timetogoal

Timetogoal, often abbreviated TTG, is a metric used in planning, control, and reinforcement learning to quantify the expected time required to reach a designated goal state from a given state under a specified policy.

Formally, in a Markov decision process with state space S, goal state g, and policy π, TTG(s, π) is the expected number of steps to reach g starting from state s and following π. The optimal timetogoal, TTG*(s), is the minimum expected time over all policies. If the goal is absorbing and time is discrete, TTG*(g) = 0, and for s ≠ g, TTG*(s) satisfies the Bellman equation T*(s) = 1 + min_a ∑_{s'} P(s'|s, a) T*(s').

Computation can be performed using model-based dynamic programming methods such as value iteration or policy iteration, given a known transition model. In environments where the model is unknown or large, model-free techniques like temporal-difference learning or Monte Carlo estimation can be applied to approximate TTG values, often with function approximation in high-dimensional spaces.

Applications of timetogoal span robotics, autonomous navigation, video game AI, and logistics optimization, where TTG serves both as a performance measure and as an objective during policy design. It helps compare strategies, set timing targets, and evaluate the efficiency of different policies in reaching desired outcomes.

Variants and extensions include continuous-time timetogoal, discounted versions for stochastic processes with termination probabilities, and TTG to intermediate subgoals. In practice, TTG requires careful handling of model accuracy and the possibility that a goal may be unreachable under certain policies.