Raising expectations: automating expected cost analysis with types

Abstract

This article presents a type-based analysis for deriving upper bounds on the expected execution cost of probabilistic programs. The analysis is naturally compositional, parametric in the cost model, and supports higher-order functions and inductive data types. The derived bounds are multivariate polynomials that are functions of data structures. Bound inference is enabled by local type rules that reduce type inference to linear constraint solving. The type system is based on the potential method of amortized analysis and extends automatic amortized resource analysis (AARA) for deterministic programs. A main innovation is that bounds can contain symbolic probabilities, which may appear in data structures and function arguments. Another contribution is a novel soundness proof that establishes the correctness of the derived bounds with respect to a distribution-based operational cost semantics that also includes nontrivial diverging behavior. For cost models like time, derived bounds imply termination with probability one. To highlight the novel ideas, the presentation focuses on linear potential and a core language. However, the analysis is implemented as an extension of Resource Aware ML and supports polynomial bounds and user defined data structures. The effectiveness of the technique is evaluated by analyzing the sample complexity of discrete distributions and with a novel average-case estimation for deterministic programs that combines expected cost analysis with statistical methods.

On continuation-passing transformations and expected cost analysis

“…Notable also, in [Wang et al 2020], the amortized cost analysis underlying resource aware ML [Hoffmann et al 2012] has been suited to probabilistic programs. This work sets itself apart by having a fully automated inference machinery, differently from what we do here.…”

“…This work sets itself apart by having a fully automated inference machinery, differently from what we do here. The two particular instances of walk given in Example 3.1 lie within the scope of [Wang et al 2020]'s methodology, when specialized to two first-order programs. However, depending on its functional arguments, the expected cost of walk can vary between constant, to linear, or even be infinite.…”

“…Some of the approaches used in the deterministic setting extend to the probabilistic setting, namely type systems and program logics [Kaminski and Katoen 2017;. In a higher-order setting, the only technique which has been studied is the one based on types, which have indeed been proved to be applicable to higher-order randomized languages [Avanzini et al 2019a;Dal Lago and Grellois 2017;Wang et al 2020]. The main advantage of adopting type systems is the inherent compositionality of the obtained methodology.…”

“…At least for programs whose cost analysis is not relying on too complicated functional properties, it seems feasible to incorporate template based approach resting on SMT-solvers Ð the pre-dominant approach underlying automated tools (see e.g. [Avanzini et al 2015;Wang et al 2020]). Ideally, one would arrive at a semi-automated analysis tool, where simple proof steps are automatically discharged and the user can focus on the non-trivial aspects of the cost analysis.…”

Work Analysis with Resource-Aware Session Types

“…• Trace-based [Borgström et al 2016;Castellan and Paquet 2019]: The state-to-state transition relation is interpreted under a fixed trace of random sources (e.g., a trace of coin flips), where the random constructs (e.g., coin flips) are resolved as deterministic readouts from the trace. In our setting, however, this approach lacks a mechanism for tracking the correlations among traces, which have been shown to be important for expected-cost analysis [Wang et al 2020b]. With the presence of probabilistic internal-and external-choice types, it is also unclear how to guarantee type preservation for the transitions guided by a fixed trace.…”