Game

In order to capture the interactions between model and policy learning, we formulate MBRL as the following two-player general sum game (ref. as MBRL game)

$$\begin{array}{cccc}& \stackrel{\text{p o l i c y - p l a y e r}}{\overbrace{\underset{\pi }{\max }J(\pi ,\widehat{M})}},\stackrel{\text{m o d e l - p l a y e r}}{\overbrace{\underset{\widehat{M}}{\min }\mathrm{\ell }(\widehat{M},{\mu }_M^{\pi })}}& & \text{(4)}\end{array}\backslash overbrace\; \{\backslash max\; \_\; \{\backslash pi\}\; J\; (\backslash pi\; ,\; \backslash widehat\; \{M\})\}\; ^\; \{\backslash text\; \{p\; o\; l\; i\; c\; y\; -\; p\; l\; a\; y\; e\; r\}\},\; \backslash overbrace\; \{\backslash min\; \_\; \{\backslash widehat\; \{M\}\}\; \backslash ell\; (\backslash widehat\; \{M\}\; ,\; \backslash mu\_\; \{M\}\; ^\; \{\backslash pi\})\}\; ^\; \{\backslash text\; \{m\; o\; d\; e\; l\; -\; p\; l\; a\; y\; e\; r\}\}\; \backslash tag\; \{4\}$$

We use $\mu_{M}^{\pi} = \frac{1}{T}\sum_{t=0}^{T}P(s_{t} = s,a_{t} = a)$ to denote the average state visitation distribution. The policy player maximizes performance in the learned model, while the model player minimizes prediction error under policy player's induced state distribution. This is a game since the objective of each player depends on the parameters of both players.

The above formulation separates MBRL into the constituent components of policy learning (planning) and generative model learning. At the same time, it exposes that the two components are closely intertwined and must be considered together in order to succeed in MBRL. We discuss algorithms for solving the game in Section 4, and first focus on the equilibrium properties of the MBRL game. Our results establish that at (approximate) Nash equilibrium of the MBRL game: (1) the model can accurately simulate and predict the performance of the policy; (2) the policy is near-optimal.

Theorem 1. (Global perf. of equilibrium pair; informal) Suppose we have a pair of policy and model, $(\pi, \widehat{M})$ , such that simultaneously

$$\mathrm{\ell }(\widehat{M},{\mu }_M^{\pi })\le {ϵ}_{M}andJ(\pi ,\widehat{M})\ge J({\pi }^{\mathrm{\prime }},\widehat{M})-{ϵ}_{\pi }\mathrm{\forall }{\pi }^{\mathrm{\prime }}\mathrm{.}\backslash ell\; (\backslash widehat\; \{M\},\; \backslash mu\_\; \{M\}\; ^\; \{\backslash pi\})\; \backslash leq\; \backslash epsilon\_\; \{M\}\; a\; n\; d\; J\; (\backslash pi\; ,\; \backslash widehat\; \{M\})\; \backslash geq\; J\; (\backslash pi^\; \{\backslash prime\},\; \backslash widehat\; \{M\})\; -\; \backslash epsilon\_\; \{\backslash pi\}\; \backslash forall\; \backslash pi^\; \{\backslash prime\}.$$

For an optimal policy $\pi^{*}$ , we have

$$\begin{array}{cccc}& \begin{array}{c}J({\mathit{\pi }}^{\ast },\mathit{M})-J(\mathit{\pi },\mathit{M})\le \\ O({ϵ}_{\pi }+\frac{\sqrt{{ϵ}_M}}{(1-\gamma {)}^2}+\frac{1}{1-\gamma }{D}_{TV}({\mu }_M^{{\pi }^{\ast }},{\mu }_{\widehat{M}}^{{\pi }^{\ast }}))\mathrm{.}\end{array}& & \text{(5)}\end{array}\backslash begin\{array\}\{l\}\; J\; (\backslash boldsymbol\; \{\backslash pi\}\; ^\; \{*\},\; \backslash boldsymbol\; \{M\})\; -\; J\; (\backslash boldsymbol\; \{\backslash pi\},\; \backslash boldsymbol\; \{M\})\; \backslash leq\; \backslash \backslash \; O\; \backslash left(\backslash epsilon\_\; \{\backslash pi\}\; +\; \backslash frac\; \{\backslash sqrt\; \{\backslash epsilon\_\; \{M\}\}\}\{(1\; -\; \backslash gamma)\; ^\; \{2\}\}\; +\; \backslash frac\; \{1\}\{1\; -\; \backslash gamma\}\; D\; \_\; \{T\; V\}\; \backslash left(\backslash mu\_\; \{M\}\; ^\; \{\backslash pi^\; \{*\}\},\; \backslash mu\_\; \{\backslash widehat\; \{M\}\}\; ^\; \{\backslash pi^\; \{*\}\}\backslash right)\backslash right).\; \backslash tag\; \{5\}\; \backslash \backslash \; \backslash end\{array\}$$

Proof. A more formal version of the theorem and proof is provided in appendix ??

We now make some remarks about the above result.

