Suppose that you have trained a logistic regression classifier, and it outputs on a new example x a prediction hθ(x) = 0.4. This means (check all that apply):

Our estimate for P(y=0|x;θ) is 0.4.

Our estimate for P(y=0|x;θ) is 0.6.

Our estimate for P(y=1|x;θ) is 0.4.

Our estimate for P(y=1|x;θ) is 0.6.

Suppose you have the following training set, and fit a logistic regression classifier hθ(x)=g(θ0+θ1x1+θ2x2).

Which of the following are true? Check all that apply.

Adding polynomial features (e.g., instead using hθ(x)=g(θ0+θ1x1+θ2x2+θ3x21+θ4x1x2+θ5x22) ) could increase how well we can fit the training data.

At the optimal value of θ (e.g., found by fminunc), we will have J(θ)≥0.

Adding polynomial features (e.g., instead using hθ(x)=g(θ0+θ1x1+θ2x2+θ3x21+θ4x1x2+θ5x22) ) would increase J(θ)because we are now summing over more terms.

If we train gradient descent for enough iterations, for some examples x(i) in the training set it is possible to obtain hθ(x(i))>1.

For logistic regression, the gradient is given by ∂∂θjJ(θ)=∑mi=1(hθ(x(i))−y(i))x(i)j. Which of these is a correct gradient descent update for logistic regression with a learning rate of α? Check all that apply.

θ:=θ−α1m∑mi=1(hθ(x(i))−y(i))x(i).

θj:=θj−α1m∑mi=1(θTx−y(i))x(i)j (simultaneously update for all j).

θ:=θ−α1m∑mi=1(11+e−θTx(i)−y(i))x(i).

θ:=θ−α1m∑mi=1(θTx−y(i))x(i).

Which of the following statements are true? Check all that apply.

Linear regression always works well for classification if you classify by using a threshold on the prediction made by linear regression.

For logistic regression, sometimes gradient descent will converge to a local minimum (and fail to find the global minimum). This is the reason we prefer more advanced optimization algorithms such as fminunc (conjugate gradient/BFGS/L-BFGS/etc).

The cost function J(θ) for logistic regression trained with m≥1 examples is always greater than or equal to zero.

The sigmoid function g(z)=11+e−z is never greater than one (>1).

Suppose you train a logistic classifier hθ(x)=g(θ0+θ1x1+θ2x2). Suppose θ0=6,θ1=−1,θ2=0. Which of the following figures represents the decision boundary found by your classifier?

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