Curve-Fitting Policy

Our first policy is based on fitting divergence curves over the sampled divergence values found in the change profiles. It works as follows. Immediately after each refresh of page $P$, the policy updates its change profiles and then computes a new refresh period $\phi_P$, i.e., the period of time to wait before refreshing $P$ again. The process for computing $\phi_P$ has three steps:

1. Combine change profiles. Combine the observations recorded in the $h$ different change profiles, into a single profile. In particular, first normalize each change profile by subtracting the base version from each recorded timestamp, so that each profile starts from $t=0$. Then take the union of the ($t$, $D$) pairs across all profiles, while combining entries that have the same $t$ value by taking the average $D$ value.
   
2. Fit divergence curve. Fit a continuous divergence curve to the set of points in the combined change profile, based on our generative model of Section 3.3. In particular, fit one curve that corresponds to churn behavior, and a second curve for scroll behavior, and select the one that yields the closest fit.
   
3. Compute refresh period. Using Equation 4, set the refresh period $\phi_P$ equal to the $t$ value that causes $U_P(t) = T$, where $T$ is the utility threshold parameter.

To ensure that enough observations are collected at the outset, for each newly-discovered page $P$ there is an initial learning phase in which the maximum refresh period is bounded by $l \cdot n$ where $n$ denotes the total number of past refreshes of $P$ and $l > 0$ is a learning parameter. Hence initially the policy is required to perform refreshes in rapid succession, but the maximum refresh period grows linearly over time. (Other functions to control the learning phase are also possible, and may be worth studying in future work.)