Online Algorithms with Multiple Advice - PowerPoint Presentation

1. Online Algorithms with Multiple AdviceDebmalya Panigrahi

2. Online Algorithms with Multiple AdviceSreenivas Gollapudi, GoogleAmit Kumar, IIT DelhiRong Ge, Duke UniversityKeerti Anand, Duke UniversityMY PARTNERS-IN-CRIME

3. Online Algorithms with AdviceOnline Algorithm is provided with adviceAlgorithm has no information about quality of adviceFor instance, in set cover, in each online step, we get an element to be covered and the advice is a set that covers this elementConsistency: Obtain a learning-augmented online algorithm whose performance matches that of the adviceTrivial: simply follow the adviceConsistency + Robustness: Obtain a learning-augmented online algorithm whose performance matches that of the advice and is no worse than an online algorithm without adviceOnline Algorithms with Multiple Advice

4. Online Algorithms with Multiple AdviceOnline Algorithm is provided with multiple adviceAlgorithm has no information about quality of adviceThe advice may be completely different from one anotherMultiple ML models, human experts, etc.Goal: Obtain a learning-augmented online algorithm whose performance matches that of the best adviceOnline Algorithms with Multiple Advice

5. The Case of Two ExpertsSuppose one expert is a learning-augmented solution ()… and the other expert is given by a worst-case online algorithm ()Obtaining an algorithm that is competitive with both implies consistency and robustnessA simple algorithm [Azar-Broder-Manasse ’93]: follow one expert until it is twice as expensive as the other, then switch and repeatKey observation: between any two consecutive switches, better of the two experts doubles in cost; hence, total cost of algorithm is minimization problem, monotone cost, metric switching costThis simultaneously gives -consistency, -robustness where , are the best consistency and robustness bounds that are separately achievableWhat if constants are important? Online Algorithms with Multiple Advice

6. Generalization to Experts  experts The Azar-Broder-Manasse algorithm generalizes to experts Round runs from to = best expert at time In round discard any expert with cost follow current expert until it is discarded or round ends; if round hasn’t ended, switch to a random expert that hasn’t been discarded in this roundPrevious analysis across rounds + “randomized marking” for single round: total cost of algorithm is Matching lower bound of from online set cover lower bound Online Algorithms with Multiple Advice

7. Static and Dynamic ExpertsSo far, algorithm competes with best static/fixed expert, i.e., opt chooses a single expert at the beginning and sticks to it throughoutNow, we consider best dynamic/moving expert, i.e., opt can choose any expert in each stepIn online set cover, each step is an element and (possibly non-distinct) sets covering this element suggested by the expertsStatic opt is the best solution that chooses all the sets suggested by one of the expertsDynamic opt is the best solution that chooses at least one of these sets in each step Online Algorithms with Multiple Advice

8. The Online Set Cover ProblemMinimize subject to for all where for all variables, constraintsFor constraint , expert suggests = 1 where  Online Algorithms with Multiple AdviceOFFLINE VARIABLESONLINE CONSTRAINTS  Step 1 ()Step 2 ()Step 3 ()Expert 1 (Expert 2 (Expert 3 (Expert 4 (Dynamic opt is at least as large as any one of the advice in each iterationDynamic optStatic opt 

9. The Online Covering ProblemMinimize subject to for all where for all variables, constraintsFor constraint , expert suggests = 1 where such that  Online Algorithms with Multiple AdviceOFFLINE VARIABLESONLINE CONSTRAINTS   weights?Step 1 ()Step 2 ()Step 3 ()Expert 1 (Expert 2 (Expert 3 (, Expert 4 (Dynamic opt is at least as large as any one of the advice in each iterationDynamic optStatic opt 

10. Static and Dynamic Experts – experts, – constraints, – variables Static opt = Dynamic opt = min{}Clearly, Dynamic opt < Static optDynamic opt is too strong for regret in online learning, but what about competitive ratio for combinatorial optimization?Competing against dynamic optOnly for (online) covering problemsOnly fractional solution (can combine with any online rounding algorithm)Multiplicative (rather than additive) bounds Online Algorithms with Multiple Advice

11. Online Set Cover with AdviceMinimize subject to for all where for all variables, constraintsFor constraint , expert suggests = 1 where Instance transformation: set for all not suggested by any expert for constraint Dynamic opt is unchangedFrequency () at most (degree of online vertex )Online set cover algorithm [Alon, Awerbuch, Azar, Buchbinder, Naor] on transformed instance gives competitive ratio of  Online Algorithms with Multiple AdviceOFFLINE VARIABLESONLINE CONSTRAINTS  

12. Online Set Cover with AdviceMinimize subject to for all where for all variables, constraintsFor constraint , expert suggests = 1 where  Online Algorithms with Multiple AdviceOFFLINE VARIABLESONLINE CONSTRAINTS           01Algorithm:For every suggested variable , until constraint satisfied Analysis: Each iteration has cost and doubles some variable in optAt most opt iterations 

13. Online Covering with AdviceMinimize subject to for all where for all variables, constraintsFor constraint , expert suggests = 1 where such that  Online Algorithms with Multiple AdviceOFFLINE VARIABLESONLINE CONSTRAINTS           01Algorithm:For every suggested variable , until constraint satisfied Analysis: Each iteration has cost and doubles some variable in optAt most opt iterations Need to scale to  Advice/Opt can be spread over many variablesHow many times does the variable double?

