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Online convex optimization Gradient descent without a gradient. Abie Flaxman CMU Adam Tauman Kalai TTI Brendan McMahan CMU. Goal: find x f(x) ¸ max z 2 S f(z) – . High-dimensional. } . Standard convex optimization. Convex feasible set S ½ < d Concave function f : S ! <.

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online convex optimization gradient descent without a gradient

Online convex optimizationGradient descent without a gradient

Abie Flaxman CMU

Adam Tauman Kalai TTI

Brendan McMahan CMU

standard convex optimization

Goal: find x

f(x) ¸ maxz2Sf(z) – 

High-dimensional

} 

Standard convex optimization

Convex feasible set S ½<d

Concave function f : S !<

steepest ascent
Steepest ascent
  • Move in the direction of steepest ascent
  • Compute f’(x) (rf(x) in higher dimensions)
  • Works for convex optimization
  • (and many other problems)

x1

x2

x3

x4

typical application
Typical application
  • Toyota produces certain numbers of cars per month
  • Vector x 2<d (#Corollas, #Camrys, …)
  • Profit of Toyota is concave function of production vector
  • Maximize total (eq. average) profit

PROBLEMS

problem definition and results
Problem definition and results
  • Sequence of unknown (concave) functions
  • On tth month, choose production xt2 S ½<d
  • Receive profit ft(xt)
  • Maximize avg. profit
  • No assumption on distribution of ft

8 z, z’ 2 S

|| z – z’ || · D

||rf(z)|| · G

first try
First try

Zinkevich ’03:

If we could only compute gradients…

f4(x4)

f3(x3)

f2(x2)

f4

PROFIT

f1(x1)

f3

f2

f1

x4

x3

x2

x1

#CAMRYS

idea 1 point gradient estimate
Idea: 1-point gradient estimate

With probability ½, estimate = f(x + )/

With probability ½, estimate = –f(x – )/

PROFIT

E[ estimate ] ¼ f’(x)

x

x-

x+

#CAMRYS

analysis

In expectation, gradient ascent on

For online optimization, use Zinkevich’s

analysis of online gradient ascent [Z03]

Analysis

PROFIT

x-

x+

#CAMRYS

d dimensional online algorithm
d-dimensional online algorithm
  • Choose u 2<d, ||u||=1
  • Choose xt+1 = xt +  u ft(xt+u)/
  • Repeat

x3

x4

x1

x2

S

hidden complication
Hidden complication…

Dealing with steps outside set is difficult!

S

hidden complication11
Hidden complication…

Dealing with steps outside set is difficult!

S’

…reshape into

Isotropic position

related work conclusions
Related work & conclusions
  • Related work
    • Online convex gradient descent [Zinkevich03]
    • One point gradient estimates [Spall97,Granichin89]
    • Same exact problem [Kleinberg04] (different solution)
  • Conclusions
    • Can estimate gradient of a function from single evaluation (using randomness)
    • Adaptive adversary ft(xt)=ft(x1,x2,…,xt)
    • Useful for situations with a sequence of different functions, no gradient info, one evaluation per function, and others