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Learning-Augmented Algorithms: The Case of Multiple Experts

Learning-Augmented Algorithms: The Case of Multiple Experts. Debmalya Panigrahi. (based on joint work with Sreenivas Gollapudi, ). How to optimize for an unknown future?. How to optimize for an unknown future?. Online Algorithms. Machine Learning.

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Learning-Augmented Algorithms: The Case of Multiple Experts

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  1. Learning-Augmented Algorithms: The Case of Multiple Experts Debmalya Panigrahi (based on joint work with Sreenivas Gollapudi, )

  2. How to optimize for an unknown future?

  3. How to optimize for an unknown future? Online Algorithms Machine Learning Use the past to predict the future, and optimize for the predicted future Approximation ratio = Offline Algorithm / Offline Optimum + Optimistic (approx. ratio << comp. ratio for most problems) - Not robust (no guarantees if predictions are inaccurate) • Optimize for the worst possible (adversarial) future • Competitive ratio = Online Algorithm / Offline Optimum + Very robust (guarantees hold no matter what) - Pessimistic (nature is not adversarial!)

  4. Online Algorithms with Predictions • A. M. Medina and S. Vassilvitskii. Revenue optimization with approximate bid predictions. NeurIPS 2017. • T. Kraska, A. Beutel, E. H. Chi, J. Dean, and N. Polyzotis. The case for learned index structures. SIGMOD 2018. • T. Lykouris and S. Vassilvitskii. Competitive caching with machine learned advice. ICML 2018. • M. Mitzenmacher. A model for learned bloom filters and optimizing by sandwiching. NeurIPS 2018. • M. Purohit, Z. Svitkina, and R. Kumar. Improving online algorithms via ML predictions. NeurIPS 2018. • C.-Y. Hsu, P. Indyk, D. Katabi, and A. Vakilian. Learning-based frequency estimation algorithms. ICLR 2019. Consistency: If the prediction is correct, then the algorithm should perform as well as the best offline solution Robustness: Irrespective of the accuracy of the prediction, the algorithm should perform as well as the best online solution Graceful degradation: The performance of the algorithm should gracefully degrade with the accuracy of the prediction

  5. Online Algorithms with Multiple Predictions • Multiple ML models/human experts make predictions about the future • The predictions may be completely different from one another • The algorithm has no information about the absolute or relative quality of the predictions Consistency: If any of the predictions is correct, then the algorithm should perform as well as the best offline solution Robustness: Irrespective of the accuracy of the predictions, the algorithm should perform as well as the best online solution Graceful degradation: The performance of the algorithm should gracefully degrade with the accuracy of the best prediction

  6. A Single Parameter Problem: Rent or Buy (a.k.a. Ski-rental) • It costs • $1 to rent skis for a day • $B to buy skis for the season • Length of ski season is x • Offline optimum • If x ≥ B, buy on day 1 • If x < B, rent every day • Unknown future: The algorithm gets to know x only when the ski season ends • Online algorithm (existing results) • Competitive ratio of 2 for deterministic algorithms • Competitive ratio of for randomized algorithms

  7. A Single Parameter Problem: Rent or Buy (a.k.a. Ski-rental) • Online algorithm with multiple predictions (achieving consistency only) • k predictions, consistency is the worst case competitive ratio of the algorithms assuming any one of the predictions is correct • k=1: consistency of 1 achieved by assuming the expert is accurate and using the offline algorithm [Purohit et al. ’18 show how to achieve robustness in this setting] • k=∞: experts can make all possible predictions, hence it reduces to the classical online setting (without predictions): best consistency achievable is 2 by a deterministic algorithm and e/(e-1) by a randomized algorithm • What consistency can we achieve for finite k > 1? • What is a good value of k? • For independent Gaussian error, we show that k between 2 and 4 is a good empirical choice

  8. Rent or Buy with Multiple Predictions: A Deterministic, Consistent Algorithm • Suppose k=2, i.e., there are only 2 predictions: x1 and x2 • Case 1: If both x1 and x2 are at least B, then buy at time 0 • Case 2: If both x1 and x2 are smaller than B, then buy at time B (or never buy!) • Case 3: If x1 < B and x2 ≥ B, then • Buying at time B is only 2-consistent • Buying at time 0 is B/x1-consistent, which can be arbitrarily large! • Buying at time x1 is better idea, but still only (1 + x1/B)-consistent, which can be arbitrarily close to 2

