Following this instruction :
1. Strictly adhere to PEP-8 standards, ensuring code is clean, well-structured, and follows Python's style guide for readability.
2. Use appropriate modules like `typing` for type hints, ensuring code clarity, maintainability, and leveraging Python's standard libraries effectively.
3. Focus on time, space, robustness, scalability; write optimized, efficient code that can handle large datasets and scale seamlessly.
4. Implement comprehensive error handling, catching exceptions gracefully, providing meaningful error messages, and ensuring code robustness under unexpected conditions.
5. Ensure the developer agent writes advanced, maintainable Python code, balancing performance and readability while adhering to best practices.

TASK :develop code ppo algorithms 

### Pseudo-Algorithm for PPO

**Initialize**:
1. Initialize the policy network `π` with parameters `θ`.
2. Set the learning rate `α`.
3. Define the discount factor `γ` for future rewards.
4. Define or initialize the penalty coefficient `λ` if using PPO-Penalty.
5. Specify the clipping range `ϵ`, if using PPO-Clip.
6. Define the maximum number of training timesteps or epochs.

### Main Training Loop

Repeat until convergence or the maximum number of steps is reached:

1. **Collect Data**:
   - Reset the environment and observe the initial state `s`.
   - Generate a batch of trajectories using the current policy `π(θ)` by interacting with the environment.
     - For each time step `t` in the trajectory:
       - Select an action `a_t` using the current policy `π(a_t|s_t; θ)`.
       - Execute the action in the environment to receive a reward `r_t` and observe the next state `s_{t+1}`.
       - Append `(s_t, a_t, r_t, s_{t+1})` to the data batch.

2. **Compute Advantages**:
   - For each trajectory, compute the advantage estimates `A_t` using a baseline, typically the value function `V(s_t; θ_v)`, and the discount factor `γ`.

3. **Policy Update**:

   For each policy update step:
   - **PPO-Penalty Variant**:
     1. Calculate the current policy's probability ratio `ρ_t = π(a_t|s_t; θ) / π(a_t|s_t; θ_old)`.
     2. Construct the surrogate loss as `L_t(θ) = ρ_t * A_t - λ * KL[π_old || π]`.
     3. If KL-divergence becomes too large, adjust penalty coefficient `λ` accordingly.
     4. Perform a gradient ascent step on `L_t(θ)` to update the policy parameters `θ`.

   - **PPO-Clip Variant**:
     1. Calculate the probability ratio `ρ_t = π(a_t|s_t; θ) / π(a_t|s_t; θ_old)`.
     2. Define the clipped surrogate objective as `L_t^{clip}(θ) = min(ρ_t * A_t, clip(ρ_t, 1 - ϵ, 1 + ϵ) * A_t)`.
     3. Perform a gradient ascent step on `L_t^{clip}(θ)` to update the policy parameters `θ`.

4. **Value Function Update**:
   - Update the value function parameter `θ_v` to minimize the error between the predicted value and the empirical return using a mean-squared error loss.

5. **Update Old Policy**:
   - Set `θ_old = θ`.

6. **End of Episode**:
   - Evaluate the updated policy intermittently on a separate validation set or predetermined policy performance metric to track improvement and ensure stability.

The key aspects of PPO ensure that the policy updates are stable and efficient. The two variants offer different approaches, each aiming to keep the new policy close to the old policy while allowing for significant improvements. Researchers often tune `λ` and `ϵ` hyperparameters based on empirical performance during training