Thread:Talk:BeepBoop/Understanding BeepBoop/Property of gradient of cross-entropy loss with kernel density estimation

I'm quite curious about the behavior of cross entropy loss between a uniform distribution and kernel density estimation with softmax weight:

where a and b is the lower and upper bound of the target uniform distribution, K is the kernel function (assume normalized), x_j is the angle of the data point, and z_j the weight before softmax.

The integral is often calculated by numerical methods, such as binning, so let's consider the gradient of the i-th data point's weight before softmax, and consider only one of the bins (with angle x) and ignore the values multiplied before integral:

It degenerates to ordinary cross entropy loss with softmax when K is either 1 or 0 (and 1 iif the "label" matches): S_i - 1 when label matches or S_i when label mismatches.

But things start to get interesting when K is different.