# Can a large discrete action space be represented using Gaussian distributions?

I have a large 1D action space, e.g. dim(A)=2000-10000. Can I use continuous action space where I could learn the mean and std of the Gaussian distributions that I would use to sample action from and round the value to the nearest integer? If yes, can I extend this idea to multi-dimensional large action space?

The answer is "it depends". Once you have arranged the actions into order, a key trait is whether the action value function has a simple enough shape that sampling from a Gaussian policy function would give consistent expected returns, enough that learning can occur. If the underlying "true" value function has a lot of high frequency noise then learning would be slow. In the worst case, if the action value $$Q(s,a_{n})$$ and $$Q(s,a_{n+1})$$ is not correlated for any $$n$$, then it will not be possible to learn with the approximation at all.
You may have some sense of how similar actions $$a_{n}$$ and $$a_{n+1}$$ are. If the actions represent different ordinal choices, such as selecting an integer number of items to perform some task with such as buy/sell or transport, then in many environments there will often be a strong correlation between outcome of choosing e.g. $$a_{900}$$ and $$a_{901}$$. If this holds in general, then that is a good indicator that you can treat $$n$$ as being continuous, and use learned parameters of a simple distribution function to find optimal policies (and in addition this could be far more efficient than using a discrete representation).
It might not matter if for a small fraction of cases the difference in outcomes between $$a_{n}$$ and $$a_{n+1}$$ is large, provided that successive approximations to the optimal policy can improve towards optimal through adjusting mean $$\mu(s)$$ and standard deviation $$\sigma(s)$$.
There may still be difficult cases that cannot be learned by the approximation - for instance if a specific action $$a_n$$ is optimal for a given state $$s$$, but $$a_{n-1}$$ and $$a_{n+1}$$ are a lot worse, then the training process of a typical policy gradient approach may never settle upon $$\mu(s) = n$$ and $$\sigma(s) \approx 0$$, because any intermediate values between the starting policy and the optimal one will perform badly.