## 2 MAML Meta-Reinforcement Learning  
The core objective of MAML meta-reinforcement learning is to train a set of initial policy parameters such that the policy initialized with these parameters can adapt to a new task rapidly by fine-tuning the parameters with minimal interaction dates.  
The training process of MAML meta-reinforcement learning involves an inner reinforcement learning loop and an outer reinforcement learning loop.  
Within the inner loop, the agent randomly samples \(K\) different training tasks \(\mathcal{T}_{i}\) from the task distribution \(p(\mathcal{T})\) and interacts with the environment for each task \(\mathcal{T}_{i}\) to collect data \(\mathcal{D}_{i}\). Let \(\pi_{\theta}\) represent the agent's policy, where \(\theta\) represents the policy parameters. The interaction process generates an action \(a_{t}\) based on the observed current state \(a_{t}\) according to the policy. The environment moves to the next state \(s_{t}+1\) and generates a reward \(r_{t}\) after the agent executes the action. The expected reward for each task \(\mathcal{T}_{i}\) can be expressed as  
\[\mathcal{L}_{T_{i}}(f_{\theta})=E_{x_{t},a_{t}\sim f_{\theta},\mathcal{T}_{i}} \bigg{[}\sum\limits_{t=1}^{H}r_{i}(x_{t},a_{t})\bigg{]} \tag{1}\]  
The policy parameters are updated by maximizing the expected reward in the inner loop, and the update formula can be expressed as  
\[\theta_{i}^{\prime} = \theta-\alpha\
abla_{\theta}\mathcal{L}_{\mathcal{T}_{i}}(f_{ \theta}) \tag{2}\]  
Within the outer loop, the parameters of the policy are updated according to the total expected rewards collected from each task. Within the inner loop, the agent interacts with the environment for each task \(\mathcal{T}_{i}\) to collect data \(\mathcal{D}_{i}^{\prime}\) using the new policy \(\pi_{\theta^{\prime}}\), which was trained in the inner loop. Sum up the expected return \(\mathcal{L}_{\mathcal{T}_{i}^{\prime}}\Big{(}f_{\theta_{i}^{\prime}}\Big{)}\) for each task to obtain \(\sum\limits_{i=1}^{K}\mathcal{L}_{\mathcal{T}_{i}^{\prime}}\Big{(}f_{\theta_ {i}^{\prime}}\Big{)}\). The policy parameters are updated by maximizing the sum of the expected returns \(\sum\limits_{i=1}^{K}\mathcal{L}_{\mathcal{T}_{i}^{\prime}}\Big{(}f_{\theta_ {i}^{\prime}}\Big{)}\), and the update formula can be expressed as  
\[\theta=\theta-\beta\
abla_{\theta}\sum\limits_{i=1}^{K}\mathcal{L}_{\mathcal{T }_{i}^{\prime}}\Big{(}f_{\theta_{i}^{\prime}}\Big{)} \tag{3}\]  
The overall training procedure of the MAML meta-reinforcement learning algorithm is illustrated in Algorithm 1.  
```
0: Task distribution
0: Learning rates
1: Randomly initialize policy parameters
2:while not done do:
3: Sample a batch of tasks
4:if all \(\mathcal{T}_{i}\)do:
5: Interact with the environment using policy \(f_{\theta}\) to collect data \(\mathcal{D}_{i}\)
6: Calculate \(\
abla_{\theta}\mathcal{L}_{\mathcal{T}_{i}}(f_{\theta})\) based on \(\mathcal{D}_{i}\) and \(\mathcal{L}_{\mathcal{T}_{i}}\)
7: Update \(\theta_{i}^{\prime}\)=\(\theta-\alpha\
abla_{\theta}\mathcal{L}_{\mathcal{T}_{i}}(f_{\theta})\)
8: Interact with the environment using policy \(f_{\theta_{i}}\) to collect data \(\mathcal{D}_{i}^{\prime}\)
9:endfor
10: Update \(\theta=\theta-\beta\
abla_{\theta}\sum\limits_{i=1}^{K}\mathcal{L}_{\mathcal{ T}_{i}^{\prime}}\Big{(}f_{\theta_{i}^{\prime}}\Big{)}\)
11:endwhile
```  
**Algorithm 1**MAML meta-reinforcement learning