Question 4

Let f be some function so that

f(θ₀,θ₁) outputs a number. For this problem,

f is some arbitrary/unknown smooth function (not necessarily the

cost function of linear regression, so f may have local optima).

Suppose we use gradient descent to try to minimize f(θ₀,θ₁) as a function of θ₀ and θ₁. Which of the

following statements are true? (Check all that apply.)

• Even if the learning rate α is very large, every iteration of gradient descent will decrease the value of f(θ₀,θ₁).
• If the learning rate is too small, then gradient descent may take a very long time to converge.
• If θ₀ and θ₁ are initialized at a local minimum, then one iteration will not change their values.
• If θ₀ and θ₁ are initialized so that θ₀=θ₁, then by symmetry (because we do simultaneous updates to the two parameters), after one iteration of gradient descent, we will still have θ₀=θ₁.

Answers:

True or False	Statement	Explanation
True	If the learning rate is too small, then gradient descent may take a very long time to converge.	If the learning rate is small, gradient descent ends up taking an extremely small step on each iteration, and therefore can take a long time to converge
True	If θ0 and θ1 are initialized at a local minimum, then one iteration will not change their values.	At a local minimum, the derivative (gradient) is zero, so gradient descent will not change the parameters.
False	Even if the learning rate α is very large, every iteration of gradient descent will decrease the value of f(θ0,θ1).	If the learning rate is too large, one step of gradient descent can actually vastly “overshoot” and actually increase the value of f(θ0,θ1).
False	If θ0 and θ1 are initialized so that θ0=θ1, then by symmetry (because we do simultaneous updates to the two parameters), after one iteration of gradient descent, we will still have θ0=θ1.	The updates to θ0 and θ1 are different (even though we’re doing simulaneous updates), so there’s no particular reason to update them to be same after one iteration of gradient descent.

Other Options:

True or False	Statement	Explanation
True	If the first few iterations of gradient descent cause f(θ0,θ1) to increase rather than decrease, then the most likely cause is that we have set the learning rate to too large a value	if alpha were small enough, then gradient descent should always successfully take a tiny small downhill and decrease f(θ0,θ1) at least a little bit. If gradient descent instead increases the objective value, that means alpha is too large (or you have a bug in your code!).
False	No matter how θ0 and θ1 are initialized, so long as learning rate is sufficiently small, we can safely expect gradient descent to converge to the same solution	This is not true, depending on the initial condition, gradient descent may end up at different local optima.
False	Setting the learning rate to be very small is not harmful, and can only speed up the convergence of gradient descent.	If the learning rate is small, gradient descent ends up taking an extremely small step on each iteration, so this would actually slow down (rather than speed up) the convergence of the algorithm.