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Title: CAN ALTQ LEARN FASTER: EXPERIMENTS AND THEORY

Paper content:
Differently from the popular Deep Q-Network (DQN) learning, Alternating Q-learning (AltQ) does not fully fit a target Q-function at each iteration, and is generally known to be unstable and inefficient.
Limited applications of AltQ mostly rely on substantially altering the algorithm architecture in order to improve its performance.
Although Adam appears to be a natural solution, its performance in AltQ has rarely been studied before.
In this paper, we first provide a solid exploration on how well AltQ performs with Adam.
We then take a further step to improve the implementation by adopting the technique of parameter restart.
More specifically, the proposed algorithms are tested on a batch of Atari 2600 games and exhibit superior performance than the DQN learning method.
The convergence rate of the slightly modified version of the proposed algorithms is characterized under the linear function approximation.
To the best of our knowledge, this is the first theoretical study on the Adam-type algorithms in Q-learning.
Q-learning (Watkins & Dayan, 1992 ) is one of the most important model-free reinforcement learning (RL) problems, which has received considerable research attention in recent years (Bertsekas & Tsitsiklis, 1996; Even-Dar & Mansour, 2003; Hasselt, 2010; Lu et al., 2018; Achiam et al., 2019) .
When the state-action space is large or continuous, parametric approximation of the Q-function is often necessary.
One remarkable success of parametric Q-learning in practice is its combination with deep learning, known as the Deep Q-Network (DQN) learning (Mnih et al., 2013; 2015) .
It has been applied to various applications in computer games (Bhatti et al., 2016) , traffic control (Arel et al., 2010) , recommendation systems (Zheng et al., 2018; Zhao et al., 2018) , chemistry research (Zhou et al., 2017) , etc.
Its on-policy continuous variant (Silver et al., 2014) has also led to great achievements in robotics locomotion (Lillicrap et al., 2016) .
The DQN algorithm is performed in a nested-loop manner, where the outer loop follows an one-step update of the Q-function (via the empirical Bellman operator for Q-learning), and the inner loop takes a supervised learning process to fit the updated (i.e., target) Q-function with a neural network.
In practice, the inner loop takes a sufficiently large number of iterations under certain optimizer (e.g. stochastic gradient descent (SGD) or Adam) to fit the neural network well to the target Q-function.
In contrast, a conventional Q-learning algorithm runs only one SGD step in each inner loop, in which case the overall Q-learning algorithm updates the Q-function and fits the target Q-function alternatively in each iteration.
We refer to such a Q-learning algorithm with alternating updates as Alternating Q-learning (AltQ).
Although significantly simpler in the update rule, AltQ is well known to be unstable and have weak performance (Mnih et al., 2016) .
This is in part due to the fact that the inner loop does not fit the target Q-function sufficiently well.
To fix this issue, Mnih et al. (2016) proposed a new exploration strategy and asynchronous sampling schemes over parallel computing units (rather than the simple replay sampling in DQN) in order for the AltQ algorithm to achieve comparable or better performance than DQN.
As another alternative, Knight & Lerner (2018) proposed a more involved natural gradient propagation for AltQ to improve the performance.
All these schemes require more sophisticated designs or hardware support, which may place AltQ less advantageous compared to the popular DQN, even with their better performances.
This motivates us to ask the following first question.
• Q1: Can we design a simple and easy variant of the AltQ algorithm, which uses as simple setup as DQN and does not introduce extra computational burden and heuristics, but still achieves better and more stable performance than DQN?
In this paper, we provide an affirmative answer by introducing novel lightweight designs to AltQ based on Adam.
Although Adam appears to be a natural tool, its performance in AltQ has rarely been studied yet.
Thus, we first provide a solid exploration on how well AltQ performs with Adam (Kingma & Ba, 2014) , where the algorithm is referred to as AltQ-Adam.
We then take a further step to improve the implementation of AltQ-Adam by adopting the technique of parameter restart (i.e., restart the initial setting of Adam parameters every a few iterations), and refer to the new algorithm as AltQ-AdamR.
This is the first time that restart is applied for improving the performance of RL algorithms although restart has been used for conventional optimization before.
In a batch of 23 Atari 2600 games, our experiments show that both AltQ-Adam and AltQ-AdamR outperform the baseline performance of DQN by 50% on average.
Furthermore, AltQ-AdamR effectively reduces the performance variance and achieves a much more stable learning process.
In our experiments for the linear quadratic regulator (LQR) problems, AltQ-AdamR converges even faster than the model-based value iteration (VI) solution.
This is a rather surprising result given that the model-based VI has been treated as the performance upper bound for the Q-learning (including DQN) algorithms with target update .
Regarding the theoretical analysis of AltQ algorithms, their convergence guarantee has been extensively studied (Melo et al., 2008; Chen et al., 2019b) .
More references are given in Section 1.1.
However, all the existing studies focus on the AltQ algorithms that take a simple SGD step.
Such theory is not applicable to the proposed AltQ-Adam and AltQ-AdamR that implement the Adam-type update.
Thus, the second intriguing question we address here is described as follows.
• Q2: Can we provide the convergence guarantee for AltQ-Adam and AltQ-AdamR or their slightly modified variants (if these two algorithms do not always converge by nature)?
It is well known in optimization that Adam does not always converge, and instead, a slightly modified variant AMSGrad proposed in Reddi et al. (2018) has been widely accepted as an alternative to justify the performance of Adam-type algorithms.
Thus, our theoretical analysis here also focuses on such slightly modified variants AltQ-AMSGrad and AltQ-AMSGradR of the proposed algorithms.
We show that under the linear function approximation (which is the structure that the current tools for analysis of Q-learning can handle), both AltQ-AMSGrad and AltQ-AMSGradR converge to the global optimal solution under standard assumptions for Qlearning.
To the best of our knowledge, this is the first non-asymptotic convergence guarantee on Q-learning that incorporates Adam-type update and momentum restart.
Furthermore, a slight adaptation of our proof provides the convergence rate for the AMSGrad for conventional strongly convex optimization which has not been studied before and can be of independent interest.
Notations We use x := x 2 = √ x T x to denote the 2 norm of a vector x, and use x ∞ = max i |x i | to denote the infinity norm.
When x, y are both vectors, x/y, xy, x 2 , √ x are all calculated in the element-wise manner, which will be used in the update of Adam and AMSGrad.
We denote [n] = 1, 2, . . . , n, and x ∈ Z as the largest integer such that x ≤ x < x + 1.
We propose two types of the accelerated AltQ algorithms, and demonstrate their superior performance over the state-of-the-art through a linear quadratic regulator problem and a batch of 23 Atari 2600 games.
Notably, Adam is not the only scheme in the practice for general optimization.
Heavy ball (Ghadimi et al., 2015) and Nesterov (Nesterov, 2013) are also popular momentum-based methods.
When adopting such methods in AltQ-learning for RL problems, however, we tend to observe a less stable learning process than AltQ-Adam.
This is partially caused by the fact that they optimize over a shorter historical horizon of updates than Adam.
Furthermore, the restart scheme provides somewhat remarkable performance in our study.
It is thus of considerable future interest to further investigate the potential of such a scheme.
One possible direction is to develop an adaptive restart mechanism with changing period determined by an appropriately defined signal of restart.
This will potentially relieve the effort in hyper-parameter tuning of finding a good fixed period.

assistant: text: This paper provides a convergence result for traditional Q-learning with linear function approximation when using an Adam-like update.