Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: You can still fetch the coffee today if you're dead tomorrow, published by davidad (David A. Dalrymple) on December 9, 2022 on The AI Alignment Forum.
"You can't fetch the coffee if you're dead."—Stuart Russell, on the instrumental convergence of shutdown-avoidance
Note: This is presumably not novel, but I think it ought to be better-known. The technical tl;dr is that we can define time-inhomogeneous reward, and this provides a way of "composing" different reward functions; while this is not a way to build a shutdown button, it is a way to build a shutdown timer, which seems like a useful technique in our safety toolbox.
"Utility functions" need not be time-homogeneous
It's common in AI theory (and AI alignment theory) to assume that utility functions are time-homogeneous over an infinite time horizon, with exponential discounting. If we denote the concatenation of two world histories/trajectories by ⊳, the time-consistency property in this setting can be written as
This is property is satisfied, for example, by the utility-function constructions in the standard Wikipedia definitions of MDP and POMDP, which are essentially
Under such assumptions, Alex Turner's power-seeking theorems show that optimal agents for random reward functions R will systematically tend to disprefer shutting down (formalized as "transitioning into a state with no transitions out").
Exponential discounting is natural because if an agent's preferences are representable using a time-discount factor that depends only on relative time differences and not absolute time, then any non-exponential discounting form is exploitable (cf. Why Time Discounting Should Be Exponential).
However, if an agent has access to a clock, and if rewards are bounded by an integrable nonnegative function of time, the agent may be time-inhomogeneous in nearly arbitrary ways without actually exhibiting time inconsistency:
Any utility function with the above form still obeys an analogous version of our original time-consistency property that is modified to index over initial time t0:
Note that time-homogenous utility functions are a special case in which U(t,h)=γtU(0,h).
Time-bounded utility functions can be sequentially composed
We define a time-bounded utility function as a dependent tuple
i.e., a family of utility functions indexed by times within a given fixed range. The intended semantics of a time-bounded utility function in (τ,R) form is:
Given two time-bounded utility functions (in the same environment), they can be concatenated into a new time-bounded utility function:
You can check that ⊳ is a monoid, with the neutral element given by (0,∅).
How to build a shutdown timer
Let R1 be the reward function for a time-bounded task and τ1 be the time limit for the task, after which we want this agent to shut down. Assume that R1 also has bounded output, with per-stage reward always between R1––– and ˆR1. We define
We can then define τ2 to be 1 or indeed any positive integer. If an agent does not reach a shutdown state before τ1 is up, then it will realize a cost in R2 that outweighs all other rewards it could receive during the episode by a factor of C (a constant greater than 1). Therefore, optimal agents for (τ1,R1)⊳(τ2,R2) must shut down within time τ1 with probability ≥1−1/C (if the shutdown state is reachable in that time by any agent).
Proof
Suppose that the optimal policy π∗ results in a shutdown probability p<1−1/C, but there exists a policy π′ which shuts down deterministically (with probability 1). Then
which contradicts the optimality of π∗.
Comparison with the shutdown switch problem
Several years ago, MIRI's Agent Foundations group worked on how to make a reflectively stable agent with a shutdown switch, and (reportedly) gave up after failing to find a solution where the agent neither tries to manipulate the ...