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Learning to Rank (part 1) NESCAI 2008 Tutorial Yisong Yue Cornell University Booming Search Industry Goals for this Tutorial Basics of information retrieval What machine learning contributes New challenges to address New insights on developing ML algorithms (Soft) Prerequisites Basic knowledge of ML algorithms Support Vector Machines Neural Nets Decision Trees Boosting Etc… Will introduce IR concepts as needed Outline (Part 1) Conventional IR Methods (no learning) Ordinal Regression 1970s to 1990s 1994 onwards Optimizing Rank-Based Measures 2005 to present Outline (Part 2) Effectively collecting training data Beyond independent relevance E.g., interpreting clickthrough data E.g., diversity Summary & Discussion Disclaimer This talk is very ML-centric Use IR methods to generate features Learn good ranking functions on feature space Focus on optimizing cleanly formulated objectives Outperform traditional IR methods Disclaimer This talk is very ML-centric Use IR methods to generate features Learn good ranking functions on feature space Focus on optimizing cleanly formulated objectives Outperform traditional IR methods Information Retrieval Broader than the scope of this talk Deals with more sophisticated modeling questions Will see more interplay between IR and ML in Part 2 Brief Overview of IR Predated the internet Active research topic by the 1960’s As We May Think by Vannevar Bush (1945) Vector Space Model (1970s) Probabilistic Models (1980s) Introduction to Information Retrieval (2008) C. Manning, P. Raghavan & H. Schütze Basic Approach to IR Given query q and set of docs d1, … dn Find documents relevant to q Typically expressed as a ranking on d1,… dn Basic Approach to IR Given query q and set of docs d1, … dn Find documents relevant to q Typically expressed as a ranking on d1,… dn Similarity measure sim(a,b)!R Sort by sim(q,di) Optimal if relevance of documents are independent. [Robertson, 1977] Vector Space Model Represent documents as vectors One dimension for each word Queries as short documents Similarity Measures Cosine similarity = normalized dot product A B cos( A, B) A B Cosine Similarity Example Other Methods TF-IDF Okapi BM25 [Salton & Buckley, 1988] [Robertson et al., 1995] Language Models [Ponte & Croft, 1998] [Zhai & Lafferty, 2001] Machine Learning IR uses fixed models to define similarity scores Many opportunities to learn models Appropriate training data Appropriate learning formulation Will mostly use SVM formulations as examples General insights are applicable to other techniques. Training Data Supervised learning problem Document/query pairs Embedded in high dimensional feature space Labeled by relevance of doc to query Traditionally 0/1 Recently ordinal classes of relevance (0,1,2,3,…) Feature Space Use to learn a similarity/compatibility function Based off existing IR methods Can use raw values Or transformations of raw values Based off raw words Capture co-occurrence of words Training Instances xq , d TF (qi , d ) 0.1 i TF (qi , d ) 0.05 i rank IR (q, d ) in top 5 (q, d ) rank IR (q, d ) in top 10 sim IR (q, d ) w j qi w j d i wk qi wk d i Learning Problem Given training instances: Learn a ranking function (xq,d, yq,d) for q = {1..N}, d = {1 .. Nq} f(xq,1, … xq,Nq ) ! Ranking Typically decomposed into per doc scores f(x) ! R (doc/query compatibility) Sort by scores for all instances of a given q How to Train? Classification & Regression Learn f(x) ! R in conventional ways Sort by f(x) for all docs for a query Typically does not work well 2 Major Problems Labels have ordering Additional structure compared to multiclass problems Severe class imbalance Most documents are not relevant Somewhat Relevant Very Relevant Not Relevant Conventional multiclass learning does not incorporate ordinal structure of class labels Somewhat Relevant Very Relevant Not Relevant Conventional multiclass learning does not incorporate ordinal structure of class labels Ordinal Regression Assume class labels are ordered True since class labels indicate level of relevance Learn hypothesis function f(x) ! R Such that the ordering of f(x) agrees with label ordering Ex: given instances (x, 1), (y, 1), (z, 2) f(x) < f(z) f(y) < f(z) Don’t care about f(x) vs f(y) Ordinal Regression Compare with classification Similar to multiclass prediction But classes have ordinal structure Compare with regression Doesn’t necessarily care about value of f(x) Only care that ordering is preserved Ordinal Regression Approaches Learn multiple thresholds Learn multiple classifiers Optimize pairwise