Work

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

Year: 2024

Type: article

Abstract: 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 potentialbased 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.

Source: Proceedings of the ACM on Programming Languages

DOI: 10.1145/3689725

Authors: Vineet Rajani, Gilles Barthe, Deepak Garg

Institutions: University of Kent, Max Planck Institute for Security and Privacy, IMDEA Software Institute, Max Planck Institute for Software Systems

Language: English

Topic: Logic, programming, and type systems

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: diamond

Funder: Engineering and Physical Sciences Research Council

Awards: TYPDSEC: TYPe-based information Declassification and its SEcure Compilation, G2341019582