1. The first two terms are related to sub-optimality in policy optimization (planning) and model learning, and can be made small with more compute and data, assuming sufficient capacity.
2. There may be multiple Nash equilibrium for the MBRL game, and the third domain adaptation or transfer learning term in the bound captures the quality of an equilibrium. It captures the idea that model is trained under

distribution of $\pi$ , i.e. $\mu_{M}^{\pi}$ , but evaluated under the distribution of $\pi^{}$ , i.e. $\mu_{M}^{\pi^{}}$ . If the model can accurately simulate $\pi^{*}$ , we can expect to find it in the planning phase, since it would obtain high rewards. This domain adaptation term is a consequence of the exploration problem, and is unavoidable if we desire globally optimal policies. Indeed, even purely model-free algorithms suffer from an analogous divergence term (Kakade & Langford, 2002; Munos & Szepesvári, 2008). However, Theorem 1 also applies to locally optimal policies (see appendix ??) for which we may expect better model transfer.

3. The domain adaptation term can be minimized by considering a wide initial state distribution (Kakade & Langford, 2002; Rajeswaran et al., 2017). This ensures the learned model is more broadly accurate. However, in some applications, the initial state distribution may not be under our control. In such a case, we may draw upon advances in domain adaptation (Ben-David et al., 2006; Sun et al., 2015) to learn state-action representations better suited for transfer across different policies.

4. Algorithms

So far, we have established how MBRL can be viewed as a game that couples policy and model learning. We now turn to developing algorithms for solving the game. Unlike common deep learning settings (e.g. supervised learning), there are no standard workhorses for continuous games. Direct extensions of optimization workhorses (e.g. SGD) are unstable for games due to non-stationarity (Wang et al., 2019b; Fiez et al., 2019). We first review some of these extensions before presenting our final algorithms.

4.1. Independent simultaneous learners

We first consider a class of algorithms where each player individually optimize their own objectives using gradient descent. Thus, each player treats the setting as stochastic optimization unaware of potential drifts in their objectives due to the two-player nature. These algorithms are sometimes called independent learners, simultaneous learners, or naive learners (Wang et al., 2019b; Foerster et al., 2017).

Gradient Descent Ascent (GDA) In GDA, each player performs an improvement step holding the parameters of the other player fixed. The resulting updates are given below.

$$\begin{array}{cccc}& {\pi }_{k+1}={\pi }_k+{\alpha }_k{\mathrm{\nabla }}_{\pi }J({\pi }_k,{\widehat{M}}_k)& & \text{(6)}\end{array}\backslash pi\_\; \{k\; +\; 1\}\; =\; \backslash pi\_\; \{k\}\; +\; \backslash alpha\_\; \{k\}\; \backslash nabla\_\; \{\backslash pi\}\; J\; \backslash left(\backslash pi\_\; \{k\},\; \backslash widehat\; \{M\}\; \_\; \{k\}\backslash right)\; \backslash tag\; \{6\}$$

$$\begin{array}{cccc}& {\widehat{\mathit{M}}}_{k+1}={\widehat{\mathit{M}}}_k-{\beta }_k{\mathrm{\nabla }}_{\widehat{\mathit{M}}}\mathrm{\ell }({\widehat{\mathit{M}}}_k,{\mu }_{\widehat{\mathit{M}}}^{{\pi }_k})& & \text{(7)}\end{array}\backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\; \_\; \{k\; +\; 1\}\; =\; \backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\; \_\; \{k\}\; -\; \backslash beta\_\; \{k\}\; \backslash nabla\_\; \{\backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\}\; \backslash ell\; (\backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\; \_\; \{k\},\; \backslash mu\_\; \{\backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\}\; ^\; \{\backslash pi\_\; \{k\}\})\; \backslash tag\; \{7\}$$

Both the players update their parameters simultaneously from iteration $k$ to $k + 1$ . For simplicity, we consider standard gradient descent, which can be equivalently replaced with momentum, Adam, natural gradient etc. Variants of GDA have been used to solve min-max games arising in

deep learning such as GANs. However, for certain problems, it can exhibit poor convergence and require very small learning rates (Schafer & Anandkumar, 2019) or domain-specific heuristics. Furthermore, it makes sub-optimal use of data, since it is desirable to take multiple policy improvement steps to fully reap the benefits of model learning.

Best Response (BR) The BR algorithm aims to mitigate the above drawback, where each player computes the best response while fixing the parameters of other players. The best response can be approximated in practice using a large number of gradient steps.

$$\begin{array}{cccc}& {\mathit{\pi }}_{k+1}=\arg \underset{\mathit{\pi }}{\max }J(\mathit{\pi },{\widehat{\mathit{M}}}_k)& & \text{(8)}\end{array}\backslash boldsymbol\; \{\backslash pi\}\; \_\; \{k\; +\; 1\}\; =\; \backslash arg\; \backslash max\; \_\; \{\backslash boldsymbol\; \{\backslash pi\}\}\; J\; (\backslash boldsymbol\; \{\backslash pi\},\; \backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\; \_\; \{k\})\; \backslash tag\; \{8\}$$

$$\begin{array}{cccc}& {\widehat{\mathit{M}}}_{k+1}=\arg \underset{\widehat{\mathit{M}}}{\min }\mathrm{\ell }(\widehat{\mathit{M}},{\mu }_{\mathit{M}}^{{\mathit{\pi }}_k})& & \text{(9)}\end{array}\backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\; \_\; \{k\; +\; 1\}\; =\; \backslash arg\; \backslash min\; \_\; \{\backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\}\; \backslash ell\; (\backslash widehat\; \{\backslash boldsymbol\; \{M\}\},\; \backslash mu\_\; \{\backslash boldsymbol\; \{M\}\}\; ^\; \{\backslash boldsymbol\; \{\backslash pi\}\; \_\; \{k\}\})\; \backslash tag\; \{9\}$$

Again, both players simultaneously update their parameters. It is known from a large body of work in online learning that aggressive changes can destabilize learning in nonstationary settings (Cesa-Bianchi & Lugosi, 2006). Large changes to the policy can dramatically alter the sampling distribution, which renders the model incompetent. Similarly, large changes in the model can bias policy learning. In Section 5 we experimentally study the performance of GDA and BR on a suite of control tasks and verify that they inefficient (slow) or unstable.