preferences Option 1: Multiple Thresholds Maintain T thresholds (b1, … bT) b1 < b 2 < … < b T Learn model parameters + (b1, …, bT) Goal Model predicts a score on input example Minimize threshold violation of predictions Ordinal SVM Example [Chu & Keerthi, 2005] Ordinal SVM Formulation 1 2 C arg min w N w,b , , 2 i , j i , j j 1 i i T Such that for j = 0..T : wT xi b j 1 i, j , i : yi j wT xi b j 1 i, j 1 , i : yi j 1 i, j , i, j 1 0, And also: i b1 b2 ... bT [Chu & Keerthi, 2005] Learning Multiple Thresholds Gaussian Processes Decision Trees [Kramer et al., 2001] Neural Nets [Chu & Ghahramani, 2005] RankProp [Caruana et al., 1996] SVMs & Perceptrons PRank [Crammer & Singer, 2001] [Chu & Keerthi, 2005] Option 2: Voting Classifiers Use T different training sets Classifier 1 predicts 0 vs 1,2,…T Classifier 2 predicts 0,1 vs 2,3,…T … Classifier T predicts 0,1,…,T-1 vs T Final prediction is combination E.g., sum of predictions Recent work McRank [Li et al., 2007] [Qin et al., 2007] •Severe class imbalance •Near perfect performance by always predicting 0 Option 3: Pairwise Preferences Most popular approach for IR applications Learn model to minimize pairwise disagreements %(Pairwise Agreements) = ROC-Area • 2 pairwise disagreements Optimizing Pairwise Preferences Consider instances (x1,y1) and (x2,y2) Label order has y1 > y2 Optimizing Pairwise Preferences Consider instances (x1,y1) and (x2,y2) Label order has y1 > y2 Create new training instance (x’, +1) where x’ = (x1 – x2) Repeat for all instance pairs with label order preference Optimizing Pairwise Preferences Result: new training set! Often represented implicitly Has only positive examples Mispredicting means that a lower ordered instance received higher score than higher order instance. Pairwise SVM Formulation 1 2 C arg min w 2 N w, i, j i, j Such that: wT xi wT x j 1 i , j , i , j 0, i, j : yi y j i, j [Herbrich et al., 1999] Can be reduced to O(n log( n)) time [Joachims, 2005]. Optimizing Pairwise Preferences Neural Nets Boosting & Hedge-Style Methods RankNet [Burges et al., 2005] [Cohen et al., 1998] RankBoost [Freund et al., 2003] [Long & Servidio, 2007] SVMs [Herbrich et al., 1999] SVM-perf [Joachims, 2005] [Cao et al., 2006] Rank-Based Measures Pairwise Preferences not quite right Assigns equal penalty for errors no matter where in the ranking People (mostly) care about top of ranking IR community use rank-based measures which capture this property. Rank-Based Measures Binary relevance [email protected] ([email protected]) Mean Average Precision (MAP) Mean Reciprocal Rank (MRR) Multiple levels of relevance Normalized Discounted Cumulative Gain (NDCG) [email protected] Set a rank threshold K Compute % relevant in top K Ignores documents ranked lower than K Ex: [email protected] of 2/3 [email protected] of 2/4 [email protected] of 3/5 Mean Average Precision Consider rank position of each relevance doc K1, K2, … KR Compute [email protected] for each K1, K2, … KR Average precision = average of [email protected] Ex: MAP is Average Precision across multiple queries/rankings has AvgPrec of 1 1 2 3 0.76 3 1 3 5 Mean Reciprocal Rank Consider rank position, K, of first relevance doc 1 Reciprocal Rank score = K MRR is the mean RR across multiple queries NDCG Normalized Discounted Cumulative Gain Multiple Levels of Relevance DCG: contribution of ith rank position: Ex: 2 yi 1 log( i 1) has DCG score of 1 3 1 0 1 5.45 log( 2) log( 3) log( 4) log( 5) log( 6) NDCG is normalized DCG best possible ranking as score NDCG = 1 Optimizing Rank-Based Measures Let’s directly optimize these measures As opposed to some proxy (pairwise prefs) But… Objective function no longer decomposes Pairwise prefs decomposed into each pair Objective function flat or discontinuous Discontinuity Example D1 D2 D3 Retrieval Score 0.9 0.6 0.3 Rank 1 2 3 Relevance 0 1 0 NDCG = 0.63 Discontinuity Example NDCG computed using rank positions Ranking via retrieval scores D1 D2 D3 Retrieval Score 0.9 0.6 0.3 Rank 1 2 3 Discontinuity Example NDCG computed using rank positions Ranking via retrieval scores Slight changes to model parameters Slight changes to retrieval scores No change to ranking No change to NDCG D1 D2 D3 Retrieval Score 0.9 0.6 0.3 Rank 1 2 3 Discontinuity Example NDCG computed using rank positions Ranking via retrieval scores Slight changes to model parameters NDCG discontinuous Slight changes to retrieval scores No change to ranking w.r.t model parameters! No change to NDCG D1 D2 D3 Retrieval Score 0.9 0.6 0.3 Rank 1 2 3 [Yue & Burges, 2007] Optimizing Rank-Based Measures Relaxed Upper Bound Structural