Mathematica stuck on partial derivative calculation

Question

I was experimenting polynomial gradient descent on Mathematica, but it got stuck on partial derivative calculation. Formula below:

Traditional Form:
$$ mse = \frac{\sum_{i=1}^{n}\left(y[[i]] - \left(a\times x[[i]]^2+b\times x[[i]]+c\right) \right)^2}{n} \\ grad = \{D[mse, a], D[mse, b], D[mse, c]\} $$

Input Form:

mse = Sum[y[[i]] - (a x[[i]]^2 + b x[[i]] + c), {i, 1, n}]^2 / n;
grad = {D[mse, a], D[mse, b], D[mse, c]}

As the picture showed above, I'm able to get expected behavior if I write the partial derivatives by hand, Mathemathca seems can't get the partial derivatives for me for the MES loss gradient.

Is there a way to solve this issue ?

Answer 1

So, I've managed to solve this issue, it's mainly due to my malformed input which cause Mathematica stuck. I leave it here for anyone who have the similar issue.

Below is the minimal reproducible test case(still tediously long). You'll found there is a minor difference(there is an empty superscript after a[[i]]) but not visually noticeable, and my InputForm rewrote in the question was also incorrect(mse = Sum[y[[i]] - (a x[[i]]^2 + b x[[i]] + c), {i, 1, n}]^2 / n;) /facepalm.

derivativeTest[{x_, y_}] := 
  Module[{ n, mseOrig, mseStandard, mseInput, gradInput, 
    gradStandard, gradSelf},
            (*MSE loss*)
            n = Length[x];
   mseOrig = 1/n \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]
\*SuperscriptBox[\((y[\([\)\(i\)\(]\)]\  - \ \((a\ \ 
\*SuperscriptBox[\(x[\([\)\(i\)\(]\)]\), \(\ \)] x[\([i]\)] + \ 
           b\ x[\([\)\(i\)\(]\)]\  + \ c)\)\ )\), \(2\)]\)  ;
           mseStandard = 1/n \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]
\*SuperscriptBox[\((y[\([\)\(i\)\(]\)]\  - \ \((a\ \ x[\([i]\)] \
x[\([i]\)] + \ b\ x[\([\)\(i\)\(]\)]\  + \ c)\)\ )\), \(2\)]\)  ;
   mseInput = 
    1/n  * Sum[(y[[i]] - (a x[[i]]^2 + b x[[i]] + c))^2, {i, 1, n}];
   
   gradInput = {D[mseInput, a], D[mseInput, b], D[mseInput, c]};
   gradStandard = {D[mseStandard, a], D[mseStandard, b], 
     D[mseStandard, c]};
   
         gradSelf = {
               -2/n \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(
\*SuperscriptBox[\(x[\([\)\(i\)\(]\)]\), \(2\)] \((y[\([i]\)]\  - \ \
\((a\ 
\*SuperscriptBox[\(x[\([\)\(i\)\(]\)]\), \(2\)] + b\ x[\([i]\)]\  + \ 
            c)\))\)\)\),
               -2/n \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 
         1\), \(n\)]\(x[\([i]\)] \((y[\([i]\)]\  - \ \((a\ 
\*SuperscriptBox[\(x[\([\)\(i\)\(]\)]\), \(2\)] + b\ x[\([i]\)]\  + \ 
            c)\))\)\)\),
               -2/n \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 
         1\), \(n\)]\((y[\([i]\)]\  - \ \((a\ 
\*SuperscriptBox[\(x[\([\)\(i\)\(]\)]\), \(2\)] + b\ x[\([i]\)]\  + \ 
           c)\))\)\)
               };
    
    Grid[{{"MSE Original", "MSE StandardForm",  
       "MSE InputForm"}, {mseOrig // Simplify, 
       mseStandard // Simplify , mseInput // Simplify}}, Frame -> All]
    Grid[{{"Gradient by StandardForm", "Gradient by InputForm", 
       "Gradient by hand"}, {  gradStandard // Simplify, 
       gradInput // Simplify , gradSelf // Simplify }}, Frame -> All]
            ];