# Why use $\log \alpha$ instead of $\alpha$ for Entropy Loss in Soft Actor Critic

I am trying to implement Soft Actor-Critic with automatic Entropy tuning.

The original paper derives the optimal entropy temperature $$\alpha$$ as follows:

$$\alpha ^* _t = \arg\min_{\alpha_t} \mathbb{E}_{\mathbb{a}_t \sim \pi^* _t} [-\alpha _t \log \pi^* _t (\mathbb{a}_t | \mathbb{s}_t ; \alpha_t ) - \alpha_t \bar{\mathcal{H}}]$$

In practice, they use the following loss function to update $$\alpha$$ via stochastic gradient descent

$$J(\alpha) = \mathbb{E}_{\mathbb{a}_t \sim \pi _t} [-\alpha \log \pi _t (\mathbb{a}_t | \mathbb{s}_t ) - \alpha_t \bar{\mathcal{H}}]$$

However, I have noticed that many implementations of SAC actually use $$\log \alpha$$ instead of $$\alpha$$; in other words,

$$J(\alpha) = \mathbb{E}_{\mathbb{a}_t \sim \pi _t} [-\log \alpha (\log \pi _t (\mathbb{a}_t | \mathbb{s}_t ) + \bar{\mathcal{H}})]$$

I do get that the optimal $$\alpha$$ will also satisfy:

$$\alpha ^* _t = \arg\min_{\alpha_t} \mathbb{E}_{\mathbb{a}_t \sim \pi^* _t} [-\log \alpha _t (\log \pi^* _t (\mathbb{a}_t | \mathbb{s}_t ; \alpha_t ) + \bar{\mathcal{H}})]$$

since $$\log$$ function is monotonic increasing function and $$\alpha$$ should be positive.

However, I am not very convinced this is still valid under SGD setting. Especially since $$\alpha$$ decreases during training, I am concerned that the gradient of the loss might be larger. Also, what is the advantage of this modification?