We begin with a policy parameterized on anything you’d like with random seed weights. Then,

1. We sample a local set of parameters, one pertubation \(\pm \alpha\) per direction in the parameter vector (for instance, for a parameter in 4-space, up, down, left, right in latent space), and use those new parameters to seed a policy.
2. Check each policy for its utility via monte-carlo policy evaluation
3. If any of the adjacent points are better, we move there
4. If none of the adjacent points are better, we set \(\alpha = 0.5 \alpha\) (of the up/down/left/right) and try again

We continue until \(\alpha\) drops below some \(\epsilon\).

\begin{equation} \log a^{b} = b\log a \end{equation}

\begin{equation} \log (ab) = \log a + \log b \end{equation}

\begin{equation} \log (\frac{a}{b}) = \log a - \log b \end{equation}

TODO: connect Logan with a few fire departments

Consider:

\begin{equation} P’ = 2P(100-P) \end{equation}

for a motivation, see petri dish.

Solution

Assuming \(P\) never reaches 100

\begin{equation} \int \frac{\dd{P}}{P(100-P)} \dd{P}= \int 2 \dd{t} \end{equation}

Partial fractions time:

\begin{equation} \frac{1}{100} \int \qty(\frac{1}{p} + \frac{1}{100-p})\dd{P} = \frac{1}{100} \ln |p| - \ln |100-p| = 2t+C \end{equation}

Remember now log laws:

\begin{equation} \frac{1}{100} \ln \left| \frac{p}{100-p} \right| = 2t+C \end{equation}

And finally, we obtain:

\begin{equation} \qty | \frac{p}{100-p} | = e^{200t + C} \end{equation}

Goals

Motivation: it is very difficult to have an interpretable, causal trace of facts. Let’s fix that.

Facts

It is also further difficult to pull about what is a “fact” and what is a “syntactical relation”. For instance, the task of

The Apple iPhone is made by American company <mask>.

is different and arguably more of a syntactical relationship rather than factually eliciting prompt than

The iPhone is made by American company <mask>.

For our purposes, however, we obviate this problem by saying that both of these cases are a recall of the fact triplet <iPhone, made_by, Apple>. Even despite the syntactical relationship established by the first case, we define success as any intervention that edits this fact triplet without influencing other stuff of the form: