Recommender Systems Tutorial (Part 3) -- Online Components

Online Components:
Online Models, Intelligent Initialization,
Explore / Exploit

Why Online Components?

• Cold start
– New items or new users come to the system
– How to obtain data for new items/users (explore/exploit)
– Once data becomes available, how to quickly update the model
• Periodic rebuild (e.g., daily): Expensive
• Continuous online update (e.g., every minute): Cheap

• Concept drift
– Item popularity, user interest, mood, and user-to-item affinity may
change over time
– How to track the most recent behavior
• Down-weight old data
– How to model temporal patterns for better prediction
• … may not need to be online if the patterns are stationary

Deepak Agarwal & Bee-Chung Chen @ ICML’11 {dagarwal,beechun}@yahoo-inc.com 2

Big Picture

Most Popular Personalized
Recommendation Recommendation
Offline Models Collaborative filtering
(cold-start problem)

Online Models Time-series models Incremental CF,
Real systems are dynamic online regression

Intelligent Initialization Prior estimation Prior estimation,
Do not start cold dimension reduction

Explore/Exploit Multi-armed bandits Bandits with covariates
Actively acquire data

Extension: Segmented Most
Popular Recommendation


Online Components for
Most Popular Recommendation

Online models, intelligent initialization & explore/exploit

Most popular recommendation: Outline

• Most popular recommendation (no personalization, all
users see the same thing)
– Time-series models (online models)
– Prior estimation (initialization)
– Multi-armed bandits (explore/exploit)

– Sometimes hard to beat!!

• Segmented most popular recommendation
– Create user segments/clusters based on user features
– Provide most popular recommendation for each segment


Most Popular Recommendation

• Problem definition: Pick k items (articles) from a pool of N
to maximize the total number of clicks on the picked items
• Easy!? Pick the items having the highest click-through
rates (CTRs)
• But …
– The system is highly dynamic:
• Items come and go with short lifetimes
• CTR of each item changes over time
– How much traffic should be allocated to explore new items to
achieve optimal performance
• Too little → Unreliable CTR estimates
• Too much → Little traffic to exploit the high CTR items


CTR Curves for Two Days on Yahoo! Front Page

Each curve is the CTR of an item in the Today Module on www.yahoo.com over time

Traffic obtained from a controlled randomized experiment (no confounding)
Things to note:
(a) Short lifetimes, (b) temporal effects, (c) often breaking news stories


For Simplicity, Assume …

• Pick only one item for each user visit
– Multi-slot optimization later
• No user segmentation, no personalization (discussion later)
• The pool of candidate items is predetermined and is
relatively small (≤ 1000)
– E.g., selected by human editors or by a first-phase filtering method
– Ideally, there should be a feedback loop
– Large item pool problem later
• Effects like user-fatigue, diversity in recommendations,
multi-objective optimization not considered (discussion later)


Online Models

• How to track the changing CTR of an item
• For a given item, at time t, we
– Show the item nt times (i.e., nt views)
– Receive ct clicks

• Problem Definition: Given c1, n1, …, ct, nt, predict the CTR
(click-through rate) pt+1 at time t+1
• Potential solutions:
– Observed CTR at t: ct / nt → highly unstable (nt is usually small)

– Cumulative CTR: (∑all i ci) / (∑all i ni) → react to changes very slowly

– Moving window CTR: (∑i∈last K ci) / (∑i∈last K ni) → reasonable

• But, no estimation of Var[pt+1] (useful for explore/exploit)

Online Models: Dynamic Gamma-Poisson

Notation: • Show the item nt times
• Receive ct clicks
• Model-based approach • pt = CTR at time t
– (ct | nt, pt) ~ Poisson(nt pt)

– pt = pt-1 εt, where εt ~ Gamma(mean=1, var=η)

– Model parameters:
• p1 ~ Gamma(mean=µ 0, var=σ 02) is the offline CTR estimate
∀ η specifies how dynamic/smooth the CTR is over time

– Posterior distribution (pt+1 | c1, n1, …, ct, nt) ~ Gamma(?,?)
• Solve this recursively (online update rule)
c1 c2
n1 n2
µ0, σ02 p1 p2 …
η