4.2. Stackelberg formulation and algorithms

To achieve stable and sample efficient learning, we require algorithms that take the game structure into account. While good workhorses are lacking for general games, Stackelberg games (von Stackelberg, 1934) are an exception. They are asymmetric games where we impose a specific playing order and are a generalization of min-max games. We cast the MBRL game in the Stackelberg form, and derive gradient based algorithms to solve the resulting game.

First, we briefly review continuous Stackelberg games. Consider a two player game with players $A$ and $B$ . Let $\theta_A, \theta_B$ be their parameters, and $\mathcal{L}_A(\theta_A, \theta_B), \mathcal{L}_B(\theta_A, \theta_B)$ be their losses. Each player would like their losses minimized. With player $A$ as the leader, the Stackelberg game corresponds to the following nested optimization:

$$\underset{{\mathit{\theta }}_A}{\min }{\mathcal{L}}_A({\mathit{\theta }}_A,{\mathit{\theta }}_B^{\ast }({\mathit{\theta }}_A))\backslash min\; \_\; \{\backslash boldsymbol\; \{\backslash theta\}\; \_\; \{A\}\}\; \backslash mathcal\; \{L\}\; \_\; \{A\}\; \backslash left(\backslash boldsymbol\; \{\backslash theta\}\; \_\; \{A\},\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{B\}\; ^\; \{*\}\; (\backslash boldsymbol\; \{\backslash theta\}\; \_\; \{A\})\backslash right)$$

$$\begin{array}{cccc}& \text{s u b j e c t}{\theta }_B^{\ast }({\theta }_A)=\arg \underset{\tilde{\mathit{\theta }}}{\min }{\mathcal{L}}_B({\theta }_A,\tilde{\mathit{\theta }})& & \text{(10)}\end{array}\backslash text\; \{s\; u\; b\; j\; e\; c\; t\}\; \backslash theta\_\; \{B\}\; ^\; \{*\}\; (\backslash theta\_\; \{A\})\; =\; \backslash arg\; \backslash min\; \_\; \{\backslash tilde\; \{\backslash boldsymbol\; \{\backslash theta\}\}\}\; \backslash mathcal\; \{L\}\; \_\; \{B\}\; \backslash left(\backslash theta\_\; \{A\},\; \backslash tilde\; \{\backslash boldsymbol\; \{\backslash theta\}\}\backslash right)\; \backslash tag\; \{10\}$$

Since the follower chooses the best response, the follower's parameters are implicitly a function of the leader's parameters. The leader is aware of this, and can utilize this information when updating its parameters. The Stackelberg formulation has a number of appealing properties.

• Algorithm design based on optimization: From the leader's viewpoint, the Stackelberg formulation trans-

forms a game with complex interactions into a more familiar albeit complex bi-level optimization, for which we have gradient based workhorses (Colson et al., 2007).

• Notion of stability and progress: In general games, there exists no single function that can be used to check if an iterative algorithm makes progress towards the equilibrium. This makes algorithm design and diagnosis difficult. By reducing the game to an optimization, the leader's loss $\mathcal{L}_A(\pmb{\theta}_A, \pmb{\theta}_B)$ can be used to track progress.

For simplicity of exposition, we assume that the best-response is unique for the follower. We later remark on the possibility of multiple minimizers. To solve the nested optimization, it suffices to focus on $\theta_{A}$ since the follower parameters $\theta_B^* (\theta_A)$ are implicitly a function of $\theta_{A}$ . We can iteratively optimize $\theta_{A}$ as: $\theta_{A} \gets \theta_{A} - \alpha_{A}(\mathrm{d}\mathcal{L}{A}(\theta{A},\theta_{B}^{*}(\theta_{A})) / \mathrm{d}\theta_{A})$ , where the gradient is described in Eq. 11. The key to solving a Stackelberg game is to make the follower learn very quickly to approximate the best response, while the leader learns slowly.

