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

Online computation with untrusted advice Joint work with Spyros Angelopoulos, 
 Jin S., S. Kamali, M. Renault. (ITCS’2020) Christoph Dürr - August 2020

Cost OPT ALG (k=B) Ski rental problem 7 Rent or buy 6 • Renting skis cost 1 / day 5 • Buying skis costs B (once) B= 4 • Unkown number of skiing days D 3 • Optimum = min{D,B} 2 • Algorithm decides to buy on day k 1 • Deterministic competitive ratio 
 = 1+(B-1)/B ≃ 2 0 1 2 3 4 5 6 • Randomized ratio ≃ e/(e-1) Worst D case

Advice Di ff erent approaches • In theory advice can be any information May computed from the instance revealing May 1st May some of the hidden information and is 1st May 1st May tailored for the algorithm which exploits it. 1st May 1st May 1st May • In practice advice is typically a machine 12nd 1st learning based prediction and might be wrong. Advice: you will stop skiing before day B • 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) 
 Should the algorithm ignore the advice or blindly trust it? or wrong (in the adversarial sense) Our approach

Untrusted advice Untrusted ratio Trust blindly sup D / B = ∞ • Every algorithm has • A trusted ratio in case advice was right 1 + B − 1 Ignore advice B • An untrusted ratio in case advice was wrong • Identify the Pareto-optimal algorithms 1 + B − 1 1 Trusted ratio B

Optimal algorithm Untrusted ratio Parameter k 1 + B k • Advice indicates if D<B (opt rents) 1 + B − 1 k • 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 or on day k ⌈ B / k ⌉ Trusted ratio 1 + k − 1 1 + k B B

Online bidding Introduced to illustrate doubling strategies x j • Hidden value u ( ) u ≥ 1 U • Strategy = sequence x 1 , x 2 , … How much fuel x 3 is needed to escape • Cost := for j such x 1 + x 2 + … + x j gravitation from x 2 that earth ? x j − 1 < u ≤ x j x 1 x 1 + … + x j • Competitive ratio = u • Doubling strategy x i = 2 i has optimal competitive ratio 4

Online bidding Untrusted ratio Parameter w, Advice is u W • 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 4 the LP in case advice was right • Constraints hold with equality, so we can solve linear recurrence to get the optimal strategy. Then we optimise m. 1 2 Trusted ratio • The obtained strategy is Pareto optimal. w 2 − 4 w w − 2

Example for k=1 U Online bidding Advice is 0 Only advice bits k • Upper bound (we have also a result for general ) w ≥ 4 Advice is 1 • Choose the best among 2 k 
 4-competitive strategies 2 1+2 − k • Is 3 -competitive if advice was right 2.5 • Lower bound 2 • Every 4-competitive algorithm has ratio 1.5 1 at least in case advice was 2 + 1 3 ⋅ 2 k right 1 2 3 4 5 Number of advice bits k

Bin packing Asymptotic competitive ratio What was known before First-Fit: consider bins in the order they have been opened. Place 1.7 new item in first bin where it fits, opening a new bin if necessary • Reserve-Critical • Classify by size : 1.5783 • Tiny (0,1/3] Deterministic competitive ratio is between these bounds 1.54278 • Small (1/3,1/2] Boyar, Kamali, Larsen, López-Ortiz, 2016: 1.5 Reserve-Critical needs O(log n) advice bits • Critical (1/2,2/3] Angelopoulos, D. Kamali, Renault, Rosén, 2018: 
 1.4702 • Large (2/3,1] needs O(1) advice bits • Advice c = number of critical items • Reserve space for critical items in dedicated bins. 
 Other than that, follow First-Fit.

Bin packing Reserve-Critical can be fooled • Suppose advice is c>0 • But all items have size 1/6+ ε • Competitive ratio is 6 Reserved Reserved … 1/6+ ε 1/6+ ε

Bin packing Untrusted ratio Parameter , Advice 6 α γ • There is an algorithm with 4.5 trusted ratio r and untrusted ratio max{33-18r, 7/4} 3 • Maintains a proportion close to between critical and min{ α , γ } tiny bins 1.5 • We also analysed the algorithm when advice is given with 0 1.4 1.5 1.6 1.7 1.8 1/2 k precision Trusted ratio

List update Request sequence : 2 1 2 7 8 4 11 5 …. Boyar, Kamali, Larsen, López-Ortiz, 2016 Serving cost (= rank in list) • Timestamp : move requested item x in front of first item requested at most once since the last request of x 3 7 2 11 4 5 1 8 • Move-to-front-even : if total number of requests to x is even, then move to front Free exchange Paid exchange 
 • Move-to-front-odd : (for the currently requested item) (any consecutive items) • Move-to-front : always move requested item to front • Previous work: On any request sequence one of the first 3 algorithms has ratio 5/3 
 ≤ (2 bits advice su ffi ces)

List update Ignoring phase Parameter , β ∈ [0,1/2] Trusting phase Advice A {Timestamp, MTF-Even, MTF-Odd} ∈ A MTF A MTF A … • If A=Timestamp, then run A • Else, alternate between A and Move-to- front. Untrusted ratio 2.5 • When starting trusted phase, make paid exchanges to have the list as if A was run ≤ m 2 from the beginning. (cost for m=size 2.4 of list). • End trusted phase when cost 2.3 m 3 exceeds • End ignoring phase when cost exceeds β m 3 2.2 1.6 1.7 1.8 1.9 2 Trusted ratio

Future work Kraska, Beutel, Chi, Dean, Polyzotis, 2018 17? • We provide also some Machine learning predicts insertion point randomised algorithms • But are they Pareto-optimal? 
 (Lower bounds for the 2 4 5 9 13 14 17 21 randomised trusted / untrusted competitive ratios) d • Machine learning advice is also Number of queries is where d is the error of the prediction O (log d ) studied for the o ffl ine setting