Transcript Online Primal-Dual Algorithms for Covering and Packing Problems

A Primal-Dual Approach to
Online Optimization Problems
Seffi Naor
Computer Science Dept.
Technion, Haifa, Israel
Based on joint papers with Nikhil Bansal, Niv Buchbinder, and Kamal Jain
Road Map
Introducing the framework:
• Ski rental
• Online set cover
• Virtual circuit routing
The general framework:
• General covering/packing linear programs
More applications:
• The ad-auctions problem
• Weighted caching
Further research
What is an Online Algorithm?
• Input is given “in pieces” over time, each piece is called a
“request”
• Request sequence: ¾ = ¾1, … , ¾n
• Upon arrival of request ¾i :
• Online algorithm has to serve the request
• Previous decisions for requests ¾1, …, ¾i-1 cannot be
changed
Performance Evaluation: Competitive Factor
• How to evaluate performance of online algorithm A?
• For every request sequence ¾ = ¾1, … , ¾n:
compare cost of A to the cost of an optimal offline
algorithm which is given the request sequence in
advance
• Competitive factor of online algorithm A is ® if
for every request sequence ¾ = ¾1, … , ¾n:
Linear Programming
Linear constraints, linear objective
Min x1 – 3 x2 + 2 x3
x1 – x2 + 7 x3 <= 2
x2 - x3 >= 0
x1 + 3 x2 + x3 >= -3
Polynomial time solvable. Lies at the heart of optimization.
The Ski Rental Problem
• Buying costs $B.
• Renting costs $1 per day.
Problem:
• Number of ski days is not known in advance.
Goal: Minimize the total cost.
Ski Rental – Integer Program
1 - B uy
1 - R ent on day i
zi  
x
0 - D on't rent on day i
0
D
on't
B
uy


k
m in B x 
z
i
i 1
Subject to:
For each day i: x  z i  1
x , z i  {0,1}
Ski Rental – Relaxation
P: Primal Covering
D: Dual Packing
k
m in B x 
z
k
i
i 1
For each day i: x  z i  1
x , zi  0
m ax  y i
i 1
For each day i: y i  1
k

yi  B
i 1
Online setting:
• Primal: New constraints arrive one by one.
• Requirement: Upon arrival, constraints should
be satisfied.
• Monotonicity: Variables can only be increased.
Ski Rental – Algorithm
P: Primal Covering
D: Dual Packing
k
m in B x 
z
k
m ax  y i
i
i 1
For each day i: x  z i  1
i 1
For each day i: y i  1
x , zi  0
k

i 1
Initially x 0
Each new day (new constraint):
if x<1:
 zi  1-x
 x  x(1+ 1/B) + 1/(c*B)
 yi  1
- ‘c’ later.
yi  B
Analysis of Online Algorithm
Proof of competitive factor:
1. Primal solution is feasible.
2. In each iteration, ΔP ≤ (1+ 1/c)ΔD.
3. Dual is feasible.
Conclusion: Algorithm is (1+ 1/c)-competitive
Initially x 0
Each new day (new constraint):
if x<1:
 zi  1-x
 x  x(1+ 1/B) + 1/(c*B)
 yi  1
- ‘c’ later.
Analysis of Online Algorithm
1. Primal solution is feasible.
If x ≥1 the solution is feasible.
Otherwise set: zi  1-x.
2. In each iteration, ΔP ≤ (1+ 1/c)ΔD:
Algorithm:
If x≥1, ΔP =ΔD=0
When new constraint
Otherwise:
arrives, if x<1:
zi1-x
• Change in dual: 1
x x(1+ 1/B) + 1/c*B
• Change in primal:
y 1
BΔx + zi = x+ 1/c+ 1-x = 1+1/c
i
Analysis of Online Algorithm
3.
Dual is feasible:
k
Need to prove:  y i  B
i 1
We prove that after B days x≥1
x is a sum of geometric sequence
a1 = 1/(cB), q = 1+1/B
B
1 