Tracking behavior of Gamma-Poisson model

• Low click rate articles – More temporal smoothing


Intelligent Initialization: Prior Estimation

• Prior CTR distribution: Gamma(mean=µ 0, var=σ 02)
– N historical items:
• ni = #views of item i in its first time interval
• ci = #clicks on item i in its first time interval
– Model
• ci ~ Poisson(ni pi) and pi ~ Gamma(µ 0, σ 02)
⇒ ci ~ NegBinomial(µ 0, σ 02, ni)
– Maximum likelihood estimate (MLE) of (µ 0, σ 02)
 c + µ 02  −  c + µ 02  log n + µ 0 
 µ 02  +
∑
µ0
2
µ0
arg max N log − N log Γ 2  log Γ i 2   2   i σ2 
σ0
2
σ0
2
 σ0  i  σ0   i σ0   0 
µ0 , σ 0
2

• Better prior: Cluster items and find MLE for each cluster
– Agarwal & Chen, 2011 (SIGMOD)

Explore/Exploit: Problem Definition

now
clicks in the future
t –2 t –1 t
time

Item 1 x1% page views
… …
Item K xK% page views
Determine (x1, x2, …, xK) based on clicks and views observed before t
in order to maximize the expected total number of clicks in the future


Modeling the Uncertainty, NOT just the Mean

Simplified setting: Two items

If we only make a single decision,
give 100% page views to Item A
Item A
Probability density

If we make multiple decisions in the future
explore Item B since its CTR can
Item B potentially be higher

Potential = ∫ ( p − q ) ⋅ f ( p) dp
p>q
CTR CTR of item A is q
CTR of item B is p
Probability density function of item B’s CTR is f(p)

We know the CTR of Item A (say, shown 1 million times)
We are uncertain about the CTR of Item B (only 100 times)


Multi-Armed Bandits: Introduction (1)

For now, we are attacking the problem of choosing best article/arm for all users

Bandit “arms”

(unknown payoff
p1 p2 p3
probabilities)

• “Pulling” arm i yields a reward:
• reward = 1 with probability pi (success)
• reward = 0 otherwise (failure)



Bandit “arms”

(unknown payoff
p1 p2 p3 probabilities)
• Goal: Pull arms sequentially to maximize the total reward
• Bandit scheme/policy: Sequential algorithm to play arms (items)
• Regret of a scheme = Expected loss relative to the “oracle” optimal
scheme that always plays the best arm
– “best” means highest success probability
– But, the best arm is not known … unless you have an oracle
– Regret is the price of exploration
– Low regret implies quick convergence to the best



• Bayesian approach
– Seeks to find the Bayes optimal solution to a Markov decision
process (MDP) with assumptions about probability distributions
– Representative work: Gittins’ index, Whittle’s index
– Very computationally intensive

• Minimax approach
– Seeks to find a scheme that incurs bounded regret (with no or
mild assumptions about probability distributions)
– Representative work: UCB by Lai, Auer
– Usually, computationally easy
– But, they tend to explore too much in practice (probably because
the bounds are based on worse-case analysis)


Multi-Armed Bandits: Markov Decision Process (1)

• Select an arm now at time t=0, to maximize expected total number
of clicks in t=0,…,T
• State at time t: Θ t = (θ1t, …, θKt)
θit = State of arm i at time t (that captures all we know about arm i at t)
• Reward function Ri(Θ t, Θ t+1)
– Reward of pulling arm i that brings the state from Θ t to Θ t+1
• Transition probability Pr[Θ t+1 | Θ t, pulling arm i ]
• Policy π: A function that maps a state to an arm (action)
π(Θ t) returns an arm (to pull)
• Value of policy π starting from the current state Θ 0 with horizon T
Immediate reward Value of the remaining T-1 time slots
if we start from state Θ 1


Multi-Armed Bandits: MDP (2)

Immediate reward Value of the remaining T-1 time slots
if we start from state Θ 1

• Optimal policy:

• Things to notice:
– Value is defined recursively (actually T high-dim integrals)
– Dynamic programming can be used to find the optimal policy
– But, just evaluating the value of a fixed policy can be very expensive
• Bandit Problem: The pull of one arm does not change the state of
other arms and the set of arms do not change over time



• Which arm should be pulled next?
– Not necessarily what looks best right now, since it might have had a few
lucky successes
– Looks like it will be a function of successes and failures of all arms
• Consider a slightly different problem setting
– Infinite time horizon, but
– Future rewards are geometrically discounted
Rtotal = R(0) + γ.R(1) + γ2.R(2) + … (0<γ<1)
• Theorem [Gittins 1979]: The optimal policy decouples and solves a
bandit problem for each arm independently
Policy π(Θ t) is a function of (θ1t, …, θKt) One K-dimensional problem

Policy π(Θ t) = argmaxi { g(θit) } K one-dimensional problems
Still computationally expensive!!
Gittins’ Index