14. Online Covering with AdviceOnline Algorithms with Multiple AdviceExpert 1 ()Expert 2 ()Expert 3 ()         For every variable , update where (after making all suggestions tight) until  

15. Online Covering with AdviceFix a variable Suppose is the value of in optRoughly speaking, the algorithm ensures that remains in the range doubles in every iteration, i.e., only updates increases over time – many sets of updates?How do we ensure that ?How do we handle ? Online Algorithms with Multiple Advice 01   steps  steps Algorithm: For every variable , update where (after making all suggestions tight) until  

16. Online Covering with Advice increases over time: Algorithm only charges to at each stage: (i.e., only )This is sufficient because feasibility of opt ensures that   01Online Algorithms with Multiple Advice   steps  steps Algorithm: For every variable , update where (after making all suggestions tight) until  

17. Online Covering with Advice can exceed Need to stop charging to Stop updates once (and double all variables implicitly)Since but , we have   01Online Algorithms with Multiple Advice   steps  steps Algorithm: For every variable , update where (after making all suggestions tight) until   

18. Online Covering with AdviceUse exponential updates but with coefficient where   01Online Algorithms with Multiple Advice   steps  steps Algorithm: For every variable , update where (after making all suggestions tight) until   

19. AnalysisStandard analysis for online fractional set cover is via primal dual methodExtended to the advice setting by Bamas, Maggiori, and Svensson ’20Considers single integer advice available offlineNot clear how to handle multiple fractional advice in the primal dual frameworkInstead, we will use a potential function [inspired by Bansal, Buchbinder, and Naor]Online Algorithms with Multiple Advice Algorithm: For every variable , update ) where (after making all suggestions tight) until  

20. Potential FunctionFor , we have Goal: for some constant , and (since we only consider terms ) (for for all )Main Lemma: for some constant  Online Algorithms with Multiple Advice 

21. Applications and ExtensionsCan be applied to any online covering problem and combined with any online rounding algorithm(Weighted) Set Cover: -competitive algorithm against dynamic opt ( = # advice, = # elements)(Weighted) Paging: -competitive algorithm against dynamic opt Note that suggestions provide entire cache stateDoes not contradict lower bounds for weighted caching with predictions (e.g., Sun, P., and Zhang, 2021) which use weaker prediction models like next request times for pagesCan be extended to handle precedence/box-type constraintsFacility location: -competitive fractional algorithmOnline rounding for facility location? Online Algorithms with Multiple Advice

22. Recall: Competing against the best expert experts The Azar-Broder-Manasse algorithm gives competitive ratio of againstMatching lower bound of from online set coverNow, we consider a problem where this lower bound does not hold (and therefore, constants matter) Online Algorithms with Multiple Advice

23. The Rent-or-Buy (a.k.a. Ski-rental) ProblemIt costs$1 to rent skis for a day$ to buy skis for the seasonLength of ski season is Offline optimumIf , buy on day 1If , rent every dayThe algorithm gets to know only when the ski season endsBest (deterministic) online algorithm:buy after dayscompetitive ratio of 2 Online Algorithms with Multiple Advice

24. The Rent-or-Buy (a.k.a. Ski-rental) ProblemAdvice: prediction of length of ski season predictions, algorithm has to compete with the “best prediction”Assume (for simplicity) that one prediction is correctWhat competitive ratio can the algorithm achieve?: CR of 1 achieved by using offline algorithm on expert’s prediction: experts can make all possible predictions; hence, best option is to run online algorithm and obtain CR of 2Our goal: design algorithm and find competitive ratio for  Online Algorithms with Multiple Advice

25. The Case of Two PredictionsSuppose , and predictions are and Case 1: If , then buy at time 0Case 2: If , then never buyCase 3: If and , thenBuying at time is 2-competitiveBuying at time 0 is -competitive, which can be arbitrarily largeBuying at time is better than 0: this is -competitive (can be arbitrarily close to 2)Combine the last two choices to get -competitiveBuy at time 0 if Buy at time otherwise Online Algorithms with Multiple AdviceCompetitive ratio of the algorithm:which is the golden ratio  

26. Generalizing to  For create a breakpoint at buy in the first interval that does not contain a predictionFor general create breakpoints (optimize for the breakpoints)buy in the first interval that does not contain a prediction  0  : buy at  0    …Online Algorithms with Multiple Advice: buy at 0 

27. Competitive RatioFor , CR is the golden ratio is the limit of the ratio of two consecutive Fibonacci numbersFor general , CR is Every k-acci number is the sum of the previous -acci numbersThis competitive ratio is tight for any value of  Online Algorithms with Multiple Advice

28. Open ProblemsHow general can our bounds be against dynamic opt (compared to static opt for which the bound is essentially problem-agnostic)?-server with multiple advice (even with full state given by advice)Can we formalize the advantage of multiple advice over single advice? Optimal number of advice? Online Algorithms with Multiple Advice

29. Thank YouOnline Algorithms with Multiple Advice