1


 1
1 
B
x


1 
cB 
1


 1
B

B
1 

1


 1
B

c
Algorithm:
When new constraint
arrives, if x<1:
zi1-x
x x(1+ 1/B) + 1/c*B
yi1
B
1 

c  1    1  e  1
B

1
e
1 
c e 1
Randomized Algorithm
0
1
X:
X1 X2
X3
X4
•
Choose d uniformly in [0,1]
•
Buy on the day corresponding to the “bin” d falls in
•
Rent up to that day
Analysis:
•
Probability of buying on the i-th day is xi
•
Probability of renting on the i-th day is at most zi
Going Beyond the Ski Problem
• Ski problem: coefficients in the constraint matrix
belong to {0,1}
• What can be said about general constraint
matrices with coefficients from {0,1}?
The Online Set-Cover Problem
• Elements: e1, e2, …, en
• Set system: s1, s2, … sm
• Costs: c(s1), c(s2), … c(sm)
Online Setting:
• Elements arrive one by one.
• Upon arrival elements need to be covered.
• Sets that are chosen cannot be “unchosen”.
Goal: Minimize the cost of the chosen sets.
Online Set-Cover: Lower Bound
• elements: 1, … ,n
• sets: all (n choose n) subsets of cardinality n
• cost: unit cost
Adversary’s strategy:
• While possible: pick an element that is not covered
(# of elements offered ≥ n)
Competitive ratio: n
(cost of online: = n, cost of OPT = 1)
But, … n ≈ log(n choose n). polylog(m,n) is not ruled out!
Set Cover – Linear Program
P: Primal Covering
D: Dual Packing
m ax  y ( e )
m in  c ( s ) x ( s )
e E
s S
e  E

s e s
x(s)  1
s  S
 y (e)  c ( s )
e s
Online setting:
• Primal: constraints arrive one by one.
• Requirement: each constraint is satisfied.
• Monotonicity: variables can only be increased.
Set Cover – Algorithm
P: Primal Covering
D: Dual Packing
m ax  y ( e )
m in  c ( s ) x ( s )
e E
s S
e  E

x(s)  1
s  S
 y (e)  c ( s )
e s
s e s
Initially x(s) 0
When new element arrives, while  x ( s )  1 :
• y(e)  y(e)+1
s e s
• . s e  s x ( s )  x ( s )  1  1 / c ( s )   1 /  m  c ( s ) 
Analysis of Online Algorithm
Proof of competitive factor:
1. Primal solution is feasible.
2. In each iteration, ΔP ≤ 2ΔD.
3. Dual is (almost) feasible.
Conclusion: We will see later.
Initially x(S) 0
When new element e arrives, while

s e s
x(s)  1 :
• y(e)  y(e)+1
• .  s e  s x ( s )  x ( s ) 1  1 / c ( s )   1 /  m  c ( s ) 
Analysis of Online Algorithm
1. Primal solution is feasible.
We increase the primal variables until the
constraint is feasible.
Initially x(S) 0
When new element e arrives, while

s e s
x(s)  1 :
• y(e)  y(e)+1
• .  s e  s x ( s )  x ( s ) 1  1 / c ( s )   1 /  m  c ( s ) 
Analysis of Online Algorithm
2. In each iteration, ΔP ≤ 2ΔD.
In each iteration:
• ΔD = 1
 x(s)