Bandit Policy
Priority Priority Priority 1. Compute the priority
1 2 3 (Gittins’ index) of each
arm based on its state
2. Pull arm with max
priority, and observe
reward
3. Update the state of the
pulled arm



• Theorem [Gittins 1979]: The optimal policy decouples and
solves a bandit problem for each arm independently
– Many proofs and different interpretations of Gittins’ index exist
• The index of an arm is the fixed charge per pull for a game with two
options, whether to pull the arm or not, so that the charge makes
the optimal play of the game have zero net reward
– Significantly reduces the dimension of the problem space
– But, Gittins’ index g(θit) is still hard to compute
• For the Gamma-Poisson or Beta-Binomial models
θit = (#successes, #pulls) for arm i up to time t
• g maps each possible (#successes, #pulls) pair to a number
– Approximate methods are used in practice
– Lai et al. have derived these for exponential family distributions


Multi-Armed Bandits: Minimax Approach (1)

• Compute the priority of each arm i in a way that the regret
is bounded
– Lowest regret in the worst case
• One common policy is UCB1 [Auer 2002]

Number of successes Total number of pulls
of arm i of all arms

ci 2 ⋅ log n
Priority i = +
ni ni

Number of pulls Observed Factor representing
of arm i success rate uncertainty



ci 2 ⋅ log n
Priority i = +
ni ni
Observed Factor
payoff representing
uncertainty
• As total observations n becomes large:
– Observed payoff tends asymptotically towards the true payoff
probability
– The system never completely “converges” to one best arm; only
the rate of exploration tends to zero



ci 2 ⋅ log n
Priority i = +
ni ni
Observed Factor
payoff representing
uncertainty
• Sub-optimal arms are pulled O(log n) times
• Hence, UCB1 has O(log n) regret
• This is the lowest possible regret (but the constants matter )
• E.g. Regret after n plays is bounded by
 ln n   π 2   K 
8 ∑
 i:µ < µ ∆  + 1 + 3  ⋅  ∑ ∆ j , where ∆ i = µbest − µi
    
 i best i     j =1 

Classical Multi-Armed Bandits: Summary

• Bayesian approach (Markov decision process)
– Representative work: Gittins’ index [Gittins 1979]
• Gittins’ index is optimal for a fixed reward distribution
• Idea: Pull the arm currently having the highest index value
– Representative work: Whittle’s index [Whittle 1988]
• Extend Gittins’ index to a changing reward distribution
• Only near optimal; approximate by Lagrange relaxation
– Computationally intensive
• Minimax approach (providing guaranteed regret bounds)
– Representative work: UCB1 [Auer 2002]
• Index = Upper confidence bound (model agnostic)
• Heuristics
ε-Greedy: Random exploration using fraction ε of traffic
– Softmax: exp{µi / τ }
ˆ
µi = predicted CTR of item i, τ = temperature
ˆ
∑ exp{µ
j
ˆ j /τ }
– P% upper confidence bound (model-based)


Do Classical Bandits Apply to Web Recommenders?

Each curve is the CTR of an item in the Today Module on www.yahoo.com over time

Traffic obtained from a controlled randomized experiment (no confounding)
Things to note:
(a) Short lifetimes, (b) temporal effects, (c) often breaking news stories


Characteristics of Real Recommender Systems

• Dynamic set of items (arms)
– Items come and go with short lifetimes (e.g., a day)
– Asymptotically optimal policies may fail to achieve good performance
when item lifetimes are short
• Non-stationary CTR
– CTR of an item can change dramatically over time
• Different user populations at different times
• Same user behaves differently at different times (e.g., morning, lunch
time, at work, in the evening, etc.)
• Attention to breaking news stories decays over time
• Batch serving for scalability
– Making a decision and updating the model for each user visit in real time
is expensive
– Batch serving is more feasible: Create time slots (e.g., 5 min); for each
slot, decide what fraction xi of the visits in the slot should give item i
[Agarwal et al., ICDM, 2009]


Explore/Exploit in Recommender Systems

now
clicks in the future
t –2 t –1 t
time

… …
Item K xK% page views
Determine (x1, x2, …, xK) based on clicks and views observed before t
in order to maximize the expected total number of clicks in the future

Let’s solve this from first principle

Bayesian Solution: Two Items, Two Time Slots (1)

• Two time slots: t = 0 and t = 1
– Item P: We are uncertain about its CTR, p0 at t = 0 and p1 at t = 1
– Item Q: We know its CTR exactly, q0 at t = 0 and q1 at t = 1
• To determine x, we need to estimate what would happen in the future

