Optimization methods Morten Nielsen Department of Systems biology , DTU
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Transcript of Optimization methods Morten Nielsen Department of Systems biology , DTU
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Optimization methods
Morten NielsenDepartment of Systems biology,
DTU
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*Adapted from slides by Chen Kaeasar, Ben-Gurion University
The path to the closest local minimum = local minimization
Minimization
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*Adapted from slides by Chen Kaeasar, Ben-Gurion University
The path to the closest local minimum = local minimization
Minimization
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The path to the global minimum
*Adapted from slides by Chen Kaeasar, Ben-Gurion University
Minimization
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Outline
• Optimization procedures – Gradient descent– Monte Carlo
• Overfitting – cross-validation
• Method evaluation
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Linear methods. Error estimate
I1 I2w1 w2
Linear function
o
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Gradient descent (from wekipedia)
Gradient descent is based on the observation that if the real-valued function F(x) is defined and differentiable in a neighborhood of a point a, then F(x) decreases fastest if one goes from a in the direction of the negative gradient of F at a. It follows that, if
for > 0 a small enough number, then F(b)<F(a)
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Gradient descent (example)
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Gradient descent
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Gradient descent
Weights are changed in the opposite direction of the gradient of the error
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Gradient descent (Linear function)
Weights are changed in the opposite direction of the gradient of the error
I1 I2w1 w2
Linear function
o
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Gradient descent
Weights are changed in the opposite direction of the gradient of the error
I1 I2w1 w2
Linear function
o
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Gradient descent. Example
Weights are changed in the opposite direction of the gradient of the error
I1 I2w1 w2
Linear function
o
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Gradient descent. Example
Weights are changed in the opposite direction of the gradient of the error
I1 I2w1 w2
Linear function
o
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Gradient descent. Doing it your selfWeights are changed in the opposite direction of the gradient of the error
1 0
W1=0.1 W2=0.1
Linear function
o
What are the weights after 2 forward (calculate predictions) and backward (update weights) iterations with the given input, and has the error decrease (use =0.1, and t=1)?
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Fill out the table
itr W1 W2 O
0 0.1 0.1
1
2
What are the weights after 2 forward/backward iterations with the given input, and has the error decrease (use =0.1, t=1)?
1 0
W1=0.1 W2=0.1
Linear function
o
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Fill out the table
itr W1 W2 O
0 0.1 0.1 0.1
1 0.19 0.1 0.19
2 0.27 0.1 0.27
What are the weights after 2 forward/backward iterations with the given input, and has the error decrease (use =0.1, t=1)?
1 0
W1=0.1 W2=0.1
Linear function
o
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Monte Carlo
Because of their reliance on repeated computation of random or pseudo-random numbers, Monte Carlo methods are most suited to calculation by a computer. Monte Carlo methods tend to be used when it is unfeasible or impossible to compute an exact result with a deterministic algorithmOr when you are too stupid to do the math yourself?
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Example: Estimating Π by Independent
Monte-Carlo SamplesSuppose we throw darts randomly (and uniformly) at the square:
Algorithm:For i=[1..ntrials] x = (random# in [0..r]) y = (random# in [0..r]) distance = sqrt (x^2 + y^2) if distance ≤ r hits++EndOutput:
Adapted from course slides by Craig Douglas
http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html
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Estimating P
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After a long run, we want to find low-energy conformations, with high probability
Sampling Protein Conformations with MCMC(Markov Chain Monte Carlo)
Protein image taken from Chemical Biology, 2006
Markov-Chain Monte-Carlo (MCMC) with “proposals”:1. Perturb Structure to create a “proposal”2. Accept or reject new conformation with a “certain”
probability
But how?
A (physically) natural* choice is the Boltzman distribution, proportional to:
Ei = energy of state ikB = Boltzman constantT = temperatureZ = “Partition Function”
* In theory, the Boltzman distribution is a bit problematic in non-gas phase, but never mind that for now…
Ze Tk
E
B
i
Slides adapted from Barak Raveh
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The Metropolis-Hastings Criterion
• Boltzman Distribution:
• The energy score and temperature are computed (quite) easily• The “only” problem is calculating Z (the “partition function”) –
this requires summing over all states.• Metropolis showed that MCMC will converge to the true
Boltzman distribution, if we accept a new proposal with
probability
"Equations of State Calculations by Fast Computing Machines“ – Metropolis, N. et al. Journal of Chemical Physics (1953)
Ze TkE Bi
Slides adapted from Barak Raveh
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If we run till infinity, with good perturbations, we will visit every conformation according to the Boltzman distribution
Sampling Protein Conformations with Metropolis-Hastings MCMC
Protein image taken from Chemical Biology, 2006
Markov-Chain Monte-Carlo (MCMC) with “proposals”:1. Perturb Structure to create a “proposal”2. Accept or reject new conformation by the Metropolis criterion3. Repeat for many iterations
But we just want to find the energy minimum. If we do our perturbations in a smart manner, we can still cover relevant (realistic, low-energy) parts of the search space
Slides adapted from Barak Raveh
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Monte Carlo (Minimization)
dE<0dE>0
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The Traveling Salesman
Adapted from www.mpp.mpg.de/~caldwell/ss11/ExtraTS.pdf
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Adapted from www.mpp.mpg.de/~caldwell/ss11/ExtraTS.pdf
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Adapted from www.mpp.mpg.de/~caldwell/ss11/ExtraTS.pdf
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Adapted from www.mpp.mpg.de/~caldwell/ss11/ExtraTS.pdf
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Adapted from www.mpp.mpg.de/~caldwell/ss11/ExtraTS.pdf
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Adapted from www.mpp.mpg.de/~caldwell/ss11/ExtraTS.pdf
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Gibbs sampler. Monte Carlo simulations RFFGGDRGAPKRGYLDPLIRGLLARPAKLQVKPGQPPRLLIYDASNRATGIPA GSLFVYNITTNKYKAFLDKQ SALLSSDITASVNCAK GFKGEQGPKGEPDVFKELKVHHANENI SRYWAIRTRSGGITYSTNEIDLQLSQEDGQTIE
RFFGGDRGAPKRGYLDPLIRGLLARPAKLQVKPGQPPRLLIYDASNRATGIPAGSLFVYNITTNKYKAFLDKQ SALLSSDITASVNCAK GFKGEQGPKGEPDVFKELKVHHANENI SRYWAIRTRSGGITYSTNEIDLQLSQEDGQTIE
E1 = 5.4 E2 = 5.7
E2 = 5.2
dE>0; Paccept =1
dE<0; 0 < Paccept < 1
Note the sign. Maximization
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Monte Carlo Temperature
• What is the Monte Carlo temperature?
• Say dE=-0.2, T=1
• T=0.001
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MC minimization
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Monte Carlo - Examples
• Why a temperature?
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Local minima