1
 P =  c(s)   x ( s )   c(s)  


c
(
s
)
m

c
(
s
)
s|es
s|es




s|e s
x (s) 
 1/m  2
s|e s
Initially x(S) 0
When new element e arrives, while

s e s
x(s)  1 :
• y(e)  y(e)+1
• .  s e  s x ( s )  x ( s ) 1  1 / c ( s )   1 /  m  c ( s ) 
Analysis of Online Algorithm
3.
Dual is (almost) feasible:
•
We prove that:  s  S ,
 y(e)  c(s)O (log m )
e S
•
If y(e) increases, then x(s) increases (for e in S).
•
x(s) is a sum of a geometric series:
a1 = 1/[mc(s)], q = (1+ 1/c(s))
Initially x(S) 0
When new element e arrives, while

s e s
x(s)  1 :
• y(e)  y(e)+1
• .  s e  s x ( s )  x ( s ) 1  1 / c ( s )   1 /  m  c ( s ) 
Analysis of Online Algorithm
 After c(s)O(log m) rounds:
x(s) 
1
 1  1 / c ( s ) 
c ( s ) O (log m )
1
1  1 / c ( s )   1
m  c(s)
1  1 / c ( s ) 


c ( s ) O (log m )
1
 1
m
We never increase a variable x(s)>1!
Initially x(S) 0
When new element e arrives, while

s e s
x(s)  1 :
• y(e)  y(e)+1
• .  s e  s x ( s )  x ( s ) 1  1 / c ( s )   1 /  m  c ( s ) 

Conclusion
•
The dual is feasible with cost 1/O(log m) of the primal.
 The algorithm produces a fractional set cover that is
O(log m)-competitive.
•
Remark: No online algorithm can perform better in
general.
What about an integral solution?
•
Round fractional solution. (With O(log n) amplification.)
•
Can be done deterministically online [AAABN03].
•
Competitive ratio is O(log m log n).
Online Virtual Circuit Routing
Network graph G=(V, E)
capacity function u: E Z+
Requests: ri = (si, ti)
• Problem: Connect si to ti by a path, or reject the
request.
• Reserve one unit of bandwidth along the path.
• No re-routing is allowed.
• Load: ratio between reserved edge bandwidth and
edge capacity.
• Goal: Maximize the total throughput.
Routing – Linear Program
y ( ri , p )
P ( ri )
= Amount of bandwidth allocated for ri on path p
- Available paths to serve request ri
m ax 
ri

y ( ri , p )
p  P ( ri )
s.t:
For each ri:

y ( ri , p )  1
p  P ( ri )
For each edge e:

ri

p  P ( ri ) e p
y ( ri , p )  u ( e )
Routing – Linear Program
P: Primal Covering
m in  u ( e ) x ( e ) 
e E
D: Dual Packing
 z (r )
m ax 
i
ri
ri
 ri , p  P ( ri ) :
 ri
 x(e)  z ( r )  1
i
e p


y ( ri , p )
p  P ( ri )
y ( ri , p )  1
p  P ( ri )
e :

ri

y ( ri , p )  u ( e )
p  P ( ri ) e  p
Online setting:
• Dual: new columns arrive one by one.
• Requirement: each dual constraint is satisfied.
• Monotonicity: variables can only be increased.
Routing – Algorithm
P: Primal Covering
m in  u ( e ) x ( e ) 
e E
D: Dual Packing
 z (r )
m ax 
i
ri
ri
 ri , p  P ( ri ) :
 ri
 x(e)  z ( r )  1
i
e p


y ( ri , p )
p  P ( ri )
y ( ri , p )  1
p  P ( ri )
e :

ri

y ( ri , p )  u ( e )
p  P ( ri ) e  p
Initially x(e) 0
When new request arrives, if  p  P ( ri ),  x(e)  1 :
e p
 z(ri)  1

1 
1
 . e  p : x ( e )  x ( e )  1 

u (e )  n  u (e )

 y(ri,p)  1
Analysis of Online Algorithm
Proof of competitive factor:
1. Primal solution is feasible.
2. In each iteration, ΔP ≤ 3ΔD.
3. Dual is (almost) feasible.
Conclusion: We will see later.
Initially x(e) 0
When new request arrives, if  p  P ( ri ),  x(e)  1 :
e p
 z(ri)  1

