On the correctness of monadic backward induction

TL;DR

This paper formalizes the correctness of monadic backward induction in finite-horizon SDPs, broadening its applicability beyond traditional deterministic and stochastic cases, using category-theoretical conditions.

Contribution

It defines a correctness notion for monadic SDPs and proves conditions under which monadic backward induction is valid, extending classical results to a more general framework.

Findings

Backward induction is correct under certain general conditions.

The framework applies to deterministic, stochastic, and some other SDPs.

Some instances considered admissible are ruled out by the conditions.

Abstract

In control theory, to solve a finite-horizon sequential decision problem (SDP) commonly means to find a list of decision rules that result in an optimal expected total reward (or cost) when taking a given number of decision steps. SDPs are routinely solved using Bellman's backward induction. Textbook authors (e.g. Bertsekas or Puterman) typically give more or less formal proofs to show that the backward induction algorithm is correct as solution method for deterministic and stochastic SDPs. Botta, Jansson and Ionescu propose a generic framework for finite horizon, monadic SDPs together with a monadic version of backward induction for solving such SDPs. In monadic SDPs, the monad captures a generic notion of uncertainty, while a generic measure function aggregates rewards. In the present paper we define a notion of correctness for monadic SDPs and identify three conditions that allow us to prove a correctness result for monadic backward induction that is comparable to textbook correctness proofs for ordinary backward induction. The conditions that we impose are fairly general and can be cast in category-theoretical terms using the notion of Eilenberg-Moore-algebra. They hold in familiar settings like those of deterministic or stochastic SDPs but we also give examples in which they fail. Our results show that backward induction can safely be employed for a broader class of SDPs than usually treated in textbooks. However, they also rule out certain instances that were considered admissible in the context of Botta et al.'s generic framework. Our development is formalised in Idris as an extension of the Botta et al. framework and the sources are available as supplementary material.