The blog post tackles a single question: when does learning from data actually work? It explains that a hypothesis class is learnable iff it has finite VC dimension. The author builds the theory from scratch, starting with Markov’s inequality and culminating in the Fundamental Theorem of Statistical Learning in the context of binary classification settings.

To ground the discussion, the author outlines the learning problem: input space \(\mathcal{X}\), binary labels \(\mathcal{Y}\), unknown distribution \(\mathcal{D}\), and hypothesis class \(\mathcal{H}\). Training samples arrive i.i.d. from \(\mathcal{D}\). The key metrics are true risk \(L_{\mathcal{D}}(h)\) and empirical risk \(L_S(h)\), whose relationship drives generalization for models that can reliably predict unseen data across arbitrary distributions and meet industry standards.

The post then formalizes learnability through PAC learning: an algorithm must output a hypothesis within \(\epsilon\) of the best possible error with probability \(1-\delta\) for every distribution. It also introduces uniform convergence, requiring training error to track true error simultaneously for all hypotheses—a stricter condition that guarantees ERM succeeds for practical model selection in tasks.

Finally, the author promises a Part 2 that will explore Rademacher complexity, a data‑dependent refinement of VC dimension. By tying together concentration inequalities, symmetrization, and information‑theoretic lower bounds, the series offers a complete, self‑contained proof that bridges classic statistical learning theory with modern algorithmic practice for researchers and practitioners seeking rigorous foundations to guide model evaluation.