What is a non-informative choice of parameters for a Dirichlet distribution?

Dirichlet distribution is a conjugate prior for multinomial distribution. I want to impose a non-informative prior over sampling weights $$\pi$$ for a draw $$x=(x_1,…,x_N)$$ from a multinomial distribution with support $$d=(d_1,…,d_K)$$ (all the possible values that $$x_i$$ can take) and sampling weights $$\pi=(\pi_1,…,\pi_K)$$.

I was under impression that $$Dir(\alpha)$$ with $$\alpha_i=1$$ is a right choice. But I've read that (see e.g. this) $$Dir(\alpha)$$ with $$\alpha_i=0$$ yields an improper non-informative distribution.

Question:

1. Why $$Dir(\alpha)$$ with $$\alpha_i=0$$ is non-informative? Doesn't $$\alpha\to 0$$ impose higher sampling weight on a single data and zero on all others?
2. Shouldn't a uniform distribution $$Dir(\alpha)$$ with $$\alpha_i=1$$ be non-informative choice for the prior instead?

In regard to your specific questions, yes, the $$\text{Dirichlet}(\mathbf{0})$$ distribution is an improper distribution, so if you use it as a prior then it is an improper prior. As to whether this prior is better or worse than the flat prior, I will leave it to you to have a read of the literature on improper priors and see the advantages of each method. It is worth noting that they are not very different so long as you have a reasonable amount of data --- data manifests in the posterior as an increase of one in a parameter value for each observed data point. Bayesian analysis has a number of useful consistency theorems that establish that posterior beliefs converge even with different priors, and for priors like this, that are only slightly different, this convergence is quite rapid.