1 
1
 .e  p : x ( e )  x ( e )  1 

u (e )  n  u (e )

 y(ri,p)  1
Analysis of Online Algorithm
1. Primal solution is feasible.
If  p  P ( r ),  x(e)  1 : the solution is feasible.
i
e p
Otherwise: we update z(ri)  1
Initially x(e) 0
When new request arrives, if  p  P ( ri ),  x(e)  1 :
e p
 z(ri)  1

1 
1
 . e  p : x ( e )  x ( e )  1 

u (e )  n  u (e )

 y(ri,p)  1
Analysis of Online Algorithm
In each iteration: ΔP ≤ 3ΔD.
If  p  P ( ri ) :  x(e)  1 ΔP = ΔD=0
e p
Otherwise:
ΔD=1
2.
P 

e p
u ( e )  x ( e )  z ( ri )
 x (e)

1
  u (e) 

 1 3
e p
 u (e) n  u (e) 
Initially x(e) 0
When new request arrives, if  p  P ( ri ),  x(e)  1 :
e p
 z(ri)  1

1 
1
 . e  p : x ( e )  x ( e )  1 

u (e )  n  u (e )

 y(ri,p)  1
Analysis of Online Algorithm
3. Dual is (almost) feasible.
We prove:
• For each e, after routing u(e)O(log n) on e, x(e)≥1
x(e) is a sum of a geometric sequence
x(e)1 = 1/(nu(e)), q = 1+1/u(e)
 After u(e)O(log n) requests:
u ( e ) O (log n )

1 
1
1 

u (e) 
1

x (e) 


n  u (e )

1 
1 
 1
u (e) 


1 
1 

u
(
e
)


