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Drill 31 Inverted dropout

MT2 numerical practice from direct to mixed

Question

Question. Activations = [2,−1,0,3], keep probability 0.75, mask = [1,0,1,1]. Find training outputs and explain inference.

Answer — step by step

Given

Activations = [2,−1,0,3].

Keep probability q = 0.75; keep mask = [1,0,1,1].

To find

Find training outputs and state the inference rule.

Formula — which one and why

Inverted dropout. Multiply by the keep/drop mask, then divide retained activations by the keep probability. This preserves each activation’s expectation over random masks.

aᵢ = original activation; mᵢ is 1 to keep or 0 to drop; q = keep probability = 1−dropout rate. Disable this masking/scaling at inference.

a_i^{\mathrm{train}}=\frac{m_ia_i}{q}LaTeX: a_i^{\mathrm{train}}=\frac{m_ia_i}{q}
a_i^{\mathrm{test}}=a_iLaTeX: a_i^{\mathrm{test}}=a_i

Yaad rakhne ka cue: Mask se keep/drop; kept value ko keep probability se divide.

Solution — step by step

Meaning and formula

The mask entry m is 1 when an activation is kept and 0 when dropped. The keep probability q is 0.75. Inverted dropout divides the kept activations by q during training so that their expected value is preserved.

a'_j=\frac{m_j a_j}{q}LaTeX: a'_j=\frac{m_j a_j}{q}

The multiplication is entry by entry; there is no dot-product sum.

Step 1: apply the mask

[1(2),\;0(-1),\;1(0),\;1(3)]=[2,0,0,3]LaTeX: [1(2),\;0(-1),\;1(0),\;1(3)]=[2,0,0,3]

Step 2: divide each result by 0.75

a'_1=\frac2{0.75}=2.666666667LaTeX: a'_1=\frac2{0.75}=2.666666667
a'_2=\frac0{0.75}=0,\qquad a'_3=\frac0{0.75}=0LaTeX: a'_2=\frac0{0.75}=0,\qquad a'_3=\frac0{0.75}=0
a'_4=\frac3{0.75}=4LaTeX: a'_4=\frac3{0.75}=4

Training output = [2.666666667, 0, 0, 4]. A kept activation that was already zero stays zero.

During inference

Disable the random mask and the division. If the incoming activations are the same [2,−1,0,3], they pass through unchanged. The scaling has already been accounted for during training.