PrasadL09VSM-Ranking1 Vector Space Model : Ranking Revisited Adapted from Lectures by Prabhakar Raghavan (Google) and Christopher Manning (Stanford)

Recap: tf-idf weighting The tf-idf weight of a term is the product of its tf weight and its idf weight. 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 idf is a measure of the informativeness of the term. Prasad2L09VSM-Ranking

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 Prasad3L09VSM-Ranking

Recap: cosine(query,document) Dot product Unit vectors cos(q,d) is the cosine similarity of q and d … or, equivalently, the cosine of the angle between q and d. Prasad4L09VSM-Ranking

Introduction to Information Retrieval 5 Why is ranking so important?  Problems with unranked retrieval  Users want to look at a few results – not thousands.  It’s very hard to write queries that produce a few results.  Even for expert searchers  → Ranking is important because it effectively reduces a large set of results to a very small one.  Actually, in the vast majority of the cases users only examine just top few (1, 2, or 3) results, or at most 1 page of results. 5

Introduction to Information Retrieval 6 Why rank?: User Studies Summary  Viewing abstracts: Users are a lot more likely to read the abstracts of the top-ranked pages (1, 2, 3, 4) than the abstracts of the lower ranked pages (7, 8, 9, 10).  Clicking: Distribution is even more skewed for clicking.  In 1 out of 2 cases, users click on the top-ranked page.  Even if the top-ranked page is not relevant, 30% of users will click on it.  → Getting the ranking right is very important.  → Getting the top-ranked page right is most important. 6

Introduction to Information Retrieval 7 Exercise: A problem for cosine normalization  Query q: “anti-doping rules Beijing 2008 olympics”  Compare three documents  d 1 : a short document on anti-doping rules at 2008 Olympics  d 2 : a long document that consists of a copy of d 1 and 5 other news stories, all on topics different from Olympics/anti- doping  d 3 : a short document on anti-doping rules at the 2004 Athens Olympics  What ranking do we expect in the vector space model?  What can we do about this? 7

Introduction to Information Retrieval 8 Pivot normalization  Cosine normalization produces weights that are too large for short documents and too small for long documents (on average).  Adjust cosine normalization by linear adjustment: “turning” the average normalization on the pivot  Effect: Similarities of short documents with query decrease; similarities of long documents with query increase.  This removes the unfair advantage that short documents have. 8

Introduction to Information Retrieval 9 Predicted and true probability of relevance source: Lillian Lee 9

Introduction to Information Retrieval 10 Pivot normalization source: Lillian Lee 10

This lecture Speeding up vector space ranking Putting together a complete search system Will require learning about a number of miscellaneous topics and heuristics Prasad11L09VSM-Ranking

Introduction to Information Retrieval 12 Term frequencies in the inverted index  In each posting, store tf t,d in addition to docID d  As an integer frequency, not as a (log-)weighted real number... ... because real numbers are difficult to compress.  Unary/Binary codes are effective for encoding term frequencies.  Why?  Overall, additional space requirements are small: less than a byte per posting with bitwise compression.  Or a byte per posting with variable byte code. 12

Computing cosine scores

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 7.1 Prasad14L09VSM-Ranking

Faster cosine: unweighted query 7.1 Read redundant?

Efficient cosine ranking Find the K docs in the collection “nearest” to the query  K largest query-doc cosines. Efficient ranking: Computing a single cosine efficiently. Choosing the K largest cosine values efficiently. Can we do this without computing all N cosines? Prasad16L09VSM-Ranking

(cont’d) 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 Prasad17L09VSM-Ranking

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 7.1 Prasad18L09VSM-Ranking

Use heap for selecting top K Binary tree in which each node’s value > the values of children Takes O(J) operations to construct the heap, then K “winners” read off in O(K log J) steps. For J=10M, K=100, this is about 10% of the cost of sorting (O(J log J) ) Prasad19L09VSM-Ranking

Binary Max Heap Example PrasadL09VSM-Ranking20 for every node i other than the root A[ Parent( i ) ] ≥ A[ i ]

Heapify PrasadL09VSM-Ranking21

Build Heap Precondition says all subtrees of i are max heaps Postcondition says that the i subtree is a max heap. Note that leaves are in array indices greater than A.length/2. Leaves are heaps. Internal nodes need to be heapified. PrasadL09VSM-Ranking22

Build Heap Note that heap can be built in two ways. Top down approach has a complexity of O(n log n). Bottom-up approach, given earlier, has a complexity of O(n)! Smaller number of nodes travel larger distances. PrasadL09VSM-Ranking23

Complexity of Heap Sort Steps (Single) Heapify, which runs in O(log n) time. Build-Heap, which runs in O(n) time (linear). Heap Sort, which runs in O(n log n) time. Binary Max Heap => ascending order sort Binary Min Heap => descending order sort Extract-Max, which runs in O(log n) time. To extract top k elements, it requires O(k log n) time. PrasadL09VSM-Ranking24

Bottlenecks Primary computational bottleneck in scoring: cosine computation Can we avoid some of 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? Prasad25L09VSM-Ranking

Cosine similarity is only a proxy User has a task and a query formulation Cosine similarity matches docs to query However, this is only a proxy for user information need If we get a list of K docs “close” to the top K by cosine measure, it should be ok Prasad26L09VSM-Ranking

