Deep Learning · MT2 · section

F22 Q1b

F22 selected MT2 practice

Question

Question (4 marks). Weights are 0.55 and 0.65; their gradients are+0.3 and−0.25. With η = 0.1, compute gradient-descent changes and new weights.

Answer — step by step

Given

Old weights = [0.55,0.65].

Corresponding supplied gradients = [0.3,−0.25]; η = 0.1.

To find

Calculate the signed weight changes and both new weights.

Formula — which one and why

Gradient descent. Evaluate the gradient at the current parameter, multiply by the learning rate and subtract. If the objective is supplied, differentiate it and recompute the gradient after each update.

θ (theta) = trainable parameter; g = derivative of the stated objective J; η = learning rate; t labels the update number.

g_t=J\prime(\theta_{t-1})LaTeX: g_t=J\prime(\theta_{t-1})
\theta_t=\theta_{t-1}-\eta g_tLaTeX: \theta_t=\theta_{t-1}-\eta g_t

Yaad rakhne ka cue: Naya = purana − learning rate × current gradient.

Solution — step by step

Step 1: calculate the change in weight 1.

\Delta w_1=-0.1(0.3)=-0.03LaTeX: \Delta w_1=-0.1(0.3)=-0.03

Step 2: add that change to old weight 1.

w_1^{\mathrm{new}}=0.55+(-0.03)=0.52LaTeX: w_1^{\mathrm{new}}=0.55+(-0.03)=0.52

Step 3: calculate the change in weight 2.

\Delta w_2=-0.1(-0.25)=0.025LaTeX: \Delta w_2=-0.1(-0.25)=0.025

Step 4: add that change to old weight 2.

w_2^{\mathrm{new}}=0.65+0.025=0.675LaTeX: w_2^{\mathrm{new}}=0.65+0.025=0.675

Final answer: changes = [−0.03, +0.025]; new weights = [0.52, 0.675]. A negative gradient increases its weight under descent.