Question $2$c: Perform Rocchio query updates $(20$ pts$)$

Finally it's time to engage with the inquisitive User. Complete the method gather$\_$feedback for inviting a user to your Rocchio relevance feedback loop. Your interaction with the user proceeds in four steps.

Step $1$: You request the user to issue queries by invoking the user's issue$\_$queries method. The user then returns to you a dict of queries represented by queries, and a number max$\_$docs indicating the maximum number of documents the user wants to retrieve for each query.

Step $2$: You retrieve the required number of documents based on queries as you did in the last question.

Step $3$: You invite the user to give feedback to the documents retrieved, ret$\_$docs, by invoking the user's give$\_$feedback method with ret$\_$docs passed along. Normally in a relevance feedback loop, the user would mark each document retrieved as either relevant or irrelevant; however, the user you are engaging with prefers giving feedback in the form of precisions at each rank.

Specifically, suppose your ret$\_$docs is as follows:

$\{$

$\text{'}$q$167\text{'}$: $[\text{'}$d$553\text{'},\text{'}$d$1100\text{'},\text{'}$d$1096\text{'},\text{'}$d$1279\text{'},\text{'}$d$1099\text{'}],$

$\text{'}$q$54\text{'}$ : $[\text{'}$d$123\text{'},\text{'}$d$554\text{'},\text{'}$d$623\text{'},\text{'}$d$398\text{'},\text{'}$d$102\text{'}],$

$\text{'}$q$197\text{'}$: $[\text{'}$d$768\text{'},\text{'}$d$884\text{'},\text{'}$d$883\text{'},\text{'}$d$909\text{'},\text{'}$d$882\text{'}]$

$\}$

from which we can deduce that the user must have issued three queries and for each query at most five documents should be retrieved. If you pass your ret$\_$docs to the user for feedback, you would get back:

$\{$

$\text{'}$q$167\text{'}$: $[0\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}0],$

$\text{'}$q$54\text{'}$ : $[0\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}0,0\mathrm{.}0],$

$\text{'}$q$197\text{'}$: $[0\mathrm{.}0,0\mathrm{.}5,0\mathrm{.}3333333333333333,0\mathrm{.}25,0\mathrm{.}2]$

$\}$

where each number indicates the precision at that rank. For example, the second number $0\mathrm{.}5$ from q$197$ indicates that the precision at rank $2$ is $0\mathrm{.}5$ for query q$197.$ Similarly, the precision at rank $4$ is $0\mathrm{.}25$ for query q$197.$ For queries q$167$ and q$54,$ the precisions at top $5$ ranks are, unfortunately, zero. The precisions at each rank are returned back to you in the variable pre$\_$at$\_$n$,$ along with another variable avg$\_$ndcg which indicates your average NDCG for all queries.

Code for the first three steps has been provided to you. Your task is to complete the code for

Step $4$: You perform a Rocchio update on all the queries the user issued to you based on the feedback you received. As a result, the query vectors in self.q$\_$vecs should be updated accordingly. In other words, the "modified query vector" for each query under consideration should be stored in the corresponding entry in self.q$\_$vecs.