$$\begin{array}{cccc}& \begin{array}{c}\frac{\mathrm{d}{\mathcal{L}}_A({\mathit{\theta }}_A,{\mathit{\theta }}_B^{\ast }({\mathit{\theta }}_A))}{\mathrm{d}{\mathit{\theta }}_A}=\frac{\mathrm{d}{\mathit{\theta }}_B^{\ast }}{\mathrm{d}{\mathit{\theta }}_A}{\frac{\mathrm{\partial }{\mathcal{L}}_A({\mathit{\theta }}_A,{\mathit{\theta }}_B)}{\mathrm{\partial }{\mathit{\theta }}_B}\mid }_{{\mathit{\theta }}_B={\mathit{\theta }}_B^{\ast }}\\ +{\frac{\mathrm{\partial }{\mathcal{L}}_A({\mathit{\theta }}_A,{\mathit{\theta }}_B)}{\mathrm{\partial }{\mathit{\theta }}_A}\mid }_{{\mathit{\theta }}_B={\mathit{\theta }}_B^{\ast }}\end{array}& & \text{(11)}\end{array}\backslash begin\{array\}\{l\}\; \backslash frac\; \{\backslash mathrm\; \{d\}\; \backslash mathcal\; \{L\}\; \_\; \{A\}\; \backslash left(\backslash boldsymbol\; \{\backslash theta\}\; \_\; \{A\}\; ,\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{B\}\; ^\; \{*\}\; (\backslash boldsymbol\; \{\backslash theta\}\; \_\; \{A\})\backslash right)\}\{\backslash mathrm\; \{d\}\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{A\}\}\; =\; \backslash frac\; \{\backslash mathrm\; \{d\}\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{B\}\; ^\; \{*\}\}\{\backslash mathrm\; \{d\}\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{A\}\}\; \backslash left.\; \backslash frac\; \{\backslash partial\; \backslash mathcal\; \{L\}\; \_\; \{A\}\; \backslash left(\backslash boldsymbol\; \{\backslash theta\}\; \_\; \{A\}\; ,\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{B\}\backslash right)\}\{\backslash partial\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{B\}\}\; \backslash right|\; \_\; \{\backslash boldsymbol\; \{\backslash theta\}\; \_\; \{B\}\; =\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{B\}\; ^\; \{*\}\}\; \backslash tag\; \{11\}\; \backslash \backslash \; +\; \backslash left.\; \backslash frac\; \{\backslash partial\; \backslash mathcal\; \{L\}\; \_\; \{A\}\; (\backslash boldsymbol\; \{\backslash theta\}\; \_\; \{A\}\; ,\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{B\})\}\{\backslash partial\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{A\}\}\; \backslash right|\; \_\; \{\backslash boldsymbol\; \{\backslash theta\}\; \_\; \{B\}\; =\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{B\}\; ^\; \{*\}\}\; \backslash \backslash \; \backslash end\{array\}$$

The implicit Jacobian term $(\mathrm{d}\pmb{\theta}_B^* /\mathrm{d}\pmb{\theta}_A)$ can be obtained using the implicit function theorem (Krantz & Parks, 2002; Rajeswaran et al., 2019). Thus, in principle, we can compute the gradient with respect to the leader parameters and solve the nested optimization (to at least a local minimizer). To develop a practical algorithm based on these ideas, we use a few relaxations and approximations. First, we approximate the best response with multiple steps of an iterative optimization algorithm. Secondly, we drop the implicit Jacobian term and use a "first-order" approximation of the gradient. Such an approximation has proven effective in applications like meta-learning (Nichol et al., 2018), GANs (Heusel et al., 2017; Metz et al., 2017), and multiple timescale actor-critic methods (Konda & Borkar, 1999). Finally, since the Stackelberg game is asymmetric, we can cast the MBRL game in two forms based on which player we choose as the leader.

Policy As Leader (PAL): Choosing the policy player as leader results in the following optimization:

$$\underset{\pi }{\max }\{J(\mathit{\pi },{\widehat{\mathit{M}}}^{\pi })s\mathrm{.}t\mathrm{.}{\widehat{\mathit{M}}}^{\pi }\in \arg \underset{\widehat{\mathit{M}}}{\min }\mathrm{\ell }(\widehat{\mathit{M}},{\mu }_{\mathit{M}}^{\pi })\}\mathrm{.}\backslash max\; \_\; \{\backslash pi\}\; \backslash left\backslash \{J\; (\backslash boldsymbol\; \{\backslash pi\},\; \backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\; ^\; \{\backslash pi\})\; s.\; t.\; \backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\; ^\; \{\backslash pi\}\; \backslash in\; \backslash arg\; \backslash min\; \_\; \{\backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\}\; \backslash ell\; (\backslash widehat\; \{\backslash boldsymbol\; \{M\}\},\; \backslash mu\_\; \{\backslash boldsymbol\; \{M\}\}\; ^\; \{\backslash pi\})\; \backslash right\backslash \}.$$

We solve this nested optimization using the first order gradient approximation, resulting in updates:

$$\begin{array}{cccc}& {\widehat{\mathit{M}}}_{k+1}\approx \mathrm{\ell }(\widehat{\mathit{M}},{\mu }_{\mathit{M}}^{{\mathit{\tau }}_k})& & \text{(12)}\end{array}\backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\; \_\; \{k\; +\; 1\}\; \backslash approx\; \backslash underset\; \{\backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\}\; \{\backslash arg\; \backslash min\; \}\; \backslash ell\; \backslash left(\backslash widehat\; \{\backslash boldsymbol\; \{M\}\},\; \backslash mu\_\; \{\backslash boldsymbol\; \{M\}\}\; ^\; \{\backslash boldsymbol\; \{\backslash tau\}\; \_\; \{k\}\}\backslash right)\; \backslash tag\; \{12\}$$

$$\begin{array}{cccc}& {\mathit{\pi }}_{k+1}={\mathit{\pi }}_k+{\alpha }_k{\mathrm{\nabla }}_{\mathit{\pi }}J(\mathit{\pi },{\widehat{\mathit{M}}}_{k+1})& & \text{(13)}\end{array}\backslash boldsymbol\; \{\backslash pi\}\; \_\; \{k\; +\; 1\}\; =\; \backslash boldsymbol\; \{\backslash pi\}\; \_\; \{k\}\; +\; \backslash alpha\_\; \{k\}\; \backslash nabla\_\; \{\backslash boldsymbol\; \{\backslash pi\}\}\; J\; (\backslash boldsymbol\; \{\backslash pi\},\; \backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\; \_\; \{k\; +\; 1\})\; \backslash tag\; \{13\}$$

We first aggressively improve the model to minimize the loss under current visitation distribution. Subsequently we take a conservative policy. The algorithmic template is described further in Algorithm 1. Note that the PAL updates are different from GDA even if a single gradient step is used to approximate the arg min. In PAL, the model is first updated using the current visitation distribution from $\widehat{M}k$ to $\widehat{M}{k + 1}$ . The policy subsequently uses $\widehat{M}{k + 1}$ for improvement. In contrast, GDA uses $\widehat{M}k$ for improving the policy. Finally, suppose we find an approximate solution to the PAL optimization such that $J(\pi ,\widehat{M}^{\pi})\geq \sup{\tilde{\pi}}J(\tilde{\pi},\widehat{M}^{\tilde{\pi}}) - \epsilon{\pi}$ . Since the model is optimal for the policy by constriction, we inherit the guarantees of Theorem 1.

