In this post, we will modify the blockwise estimation procedure we discussed last time, to reduce the constant factor appeared in our upper bound on the expected risk.

Alternative Blockwise Estimator

The issue is that the number of blocks could be very small compared to the number of observations. For example, when $n=1024$, the number of blocks $K = 10$.

Previously, we set each block size to be the power of 2. That is, $|B_k| = 2^k$.

To have a smaller blocksize, we can set $|B_k| = \lfloor (1+a)^k \rfloor$, where $0 < a < 1$.

In fact, if we set $a = \log_2 n$, we see that:

$$n = \sum_{k=1}^K (1 + \frac{1}{\log_2 n} )^k =$$

(WIP)