Introduction to Information Retrieval 27 More efficient computation of top k: Heuristics  Idea 1: Reorder postings lists  Instead of ordering according to docID... ... order according to some measure of “expected relevance”.  Idea 2: Heuristics to prune the search space  Not guaranteed to be correct... ... but fails rarely. 27

Generic approach 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 PrasadL09VSM-Ranking

Index elimination Basic algorithm of Fig 7.1 considers only docs containing at least one query term Take this further: Consider only high-idf query terms Consider only docs containing many query terms Prasad29L09VSM-Ranking

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 don’t alter rank-ordering much Benefit: Postings of low-idf terms have many docs  these (many) docs get eliminated from A Prasad30L09VSM-Ranking

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 Prasad31L09VSM-Ranking

3 of 4 query terms Brutus Caesar Calpurnia Antony Scores computed only for 8, 16 and Prasad32L09VSM-Ranking

Champion lists 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 time At query time, compute scores only for docs in the champion list of some query term Pick the K top-scoring docs from amongst these Prasad33L09VSM-Ranking

Exercises How do Champion Lists relate to Index Elimination? Can they be used together? How can Champion Lists be implemented in an inverted index? Note the champion list has nothing to do with small docIDs Prasad34L09VSM-Ranking

Quantitative 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, Y!buzzes or del.icio.us marks (Pagerank) 7.1.4

Modeling authority Assign to 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 Prasad36L09VSM-Ranking

Net score 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 rather than an equal weighting Indeed, any function of the two “signals” of user happiness – more later Now we seek the top K docs by net score Prasad37L09VSM-Ranking

Top K by net score – fast methods First idea: Order all postings by g(d) Key: this is a common ordering for all postings 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 Prasad38L09VSM-Ranking

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-idf td Seek top-K results from only the docs in these champion lists Prasad39L09VSM-Ranking

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) 7.1.4

Impact-ordered postings We only want to compute scores for docs for which wf t,d is high enough We sort each postings list by wf t,d Now: not all postings in a common order! How do we compute scores in order to pick off top K? Two ideas follow Prasad41L09VSM-Ranking

1. Early termination When traversing t’s postings, stop early after either a fixed number of r docs wf t,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 Prasad42L09VSM-Ranking

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 Prasad43L09VSM-Ranking

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 Prasad44L09VSM-Ranking

Cluster pruning: query processing Why use random sampling? Fast Leaders reflect data distribution Process a query as follows: Given query Q, find its nearest leader L. Seek K nearest docs from among L’s followers Prasad45L09VSM-Ranking

Visualization Query LeaderFollower Prasad

General variants Have each follower attached to b1=3 (say) nearest leaders. From query, find b2=4 (say) nearest leaders and their followers. Can recur on leader/follower construction Prasad47L09VSM-Ranking

Exercises 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 Prasad48L09VSM-Ranking

Parametric and zone indexes Thus far, a doc has been a sequence of terms In fact documents have multiple parts, some with special semantics: Author Title Date of publication Language Format etc. These constitute the metadata about a document 6.1 Prasad49L09VSM-Ranking

Fields 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) 6.1 Prasad50L09VSM-Ranking

Zone A zone is a region of the doc that can contain an arbitrary amount of text e.g., Title Abstract References … 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” 6.1 Prasad51L09VSM-Ranking

Example zone indexes 6.1 Encode zones in dictionary vs. postings.

Introduction to Information Retrieval 53 Tiered indexes  Basic idea:  Create several tiers of indexes, corresponding to importance of indexing terms  During query processing, start with highest-tier index  If highest-tier index returns at least k (e.g., k = 100) results: stop and return results to user  If we’ve only found < k hits: repeat for next index in tier cascade  Example: two-tier system  Tier 1: Index of all titles  Tier 2: Index of the rest of documents  Pages containing the search words in the title are better hits than pages containing the search words in the body of the text. 53

Example tiered index Prasad

Introduction to Information Retrieval 55 Tiered indexes  The use of tiered indexes is believed to be one of the reasons that Google search quality was significantly higher initially (2000/01) than that of its competitors.  (along with PageRank, use of anchor text and proximity constraints) 55

Introduction to Information Retrieval 56 Exercise  Design criteria for tiered system  Each tier should be an order of magnitude smaller than the next tier.  The top 100 hits for most queries should be in tier 1, the top 100 hits for most of the remaining queries in tier 2 etc.  We need a simple test for “can I stop at this tier or do I have to go to the next one?”  There is no advantage to tiering if we have to hit most tiers for most queries anyway.  Question 1: Consider a two-tier system where the first tier indexes titles and the second tier everything. What are potential problems with this type of tiering?  Question 2: Can you think of a better way of setting up a multitier system? Which “zones” of a document should be indexed in the different tiers (title, body of document, others?)? What criterion do you want to use for including a document in tier 1? 56

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? L09VSM-Ranking

Query parsers 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 Prasad58L09VSM-Ranking

Aggregate scores 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 ranking scheme with 1000’s of parameters effecting the final ranking Prasad59L09VSM-Ranking

Putting it all together Prasad60L09VSM-Ranking