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How do you model the future - PowerPoint PPT Presentation

How do you model the future?. Stochastic approach: The future can be modeled as a distribution over possible events. Very successful in many contexts. Alternative: Think of the future as an adversary, do well against all possible future outcomes. Toy Example: Ski Optimization.

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Presentation Transcript

• Stochastic approach: The future can be modeled as a distribution over possible events.

• Very successful in many contexts.

• Alternative: Think of the future as an adversary, do well against all possible future outcomes.

• I decide to take up skiing.

• Should I rent or buy skis?

• Uncertainty:

• Will I like skiing?

• Will there be snow?

• Will I break my leg?

• Will the government outlaw skiing?

• I want to have a good strategy against all possible outcomes In this case an outcome is the number of times I wind up going skiing.

• A pair of skis (and boots) costs \$300.

• A ski rental costs \$50.

• What should you do?

• How do you evaluate if you did the right thing?

• You give a strategy (algorithm)

• You compare against how well someone who knows that future could do.

• You take the worst case and call that the competitive ratio

• Let A be my algorithm.

• Let OPT be the behavior of someone who knows the future

• Consider any realization of the future I (number of times I actually ski)

• Competetive ratio

• We want a strategy with a small competitive ratio

Competitive ratio = 6

Competitive ratio = lots

Competitive ratio = 2

• Without knowing the future, you can guarantee that no matter what happens, you will never spend more than twice what anyone could have spent.

• A good algorithm balances different bad outcomes

• If you allow randomization, you can decrease the competetive ratio to e/(e-1), around 1.58.