Importance Sampling and Effective Number of Samples

To estimate the total number of self-avoiding walks (SAWs) for a grid size n = 10, we will use Monte Carlo integration. First we design a trial probability $g(x)$ for a SAW that is easier to sample from than the true target distribution. We then sample $M$ SAWs from $g(x)$ and estimate the total count of SAWs by the mean of $1 / g(x)$ for SAWs $x$. The crux of the issue lies in how to design $g(x)$. Here we will examine three different designs for $g(x)$, each assuming the grid size $n = 10$ and each generating a total of $M = 107$ total samples.

Three Designs for $g(x)$:
Design 1: Here $g(x)$ is equal to the product of $1 / k_j$ terms, where $k_j$ represents the number of possible choices for the $j$th move. At each step $j$, we sample uniformly from the $k_j$ choices. In this design the distribution will resemble a Gaussian because we do not constrain the length of walk.
The total estimated number $K$ of SAWs for Design 1 was $3.1804 * 10^{25}$.

Design 2: Here $g(x)$ is equal to the $g(x)$ of Design 1, but multiplied by $(1 - ε)m$, where $ε = 0.1$ is an early termination probability at each step. This has the effect of lending shorter but more walks overall than design 1. In the log-log plot below of $K$ SAWs over $M$ samples, we can see that Design 2 obviously converges the slowest. This makes sense because Design 2 terminates some paths early, preventing the algorithm from finding paths it might otherwise find. The total estimated number $K$ of SAWs for Design 2 was $3.2535*10^{25}$.

Design 3: This design is a modification of Design 1 to favor longer walks. For any walk longer than $50$, Design 3 generates $5$ more children from that branch of the walk, reweighting each of the children by $w_0 = w / 5$. This design converges the fastest. The total estimated number $K$ of SAWs for Design 3 was $2.0898 * 10^{25}$.

Plotting the total number of SAWs K against $M = 107$ examples in the log-log plot below for each of the Designs 1, 2, and 3, we can analyze whether the sequential importance sampling process has converged. As expected, Design 2 has the slowest convergence after roughly $M = 107$ samples due to its early termination probability. Design 3 has an improved convergence rate over Design 1 due to the behavior of generating 5 children each time a SAW reaches a length of $50$. Both Designs 1 and 3 converge after roughly $M = 103$ samples, but Design 3 converges sooner and is thus the optimal design in this study.

We can make a modification to the most optimal sampling procedure, Design 3, to investigate the total number of self-avoiding walks that start from $(0, 0)$ and reach a specific point $(n, n)$. Namely, we now only count the SAWs that successfully reach $(n, n)$ during their walk instead of all the SAWs. Generating $106$ samples, the true value for the total number of SAWs from $(0, 0)$ to $(n, n)$ is $1.5687 * 10^{24}$.

In an effort to approximate this true value as closely as possible, I experimented with generating $3, 5$, or $7$ children for paths that had grown to a checkpoint length of $25, 50, 70, 80, 9$, or $100$ and found that it generally helps to spawn more children at a given checkpoint length, and it also helps to increase the checkpoint length at which this happens. These conclusions are reasonable, as the longer a SAW path is, the more “special” it is and the more unlikely it is that we have already found SAWs that stem from this one. Of course, if the checkpoint length is too high, e.g. $200$, then of course we will never even get the opportunity to generate children (because a path will never reach that length).

Ultimately, the closest approximation to the true value of $1.5687 * 10^{24}$ reached was for generating $7$ children once a path had reached a length of $90$. For these parameters, the estimated number of SAWs was $1.2224 * 10^{24}$.

Below is a visualization of the longest SAWs for each of the Designs 1, 2, and 3.

Eric M. Fischer
Ph.D. Statistics with specialization in Artificial Intelligence

My research interests are in natural language processing and generative modeling.