Online Collec*ve Inference Jay Pujara U. Maryland, College Park U. - - PowerPoint PPT Presentation

Online Collec*ve Inference

Jay Pujara

• U. Maryland, College Park

Ben London

• U. Maryland, College Park

Lise Getoor

• U. California, Santa Cruz

BIRS Workshop: New Perspec*ves for Rela*onal Learning 4/23/2015

Real-world problems…

…benefit from rela*onal models

Genre(M1, G)∧ Genre(M2, G)∧ Likes(U1, M1) → Likes(U1, M2)

Collabora*ve Filtering

Friends Genre

Likes(U1, M1)∧ Friends(U1, U2) → Likes(U2, M1)

Users Items

Coworkers(U1, C)∧ Coworkers(U2, C)∧ → Coworkers(U1, U2)

Link Predic*on

Knowledge Graph Iden*fica*on

MutEx(L1, L2)∧ Label(E, L1)∧ → ¬Label(E, L2) Rel(R, E1, T)∧ Rel(R, E2, T)∧ Label(E1, L)∧ → Label(E2, L)

Genre Ar*st

(Jiang et al., ICDM12; Pujara et al., ISWC13)

…benefit from rela*onal models

Real-world problems are big!

Millions of users, thousands movies Millions of facts, thousands of

• ntological constraints

Millions of users Millions of users, thousands of genes

What happens when?

A user rates a new movie? New facts are extracted from the Web? New user links form? A new gene=c similarity is discovered?

What happens when?

A user rates a new movie? New facts are extracted from the Web? New user links form? A new gene=c similarity is discovered?

Repeat Inference!

Why can’t we repeat inference?

• We want rich, collec*ve models!
• But, 10M-1B factors = 1-100s hours*
• Ideal: Inference *me balances update cycle
• Insanity is doing the same thing over and over…

PROBLEM SETTING

Online Collec*ve Inference

Key Problem

• Real-world problems -> large graphical models
• Changing evidence -> repeat inference

Key Problem

• Real-world problems -> large graphical models
• Changing evidence -> repeat inference
• What happens when par*ally upda*ng inference?
• Can we scalably approximate the MAP state

without recompu*ng inference?

Generic Answer: NO!

• Nodes can be or
• Model has prob. mass only when nodes same
• Fix some nodes to then observe evidence for

Generic Answer: NO!

• Nodes can be or
• Model has prob. mass only when nodes same
• Fix some nodes to then observe evidence for

Full Inference

Generic Answer: NO!

• Nodes can be or
• Model has prob. mass only when nodes same
• Fix some nodes to then observe evidence for

Full Inference

Previous Work

• Belief Revision

– e.g. Gardenfors, 1992

• Bayesian Network Updates

– e.g. Bun*ne, 1991; Friedman & Goldszmidt, 1997

• Dynamic / Sequen*al Models

– e.g. Murphy, 2002 / Fine et al., 1998

• Adap*ve Inference

– e.g. Acar et al., 2008

• BP Message Passing

– e.g. Nath & Domingos, 2010

• Collec*ve Stability

– e.g. London et al., 2013

Problem Seing

• Fixed model: dependencies & weights known
• Online: changing evidence or observa*ons
• Closed world: all variables iden*fied
• Budget: infer only m variables in each epoch
• Strongly-convex inference objec=ve (e.g. PSL)

Ques*ons:

• What guarantees can we offer?
• Which m variables should we infer?

Approach

• Define “regret” for online collec*ve inference
• Introduce regret bounds for strongly convex

inference objec*ves (like PSL!)

• Develop algorithms to ac%vate a subset of the

variables during inference, given budget

REGRET BOUNDS

Online Collec*ve Inference

Inference Regret

• General inference problem: es*mate
• In online collec*ve inference: fix , infer
• Regret (learning): captures distance to op*mal
• Regret (inference): the distance between the

full inference result and the par*al inference update (when condi*oning on )

YS YS

?

YS P(Y |X)

Defining Regret

• Regret: distance between full & approximate

inference where

h(x; ˙ w) = arg min

y

w · f(x, y) + wp 2 kyk2

2 .