Now End
N0 views N1 views
time
Question: t=0 t=1
What fraction x of N0 views to item P • Assume we observe c; we can update p1
(1-x) to item Q • If x and c are given, optimal solution:
Give all views to Item P iff
E[ p1(x,c) I x, c ] > q1
Item Q
density

ˆ
p1 ( x, c)
Item P
p0 Item Q

density
Item P
q0 CTR
p1(x,c)
Obtain c clicks after serving x CTR
q1
(not yet observed; random variable) ˆ
p1 ( x, c)


• Expected total number of clicks in the two time slots
E[#clicks] at t = 0 E[#clicks] at t = 1

N 0 xp0 + N 0 (1 − x )q0 + N1 Ec [max{ p1 ( x, c), q1}]
ˆ ˆ
Item P Item Q ˆ
Show the item with higher E[CTR]: max{ p1 ( x, c ), q1}

= N 0 q0 + N1q1 + N 0 x ( p0 − q0 ) + N1 Ec [max{ p1 ( x, c ) − q1 , 0}]
ˆ ˆ

E[#clicks] if we Gain(x, q0, q1)
always show Gain of exploring the uncertain item P using x
item Q

Gain(x, q0, q1) = Expected number of additional clicks if we explore
the uncertain item P with fraction x of views in slot 0, compared to
a scheme that only shows the certain item Q in both slots
Solution: argmaxx Gain(x, q0, q1)


• Approximate p1 ( x, c) by the normal distribution
ˆ
– Reasonable approximation because of the central limit theorem

  q1 − p1  
ˆ  q1 − p1  
ˆ 
Gain( x, q0 , q1 ) = N 0 x( p0 − q0 ) + N1 σ 1 ( x) ⋅ φ   
ˆ  σ ( x)  + 1 − Φ σ ( x)  ( p1 − q1 )
   ˆ

  1    1  

Prior of p1 ~ Beta(a, b)
p1 = Ec [ p1 ( x, c)] = a /( a + b)
ˆ ˆ
xN 0 ab
σ 12 ( x) = Var[ p1 ( x, c)] =
ˆ
(a + b + xN 0 ) (a + b) 2 (1 + a + b)

• Proposition: Using the approximation,
the Bayes optimal solution x can be
found in time O(log N0)



• Quiz: Is it correct that the more we are uncertain about the
CTR of an item, the more we should explore the item?

(Fraction of views to give to the item)

Different curves are
for different prior
mean settings

Uncertainty: High Uncertainty: Low


Bayesian Solution: General Case (1)

• From two items to K items
– Very difficult problem: max ( N 0 ∑ i xi pi 0 + N1 Ec [max i { pi1 ( xi , ci )}] )
ˆ ˆ
x ≥0
∑ i xi = 1
Note: c = [c1, …, cK] max Ec [∑ i zi (c) pi1 ( xi , ci )]
ˆ
ci is a random variable representing z ≥0

the # clicks on item i we may get ∑ i zi (c) = 1, for all possible c

– Apply Whittle’s Lagrange relaxation (1988) to this problem setting
• Relax ∑i zi(c) = 1, for all c, to Ec [∑i zi(c)] = 1
• Apply Lagrange multipliers (q1 and q2) to enforce the constraints

min ( N 0 q0 + N1q1 + ∑i max Gain(xi , q0 , q1 ) )
q0 , q1 xi

– We essentially reduce the K-item case to K independent two-item
sub-problems (which we have solved)


Bayesian Solution: General Case (2)

• From two intervals to multiple time slots
– Approximate multiple time slots by two stages
• Non-stationary CTR
– Use the Dynamic Gamma-Poisson model to estimate the CTR
distribution for each item


Simulation Experiment: Different Traffic Volume

• Simulation with ground truth estimated based on Yahoo! Front Page data
• Setting:16 live items per interval
• Scenarios: Web sites with different traffic volume (x-axis)


Simulation Experiment: Different Sizes of the Item Pool

• Simulation with ground truth estimated based on Yahoo! Front Page data
• Setting: 1000 views per interval; average item lifetime = 20 intervals
• Scenarios: Different sizes of the item pool (x-axis)


Characteristics of Different Explore/Exploit Schemes (1)

• Why the Bayesian solution has better performance
• Characterize each scheme by three dimensions:
– Exploitation regret: The regret of a scheme when it is showing the item
which it thinks is the best (may not actually be the best)
• 0 means the scheme always picks the actual best
• It quantifies the scheme’s ability of finding good items
– Exploration regret: The regret of a scheme when it is exploring the items
which it feels uncertain about
• It quantifies the price of exploration (lower → better)
– Fraction of exploitation (higher → better)
• Fraction of exploration = 1 – fraction of exploitation
All traffic to a web site

