# ECE 4200 Assignment Seven solution

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Problem 1. (10 points). Suppose AdaBoost is run on n training examples, and suppose on each
round that the weighted training error εt of the tth weak hypothesis is at most 1
2 − γ, for some
number γ > 0. Show that after T > ln n

2 rounds of AdaBoost the final combined classifier has zero
training error!
Problem 2. (10 points). Recall bagging. Starting from a training set S of size n, we created m
bootstrap training sets S1, . . . , Sm, each of size n each by sampling with replacement from S.
1. For a bootstrap sample Si
, what is the expected fraction of the training set that does not
appear at all in Si? As n → ∞, what does this fraction approach?
2. Let m > 2 ln n, and n → ∞. Show that the expected number of training examples in S that
appear in at least one Si
is more than n − 1.
Problem 3. (10 points). The tanh function is tanh(y) = (e
y − e
−y
)/(e
y + e
−y
). Consider the
function tanh(w0 + w1x1 + w2x2), with five inputs, and a scalar output.
1. Draw the computational graph of the function (you can use tanh in your computation graph).
2. What is the derivative of tanh(y) with respect to y.
3. Suppose (w0, w1, w2, x1, x2) = (−2, −3, 1, 2, 3). Compute the forward function values, and