Interpretation of Convergence plot in terms of mean square error
0 dB corresponds to a MSE of 1. So, if your method gives MSE that approaches 0 dB as the number of iterations increases, then your method has an error floor; it cannot drive the MSE to any value smaller than 1 no matter how many iterations it takes (or how many data points you have) etc. A good method should drive the MSE to 0 (corresponding to $-\infty$ dB), that is, the MSE should decrease continually as the number of iterations increases, and not saturate at a value of 1 (0 dB). We can then argue as to whether the approach to 0 MSE ($-\infty$ dB) is exponential (MSE $= O(e^{-n})$) or polynomial (MSE $= O(n^{-k})$) including as special cases quadratic (MSE $=O(n^{-2})$) and
linear (MSE $= O(n^{-1})$), or sublinear 
(e.g., MSE $= O(n^{-1/2})$ or MSE $= O((\ln n)^{-1})$ or similar)
where $n$ is the number of iterations (or number of data points).
Naturally, the faster the decrease, the better (and usually
more expensive) the method. 

On the other hand, if the MSE of the best currently
known method is approaching
an error floor of 0 dB, there is a fantastic research opportunity (and
possibly a get-rich-quick opportunity too): devise a method that
drives the MSE to 0 instead of 1 and gain instant fame 
(and if you patent the method and found a start-up company
to further develop it, possibly fortune RSN as well).