Deep Learning · TEE · section

Drill 2 Half sum versus mean

Numerical practice from direct to mixed

Question

Question. Predictions = [0.2,0.7,0.4], targets = [0,1,1]. Find SSE, half SSE and MSE.

Answer — step by step

Given

Predictions = [0.2, 0.7, 0.4].

Targets = [0, 1, 1]; number of outputs n = 3.

To find

Find SSE, half SSE, MSE and the prediction gradients requested in the question.

Formula — which one and why

Squared-error objectives. Calculate signed errors first and square them. SSE adds the squares; half SSE divides that sum by 2; MSE divides by the number of scalar predictions. These denominators are different conventions.

ŷᵢ = prediction; yᵢ = target; eᵢ = signed error; n = number of scalar predictions included in the stated loss.

$e_i=\hat y_i-y_i$ LaTeX: e_i=\hat y_i-y_i

$\mathrm{SSE}=\sum_i e_i^2$ LaTeX: \mathrm{SSE}=\sum_i e_i^2

$L_{\mathrm{half}}=\frac12\sum_i e_i^2$ LaTeX: L_{\mathrm{half}}=\frac12\sum_i e_i^2

$\mathrm{MSE}=\frac1n\sum_i e_i^2$ LaTeX: \mathrm{MSE}=\frac1n\sum_i e_i^2

Yaad rakhne ka cue: Pehle error square; phir question ke hisaab se sum, half ya mean.

Solution — step by step

Given values and loss definitions

There are n = 3 prediction–target pairs. Use the supplied predictions directly; no rounding to class labels is required. Let eᵢ = ŷᵢ − yᵢ be the signed residual for pair i.

$\mathrm{SSE}=\sum_{i=1}^{n}e_i^2$ LaTeX: \mathrm{SSE}=\sum_{i=1}^{n}e_i^2

$L_{\mathrm{half}}=\frac12\sum_{i=1}^{n}e_i^2=\frac{\mathrm{SSE}}2$ LaTeX: L_{\mathrm{half}}=\frac12\sum_{i=1}^{n}e_i^2=\frac{\mathrm{SSE}}2

$\mathrm{MSE}=\frac1n\sum_{i=1}^{n}e_i^2=\frac{\mathrm{SSE}}n$ LaTeX: \mathrm{MSE}=\frac1n\sum_{i=1}^{n}e_i^2=\frac{\mathrm{SSE}}n

SSE is the sum of squared errors. Half SSE and MSE rescale that same sum by different denominators: 2 and n, respectively.

Step 1: calculate each residual and its square

Pair	Prediction ŷᵢ	Target yᵢ	Residual ŷᵢ − yᵢ	Squared residual
1	0.2	0	0.2 − 0 = 0.2	(0.2)² = 0.04
2	0.7	1	0.7 − 1 = −0.3	(−0.3)² = 0.09
3	0.4	1	0.4 − 1 = −0.6	(−0.6)² = 0.36

Step 2: sum the squared residuals

Square each residual before summing. Squaring the sum of signed residuals would be a different quantity, because positive and negative residuals could cancel.

Step 3: apply each requested normalization

For half SSE, the denominator is 2:

$L_{\mathrm{half}}=\frac{0.49}{2}=0.245$ LaTeX: L_{\mathrm{half}}=\frac{0.49}{2}=0.245

For MSE, the denominator is the number of pairs, n = 3:

Final answer: SSE = 0.49; half SSE = 0.245; MSE ≈ 0.163333. They represent the same residuals at different scales. Half SSE and MSE coincide when n = 2; they do not coincide for these three nonzero residuals. No derivative is required by this question.