Online Metric Algorithms with Untrusted Predictions Antonios - PowerPoint PPT Presentation

Online Metric Algorithms with Untrusted Predictions Antonios Antoniadis 1 Christian Coester 2 Marek Elias 3 Adam Polak 4 Bertrand Simon 5 1: MPI, Saarbrucken (Germany). 2: CWI, Amsterdam (Netherlands). 3: EPFL, Lausanne (Switzerland). 4: Jagiellonian University, Kraków (Poland). 5: University of Bremen (Germany). A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 1 / 12

Motivation Online algorithms ◮ Optimization with incomplete information ◮ Guaranteed competitive ratio: r s.t. ∀ I : Alg ( I ) ≤ r · Opt ( I ) ◮ Bad performance on easy instances, overly pessimistic A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 2 / 12

Motivation Online algorithms ◮ Optimization with incomplete information ◮ Guaranteed competitive ratio: r s.t. ∀ I : Alg ( I ) ≤ r · Opt ( I ) ◮ Bad performance on easy instances, overly pessimistic A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 2 / 12

Motivation Online algorithms Machine learning predictions ◮ Optimization with incomplete ◮ Often very powerful information ◮ Guaranteed competitive ratio: r s.t. ∀ I : Alg ( I ) ≤ r · Opt ( I ) ◮ Bad performance on easy instances, overly pessimistic ◮ No guarantee, can be arbitrarily bad A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 2 / 12

Motivation Online algorithms Machine learning predictions ◮ Optimization with incomplete ◮ Often very powerful information ◮ Guaranteed competitive ratio: r s.t. ∀ I : Alg ( I ) ≤ r · Opt ( I ) ◮ Bad performance on easy instances, overly pessimistic ◮ No guarantee, can be arbitrarily bad ◮ prediction error η A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 2 / 12

Motivation Online algorithms Machine learning predictions ◮ Optimization with incomplete ◮ Often very powerful information ◮ Guaranteed competitive ratio: r s.t. ∀ I : Alg ( I ) ≤ r · Opt ( I ) ◮ Bad performance on easy instances, overly pessimistic ◮ No guarantee, can be arbitrarily bad ◮ prediction error η Prediction-augmented algorithms ◮ Target competitive ratio: O ( min { ONLINE , f ( η/ Opt ) } ) A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 2 / 12

Some previously studied problems ◮ Ski rental: predict #days we will ski [PurohitSK’18] ◮ Non-clairvoyant scheduling: predict processing times [PurohitSK’18] ◮ Restricted assignment: predict machine weights [LattanziLMV’20] ◮ Caching: predict next arrival time [LykourisV’18,Rohatgi’20,Wei’20] ◮ Weighted caching: predict all requests until next arrival [JiangPS’20] Issue: lack of generality, predictions tailored to specific problems Proposition (Antoniadis, Coester, Elias, Polak, Simon; ICML’20) Previously used caching predictions not useful for weighted caching. = ⇒ need more general prediction setup A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 3 / 12

Metrical Task Systems (MTS) ◮ Given: metric space ( M , d ), x 0 ∈ M 4 3 ◮ At time t : 2 2 1. cost function ℓ t : M �→ R + revealed 2. algo chooses x t ∈ M 6 2 3. pays d ( x t − 1 , x t ) + ℓ t ( x t ) Cost: 0 MTS generalizes caching, k -server, convex body/function chasing, layered graph traversal. . . A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 4 / 12

Metrical Task Systems (MTS) ◮ Given: metric space ( M , d ), x 0 ∈ M 1 1 4 3 ◮ At time t : 2 10 2 1. cost function ℓ t : M �→ R + revealed 2. algo chooses x t ∈ M 6 2 64 45 3. pays d ( x t − 1 , x t ) + ℓ t ( x t ) Cost: 0 MTS generalizes caching, k -server, convex body/function chasing, layered graph traversal. . . A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 4 / 12

Metrical Task Systems (MTS) ◮ Given: metric space ( M , d ), x 0 ∈ M 1 1 4 3 ◮ At time t : 2 10 2 1. cost function ℓ t : M �→ R + revealed 2. algo chooses x t ∈ M 6 2 64 45 3. pays d ( x t − 1 , x t ) + ℓ t ( x t ) Cost: 3+1 MTS generalizes caching, k -server, convex body/function chasing, layered graph traversal. . . A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 4 / 12

Metrical Task Systems (MTS) ◮ Given: metric space ( M , d ), x 0 ∈ M 2 18 4 3 ◮ At time t : 2 13 2 1. cost function ℓ t : M �→ R + revealed 2. algo chooses x t ∈ M 6 2 17 15 3. pays d ( x t − 1 , x t ) + ℓ t ( x t ) Cost: 3+1 MTS generalizes caching, k -server, convex body/function chasing, layered graph traversal. . . A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 4 / 12

