Hi,
probably not.
I would venture that what you are observing is a direct result of the problem being “too difficult” (insert your favourite mathematical definition of that here).
You will see that you often do not get much of a gradient for your parameters. Intuitively, if amplitude, frequency and phase are off, changing any one of these parameters just a bit will not help much with the MSE.
Of course, if you have this specific problem and know that your model is actually a good representation of the data, there are ways to estimate the three quantities you look for that work better than minimizing MSE.
Best regards
Thomas