Information Retrieval and Web Search
Pandu Nayak and Prabhakar Raghavan
Lecture 7: Scoring and results assembly
Lecture 6 – I introduced a bug
In my anxiety to avoid taking the log of zero, I rewrote
In fact this was unnecessary, since the zero case is treated
specially above; net the FIRST version above is right.
Recap: tf-idf weighting
The tf-idf weight of a term is the product of its tf weight and its idf weight.
wt,d= (1+log10 tft,d) x log10(N/dft)
Best known weighting scheme in information retrieval
Increases with the number of occurrences within a document
Increases with the rarity of the term in the collection
Recap: Queries as vectors
Key idea 1: Do the same for queries: represent them as vectors in the space
Key idea 2: Rank documents according to their proximity to the query in this space
proximity = similarity of vectors
cos(q,d) is the cosine similarity of q and d … or,
equivalently, the cosine of the angle between q and d.
Speeding up vector space ranking
Putting together a complete search system
Will require learning about a number of miscellaneous topics and heuristics
Computing cosine scores
Efficient cosine ranking
Find the K docs in the collection “nearest” to the query ⇒ K largest query-doc cosines.
Computing a single cosine efficiently.
Choosing the K largest cosine values efficiently.
Can we do this without computing all N cosines?
Efficient cosine ranking
What we’re doing in effect: solving the K-nearest neighbor problem for a query vector
In general, we do not know how to do this efficiently for high-dimensional spaces
But it is solvable for short queries, and standard indexes support this well
Special case – unweighted queries
No weighting on query terms
Assume each query term occurs only once
Then for ranking, don’t need to normalize query vector
Slight simplification of algorithm from Lecture 6
Computing the K largest cosines: selection vs. sorting
Typically we want to retrieve the top K docs (in the cosine ranking for the query)
not to totally order all docs in the collection
Can we pick off docs with K highest cosines?
Let J = number of docs with nonzero cosines
We seek the K best of these J
Use heap for selecting top K
Binary tree in which each node’s value > the values of children
Takes 2J operations to construct, then each of K “winners” read off in 2log J steps.
For J=1M, K=100, this is about 10% of the cost of sorting.
Primary computational bottleneck in scoring: cosine computation
Can we avoid all this computation?
Yes, but may sometimes get it wrong
a doc not in the top K may creep into the list of K output docs
Is this such a bad thing?
Cosine similarity is only a proxy
User has a task and a query formulation
Cosine matches docs to query
Thus cosine is anyway a proxy for user happiness
If we get a list of K docs “close” to the top K by cosine measure, should be ok
Find a set A of contenders, with K < |A| << N
A does not necessarily contain the top K, but has many docs from among the top K
Return the top K docs in A
Think of A as pruning non-contenders
The same approach is also used for other (non-cosine) scoring functions
Will look at several schemes following this approach
Basic algorithm cosine computation algorithm only considers docs containing at least one query term
Take this further:
Only consider high-idf query terms
Only consider docs containing many query terms
High-idf query terms only
For a query such as catcher in the rye
Only accumulate scores from catcher and rye
Intuition: in and the contribute little to the scores and so don’t alter rank-ordering much
Postings of low-idf terms have many docs these (many) docs get eliminated from set A of contenders
Docs containing many query terms
Any doc with at least one query term is a candidate for the top K output list
For multi-term queries, only compute scores for docs containing several of the query terms
Say, at least 3 out of 4
Imposes a “soft conjunction” on queries seen on web search engines (early Google)
Easy to implement in postings traversal
3 of 4 query terms
Scores only computed for docs 8, 16 and 32.
Precompute for each dictionary term t, the r docs of highest weight in t’s postings
Call this the champion list for t
(aka fancy list or top docs for t)
Note that r has to be chosen at index build time
Thus, it’s possible that r < K
At query time, only compute scores for docs in the champion list of some query term
Pick the K top-scoring docs from amongst these
How do Champion Lists relate to Index Elimination? Can they be used together?
How can Champion Lists be implemented in an inverted index?
Note that the champion list has nothing to do with small docIDs
Static quality scores
We want top-ranking documents to be both relevant and authoritative
Relevance is being modeled by cosine scores
Authority is typically a query-independent property of a document
Examples of authority signals
Wikipedia among websites
Articles in certain newspapers
A paper with many citations
Many bitly’s, diggs or del.icio.us marks
Assign to each document a query-independent quality score in [0,1] to each document d
Denote this by g(d)
Thus, a quantity like the number of citations is scaled into [0,1]
Exercise: suggest a formula for this.