Exploitation traffic Exploration
traffic


Characteristics of Different Explore/Exploit Schemes (2)

• Exploitation regret: Ability of finding good items (lower → better)
• Exploration regret: Price of exploration (lower → better)
• Fraction of Exploitation (higher → better)

Exploitation Regret
Exploitation Regret

Good Good

Exploration Regret Exploitation fraction

Discussion: Large Content Pool

• The Bayesian solution looks promising
– ~10% from true optimal for a content pool of 1000 live items
• 1000 views per interval; item lifetime ~20 intervals
• Intelligent initialization (offline modeling)
– Use item features to reduce the prior variance of an item
• E.g., Var[ item CTR | Sport ] < Var[ item CTR ]
– Linear models that estimate CTR distributions
– Hierarchical smoothing: Estimate the CTR distribution of a
random article of a item category for a user segment
• Use existing hierarchies of items and users
• Create supervised clusters via extended version of LDA
• Feature-based explore/exploit
– Estimate model parameters, instead of per-item CTR
– More later

Discussion: Multiple Positions, Ranking

• Feature-based approach
– reward(page) = model(φ(item 1 at position 1, … item k at position k))
– Apply feature-based explore/exploit
• Online optimization for ranked list
– Ranked bandits [Radlinski et al., 2008]: Run an independent bandit
algorithm for each position
– Dueling bandit [Yue & Joachims, 2009]: Actions are pairwise
comparisons
• Online optimization of submodular functions
∀ S1, S2 and a, fa(S1 ⊕ S2) ≤ fa(S1)
• where fa(S) = fa(S ⊕ 〈a〉) – fa(S)
– Streeter & Golovin (2008)


Discussion: Segmented Most Popular

• Partition users into segments, and then for each segment,
provide most popular recommendation
• How to segment users
– Hand-created segments: AgeGroup × Gender
– Clustering based on user features
• Users in the same cluster like similar items
• Segments can be organized by taxonomies/hierarchies
– Better CTR models can be built by hierarchical smoothing
• Shrink the CTR of a segment toward its parent
• Introduce bias to reduce uncertainty/variance
– Bandits for taxonomies (Pandey et al., 2008)
• First explore/exploit categories/segments
• Then, switch to individual items


Most Popular Recommendation: Summary

• Online model:
– Estimate the mean and variance of the CTR of each item over time
– Dynamic Gamma-Poisson model
• Intelligent initialization:
– Estimate the prior mean and variance of the CTR of each item
cluster using historical data
• Cluster items → Maximum likelihood estimates of the priors
• Explore/exploit:
– Bayesian: Solve a Markov decision process problem
• Gittins’ index, Whittle’s index, approximations
• Better performance, computation intensive
– Minimax: Bound the regret
• UCB1: Easy to compute
• Explore more than necessary in practice
ε-Greedy: Empirically competitive for tuned ε

Online Components for
Personalized Recommendation

Online models, intelligent initialization & explore/exploit

Personalized recommendation: Outline

• Online model
– Methods for online/incremental update (cold-start problem)
• User-user, item-item, PLSI, linear model
– Methods for modeling temporal dynamics (concept drift problem)
• State-space model, tensor factorization
• timeSVD++ [Koren 2009] for Netflix, (not really online)

• Intelligent initialization (cold-start problem)
– Feature-based prior + reduced rank regression (for linear model)

• Explore/exploit
– Bandits with covariates


Online Update for Similarity-based Methods

• User-user methods
– Key quantities: Similarity(user i, user j)
– Incremental update (e.g., [Papagelis 2005])
Bij
Incrementally maintain three
∑ k (rik − ri )(rjk − rj ) sets of counters: B, C, D
corr (i, j ) =
∑ k (rik − ri ) ∑ k (rjk − rj )
Ci Dj
– Clustering (e.g., [Das 2007])
• MinHash (for Jaccard similarity)
• Clusters(user i) = (h1(ri), …, hK(ri)) ← fixed online (rebuilt periodically)
• AvgRating(cluster c, item j) ← updated online
score(user i, item j ) ∝ ∑ k
AvgRating (hk (ri ), j )
• Item-item methods (similar ideas)

Online Update for PLSI

• Online update for probabilistic latent semantic indexing
(PLSI) [Das 2007]
p (item j | user i ) = ∑ k
p (cluster k | i ) p ( j | cluster k )

Fixed online Updated online
(rebuilt Periodically)
∑ user u
I (u clicks j ) p(k | u )
∑ user u
p(k | u )


