Reference · The Applied Higher-Order Functions catalogue

rainbow.algorithms.proofs.ahof.perceptron

The Applied Higher-Order Functions cataloguesrc/rainbow/algorithms/proofs/ahof/perceptron.clj1 definitions, 3 theorems

AHoF Perceptron (Rosenblatt): the foundational linear classifier and the atom of neural networks. It predicts a class from the sign of a weighted sum of inputs (the activation) relative to a threshold, and learns by the mistake-driven rule w <- w + eta*(y - y_hat)*x, which -- by the perceptron convergence theorem -- finds a separating hyperplane in finitely many updates when the data are linearly separable. The firing (activation) decision is: fires? activation threshold = decide (threshold <= activation) over exact-rational activations. Proved native and axiom-free: the firing test and worked instances on concrete values -- activation 3 against threshold 2 fires; the worked certificate 2 <= 3 as a native nnrat/at-most; and activation 1 against threshold 2 does not fire. The weight vector, the update rule, and the dot product are executable; what is proved is the firing decision at these instances. The perceptron convergence theorem is not proved here. Requires nnrat/at-most, audit/decide-nnrat-at-most, nnrat/decide-at-most-true-sound (all in the D63 checkpoint). Scope (measured by the kernel): this module has 3 worked instances (statements about specific values). None of its theorems is a guarantee about the algorithm for all inputs; `lein verify rainbow.algorithms.proofs.ahof.perceptron` prints this classification.

perceptron/fires? :
  (x₁ x₂ : ℚ≥0)
  → data/Bool

perceptron/activation-fires :
  → bo/is-safe 3 2 = data/true

perceptron/firing-meets-threshold :
  → 2 ≤ 3

perceptron/subthreshold-does-not-fire :
  → bo/is-safe 1 2 = data/false