Module 1 Discussion Topics

These questions are designed for in-class discussion or written response. Strong answers are precise: they identify the specific structural failure, not just "the model didn't generalize."

Discussion 1: A Real Extrapolation Failure

Find a documented case of a deployed ML system failing on inputs that were structurally novel (not just statistically rare). Examples include autonomous vehicles encountering unusual road markings, language models producing confidently wrong answers on novel compositional queries, or medical AI failing on populations underrepresented in training data.

• Describe the failure precisely: what was the input, what was the output, what should the output have been?
• Identify the structural property of the novel input that the model failed to handle.
• What inductive bias would have prevented the failure? Be specific — "more data" is not an answer.
• Could a morphism structure encode the relevant constraint? What would the objects and morphisms be?

Discussion 2: The No Free Lunch Theorem

The No Free Lunch Theorem states that no learning algorithm outperforms random guessing when averaged uniformly over all possible target functions. This is sometimes cited as evidence that inductive bias cannot be chosen principally — that all biases are equally arbitrary.

• State the No Free Lunch Theorem precisely. What is it actually saying?
• The theorem assumes a uniform distribution over target functions. Is this assumption reasonable for physical or structured domains? Why or why not?
• How does the structural constraint paradigm respond to the NFL argument? Does it concede anything to it?
• Is the choice of morphism structure arbitrary in the same way that the choice of a regularizer is arbitrary?

Discussion 3: LeCun's Diagnosis vs. the Categorical Diagnosis

LeCun argues in the required reading that the core problem with current AI is the inability to reason and plan. His proposed remedy (JEPA, non-contrastive learning) avoids the partition function by predicting in latent space.

• Summarize LeCun's diagnosis of the failure of current models in one paragraph.
• The course thesis offers a different diagnosis: the model lacks morphism structure. Are these two diagnoses compatible, competing, or addressing different problems?
• LeCun argues that normalizing over all possible states is computationally wasteful. Is this an argument against probability, or against a particular choice of category over which to normalize?
• If you had to design an experiment that would distinguish between LeCun's remedy and the categorical remedy, what would it look like?

Discussion 4: What Is a Model?

The course thesis claims "the model is the category." Before accepting or rejecting this, it is worth interrogating what a model is in the statistical learning sense.

• In standard ML: a model is a parameterized function class. What are the parameters, and what do they represent?
• In the categorical view: the model is a category (objects + morphisms). What are the degrees of freedom, and what do they represent?
• What is lost in the categorical view compared to the statistical view? What is gained?
• Is there a setting in which both views are simultaneously correct? What would that look like?