Algorithm 1 Policy as Leader (PAL) meta-algorithm

1: Initialize: policy $\pi_0$ , model $\widehat{M}0$ , data buffer $\mathcal{D} = {}$
2: for $k = 0,1,2,\ldots$ forever do
3: Collect data $\mathcal{D}k$ by executing $\pi_k$ in the environment
4: Build local (policy-specific) dynamics model: $\widehat{\pmb{M}}{k + 1} = \arg \min \ell (\widehat{\pmb{M}},\mathcal{D}k)$
5: Improve policy: $\pi{k + 1} = \pi_k + \alpha \nabla_\pi J(\pi_k,\widehat{M}{k + 1})$ with a conservative algorithm like NPG or TRPO.

6: end for

Model as Leader (MAL): Conversely, choosing model as the leader results in the optimization

$$\begin{array}{cccc}& \underset{\widehat{M}}{\min }\{\mathrm{\ell }(\widehat{M},{\mu }_{\widehat{M}}^{\pi }\text{s . t .}{\pi }_{\widehat{M}}\in \arg \underset{\pi }{\max }J(\pi ,\widehat{M})\}\mathrm{.}& & \text{(14)}\end{array}\backslash min\; \_\; \{\backslash widehat\; \{M\}\}\; \backslash left\backslash \{\backslash ell\; \backslash left(\backslash widehat\; \{M\},\; \backslash mu\_\; \{\backslash widehat\; \{M\}\}\; ^\; \{\backslash pi\}\; \backslash right.\; \backslash right.\; \backslash text\; \{s\; .\; t\; .\}\; \backslash pi\_\; \{\backslash widehat\; \{M\}\}\; \backslash in\; \backslash arg\; \backslash max\; \_\; \{\backslash pi\}\; J\; (\backslash pi\; ,\; \backslash widehat\; \{M\})\; \backslash \}.\; \backslash tag\; \{14\}$$

Similar to the PAL formulation, using first order approximation to the bi-level gradient results in:

$$\begin{array}{cccc}& {\pi }_{k+1}\approx \arg \underset{\pi }{\max }J(\pi ,{\widehat{M}}_k)& & \text{(15)}\end{array}\backslash pi\_\; \{k\; +\; 1\}\; \backslash approx\; \backslash arg\; \backslash max\; \_\; \{\backslash pi\}\; J\; (\backslash pi\; ,\; \backslash widehat\; \{M\}\; \_\; \{k\})\; \backslash tag\; \{15\}$$

$$\begin{array}{cccc}& {\widehat{\mathit{M}}}_{k+1}={\widehat{\mathit{M}}}_k-{\beta }_k{\mathrm{\nabla }}_{\widehat{\mathit{M}}}\mathrm{\ell }(\widehat{\mathit{M}},{\mu }_{\mathit{M}}^{{\mathit{\pi }}_{k+1}})& & \text{(16)}\end{array}\backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\; \_\; \{k\; +\; 1\}\; =\; \backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\; \_\; \{k\}\; -\; \backslash beta\_\; \{k\}\; \backslash nabla\_\; \{\backslash widehat\; \{\backslash boldsymbol\; \{M\}\}\}\; \backslash ell\; (\backslash widehat\; \{\backslash boldsymbol\; \{M\}\},\; \backslash mu\_\; \{\backslash boldsymbol\; \{M\}\}\; ^\; \{\backslash boldsymbol\; \{\backslash pi\}\; \_\; \{k\; +\; 1\}\})\; \backslash tag\; \{16\}$$

We first optimize a policy for the current model. Subsequently, we conservatively improve the model using the data collected with the optimized policy. In practice, instead of a single conservative model improvement step, we aggregate all the historical data and perform a few epochs of training. This has an effect similar to conservative model improvement in a follow the regularized leader interpretation (Shalev-Shwartz, 2012; Ross & Bagnell, 2012; McManan, 2011). The algorithmic template is described in Algorithm 2. Similar to the PAL case, we again inherit the guarantees from Theorem 1.

On distributionally robust models and policies Finally, we illustrate how the Stackelberg framework is consistent with commonly used robustification heuristics. We now consider the case where there could be multiple best responses to the leader eq. 10. For instance, in PAL, there could be

Algorithm 2 Model as Leader (MAL) meta-algorithm

1: Initialize: policy $\pi_0$ , model $\widehat{M}0$ , data buffer $\mathcal{D} = {}$
2: for $k = 0,1,2,\ldots$ forever do
3: Optimize $\pi{k + 1} = \arg \max_{\pi}J(\pi ,\widetilde{M}k)$ using any algorithm (RL, MPC, planning etc.)
4: Collect environment data $\mathcal{D}{k + 1}$ using $\pi_{k + 1}$
5: Improve model $\widehat{M}{k + 1} = \widehat{M}k - \beta \nabla{\widehat{M}}\ell (\widehat{M},\mathcal{D}{k + 1})$ using any conservative algorithm like mirror descent, data aggregation etc.