SVMs for hinge loss relaxation Boosting for exponential loss relaxation SVM-map [Yue et al., 2007] [Chapelle et al., 2007] [Zheng et al., 2007] AdaRank [Xu et al., 2007] Smooth Approximations for Gradient Descent LambdaRank [Burges et al., 2006] SoftRank GP [Snelson & Guiver, 2007] Structural SVMs Let x denote the set of documents/query examples for a query Let y denote a (weak) ranking Same objective function: 1 2 C w 2 N i i Constraints are defined for each incorrect labeling y’ over the set of documents x. y' y : w ( y, x) w ( y' , x) ( y' ) T T After learning w, a prediction is made by sorting on wTxi [Tsochantaridis et al., 2007] Structural SVMs for MAP Maximize subject to where 1 2 C w 2 N i i y' y : w ( y, x) w ( y' , x) ( y' ) T T ( y, x) yij ( xi x j ) ( yij = {-1, +1} ) i:rel j:!rel and ( y) 1 Avgprec( y) Sum of slacks upper bound MAP loss. [Yue et al., 2007] Too Many Constraints! For Average Precision, the true labeling is a ranking where the relevant documents are all ranked in the front, e.g., An incorrect labeling would be any other ranking, e.g., This ranking has Average Precision of about 0.8 with (y,y’) ¼ 0.2 Intractable number of rankings, thus an intractable number of constraints! Structural SVM Training STEP 1: Solve the SVM objective function using only the current working set of constraints. STEP 2: Using the model learned in STEP 1, find the most violated constraint from the exponential set of constraints. STEP 3: If the constraint returned in STEP 2 is more violated than the most violated constraint the working set by some small constant, add that constraint to the working set. Repeat STEP 1-3 until no additional constraints are added. Return the most recent model that was trained in STEP 1. STEP 1-3 is guaranteed to loop for at most a polynomial number of iterations. [Tsochantaridis et al., 2005] Illustrative Example Original SVM Problem Exponential constraints Most are dominated by a small set of “important” constraints Structural SVM Approach Repeatedly finds the next most violated constraint… …until set of constraints is a good approximation. Illustrative Example Original SVM Problem Exponential constraints Most are dominated by a small set of “important” constraints Structural SVM Approach Repeatedly finds the next most violated constraint… …until set of constraints is a good approximation. Illustrative Example Original SVM Problem Exponential constraints Most are dominated by a small set of “important” constraints Structural SVM Approach Repeatedly finds the next most violated constraint… …until set of constraints is a good approximation. Illustrative Example Original SVM Problem Exponential constraints Most are dominated by a small set of “important” constraints Structural SVM Approach Repeatedly finds the next most violated constraint… …until set of constraints is a good approximation. Finding Most Violated Constraint Required for structural SVM training Depends on structure of loss function Depends on structure of joint discriminant Efficient algorithms exist despite intractable number of constraints. More than one approach [Yue et al., 2007] [Chapelle et al., 2007] Gradient Descent Objective function is discontinuous We only need the gradient! Difficult to define a smooth global approximation Upper-bound relaxations (e.g., SVMs, Boosting) sometimes too loose. But objective is discontinuous… … so gradient is undefined Solution: smooth approximation of the gradient Local approximation LambdaRank Assume implicit objective function C Goal: compute dC/dsi si = f(xi) denotes score of document xi Given gradient on document scores Use chain rule to compute gradient on model parameters (of f) [Burges et al., 2006] [Burges, 2007] Intuition: •Rank-based measures emphasize top of ranking •Higher ranked docs should have larger derivatives (Red Arrows) •Optimizing pairwise preferences emphasize bottom of ranking (Black Arrows) LambdaRank for NDCG The pairwise derivative of pair i,j is 1 ij NDCG (i, j ) 1 exp( s s ) i j Total derivative of output si is C i si jDi ij jDi ji Properties of LambdaRank 1There exists a cost function C if i j i, j : s j si Amounts to the Hessian of C being symmetric If Hessian also positive semi-definite, then C is convex. 1Subject to additional assumptions – see [Burges et al., 2006] Summary (Part 1) Machine learning is a powerful tool for designing information retrieval models Requires clean formulation of objective Advances Ordinal regression Dealing with severe class imbalances Optimizing rank-based measures via relaxations Gradient descent on non-smooth objective functions