Online Update for Linear/Factorization Model

• Linear model: the regression weight of item j on the
yij ~ ∑ k xik β = x′β
jk i j
kth user feature

rating that user i the kth feature of user i
gives item j
– xi can be user factor vector (estimated periodically, fixed online)
βj is an item factor vector (updated online)
– Straightforward to fix item factors and update user factors
• Gaussian model (use vector notation)
yij ~ N ( xi′β j , σ 2 ) E[ β j | y ] = Var[ β j | y ](V j−1µ j + ∑i yij xi / σ 2 )
β j ~ N ( µ j ,V j ) Update Var[ β j | y ] = (V j−1 + ∑i xi xi′ / σ 2 ) −1
E[βj] and Var[βj]
(current estimates)


Temporal Dynamics: State-Space Model

• Item factors βj,t change over time t
– The change is smooth: βj,t should be close to βj,t-1
Dynamic model Static model
yij ,t ~ N ( xi′,t β j ,t , σ 2 ) yij ~ N ( xi′β j , σ 2 )
β j ,t ~ N ( β j ,t −1 , V ) β j ~ N ( µ j ,V j )
random variable constants

β j ,1 ~ N ( µ j ,0 , V0 ) yij,1 yij,2
xi,1 xi,2
µj,0, V0 βj,1
V
βj,2 …
– Use standard Kalman filter update rule Subscript:
user i,
– It can be extended to Logistic (for binary data), item j
Poisson (for count data), etc. time t

Temporal Dynamics: Tensor Factorization

• Decompose ratings into three components [Xiong 2010]
– User factors uik: User i ’s membership to type k
– Item factors vjk: Item j ’s affinity to type k
– Time factors ztk: Importance/weight of type k at time t
Regular matrix factorization
yij ~ ∑k uik v jk = ui1v j1 + ui 2v j 2 + ... + uiK v jK
Tensor factorization
yij ,t ~ ∑k uik v jk ztk = ui1v j1zt1 + ui 2v j 2 zt 2 + ... + uiK v jK ztK
time-varying weights on different types/factors
Subscript:
zt , k ~ N ( zt −1, k , σ )
2
Time factors are smooth over time user i,
item j
time t

Temporal Dynamics: timeSVD++

• Explicitly model temporal patterns on historical data to remove bias
• Part of the winning method of Netflix contest [Koren 2009]

item popularity
yij ,t ~ µ + bi (t ) + b j (t ) + ui (t )′v j
user bias user factors (preference)
middle
bi (t ) = bi + α i devi (t ) + bit
distance to the middle rating time of i t
b j (t ) = b j + b j , bin (t )
time bin
ui (t ) k = uik + α ik devu (t ) + uikt Subscript:
user i,
Model parameters: µ, bi, αi, bit, bj, bjd, uik, αik, uikt, item j
for all user i, item j, factor k, time t, time bin d time t

Online Models: Summary

• Why online model? Real systems are dynamic!!
– Cold-start problem: New users/items come to the system
• New data should be used a.s.a.p., but rebuilding the entire
model is expensive
• How to efficiently, incrementally update the model
– Similarity-based methods, PLSI, linear and factorization models
– Concept-drift problem: User/item behavior changes over time
• Decay the importance of old data
– State-space model
• Explicitly model temporal patterns
– timeSVD++ for Netflix, tensor factorization Not really
online models!!
• Next
– Initialization methods for factorization models (for cold start)
• Start from linear regression models


Intelligent Initialization for Linear Model (1)

• Linear/factorization model
rating that user i factor vector of item j
gives item j
yij ~ N (ui′β j , σ 2 )
feature/factor vector of user i

β j ~ N ( µ j , Σ)

– How to estimate the prior parameters µj and Σ
• Important for cold start: Predictions are made using prior
• Leverage available features
– How to learn the weights/factors quickly
Subscript:
• High dimensional βj → slow convergence user i,
• Reduce the dimensionality item j


FOBFM: Fast Online Bilinear Factor Model
Subscript:
Per-item
online model
yij ~ ui′β j , β j ~ N ( µ j , Σ) user i
item j
Data:
• Feature-based model initialization yij = rating that
β j ~ N ( Ax j , Σ) ⇔ yij ~ ui′ Ax j + ui′v j user i gives item j
ui = offline factor vector
predicted by features v j ~ N (0, Σ) of user i
xj = feature vector
of item j
• Dimensionality reduction for fast model convergence
v j = Bθ j B is a n×k linear projection matrix (k << n)
project: high dim(vj) → low dim(θj)
θ j ~ N (0, σ θ2 I ) low-rank approx of Var[βj]: β j ~ N ( Ax j , σ θ BB′)
2

vj B θj
Offline training: Determine A, B, σθ2
= through the EM algorithm
(once per day or hour)