6: end for

multiple models that achieve low error for the policy. Similarly, in MAL, there could be multiple policies that achieve high rewards for the specified model. In such cases, the standard notion of Stackelberg equilibrium is to optimize under the worst case realization (Fiez et al., 2019), which results in:

$$\begin{array}{cccc}& \begin{array}{c}{\min }_{{\theta }_A\in B(C)}{\max }_{a=0,b=0}{\mathcal{L}}_A({\theta }_A,{\theta }_B),\text{w h e r e}\\ \begin{array}{c}{\theta }_A{\theta }_B\in R({\theta }_A)\\ R({\theta }_A)\{\tilde{\mathit{\theta }}\mid {\mathcal{L}}_B({\mathit{\theta }}_A,\tilde{\mathit{\theta }})\le {\mathcal{L}}_B({\mathit{\theta }}_A,{\mathit{\theta }}_B)\mathrm{\forall }{\mathit{\theta }}_B\}\mathrm{.}\end{array}\end{array}& & \text{(17)}\end{array}\backslash begin\{array\}\{l\}\; \backslash min\; \_\; \{\backslash theta\_\; \{A\}\; \backslash in\; B\; (C)\}\; \backslash max\; \_\; \{a\; =\; 0,\; b\; =\; 0\}\; \backslash mathcal\; \{L\}\; \_\; \{A\}\; (\backslash theta\_\; \{A\},\; \backslash theta\_\; \{B\}),\; \backslash text\; \{w\; h\; e\; r\; e\}\; \backslash \backslash \; \backslash begin\{array\}\{l\}\; \backslash theta\_\; \{A\}\; \backslash quad\; \backslash theta\_\; \{B\}\; \backslash in\; R\; (\backslash theta\_\; \{A\})\; \backslash \backslash \; R\; (\backslash theta\_\; \{A\})\; \backslash stackrel\; \{\backslash text\; \{d\; e\; f\}\}\; \{=\}\; \backslash left\backslash \{\backslash tilde\; \{\backslash boldsymbol\; \{\backslash theta\}\}\; \backslash mid\; \backslash mathcal\; \{L\}\; \_\; \{B\}\; \backslash left(\backslash boldsymbol\; \{\backslash theta\}\; \_\; \{A\},\; \backslash tilde\; \{\backslash boldsymbol\; \{\backslash theta\}\}\backslash right)\; \backslash leq\; \backslash mathcal\; \{L\}\; \_\; \{B\}\; \backslash left(\backslash boldsymbol\; \{\backslash theta\}\; \_\; \{A\},\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{B\}\backslash right)\; \backslash forall\; \backslash boldsymbol\; \{\backslash theta\}\; \_\; \{B\}\; \backslash right\backslash \}.\; \backslash end\{array\}\; \backslash tag\; \{17\}\; \backslash \backslash \; \backslash end\{array\}$$

In PAL, model ensemble approaches correspond to approximating the best response set with a finite collection (ensemble) of models. Algorithms inspired by robust or risk-averse control (Zhou et al., 1996; Garcia & Fernandez, 2015; Rajeswaran et al., 2016) explicitly improve against the adversarial choice in the ensemble, consistent with the Stackelberg setting. Similarly, in the MAL formulation, entropy regularization (Haarnoja et al., 2018; Hazan et al., 2018) and disagreement based reward bonuses (Pathak et al., 2017; 2019) lead to adversarial best response by encouraging the policy to visit parts of the state space where the model is likely to be inaccurate. Our Stackelberg formulation provides a principled foundation for these important components, which have thus far been viewed as heuristics.

5. Experiments

In our experimental evaluation, we aim to primarily answer the following questions:

1. Do independent learning algorithms (GDA and BR) learn slowly or suffer from instabilities?
2. Do the Stackelberg-style algorithms (PAL and MAL) enable stable and sample efficient learning?
3. Do MAL and PAL exhibit different learning characteristics and strengths? Can we characterize the situations where one is more preferable than the other?

Task Suite We study the behavior of algorithms on a suite of continuous control tasks consisting of: DClaw-Turn, DKitty-Orient, 7DOF-Reacher, and InHand-Pen. The tasks are illustrated in Figure 2 and

Figure 1. Comparison of the learning algorithms. We report results based on 5 random seeds, with solid lines representing the average performance, and shaded regions indicate standard deviation across seeds. PAL and MAL exhibit stable and sample efficient learning. GDA learns very slowly due to sub-optimal use of data. BR does not lead to stable learning due to aggressive changes to both policy and model. For the ROBEL tasks, as a point of comparison, we also include results of SAC a state of the art model-free algorithm.

further details are provided in Appendix ??. The DClaw and DKitty tasks use physically accurate models of robots (Zhu et al., 2018; Ahn et al., 2019). The Reacher task is a representative whole arm manipulation task, while the in-hand dexterous manipulation task (Rajeswaran et al., 2018) serves as a representative high-dimensional control task. In addition, we also present results with our algorithms in the OpenAI gym tasks in Appendix ??.

Algorithm Details For all the algorithms of interest (GDA, BR, PAL, MAL), we represent the policy as well as the dynamics model with fully connected neural networks. We instantiate all of these algorithm families with model-based natural policy gradient. Details about the implementation are provided in Appendix ??. We use ensembles of dynamics models and entropy regularization to encourage robustness.