  9. Rent or Buy with Multiple Predictions: A Deterministic, Consistent Algorithm • Case 3: If x1 < B and x2 ≥ B, then • Buying at time B is only 2-consistent • Buying at time 0 is B/x1-consistent, which can be arbitrarily large! • Buying at time x1 is better idea, but still only (1 + x1/B)-consistent, which can be arbitrarily close to 2 • Combine the last two choices: • We obtain a consistency of min(1 + x1/B, B/x1) by: • Buying at time 0 if B/x1 < 1 + x1/B • Buying at time x1 otherwise

  10. Consistency ratio of the algorithm • Combine the last choices: • We obtain a consistency of min(1 + x1/B, B/x1) by: • Buying at time 0 if B/x1 < 1 + x1/B • Buying at time x1 otherwise • The algorithm achieves consistency of: argmax 0<x1< B min(1+x1/B, B/x1) which is 1/y where 1 + y = 1/y which solves to the golden ratio Φ

  11. Is this consistency ratio tight? A tight example: • x1 = y = B/Φ, x2 > B • If the algorithm buys before time y, then x1 realizes and the ratio is at least B/y = Φ • If the algorithm buys after time y, then x2 realizes and the ratio is at least (B+y)/B = 1 + 1/Φ • Overall, the ratio is no better than Φ (which is equal to 1 + 1/Φ)! x1 ≥ B/Φ: buy at 0 x1 < B/Φ: buy at B/Φ y = B/Φ 0 B

  12. How do we generalize to k > 2? • For k = 2 • create intervals (0, B/Φ) and (B/Φ) • buy in the first interval that does not contain a prediction • For general k • create k intervals (0, z1), (z1, z2), …, (zk-1, B) • buy in the first interval that does not contain a prediction x1 ≥ B/Φ: buy at 0 x1 < B/Φ: buy at B/Φ B/Φ 0 B … B 0 z1 z2 zk-1

  13. What is the consistency ratio for k > 2? • For k = 2, the consistency ratio is the golden ratio Φ, which is the limit of the ratio of two consecutive Fibonacci numbers • For general k, the consistency ratio is the limit of the ratio of two consecutive k-acci numbers, where in the k-acci sequence, every number is the sum of the previous k numbers in the sequence (For k=2, this is the Fibonacci sequence) • This consistency ratio is tight for any value of k!

  14. Rent or Buy with Multiple Predictions • Consistency: For k predictions, we give an ηk-consistent deterministic algorithm where: • η1 = 1 • limk∞ ηk= 2 • ηk is an increasing sequence • No deterministic algorithm can achieve consistency better than ηk for k predictions Deterministic Algorithms k>2 k=∞ k=2 k=1 (golden ratio) 1 2 • Consistency: For k predictions, we give a µk-consistent • randomized algorithm where: • µ1 = 1 • limk∞ µk = • µk is an increasing sequence • No randomized algorithm can achieve consistency better than µk for k predictions Randomized Algorithms k>2 k=1 k=∞ k=2 2 1

  15. Rent or Buy with Multiple Predictions • Graceful degradation: For k predictions, we give a deterministic algorithm with alg ≤ γk (opt + err) where: • γ1 = • limk∞ γk = 2 • γk is an increasing sequence, γk > ηkfor finite k • No deterministic algorithm can achieve a ratio better than γk for for k predictions • Consistency: For k predictions, we give an ηk-consistent deterministic algorithm where: • η1 = 1 • limk∞ ηk= 2 • ηk is an increasing sequence • No deterministic algorithm can achieve consistency better than ηk for k predictions • Consistency: For k predictions, we give a µk-consistent • randomized algorithm where: • µ1 = 1 • limk∞ µk = • µk is an increasing sequence • No randomized algorithm can achieve consistency better than µk for k predictions • Robustness: For k predictions, we give a deterministic algorithm such for that any 0 < : • alg ≤ (1 + ) opt in all situations • alg ≤ ρk, opt if the best prediction has 0 error • ρk, is an increasing sequence, ρk,> ηkfor finite k • No deterministic algorithm can simultaneously achieve consistency ratio ≤ ρk, and robustness ratio ≤ (1 + ) for k predictions

  16. Future Work • Multiple predictions in other online optimization problems • Caching • Lykouris and Vassilvitskii study unweighted caching for single prediction • Some recent results with Zhihao Jiang and Kevin Sun on weighted caching for single prediction • k-server is completely open even for single prediction • Scheduling/Load Balancing • Purohit et al. consider one variant for single prediction • Several basic variants are open even for single prediction • How do we incorporate prediction costs? • Other interfaces between online algorithms and machine learning • Some recent results with Keerti Anand and Rong Ge on • Optimizing movement/recourse and loss/regret simultaneously in convex optimization • Learning for optimization in online problems

  17. thank youquestions?

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