FOBFM: Fast Online Bilinear Factor Model
Subscript:
Per-item
online model
yij ~ ui′β j , β j ~ N ( µ j , Σ) user i
item j
Data:
• Feature-based model initialization yij = rating that
β j ~ N ( Ax j , Σ) ⇔ yij ~ ui′ Ax j + ui′v j user i gives item j
ui = offline factor vector
predicted by features v j ~ N (0, Σ) of user i
xj = feature vector
of item j
• Dimensionality reduction for fast model convergence
v j = Bθ j B is a n×k linear projection matrix (k << n)
project: high dim(vj) → low dim(θj)
θ j ~ N (0, σ θ2 I ) low-rank approx of Var[βj]: β j ~ N ( Ax j , σ θ BB′)
2

• Fast, parallel online learning
yij ~ ui′ Ax j + (ui′B )θ j , where θj is updated in an online manner

offset new feature vector (low dimensional)
• Online selection of dimensionality (k = dim(θj))
– Maintain an ensemble of models, one for each candidate dimensionality

Experimental Results: My Yahoo! Dataset (1)

• My Yahoo! is a personalized news reading site
– Users manually select news/RSS feeds
• ~12M “ratings” from ~3M users on ~13K articles
– Click = positive
– View without click = negative


Methods:
• No-init: Standard online
regression with ~1000
parameters for each item
• Offline: Feature-based
model without online
update
• PCR, PCR+: Two
principal component
methods to estimate B
• FOBFM: Our fast online
method

• Item-based data split: Every item is new in the test data
– First 8K articles are in the training data (offline training)
– Remaining articles are in the test data (online prediction & learning)
• Supervised dimensionality reduction (reduced rank regression)
significantly outperforms other methods



# factors =
Number of
parameters per
item updated
online

• Small number of factors (low dimensionality) is better when the amount of data
for online leaning is small
• Large number of factors is better when the data for learning becomes large
• The online selection method usually selects the best dimensionality


Intelligent Initialization: Summary

• Online models are useful for cold start and concept drift
• Whenever historical data is available, do not start cold
• For linear/factorization models
– Use available features to setup the starting point
– Reduce dimensionality to facilitate fast learning

• Next
– Explore/exploit for personalization
– Users are represented by covariates
• Features, factors, clusters, etc
– Covariate bandits


Explore/Exploit for Personalized Recommendation

• One extreme problem formulation
– One bandit problem per user with one arm per item
– Bandit problems are correlated: “Similar” users like similar items
– Arms are correlated: “Similar” items have similar CTRs
• Model this correlation through covariates/features
– Input: User feature/factor vector, item feature/factor vector
– Output: Mean and variance of the CTR of this (user, item) pair
based on the data collected so far
• Covariate bandits
– Also known as contextual bandits, bandits with side observations
– Provide a solution to
• Large content pool (correlated arms)
• Personalized recommendation (hint before pulling an arm)


Methods for Covariate Bandits

• Priority-based methods
– Rank items according to the user-specific “score” of each item;
then, update the model based on the user’s response
– UCB (upper confidence bound)
• Score of an item = E[posterior CTR] + k StDev[posterior CTR]
– Posterior draw
• Score of an item = a number drawn from the posterior CTR distribution
– Softmax
exp{µi / τ }
ˆ
• Score of an item = a number drawn according to
∑ j exp{µ j / τ }
ˆ
∀ ε-Greedy
– Allocate ε fraction of traffic for random exploration (ε may be adaptive)
– Robust when the exploration pool is small
• Bayesian scheme
– Close to optimal if can be solved efficiently


Covariate Bandits: Some References

• Just a small sample of papers
– Hierarchical explore/exploit (Pandey et al., 2008)
• Explore/exploit categories/segments first; then, switch to individuals
– Variants of ε-greedy
• Epoch-greedy (Langford & Zhang, 2007): ε is determined based on
the generalization bound of the current model
• Banditron (Kakade et al., 2008): Linear model with binary response
• Non-parametric bandit (Yang & Zhu, 2002): ε decreases over time;
example model: histogram, nearest neighbor
– Variants of UCB methods
• Linearly parameterized bandits (Rusmevichientong et al., 2008):
minimax, based on uncertainty ellipsoid
• LinUCB (Li et al., 2010): Gaussian linear regression model
• Bandits in metric spaces (Kleinberg et al., 2008; Slivkins et al., 2009):
– Similar arms have similar rewards: | reward(i) – reward(j) | ≤ distance(i,j)