Comparison of learning algorithms We first study the performance of Stackelberg-style algorithms (PAL, MAL) and compare against the performance of independent algorithms (GDA and BR). Our results, summarized in Figure 1, suggest that PAL and MAL can learn all the tasks efficiently. We observe near monotonic improvement, suggesting that the Stackelberg formulation enables stable learning. We also observe that PAL learns faster than MAL for the tasks we study. While GDA eventually achieves near- $100%$ success

Figure 2. (a) Reacher task with a 7DOF arm. (b) In-hand manipulation task with a 24DOF dexterous hand. (c) DClaw-Turn task with a 3 fingered "claw". (d) DKitty-Orient task with a quadrupedal robot. In all the tasks, the desired goal configurations are randomized every episode, which forces the RL agent to learn generalizable policies. We measure and use success rate for our experimental evaluations.

Figure 3. Comparison of results on the OpenAI gym benchmark tasks. Results for the baselines are reproduced from Janner et al. (2019). Solid lines are the average performance curves over 5 random seeds, while shaded region represents the standard deviation over these 5 runs. We observe that PAL and MAL show near-monotonic improvement, and substantially outperform the baselines.

rate, it leads to considerably slower learning. As outlined in Section 4, this is likely due to conservative nature of updates for both the policy and the model. Furthermore, the performance fluctuates rapidly during course of learning, since it does not correspond to stable optimization of any objective. Finally, we observe that BR is unable to make consistent progress. As suggested earlier in Section 4, BR makes rapid changes to both model and policy which exacerbates the challenge of distribution mismatch.

As a point of comparison, we also plot results of SAC (Haarnoja et al., 2018), a leading model-free algorithm for the ROBEL tasks (results taken from Ahn et al. (2019)). Although SAC is able to solve these tasks, its sample efficiency is comparable to GDA, and substantially slower than PAL and MAL. To compare against other model-based algorithms, we turn to published results from prior work on OpenAI gym tasks. In Figure 3, we show that PAL and MAL significantly outperforms prior algorithms. In particular, PAL and MAL are 10 times as efficient as other model-based and model-free methods. PAL is also twice as efficient as MBPO (Janner et al., 2019), a state of the art hybrid model-based and model-free algorithm. Further details about this comparison are provided in Appendix ??

Overall our results indicate that PAL and MAL: (a) are substantially more sample efficient than prior model-based and model-free algorithms; (b) achieve the asymptotic performance of their model-free counterparts; (c) can scale to high-dimensional tasks with complex dynamics like dexterous manipulation; (d) can scale to tasks requiring extended rollout horizons (e.g. the OpenAI gym tasks).

Choosing between PAL and MAL Finally, we turn to studying relative strengths of PAL and MAL. For this, we consider two variations of the 7DOF reacher task (from Figure 2) corresponding to environment perturbations at an intermediate point of training. In the first case, we perturb

the dynamics by changing the length of the forearm. In the second case, halfway through the training, we change the goal distribution to a different region of 3D space. Training curves are presented in Figure 4. Note that there is a performance drop at the time of introducing the perturbation.

For the first case of dynamics perturbation, we observe that PAL recovers faster. Since PAL learns the model aggressively using recent data, it can forget old inconsistent data and improve the policy using an accurate model. In contrast, MAL adapts the model conservatively, taking longer to forget old inconsistent data, ultimately biasing and slowing the policy learning. In the second experiment, the dynamics is stationary but the goal distribution changes midway. Note that the policy does not generalize zero-shot to the new goal distribution, and requires additional learning or fine-tuning. Since MAL learns a more broadly accurate model, it quickly adapts to the new goal distribution. In contrast, PAL conservatively changes the policy and takes longer to adapt to the

Figure 4. PAL vs MAL in non-stationary learning environments. Y axis is the distance between end effector and goal, averaged over the trajectory (lower is better). The left plot corresponds to the case where the dynamics of $M$ is changed after $10^{4}$ samples, while the right plot corresponds to the case where we change the goal distribution after $8 \times 10^{3}$ samples. We observe that PAL recovers quickly from dynamics perturbations, while MAL recovers quickly from goal perturbations.

new goal distribution.

Thus, in summary, we find that PAL is better suited for situations where the dynamics of the world can drift over time. In contrast, MAL is better suited for situations where the task or goal distribution can change over time, and related settings like multi-task learning.

6. Related Work

MBRL and the closely related fields of adaptive control and system identification have a long and rich history (see Åström & Murray (2004); Ljung (1987) for overview). Early works in MBRL primarily focused on tabular reinforcement learning in a known generative model setting (Kearns & Singh, 1998a; Agarwal et al., 2019a). However, this setting assumes access to a highly exploratory policy to collect data, which is often not available in practice. Subsequent works like E3 (Kearns & Singh, 1998b) and R-MAX (Brafman & Tennenholtz, 2001) attempt to lift this limitation, but rely heavily on tabular representations which are inadequate for modern applications like robotics. Coupled with advances in deep learning, there has been a surge of interest in incremental MBRL algorithms with rich function approximation. They generally fall into two sets of approaches, as we outline below.

