WUT16Jarosław Arabas

Optimization as a learning process

Optimization by sampling in R1

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Optimization by sampling in R1

Optimization by sampling in R1

Learning where to sample

● Using a series of previously sampled points it is possible to predict ● a possible position of the optimum● the most promising area to sample

Learning the optimum position

Linear modelof the objective function

Learning the optimum position

Linear modelof the objective function

Expected position of the maximum

Learning the optimum position

Learning the optimum position

Parabola modelof the objective function

Learning the optimum position

Parabola modelof the objective function

Learning the optimum position

Parabola modelof the objective function

Learning where to sample

gradient - direction of expected improvement

Recall steepest ascent method

algorithm steepest ascentinitialize x(0)t ←0while ! stop

d (t)←∇ q(x(t))x (t+1)← x (t )+α d (t )

Learning where to sample

another direction of expected improvement

Recall (quasi)Newton methods

algorithmquasi−Newtoninitialize x(0)t ←0while ! stop

d (t)←H−1∇ q (x (t ))

x (t+1)← x (t )+α d (t )

When models are easy to build

● Objective function is linear

possible to predict the optimum position

● Objective function is a parabola

possible to predict the optimum position

● Objective function is convex (concave)

possible to correctly predict the direction of improvement

When models are easy to build

● It is possible to conclude on global properties of the objective function by looking at its local behavior (ability to properly extrapolate)

● Series of improvements of the best-so-far result will yield the absolute best in the end

● An analogy to the “best practices”, “quality assurance”

● When you climb up a single hill, you are guaranteed to reach the topmost peak

When models are easy to build

● It is possible to conclude on global properties of the objective function by looking at its local behavior (ability to properly extrapolate)

● Series of improvements of the best-so-far result will yield the absolute best in the end

● An analogy to the “best practices”, “quality assurance”

● When you climb up a single hill, you are guaranteed to reach the topmost peak

When local models may be misleading

● When you want to reach the topmost peak, you should decide which to climb prior to getting on the top

● When you recognize that you have chosen a wrong hill then you should go down and try to find a better goal

● But remember that you are walking without a map and in a fog

Example model probable location of the maximum

Example model probable location of the maximum

Example model probable location of the maximum

Example model probable location of the maximum