u ( e ) O (log n )
1
1
n
D: Dual Packing
m
n
min
cx
i
i
i 1
a
i 1
b
j
yj
j 1
m
n
j, 1  j  m
max
ij
xi  b j
 i, 1  i  n
a
ij
y j  ci
j1
P* = D*
•
ci, aij, bj and the variables xi are all non-negative
•
Many approximation/exact algorithms are actually
primal-dual: shortest path, matching, flows, cuts ..
Dual solutions
P: Primal Covering
Primal solutions
Covering/Packing Linear Programs
Key Idea for Online Primal-Dual
Primal: Min i ci xi
Dual: Max t bt yt
Step t, new constraint:
a1x1 + a2x2 + … + ajxj ≥ bt
New variable yt
+ bt yt in dual objective
xi  (1+ ai/ci) xi
yt  yt + 1
(mult. update)
(additive update)
 primal cost =
=  Dual Cost
Online Primal-Dual Approach
• Can the offline problem be cast as a linear
covering/packing program?
• Can the online process be described as:
– New rows appearing in a covering LP?
– New columns appearing in a packing LP?
Yes ??
• Upon arrival of a new request:
– Update primal variables in a multiplicative way.
– Update dual variables in an additive way.
Online Primal Dual Approach
Next Prove:
1. Primal solution is feasible (or nearly feasible).
2. In each round, ΔP ≤ c ΔD.
3. Dual is feasible (or nearly feasible).
Got a fractional solution, but need an integral
solution ??
•
•
Randomized rounding techniques might work.
Sometimes, even derandomization (e.g., method of
conditional probabilities) can be applied online!
Simple Lower Bound
min x1 + x2 + … + xn
x1 + x2 + x3 + … + xn ≥ 1
x2 + x3 + … + xn ≥ 1
x3 + … + xn ≥ 1
…
xn ≥ 1
Set all xi to 1/n
Increase x2 ,x3,…,xn to 1/n-1
…
Increase xn to 1
Online = H(n) ≈ lnn
(1+1/2+ 1/3+ … + 1/n)
Opt = 1 ( xn=1 suffices)
General Covering/Packing Results
{0,1} covering/packing matrix:
•
[Buchbinder Naor 05]
Competitive ratio O(log D)
(D: max number of non-zero entries in a constraint).
Remarks:
•
•
•
Fractional solutions  randomized algorithm (online rounding).
Number of constraints/variables can be exponential.
Possible tradeoff: Competitive ratio/violation of constraints.
General Covering/Packing Results
For a general covering/packing matrix [BN05] :
Covering:
• Competitive ratio O(log n)
(n – number of variables).
Packing:
• Competitive ratio O(log n + log [a(max)/a(min)])
a(max), a(min) – max/min non-zero entry
Remarks:
• Results are tight.
• Can add “box” constraints to covering LP (e.g. x≤1)
Online Primal-Dual Approach
Advantages:
1. Generic ideas and algorithms applicable to many
online problems.
2. Linear Program helps detecting the difficulties of the
online problem.
3. General recipe for the design and analysis of online
algorithms.
4. No potential function appearing “out of nowhere”.
5. Competitiveness with respect to a fractional optimal
solution.
Known Results via P-D Approach
Covering Online Problems (Minimization):
•
O(log k)-algorithm for weighted caching [BBN07]
•
Ski rental, Dynamic TCP Acknowledgement
•
Parking Permit Problem [Meyerson 05]
•
•
Online Set Cover [AAABN03]
Online Graph Covering Problems [AAABN04]:
–
–
–
–
Non-metric facility location
Generalized connectivity: pairs arrive online
Group Steiner: groups arrive online
Online multi-cut: (s,t)--pairs arrive online
Known Results via P-D Approach
Packing Online Problems (maximization):
•
Online Routing/Load Balancing Problems [AAP93,
AAPFW93, BN06].
•
•
•
General Packing/routing e.g. Multicast trees.
Online Matching [KVV91] – Nodes arrive one-by-one.
Ad-Auctions Problem [MSVV05] – In a bit …
What are Ad-Auctions?
You type in a search engine: Vacation Eilat
You get:
And …
Advertisements
Algorithmic
Search results
How do search engines sell ads?
• Each advertiser:
– Sets a daily budget
– Provides bids on interesting keywords
• Search Engine (on each keyword):
– Selects ads
– Advertiser pays bid if user clicks on ad.
Goal (of Search engine):
Maximize revenue
Mathematical Model
• Buyer i:
– Has a daily budget B(i)
• Online Setting:
– Items (keywords) arrive one-by-one.
– Each buyer gives a bid on each of the items
(can be zero)
• Algorithm:
– Assigns each item to some interested buyer.
Assumption:
Bids are small compared to the daily budget.
Results
[MSVV FOCS 05]:
• (1-1/e)-competitive online algorithm.
• Bound is tight.
• Analysis uses tradeoff revealing family of LP’s - not
very intuitive.
Our Results [Buchbinder, Jain, N, 2007]:
• A different approach based on the primal-dual method.
• Techniques are also applicable to further extensions of
the problem.
Ad-Auctions – Linear Program
B(i) – Budget of buyer i
b(i,j) – bid of buyer i on item j
I - Set of buyers.
J - Set of items.
y (i , j )
= 1  j-th ad-auction is sold to buyer i.
m ax 
i I
s.t:
For each item j:
j J
 y (i , j )  1
i I
For each buyer i:
 b (i, j ) y (i, j )
Each item is
Buyers
do not
sold once.
exceed their
budget
 b (i , j ) y (i , j )  B (i )
j J
Ad-auctions Primal and Dual
P: Primal Covering
m in  B ( i ) x ( i ) 
i I
 z( j)
j J
For each item j and buyer i: b ( i , j ) x ( i )  z ( j )  b ( i , j )
D: Dual Packing
m ax 
i I
For each item j:
 b (i, j ) y (i, j )
j J
 y (i , j )  1
i I
For each buyer i:
 b (i , j ) y (i , j )  B (i )
j J
The Primal-Dual Algorithm
• Initially: for each buyer i: x(i) 0
• When new item j arrives:
• Assign the item to the buyer i that maximizes:
b ( i , j ) 1  x ( i ) 
•
if x(i)≥1 do nothing, otherwise:
 y (i , j )  1
 z ( j )  b ( i , j ) 1  x ( i ) 

b (i , j ) 
b (i , j )
x
(
i
)

x
(
i
)
1





B
(
i
)