The first set of approaches are largely inspired by trust region methods, and are similar to the PAL family from our work. A highly accurate "local" model is constructed around the visitation distribution of the current policy, which is subsequently used to conservatively improve the policy. The trust region is intended to ensure that the model is accurate for all policies within it, thereby enabling monotonic performance improvement. GPS (Levine & Abbeel, 2014; Mordatch & Todorov, 2014), DPI (Sun et al., 2018b), and related approaches (Kumar et al., 2016) learn a time varying linear model and perform a KL-constrained policy improvement step. Such a model representation is convenient for an iLQG based policy update (Todorov & Li, 2005), but might be restrictive for complex dynamics beyond trajectory-centric RL. To remove these limitations, recent works have started to consider neural networks to represent both policy and the dynamics model. However, somewhat surprisingly, a clean version from the PAL family has not been studied with neural network models. The motivations presented by Xu et al. (2018) and Kurutach et al. (2018) resemble PAL, however their practical implementations do not strongly enforce the conservative nature of the policy update.

An alternate set of MBRL approaches take a view similar to MAL. Models are updated conservatively through data aggregation, while policies are aggressively optimized. Ross & Bagnell (2012) explicitly studied the role of data aggregation in MBRL. They presented an agnostic online learning

view of MBRL and showed that data aggregation can lead to a no-regret algorithm for learning the model, even with aggressive policy optimization. Subsequent works have used data augmentation and proposed additional components to enhance efficiency and stability, such as the use of model predictive control (Williams et al., 2017; Lowrey et al., 2019; Nagabandi et al., 2019), uncertainty quantification through Bayesian models (Deisenroth & Rasmussen, 2011), and ensembles of dynamics models (Rajeswaran et al., 2016; Chua et al., 2018; Nagabandi et al., 2019). We refer readers to Wang et al. (2019a) for overview of recent MBRL advances.

While specific instances of PAL and MAL have been studied in the past, an overarching framework around them has been lacking. Our descriptions of the PAL and MAL families generalize and unify core insights from prior work and simplify them from the lens of abstraction. Furthermore, the game theoretic formulation enables us to form a connection between the PAL and MAL frameworks. We also note that the PAL and MAL families have similarities to multiple timescale algorithms (Konda & Borkar, 1999; Konda & Tsitsiklis, 2004; Karmakar & Bhatnagar, 2015) studied for actor-critic temporal difference learning. These ideas have also been extended to study min-max games like GANs (Heusel et al., 2017). However, they have not been extended to study model-based RL.

We presented a model-based setting where the model is used to directly improve the policy through rollout based optimization. However, models can be utilized in other ways too. Dyna (Sutton, 1990) and MBPO (Janner et al., 2019) use a learned model to provide additional learning targets for an actor-critic algorithm through short-horizon synthetic trajectories. MBVE (Feinberg et al., 2018), STEVE (Buckman et al., 2018), and doubly-robust methods (Jiang & Li, 2015; Thomas & Brunskill, 2016; Farajtabar et al., 2018) use model-based rollouts to obtain more favorable bias-variance trade-offs for off-policy evaluation. Some of these works have noted that long horizon rollouts can exacerbate model bias. However, in our experiments, we were able to successfully perform rollouts of hundreds of steps. This is likely due to our practical implementation closely following the game theoretic algorithms designed explicitly to mitigate distribution shift and enable effective simulation. It is straightforward to extend PAL and MAL to a hybrid model-based and model-free algorithm, which is likely to provide further performance gains. Similarly, approaches that bootstrap from the model's predictions can improve multi-step simulation (Venkatraman et al., 2015; Bengio et al., 2015). We leave exploration of these directions for future work.

7. Summary and Conclusion

In this work, we developed a new framework for MBRL that casts it as a game between a policy player and a model player. We established that at equilibrium: (1) the model accurately simulates the policy and predicts its performance; (2) the policy is near-optimal. We derived sub-optimality bounds and made a connection to domain adaptation to characterize the equilibrium quality.

In order to solve the MBRL game, we constructed the Stackelberg version of the game. This has the advantage of: (1) effective gradient based workhorses to solve the Stackelberg optimization problem; (2) an effective objective function to track learning progress towards equilibrium. General continuous games possess neither of these characteristics. The Stackelberg game can take two forms based on which player we choose as the leader, resulting in two natural algorithm families, which we named PAL and MAL. Together they encompass, generalize, and unify a large collection of prior MBRL works. This greatly simplifies MBRL and particularly algorithm design from the lens of abstraction.

We developed practical versions of PAL and MAL using model-based natural policy gradient. We demonstrated stable and sample efficient learning on a suite of control tasks, including state of the art results on OpenAI gym benchmarks. These results suggest that our practical variants of PAL and MAL:

• are substantially more sample efficient compared to prior model-based and model-free algorithms,
• can achieve the same asymptotic performance as model-free counterparts,
• can scale to high-dimensional tasks with complex dynamics like dexterous manipulation,
• can scale to tasks requiring long horizon rollouts (e.g. OpenAI gym tasks which have a 1000 timestep horizon).

More broadly, our work adds to a growing body of recent work which suggests that MBRL can be stable, sample efficient, and more generalizable or adaptable to new tasks and non-stationary settings. For future work, we hope to study alternate ways to solve the Stackelberg optimization; such as using the full implicit gradient term and unrolled optimization. Finally, although we presented our game theoretic framework in the context of MBRL, it is more broadly applicable for any surrogate based optimization including actor-critic methods. It would make for interesting future work to study broader extensions and implications.

Acknowledgements

We thank Profs. Emo Todorov, Sham Kakade, Drew Bagnell, and Sergey Levine for valuable feedback and discussions.

We thank Michael Ahn and Michael Janner for sharing the baseline learning curves; and Ben Eysenbach and Anirudh Vemula for feedback on the paper draft. The work was done by Aravind Rajeswaran during internships at Google Brain, MTV. Aravind thanks Google for providing a highly encouraging research environment.

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