Online Components: Summary

• Real systems are dynamic
• Cold-start problem
– Incremental online update (online linear regression)
– Intelligent initialization (use features to predict initial factor values)
– Explore/exploit (pick posterior mean + k posterior standard dev)

• Concept-drift problem
– Tracking the current behavior (state-space models, Kalman filter)
– Modeling temporal patterns


Intelligent Initialization for Factorization Model (1)

• Online update for item cold start (no temporal dynamics)
Offline model Online model
yij ~ N (ui′ v j , σ 2 I ) (periodic) Offset Feature vector
offline training
Factorization output: yij ,t ~ N (ui′ Ax j + ui′ Bθ j ,t , σ 2 I )
ui, A, B, σθ2
ui ~ N (Gxi , σ u I )
2
θ j ,t = θ j ,t −1 Updated online

Feature-based init θ j ,1 ~ N (0, σ θ I )
2

Dim reduction
Scalability:
v j = Ax j + Bθ j • θj,t is low dimensional
Feature-based init • θj,t for each item j can be updated
θ j ~ N (0, σ θ I )
2 independently in parallel



Offline Online
yij ~ N (ui′ v j , σ 2 I ) yij ,t ~ N (ui′ Ax j + ui′ Bθ j ,t , σ 2 I )
ui ~ N (Gxi , σ u I )
2 θ j ,t = θ j ,t −1
v j = Ax j + Bθ j θ j ,1 ~ N (0, σ θ I )
2

θ j ~ N (0, σ θ I )
2

• Our observation so far
– Dimension reduction (ui′ B) does not improve much if factor
regressions are based on good covariates (σθ2 is small)
• Small σθ2 → strong shrinkage → small effective dimensionality
(soft dimension reduction)

– Online updates help significantly: In MovieLens (time-based split),
reduced RMSE from .93 to .86



• Include temporal dynamics
Offline computation Online computation
(rebuilt periodically)
Fix ui,t and update θj,t
yij ,t ~ N (ui′,t v j ,t , σ I )
2
yij ,t ~ N (ui′,t Dx j ,t + ui′,t Bθ j ,t , σ 2 I )
ui ,t = Gxi ,t + Hδ i ,t , θ j ,t ~ N (θ j ,t −1 , σ θ2 I )
δ i ,t ~ N (δ i ,t −1 , σ δ I )
2
Fix vj,t and update δi,t
δ i ,1 ~ N (0, sδ I )
2
yij ,t ~ N (v′j ,t Gxi ,t + v′j ,t Hδ i ,t , σ 2 I )
v j ,t = Dx j ,t + Bθ j ,t δ i ,t ~ N (δ i ,t −1 , σ δ I )
2

θ j ,t ~ N (θ j ,t −1 , σ θ2 I ) Repeat the above two steps a few times

θ j ,1 ~ N (0, sθ I )
2


Experimental Results: MovieLens Dataset

• Training-test data split
– Time-split: First 75% ratings in training; rest in test
– Movie-split: 75% randomly selected movies in training; rest in test

ROC for Movie-split
Model RMSE RMSE
Time-split Movie-split
FOBFM 0.8429 0.8549
RLFM 0.9363 1.0858
Online-UU 1.0806 0.9453
Constant 1.1190 1.1162

FOBFM: Our fast online method
RLFM: [Agarwal 2009]
Online-UU: Online version of user-user
collaborative filtering
Online-PLSI: [Das 2007]


Experimental Results: Yahoo! Front Page Dataset

• Training-test data split
– Time-split: First 75% ratings in training; rest in test

–~2M “ratings” from ~30K frequent
users to ~4K articles
•Click = positive
•View without click = negative
–Our fast learning method
outperforms others


Are Covariate Bandits Difficult?

• When features are predictive and different users/items have
different features, the myopic scheme is near optimal
– Myopic scheme: Pick the item having the highest predicted CTR (without
considering the explore/exploit problem at all)
– Sarkar (1991) and Wang et al. (2005) studied this for the two-armed bandit
case
• Simple predictive upper confidence bound gave good empirical
results
– Pick the item having highest E[CTR | data] + k Std[CTR | data]
– Pavlidis et al. (2008) studied this for Gaussian linear models
– Preliminary experiments (Gamma linear model)
• Bayesian scheme is better when features are not very predictive
• Simple predictive UCB is better when features are predictive


Recommender Systems Tutorial (Part 3) -- Online Components

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