A Modal Type Theory of Expected Cost in Higher-Order Probabilistic Programs

The design of online learning algorithms typically aims to optimise the incurred loss or cost , e.g., the number of classification mistakes made by the algorithm. The goal of this paper is to build a type-theoretic framework to prove that a certain algorithm achieves its stated bound on the cost. Online learning algorithms often rely on randomness, their loss functions are often defined as expectations, precise bounds are often non-polynomial (e.g., logarithmic) and proofs of optimality often rely on potential-based arguments. Accordingly, we present pλ-amor, a type-theoretic graded modal framework for analysing (expected) costs of higher-order probabilistic programs with recursion. pλ-amor is an effect-based framework which uses graded modal types to represent potentials, cost and probability at the type level. It extends prior work (λ-amor) on cost analysis for deterministic programs. We prove pλ-amor sound relative to a Kripke step-indexed model which relates potentials with probabilistic coupling. We use pλ-amor to prove cost bounds of several examples from the online machine learning literature. Finally, we describe an extension of pλ-amor with a graded comonad and describe the relationship between the different modalities.