 B ( i )  c  1
- ‘c’ later.
The Paging/Caching Problem
Universe of of n pages
Cache of size k ¿ n
Request sequence of pages: 1, 6, 4, 1, 4, 7, 6, 1, 3, …
If requested page is in cache: no penalty.
Else, cache miss! load requested page into cache, evicting
some other page.
Goal: minimize number of cache misses.
Question: which page to evict in case of a cache miss?
Previous Results: Paging
Paging (Deterministic) [Sleator Tarjan 85]:
• Any det. algorithm ≥ k-competitive.
• LRU is k-competitive (also other algorithms)
Paging (Randomized):
• Rand. Marking O(log k) [Fiat, Karp, Luby, McGeoch, Sleator,
Young 91].
• Lower bound Hk [Fiat et al. 91], tight results known.
The Weighted Paging Problem
One small change:
• Each page i has a different load cost w(i).
• Models scenarios in which the cost of bringing pages is
not uniform:
Main memory, disk, internet …
web
Goal
• Minimize the total cost of cache misses.
Weighted Paging (Previous Work)
Randomized
Deterministic
Paging
Lower bound k
LRU
k competitive
O(log k) Randomized
Marking [Fiat et al. 91]
Weighted Paging
k-competitive
[Chrobak, Karloff, Payne,
Vishwanathan 91]
[Young 94] using duality!
O(log k) for two weight classes
[Irani 02]
No o(k) algorithm known even
for 3 weight classes.
The k-server Problem
• k servers (fire trucks) lie in an n-point metric space.
• Requests arrive at points.
• Serving a request: move a server to request point.
Goal: Minimize total distance traveled by servers.
Example:
The k-server Problem
• Paging = k-server on a uniform metric
– every page is a point
– A page is in the cache iff a server is at the point
• Weighted paging = k-server on a weighted star metric
Deterministic Results:
• General metric spaces: (2k-1)-competitive work function
algorithm [Koutsoupias-Papadimitriou 95]
• Tree metric: k-competitive algorithm [Chrobak et al. 91]
Randomized Results:
• No o(k) algorithm known (even for very simple spaces).
• Best lower bound: (log k)
Fractional Weighted Paging
Model:
•
Fractions of pages are kept in cache:
probability distribution over pages p1,…,pn
•
The total sum of fractions of pages in the cache is
at most k.
•
If pi changes by , cost =  w(i)
k units of cache
Randomized Algorithm
[Bansal, Buchbinder, N., FOCS 07] - O(log k)-competitive
for weighted paging
High level idea:
1. Design a primal-dual O(log k)-competitive algorithm
for fractional weighted paging.
2. Obtain a randomized algorithm while losing only a
constant factor.
Setting up the Linear Program
time line
Page i
Page j Page p
Page i
Page j
t
B(t): Set of pages requested until
time t (including pt)
We can only keep k pages out of the B(t) pages
Evict ≥ [|B(t)| - 1 - (k- 1)] = [|B(t)|-k] pages from B(t)\{pt}
Weighted paging – Linear Program
n
m in 
i 1
t

i B (t)\{p t }
r ( i ,t )

w (i ) x (i , j )
j 1
x ( i , r ( i , t ))  B ( t )  k
0  x (i, j )  1
• Idea: charge for evicting pages instead of fetching pages
x(i,j) – indicator for the event that page i is evicted from the
cache between the j-th and (j+1)-st times it is requested
r(i,t) - number of times page i is requested till time t,
including t
Primal and Dual Programs
P: Primal Covering
n
m in 
i 1
t

r ( i ,t )

w (i ) x (i , j )
j 1
x ( i , r ( i , t ))  B ( t )  k
i B (t)\{p t }
0  x (i, j )  1
D: Dual Packing
m ax
  B (t )
 k  y (t ) 
t
n
r ( i ,t )