Consider a simple total score combining cosine relevance and authority
net-score(q,d) = g(d) + cosine(q,d)
Can use some other linear combination
Indeed, any function of the two “signals” of user happiness – more later
Now we seek the top K docs by net score
Top K by net score – fast methods
First idea: Order all postings by g(d)
Key: this is a common ordering for all postings
Thus, can concurrently traverse query terms’ postings for
Cosine score computation
Exercise: write pseudocode for cosine score computation if postings are ordered by g(d)
Why order postings by g(d)?
Under g(d)-ordering, top-scoring docs likely to appear early in postings traversal
In time-bound applications (say, we have to return whatever search results we can in 50 ms), this allows us to stop postings traversal early
Short of computing scores for all docs in postings
Champion lists in g(d)-ordering
Can combine champion lists with g(d)-ordering
Maintain for each term a champion list of the r docs with highest g(d) + tf-idftd
Seek top-K results from only the docs in these champion lists
High and low lists
For each term, we maintain two postings lists called high and low
Think of high as the champion list
When traversing postings on a query, only traverse high lists first
If we get more than K docs, select the top K and stop
Else proceed to get docs from the low lists
Can be used even for simple cosine scores, without global quality g(d)
A means for segmenting index into two tiers
We only want to compute scores for docs for which wft,d is high enough
We sort each postings list by wft,d
Now: not all postings in a common order!
How do we compute scores in order to pick off top K?
Two ideas follow
1. Early termination
When traversing t’s postings, stop early after either
a fixed number of r docs
wft,d drops below some threshold
Take the union of the resulting sets of docs
One from the postings of each query term
Compute only the scores for docs in this union
2. idf-ordered terms
When considering the postings of query terms
Look at them in order of decreasing idf
High idf terms likely to contribute most to score
As we update score contribution from each query term
Stop if doc scores relatively unchanged
Can apply to cosine or some other net scores
Cluster pruning: preprocessing
Pick √N docs at random: call these leaders
For every other doc, pre-compute nearest leader
Docs attached to a leader: its followers;
Likely: each leader has ~ √N followers.
Cluster pruning: query processing
Process a query as follows:
Given query Q, find its nearest leader L.
Seek K nearest docs from among L’s followers.
Why use random sampling
Leaders reflect data distribution
Have each follower attached to b1=3 (say) nearest leaders.
From query, find b2=4 (say) nearest leaders and their followers.
Can recurse on leader/follower construction.
To find the nearest leader in step 1, how many cosine computations do we do?
Why did we have √N in the first place?
What is the effect of the constants b1, b2 on the previous slide?
Devise an example where this is likely to fail – i.e., we miss one of the K nearest docs.
Likely under random sampling.
Parametric and zone indexes
Thus far, a doc has been a sequence of terms
- In fact documents have multiple parts, some with special semantics:
- Date of publication
These constitute the metadata about a document
We sometimes wish to search by these metadata
E.g., find docs authored by William Shakespeare in the year 1601, containing alas poor Yorick
Year = 1601 is an example of a field
Also, author last name = shakespeare, etc.
Field or parametric index: postings for each field value
Sometimes build range trees (e.g., for dates)
Field query typically treated as conjunction
(doc must be authored by shakespeare)
A zone is a region of the doc that can contain an arbitrary amount of text, e.g.,
Build inverted indexes on zones as well to permit querying
E.g., “find docs with merchant in the title zone and matching the query gentle rain”
Example zone indexes
Encode zones in dictionary vs. postings.
Break postings up into a hierarchy of lists
Can be done by g(d) or another measure
Inverted index thus broken up into tiers of decreasing importance
At query time use top tier unless it fails to yield K docs
If so drop to lower tiers
Example tiered index
Query term proximity
Free text queries: just a set of terms typed into the query box – common on the web
Users prefer docs in which query terms occur within close proximity of each other
Let w be the smallest window in a doc containing all query terms, e.g.,
For the query strained mercy the smallest window in the doc The quality of mercy is not strained is 4 (words)
Would like scoring function to take this into account – how?
Free text query from user may in fact spawn one or more queries to the indexes, e.g., query rising interest rates
Run the query as a phrase query
If <K docs contain the phrase rising interest rates, run the two phrase queries rising interest and interest rates
If we still have <K docs, run the vector space query rising interest rates
Rank matching docs by vector space scoring
This sequence is issued by a query parser
We’ve seen that score functions can combine cosine, static quality, proximity, etc.
How do we know the best combination?
Some applications – expert-tuned
Increasingly common: machine-learned
See May 19th lecture
Putting it all together
IIR 7, 6.1