Online Vector Balancing and Geometric Discrepancy
Description: Online Vector Balancing and Geometric Discrepancy Sahil singla IAS Princeton Nikhil Bansal Haotian Jiang Janardhan Kulkarni Makrand Sinha Paper 1: Online Geometric Discrepancy for Stochastic Arrivals with Applications to Envy
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slide1. Online Vector Balancing and Geometric Discrepancy Sahil singla
IAS & Princeton Nikhil
Bansal Haotian
Jiang Janardhan
Kulkarni Makrand
Sinha Paper 1: “Online Geometric Discrepancy for Stochastic Arrivals with Applications to Envy Minimization” with Haotian and Janardhan
Paper 2: “Online Vector Balancing and Geometric Discrepancy” with Nikhil, Haotian, and Makrand<br>
slide2. Interval Balancing 2 Other Geometric Problems? Just alternate Cannot change<br>
slide3. Tusnady’s Problem 3 Linear Algebraic View: Vector Balancing<br>
slide4. Offline Vector Balancing 4<br>
slide5. Motivation and Background 5 Variants:
Some norm-1 to norm-2
Multiple colors
Find an order What can we do Online? Vectors = items
Coordinates = values Items/subjects come one-by-one<br>
slide6. Online Vector Balancing 6 Why does it capture Interval Balancing? Hope: Not always orthogonal<br>
slide7. 7 Why a Generalization?<br>
slide8. Our Results 8<br>
slide9. OUTLINE 9<br>
slide10. Online Vector Balancing via a Potential 10 Hope: Not always orthogonal<br>
slide11. Bansal-Spencer for Indep Coordinates 11 Linear and
Quadratic (helping) (hurting) Khintchine Cauchy-Schwarz<br>
slide12. Takeaways 12<br>
slide13. OUTLINE 13<br>
slide14. Vector Balancing with Dependencies 14 WLOG assume symmetric: v & –v equally likely Anti-Concentration?<br>
slide15. Anti-concentration for Uncorrelated R.V.s 15<br>
slide16. Wrapping Up: Vector Balancing 16<br>
slide17. OUTLINE 17<br>
slide18. Online Interval Discrepancy 18<br>
slide19. Attempt 1 19 How can we do better? Fractal Structure Linear vs
Quadratic<br>
slide20. Attempt 2 20 Tusnady’s Problem?
What’s Really Happening?<br>
slide21. OUTLINE 21<br>
slide22. Haar Basis View 22<br>
slide23. Tensor product of 1-d Haar-basis
Still gives uncorrelation
Preserves sparsity
Don’t lose while going back to the original basis 23 Tusnady’s Problem<br>
slide24. Conclusion 24 Questions?<br>
slide25. Further SLIDES 25<br>
slide26. Anti-concentration for Uncorrelated R.V.s 26<br>
slide27. Fractal Structure Tree
Fractal 27<br>
slide28. 28 Attempt 3: Haar Basis Uncorrelation<br>
slide29. EXTENSIONS and Lower Bounds 29<br>
IAS & Princeton Nikhil
Bansal Haotian
Jiang Janardhan
Kulkarni Makrand
Sinha Paper 1: “Online Geometric Discrepancy for Stochastic Arrivals with Applications to Envy Minimization” with Haotian and Janardhan
Paper 2: “Online Vector Balancing and Geometric Discrepancy” with Nikhil, Haotian, and Makrand<br>
slide2. Interval Balancing 2 Other Geometric Problems? Just alternate Cannot change<br>
slide3. Tusnady’s Problem 3 Linear Algebraic View: Vector Balancing<br>
slide4. Offline Vector Balancing 4<br>
slide5. Motivation and Background 5 Variants:
Some norm-1 to norm-2
Multiple colors
Find an order What can we do Online? Vectors = items
Coordinates = values Items/subjects come one-by-one<br>
slide6. Online Vector Balancing 6 Why does it capture Interval Balancing? Hope: Not always orthogonal<br>
slide7. 7 Why a Generalization?<br>
slide8. Our Results 8<br>
slide9. OUTLINE 9<br>
slide10. Online Vector Balancing via a Potential 10 Hope: Not always orthogonal<br>
slide11. Bansal-Spencer for Indep Coordinates 11 Linear and
Quadratic (helping) (hurting) Khintchine Cauchy-Schwarz<br>
slide12. Takeaways 12<br>
slide13. OUTLINE 13<br>
slide14. Vector Balancing with Dependencies 14 WLOG assume symmetric: v & –v equally likely Anti-Concentration?<br>
slide15. Anti-concentration for Uncorrelated R.V.s 15<br>
slide16. Wrapping Up: Vector Balancing 16<br>
slide17. OUTLINE 17<br>
slide18. Online Interval Discrepancy 18<br>
slide19. Attempt 1 19 How can we do better? Fractal Structure Linear vs
Quadratic<br>
slide20. Attempt 2 20 Tusnady’s Problem?
What’s Really Happening?<br>
slide21. OUTLINE 21<br>
slide22. Haar Basis View 22<br>
slide23. Tensor product of 1-d Haar-basis
Still gives uncorrelation
Preserves sparsity
Don’t lose while going back to the original basis 23 Tusnady’s Problem<br>
slide24. Conclusion 24 Questions?<br>
slide25. Further SLIDES 25<br>
slide26. Anti-concentration for Uncorrelated R.V.s 26<br>
slide27. Fractal Structure Tree
Fractal 27<br>
slide28. 28 Attempt 3: Haar Basis Uncorrelation<br>
slide29. EXTENSIONS and Lower Bounds 29<br>