johnswentworth comments on Coherence of Caches and Agents

[johnswentworth]

Here’s what it would typically look like in a control theory problem.

There’s a long term utility $u^L$ which is a function of the final state $x\left(T\right)$, and a short term utility $u^S$ which is a function of time $t$, the state $x\left(t\right)$ at time $t$, and the action $a\left(t\right)$ at time $t$. (Often the problem is formulated with a discount rate $\gamma$ , but in this case we’re allowing time-dependent short-term utility, so we can just absorb the discount rate into $u^S$). The objective is then to maximize

$u^L\left(x\right(T\left)\right)+{\sum }_t{u}^S(t,x(t),a(t\left)\right)$

In that case, the value function $V(x,t)$ is a max over trajectories starting at $(x,t)$:

$V(x,t)={\text{max}}_{\text{trajectory}}{u}^L\left(x\right(T\left)\right)+{\sum }_{\tau \ge t}{u}^S(\tau ,x(\tau ),a(\tau \left)\right)$

The key thing to notice is that we can solve that equation for $u^S(t,x(t),a(t\left)\right)$:

$u^S(t,x(t),a(t\left)\right)={\text{max}}_{\text{trajectory}}\left(u^L\right(x\left(T\right))+{\sum }_{\tau >t}{u}^S(\tau ,x\left(\tau \right),a\left(\tau \right)\left)\right)-V(x,t)$

So given an arbitrary value function $V$, we can find a short-term utility function $u^S$ which produces that value function by using that equation to compute $u^S$ starting from the last timestep and working backwards.

Thus the claim from the post: for any value function, there exists a short-term utility function which induces that value function.

What if we restrict to only consider long-term utility, i.e. set $u^S=0$? Well, then the value function is no longer so arbitrary. That’s the case considered in the post, where we have constraints which the value function must satisfy regardless of $u^L$.

Did that clarify?

[Sheikh Abdur Raheem Ali]

I had to mull over it for five days, hunt down some background materials to fill in context, write follow up questions to a few friends (reviewing responses over phone while commuting), and then slowly chew through the math on pencil and paper when I could get spare time… but yes I understand now!