i 1
z (i , j )
j 1
For each page i and the j th time it is asked:
 t ( i , j  1)  1

  y (t )   z (i , j )  w (i )
 t  t ( i , j ) 1

Sketch of Primal-Dual algorithm
New primal constraint:

i B (t)\{p t }
Dual variable y(t) & constraint
x ( i , r ( i , t ))  B ( t )  k
 t ( i , j  1)  1

  y (t )   z (i , j )  w (i )
 t  t ( i , j ) 1

While primal constraint infeasible: Increase y(t)
When dual constraint tight, x(i,j) = 1/k
From then on, increase exponentially (until x(i,j)=1)
The growth function of x(i,j)
x (i , j )
1
1/k
0
Dual is tight
Page fully
in memory
(marked)
Dual violated
by O(log k)
Page is
“unmarked”
Corresponding
Dual constraint
Page fully
evicted
Analysis of Online Algorithm
Proof of competitive factor:
1. Primal solution is feasible.
2. Primal ≤ 2 Dual.
3. Dual is feasible up to O(log k) factor
Conclusion (weak duality):
algorithm is O(log k)-competitive
Generalized Caching
Pages have both sizes and fetching costs.
• Motivation: Web-Caching
Special models:
• Bit model: Fetching cost proportional to size (minimize
traffic)
• Fault model: Fetching cost is uniform (minimize number
of times a user has to wait for a page)
Results (Bansal, Buchbinder, N., 2008):
• O(log k)-competitive algorithms for Bit and Fault models.
• O(log2k)-competitive algorithm for the general model.
• Requires new ideas and interesting analysis in the
rounding phase.
Further Research (1)
• More applications.
• Extending the general framework
beyond packing/covering.
Further Research (2)
•
Are these techniques strong enough to resolve the
randomized k-server conjecture?
Answer: maybe … but we don’t know how.
•
certainly looks like the right direction.
Main Difficulty:
•
k-server = Online Min Cost Flow.
•
Might need to be able to solve more general LPs …
Revisiting the Framework
•
•
Positive objective function.
New covering constraints arrive one-by-one.
•
Feasible region is not known in advance (gradually
exposed as more covering constraints appear).
When out of feasible region - should move back inside
and pay movement cost.
•
Cost of “infinity” for staying outside the feasible region.
Learning from expert advice
Problem:
•
N experts (TV channels) trying to predict “rain/sun”
every day.
•
Decide whether to take umbrella or sunglasses?
•
Goal: Predict (almost) as good as the best expert.
Learning from expert advice
Feasible region: distribution yt on n points (simplex)
Each round: Cost vectors ct define cost in every point
(right/wrong)
Algorithm:
can change the distribution yt-1  yt
Payments:
•
Service cost: ct yt-1 (cost of staying in a point)
•
Movement cost: None.
[BBN09]: Primal-dual algorithm: same guarantees as
weighted majority.
Metrical Task System (uniform)
Feasible region: distribution yt on n points (simplex)
Each round: cost vectors ct defines cost in every point.
Algorithm:
can change the distribution yt-1  yt
Payments:
•
service cost: ct yt (cost of staying in a point)
•
movement cost: Earthmover distance metric between
distributions.
Metrical task system (MTS)
[BBN09]: A new algorithm for MTS that for any 0<ɛ<1:
1.
Pays service cost at most (1+ɛ)OPT
2.
Pays movement cost at most log(n/ɛ)OPT
•
Known using learning algorithms [BBBT97].
•
Leads to best known algorithm for general metrics [FM00].
•
Very intuitive primal-dual algorithm.
•
Uses a non-covering LP.
Commercial
More about this approach:
The Design of Competitive Online Algorithms
via a Primal-Dual Approach
By Niv Buchbinder and Seffi Naor
Foundations and Trends® in Theoretical Computer Science
Thank you