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Title: Algorithmic Framework for Model-based Deep Reinforcement Learning with Theoretical Guarantees

Paper content:
Model-based reinforcement learning (RL) is considered to be a promising approach to reduce the sample complexity that hinders model-free RL.
However, the theoretical understanding of such methods has been rather limited.
This paper introduces a novel algorithmic framework for designing and analyzing model-based RL algorithms with theoretical guarantees.
We design a meta-algorithm with a theoretical guarantee of monotone improvement to a local maximum of the expected reward.
The meta-algorithm iteratively builds a lower bound of the expected reward based on the estimated dynamical model and sample trajectories, and then maximizes the lower bound jointly over the policy and the model.
The framework extends the optimism-in-face-of-uncertainty principle to non-linear dynamical models in a way that requires no explicit uncertainty quantification.
Instantiating our framework with simplification gives a  variant of model-based RL algorithms Stochastic Lower Bounds Optimization (SLBO).
Experiments demonstrate that SLBO achieves the state-of-the-art performance when only 1M or fewer samples are permitted on a range of continuous control benchmark tasks.
In recent years deep reinforcement learning has achieved strong empirical success, including superhuman performances on Atari games and Go (Mnih et al., 2015; BID21 and learning locomotion and manipulation skills in robotics BID33 BID18 Lillicrap et al., 2015) .
Many of these results are achieved by model-free RL algorithms that often require a massive number of samples, and therefore their applications are mostly limited to simulated environments.
Model-based deep reinforcement learning, in contrast, exploits the information from state observations explicitly -by planning with an estimated dynamical model -and is considered to be a promising approach to reduce the sample complexity.
Indeed, empirical results BID14 Deisenroth et al., 2013; BID33 Nagabandi et al., 2017; Kurutach et al., 2018; Pong et al., 2018a) have shown strong improvements in sample efficiency.Despite promising empirical findings, many of theoretical properties of model-based deep reinforcement learning are not well-understood.
For example, how does the error of the estimated model affect the estimation of the value function and the planning?
Can model-based RL algorithms be guaranteed to improve the policy monotonically and converge to a local maximum of the value function?
How do we quantify the uncertainty in the dynamical models?It
's challenging to address these questions theoretically in the context of deep RL with continuous state and action space and non-linear dynamical models. Due
to the high-dimensionality, learning models from observations in one part of the state space and extrapolating to another part sometimes 0 * indicates equal contribution 1 The source code of this work is available at https://github.com/roosephu/slbo involves a leap of faith. The
uncertainty quantification of the non-linear parameterized dynamical models is difficult -even without the RL components, it is an active but widely-open research area. Prior
work in model-based RL mostly quantifies uncertainty with either heuristics or simpler models (Moldovan et al., 2015; BID33 BID13 .Previous
theoretical work on model-based RL mostly focuses on either the finite-state MDPs (Jaksch et al., 2010; BID4 Fruit et al., 2018; Lakshmanan et al., 2015; Hinderer, 2005; Pirotta et al., 2015; 2013) , or the linear parametrization of the dynamics, policy, or value function BID0 BID22 BID11 BID27 BID29 , but not much on non-linear models. Even with
an oracle prediction intervals 2 or posterior estimation, to the best of our knowledge, there was no previous algorithm with convergence guarantees for model-based deep RL.Towards addressing these challenges, the main contribution of this paper is to propose a novel algorithmic framework for model-based deep RL with theoretical guarantees. Our meta-algorithm
(Algorithm 1) extends the optimism-in-face-of-uncertainty principle to non-linear dynamical models in a way that requires no explicit uncertainty quantification of the dynamical models.Let V π be the value function V π of a policy π on the true environment, and let V π be the value function of the policy π on the estimated model M . We design provable
upper bounds, denoted by D π, M , on how much the error can compound and divert the expected value V π of the imaginary rollouts from their real value V π , in a neighborhood of some reference policy. Such upper bounds
capture the intrinsic difference between the estimated and real dynamical model with respect to the particular reward function under consideration.The discrepancy bounds D π, M naturally leads to a lower bound for the true value function: DISPLAYFORM0 (1.1)Our algorithm iteratively collects batches of samples from the interactions with environments, builds the lower bound above, and then maximizes it over both the dynamical model M and the policy π. We can use any RL
algorithms to optimize the lower bounds, because it will be designed to only depend on the sample trajectories from a fixed reference policy (as opposed to requiring new interactions with the policy iterate.)We show that the performance of the policy is guaranteed to monotonically increase, assuming the optimization within each iteration succeeds (see Theorem 3.1.) To the best of our knowledge, this is the first theoretical guarantee of monotone improvement for model-based deep RL.Readers may have realized that optimizing a robust lower bound is reminiscent of robust control and robust optimization. The distinction is
that we optimistically and iteratively maximize the RHS of (1.1) jointly over the model and the policy. The iterative approach
allows the algorithms to collect higher quality trajectory adaptively, and the optimism in model optimization encourages explorations of the parts of space that are not covered by the current discrepancy bounds.To instantiate the meta-algorithm, we design a few valid discrepancy bounds in Section 4. In Section 4.1, we recover
the norm-based model loss by imposing the additional assumption of a Lipschitz value function. The result suggests a norm
is preferred compared to the square of the norm. Indeed in Section 6.2, we
show that experimentally learning with 2 loss significantly outperforms the mean-squared error loss ( 2 2 ). In Section 4.2, we design
a discrepancy bound that is invariant to the representation of the state space. Here we measure the loss
of the model by the difference between the value of the predicted next state and the value of the true next state. Such a loss function is
shown to be invariant to one-to-one transformation of the state space. Thus we argue that the
loss is an intrinsic measure for the model error without any information beyond observing the rewards. We also refine our bounds
in Section A by utilizing some mathematical tools of measuring the difference between policies in χ 2 -divergence (instead of KL divergence or TV distance).Our analysis also sheds light
on the comparison between model-based RL and on-policy model-free RL algorithms such as policy gradient or TRPO BID17 . The RHS of equation (1.1) is
likely to be a good approximator of V π in a larger neighborhood than the linear approximation of V π used in policy gradient is (see Remark 4.5.)Finally, inspired by our framework and analysis, we design a variant of model-based RL algorithms Stochastic Lower Bounds Optimization (SLBO). Experiments demonstrate that
SLBO achieves state-of-the-art performance when only 1M samples are permitted on a range of continuous control benchmark tasks.
We devise a novel algorithmic framework for designing and analyzing model-based RL algorithms with the guarantee to convergence monotonically to a local maximum of the reward.
Experimental results show that our proposed algorithm (SLBO) achieves new state-of-the-art performance on several mujoco benchmark tasks when one million or fewer samples are permitted.A compelling (but obvious) empirical open question then given rise to is whether model-based RL can achieve near-optimal reward on other more complicated tasks or real-world robotic tasks with fewer samples.
We believe that understanding the trade-off between optimism and robustness is essential to design more sample-efficient algorithms.
Currently, we observed empirically that the optimism-driven part of our proposed meta-algorithm (optimizing V π, M over M ) may lead to instability in the optimization, and therefore don't in general help the performance.
It's left for future work to find practical implementation of the optimism-driven approach.In our theory, we assume that the parameterized model class contains the true dynamical model.
Removing this assumption is also another interesting open question.
It would be also very interesting if the theoretical analysis can be applied other settings involving model-based approaches (e.g., model-based imitation learning).

assistant: text: We design model-based reinforcement learning algorithms with theoretical guarantees and achieve state-of-the-art results on Mujuco benchmark tasks when one million or fewer samples are permitted.