Metrical Task Systems (MTS) ◮ Given: metric space ( M , d ), x 0 ∈ M 2 18 4 3 ◮ At time t : 2 13 2 1. cost function ℓ t : M �→ R + revealed 2. algo chooses x t ∈ M 6 2 17 15 3. pays d ( x t − 1 , x t ) + ℓ t ( x t ) Cost: 3+1+7+2 MTS generalizes caching, k -server, convex body/function chasing, layered graph traversal. . . A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 4 / 12

Metrical Task Systems (MTS) ◮ Given: metric space ( M , d ), x 0 ∈ M 2 18 4 3 ◮ At time t : 2 13 2 1. cost function ℓ t : M �→ R + revealed 2. algo chooses x t ∈ M 6 2 17 15 3. pays d ( x t − 1 , x t ) + ℓ t ( x t ) Cost: 3+1+7+2 MTS generalizes caching, k -server, convex body/function chasing, layered graph traversal. . . What to predict? ◮ Next costs? only ℓ t +1 ? useless with dummy rounds entire future? too much information ◮ Single state per round : recommendation for x t A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 4 / 12

Definition of the error and a simple algorithm Definition: error η ◮ Fix Off : offline algorithm (e.g., Opt ), who goes to states o 1 , o 2 , . . . ◮ At time t , p t := prediction of o t . ◮ Error: η := � t d ( o t , p t ) Goal: Small cost (rel. to Off ) if η small o t p t A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 5 / 12

Definition of the error and a simple algorithm Definition: error η ◮ Fix Off : offline algorithm (e.g., Opt ), who goes to states o 1 , o 2 , . . . ◮ At time t , p t := prediction of o t . ◮ Error: η := � t d ( o t , p t ) Goal: Small cost (rel. to Off ) if η small ǫ ◮ Naive algo: x t := p t 0 99 o t p t A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 5 / 12

Definition of the error and a simple algorithm Definition: error η ◮ Fix Off : offline algorithm (e.g., Opt ), who goes to states o 1 , o 2 , . . . ◮ At time t , p t := prediction of o t . ◮ Error: η := � t d ( o t , p t ) Goal: Small cost (rel. to Off ) if η small ǫ ◮ Naive algo: x t := p t 0 99 ◮ Better: x t := arg min x ∈ X { cost t ( x ) + 2 d ( p t , x ) } o t p t ◮ Call this algo FtP (Follow the Prediction) A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 5 / 12

Definition of the error and a simple algorithm Definition: error η ◮ Fix Off : offline algorithm (e.g., Opt ), who goes to states o 1 , o 2 , . . . ◮ At time t , p t := prediction of o t . ◮ Error: η := � t d ( o t , p t ) Goal: Small cost (rel. to Off ) if η small ǫ ◮ Naive algo: x t := p t 0 99 ◮ Better: x t := arg min x ∈ X { cost t ( x ) + 2 d ( p t , x ) } o t p t ◮ Call this algo FtP (Follow the Prediction) Lemma (Antoniadis, Coester, Elias, Polak, Simon; ICML’20) FtP has cost ≤ Off + 4 η . So competitive ratio vs Off is 1 + 4 η/ Off . Guarantee holds simultaneously for all offline algos ( η depending on it) Issue: FtP not robust A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 5 / 12

Robustifying FtP Combine online algorithms A and B: comb ( A , B ) [BlumB’00] ◮ E ( cost comb ( A , B ) ) ≤ (1+ ε ) · min { E ( cost A ) , E ( cost B ) } + O (diameter /ε ) Let RobustFtP := comb ( ONLINE , FtP ) A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 6 / 12

Robustifying FtP Combine online algorithms A and B: comb ( A , B ) [BlumB’00] ◮ E ( cost comb ( A , B ) ) ≤ (1+ ε ) · min { E ( cost A ) , E ( cost B ) } + O (diameter /ε ) Let RobustFtP := comb ( ONLINE , FtP ) Theorem (Antoniadis, Coester, Elias, Polak, Simon; ICML’20) RobustFtP has cost O (min { Off + η, ONLINE } ) . (Recall: η = prediction error wrt. Off ) This is asymptotically optimal. ◮ Lower bound holds for some MTS ◮ For caching (a special case of MTS), we can do better A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 6 / 12

Caching (aka Paging) ◮ Maintain a cache of k pages, pay 1 per cache miss D k = 4 C E B misses: 1 A A. Antoniadis, C. Coester , M. Elias, A. Polak, B. Simon Online Metric Algorithms with Untrusted Predictions 7 / 12