Prior weight

Rn(x, yS; ˙ w) , 1 n kh(x; ˙ w) h(x, yS; ˙ w)k1

Regret Bound

Regret Ingredients: Lipschitz constant 2-norm of model weights Weight of L2 prior L1 distance fixed variables and values in full inference

Key Takeaway: Regret depends on L1 distance between fixed variables & their “true” values in the MAP state

Rn(x, yS; ˙ w)  O s Bkwk2 n · wp kyS ˆ ySk1 !

Valida*ng Regret Bounds

5 10 15 20 25 30 35 40 45 50 0.05 0.1 0.15 0.2 0.25

# epochs inference regret

scaled regret bound HighLocal Balanced HighRelational

Measure regret of no updates versus full inference, varying the importance of rela*onal features

ACTIVATION ALGORITHMS

Online Collec*ve Inference

Which variables to fix?

• Knapsack: combinatorial, regrets/costs, budget
• Theory: fix variables that won’t change
• Prac*ce: how can we know what will change?
• Idea: Can we use features of past inferences?
• Explore op*miza*on (case study ADMM & PSL)

ADMM Inference in PSL

(Boyd et al., 2011; Bach et al. 2012) y1 y2 y3

f2 f1 f4 f3

Poten*als Variables

ADMM Inference in PSL

y12 y23 y33

f2 f1 f4 f3

y11 y22 y34

Variable Copies Poten*als

ADMM Inference in PSL

y12 y23 y33

f2 f1 f4 f3

y11 y22 y34

y1 y2 y3

Poten*als Consensus Es*mates

ADMM Inference in PSL

y12 y23 y33

f2 f1 f4 f3

y11 y22 y34

y1 y2 y3

α11 α12 α22 α23 α33 α34

Poten*als Consensus Es*mates

ADMM Features

• Weight: how important is the poten*al?
• Poten*al: what loss do we incur?
• Consensus: what is the variable’s value?
• Lagrange Mul*plier: how much

disagreement is there across poten*als? min

˜ yg

wg fg(x, ˜ yg)+ρ 2

• ˜

yg − yg + 1 ρ αg

2

• 2

Two heuris*cs for ac*va*on

• Truth-Value: Variable value near 0.5
• Weighted Lagrangian: rule weight x Lagrange

mul*pliers high

Using Model Structure

• Variable dependencies marer!
• Perform BFS, star*ng with new evidence
• Use heuris*cs + decay to priori*ze explora*on

EXPERIMENTAL EVALUATION

Two Online Inference Tasks

• Collec*ve Classifica*on (Synthe*c)

– Infer arributes of users in a social network as progressively more informa*on is shared

• Collabora*ve Filtering (Jester; Goldberg et al. 2001)

– Infer user ra*ngs of jokes as users provide ra*ngs for an increasing number of jokes

Two Online Inference Tasks

• Collec*ve Classifica*on (Synthe*c)
• 100 total trials (10 networks x 10 series)
• Network evolves from 10% to 60% observed
• Fix 50% of variables at each epoch
• Collabora*ve Filtering (Jester)
• 10 trials, 100 users, 100 jokes
• Evolve from 25% to 75% revealed ra*ngs
• Fix {25,50,75}% of variables at each epoch

Collec*ve Classifica*on: Approximate Inference

regret vs. epochs error vs. epochs

• Regret diminishes over *me
• Error decreases, approaching full inference
• 69% reduc*on in inference *me

Collabora*ve Filtering

Regret RMSE

Epochs 50% ac*vated 25% ac*vated

Collabora*ve Filtering

RMSE

Epochs 50% ac*vated 25% ac*vated

• Value: high regret, but lower

error than full inference

• Preserves polarized ra*ngs
• 66% reduc*on in *me for

approximate inference

CONCLUSION

Online Collec*ve Inference

Summary

• Extremely relevant to modern problems
• Necessity: approximate MAP state in PGMs
• Inference regret: bound approxima*on error
• Approx. algos: use op*miza*on features
• Results: low regret, low error, faster
• New possibili*es: rich models, fast inference

Future Work

• Berer bounds for approximate inference?
• Dealing with changing models/weights
• Explicitly modeling change in models
• Applica*ons:

– Drug targe*ng – Knowledge Graph construc*on – Context-aware mobile devices