Online computation with untrusted advice Joint work with Spyros - - PowerPoint PPT Presentation

Christoph Dürr - August 2020

Online computation with untrusted advice

Joint work with Spyros Angelopoulos, 
 Jin S., S. Kamali, M. Renault. (ITCS’2020)

Rent or buy

• Renting skis cost 1 / day
• Buying skis costs B (once)
• Unkown number of skiing days D
• Optimum = min{D,B}
• Algorithm decides to buy on day k
• Deterministic competitive ratio 


= 1+(B-1)/B ≃ 2

• Randomized ratio ≃ e/(e-1)

Ski rental problem

Worst case

Cost 1 2 3 4 5 6 7 D 1 2 3 4 5 6 OPT ALG (k=B)

B=

Our approach

Different approaches

• In theory advice can be any information

computed from the instance revealing some of the hidden information and is tailored for the algorithm which exploits it.

• In practice advice is typically a machine

learning based prediction and might be wrong.

• Lykouris, Vassilvitskii, 2018: online

caching with competitive ratio being function of absolute error of advice

• Kumar, Puhorit, Svitkina, 2018: advice is

either right (cooperating with algorithm) 


• r wrong (in the adversarial sense)

Advice

May 1st May 1st May 1st May 1st May 1st May 1st May 12nd May 1st

Advice: you will stop skiing before day B Should the algorithm ignore the advice or blindly trust it?

• Every algorithm has
• A trusted ratio in case advice

was right

• An untrusted ratio in case

advice was wrong

• Identify the Pareto-optimal

algorithms

Untrusted advice

1

1 + B − 1 B 1 + B − 1 B

sup D/B = ∞ Ignore advice Trust blindly Trusted ratio Untrusted ratio

Parameter k

• Advice indicates if D<B (opt rents)
• In that case our algorithm rents

until day B-1 and buys on day B

• Otherwise it buys on day k
• This is Pareto optimal.
• Puhorit-Svitkina-Kumar’s

algorithm buys either on day

• r on day k

⌈B/k⌉

Optimal algorithm

1 + B − 1 k 1 + k − 1 B 1 + k B 1 + B k

Trusted ratio Untrusted ratio

Introduced to illustrate doubling strategies

• Hidden value u (

)

• Strategy = sequence
• Cost :=

for j such that

• Competitive ratio =
• Doubling strategy

has

• ptimal competitive ratio 4

u ≥ 1 x1, x2, … x1 + x2 + … + xj xj−1 < u ≤ xj x1 + … + xj u xi = 2i

Online bidding

How much fuel is needed to escape gravitation from earth ?

U x1 x2 x3 xj

Parameter w, Advice is u

Online bidding

W 1 4 2

w − w2 − 4w 2

• We describe strategy by a minimisation

linear program such that:

• Advice u is exceeded by the m-th bid
• Competitive ratio is w in any case
• And competitive ratio is objective of

the LP in case advice was right

• Constraints hold with equality, so we

can solve linear recurrence to get the

• ptimal strategy. Then we optimise m.
• The obtained strategy is Pareto optimal.

Trusted ratio Untrusted ratio

Only advice bits

k

• Upper bound (we have also a result for

general )

• Choose the best among


 4-competitive strategies

• Is
• competitive if advice was

right

• Lower bound
• Every 4-competitive algorithm has ratio

at least in case advice was right

w ≥ 4 2k 21+2−k 2 + 1 3 ⋅ 2k

Online bidding

U Example for k=1

1 1.5 2 2.5 3 Number of advice bits k 1 2 3 4 5

Advice is 0 Advice is 1

What was known before

• Reserve-Critical
• Classify by size :
• Tiny (0,1/3]
• Small (1/3,1/2]
• Critical (1/2,2/3]
• Large (2/3,1]
• Advice c = number of critical items
• Reserve space for critical items in

dedicated bins. 
 Other than that, follow First-Fit.

Bin packing

1.5783 1.54278 1.7 1.4702 Asymptotic competitive ratio First-Fit: consider bins in the order they have been opened. Place new item in first bin where it fits, opening a new bin if necessary Deterministic competitive ratio is between these bounds Angelopoulos, D. Kamali, Renault, Rosén, 2018:
 needs O(1) advice bits 1.5 Boyar, Kamali, Larsen, López-Ortiz, 2016: Reserve-Critical needs O(log n) advice bits

Reserve-Critical can be fooled

• Suppose advice is c>0
• But all items have size 1/6+ε
• Competitive ratio is 6

Bin packing

1/6+ε Reserved 1/6+ε Reserved

…

Parameter , Advice

α γ

• There is an algorithm with

trusted ratio r and untrusted ratio max{33-18r, 7/4}

• Maintains a proportion close to

between critical and tiny bins

• We also analysed the algorithm

when advice is given with precision

min{α, γ} 1/2k

Bin packing

1.5 3 4.5 6 1.4 1.5 1.6 1.7 1.8

Trusted ratio Untrusted ratio

Boyar, Kamali, Larsen, López-Ortiz, 2016

• Timestamp: move requested item x in

front of first item requested at most

• nce since the last request of x
• Move-to-front-even: if total number of

requests to x is even, then move to front

• Move-to-front-odd:
• Move-to-front: always move requested

item to front

• Previous work: On any request sequence
• ne of the first 3 algorithms has ratio

5/3 
 (2 bits advice suffices)

≤

List update

3 7 2 11 4 5 1 8

Request sequence : 2 1 2 7 8 4 11 5 ….

Serving cost (= rank in list) Free exchange (for the currently requested item) Paid exchange
 (any consecutive items)

Parameter , Advice A {Timestamp, MTF-Even, MTF-Odd}

β ∈ [0,1/2] ∈

• If A=Timestamp, then run A
• Else, alternate between A and Move-to-

front.

• When starting trusted phase, make paid

exchanges to have the list as if A was run from the beginning. (cost for m=size

• f list).
• End trusted phase when cost

exceeds

• End ignoring phase when cost exceeds

≤ m2 m3 βm3

List update

A MTF A MTF A …

Trusting phase Ignoring phase

2.2 2.3 2.4 2.5 1.6 1.7 1.8 1.9 2

Trusted ratio Untrusted ratio

• We provide also some

randomised algorithms

• But are they Pareto-optimal?


(Lower bounds for the randomised trusted / untrusted competitive ratios)

• Machine learning advice is also

studied for the offline setting

Future work

2 4 5 9 13 14 17 21

17?

Machine learning predicts insertion point Number of queries is where d is the error of the prediction

O(log d)

d

Kraska, Beutel, Chi, Dean, Polyzotis, 2018