Probabilistic Foundations of Machine Learning

I am: Yaniv / Professor Yacoby

I use: he/they

Announcements

• HW3 Tip:
• Copy code from the MLE chapters into DeepNote
• Print things out
• Play with it
• Adapted, little by little, for HW3

Today: Optimization

So far, we:

• Used joint distributions / DGMs to represent our modeling assumptions
• Translated these into NumPyro

Next, we want to perform the MLE: but

• How do we maximize the probability of the data given the parameters?

Today:

• A simple, general-purpose optimization algorithm

Our generative process

in code.

Performing the MLE:

Checking for Convergence

Our goal: to minimize our “loss”

How can we minimize a function? Let’s get some intuition:

Idea: look for places where the derivative of the loss is 0.

Global vs. Local Optima

Looking at figure,

• Different types of optima
• Derivative is 0 at all optima
• But we only want global minima
• So is this still useful?

Answer: Yes. How?

• Get all optima
• Plug each into loss
• See which one is smallest

Initial Algorithm

…shall we try it?

Analytic MLE, Step 1: What’s our model?

Analytic MLE, Step 2: What’s our joint data likelihood?

Analytic MLE, Step 3: What’s our loss function?

Analytic MLE, Step 4: Minimize our loss

Pros and Cons of Analytic MLE

Pros:

• Fast: just compute a formula!

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Cons:

• Analytic solution doesn’t exist for most models
• It’s a lot of work…

Alternative: Use Numeric Optimization Algorithm

• Algorithm: Gradient Descent
• Gradient = derivative (in high dims)
• Gradient/derivative point in the direction of steepest ascent
• Idea: take steps following the direction of the gradient

Gradient Descent

• Repeat: 3 steps
• How are derivatives computed? Jax.

Simulations (see chapter)

Gradient Descent in Jax

Note: you should not implement/use this. Use the code we provided.

Challenges with Numeric Optimization

• It’s iterative (slow)
• It’s approximate (gets stuck in local optima)
• It’s sensitive to initialization (gets stuck in local optima)
• It adds diagnostic challenges:
• When model fits data poorly, is the model the problem? Or is the optimization too hard?

Constraining Parameters to Valid Ranges

• For initial Bernoulli example,
• Gradient descent might step outside these bounds
• How to fix this?
• Idea:
• Define
• Then, optimize with respect to psi
• That is, our original problem is:

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• Instead, we solve:

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• This is how NumPyro implements C.unit_interval

In pairs: continue implementing the IHH-ER model in NumPyro!

That’s all for today!

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Questions?

• Tell your partner: what did you appreciate about working with them today!