Safe Value Functions

TL;DR

This paper formalizes the concept of safe value functions in reinforcement learning, establishing conditions under which safety constraints can be incorporated into optimal value functions through penalties, and providing insights for designing safe reward functions.

Contribution

It introduces the formal notion of safe value functions, analyzes the relationship between penalties, rewards, and safety, and offers practical heuristics for reward design in safety-critical control tasks.

Findings

Existence of finite penalties that induce safe value functions.

Larger penalties do not compromise optimality.

Clear structure of interactions between penalties, rewards, and dynamics.

Abstract

Safety constraints and optimality are important, but sometimes conflicting criteria for controllers. Although these criteria are often solved separately with different tools to maintain formal guarantees, it is also common practice in reinforcement learning to simply modify reward functions by penalizing failures, with the penalty treated as a mere heuristic. We rigorously examine the relationship of both safety and optimality to penalties, and formalize sufficient conditions for safe value functions (SVFs): value functions that are both optimal for a given task, and enforce safety constraints. We reveal this structure by examining when rewards preserve viability under optimal control, and show that there always exists a finite penalty that induces a safe value function. This penalty is not unique, but upper-unbounded: larger penalties do not harm optimality. Although it is often not possible to compute the minimum required penalty, we reveal clear structure of how the penalty, rewards, discount factor, and dynamics interact. This insight suggests practical, theory-guided heuristics to design reward functions for control problems where safety is important.