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CS623: Introduction to Computing with Neural Nets (lecture-19)

CS623: Introduction to Computing with Neural Nets (lecture-19). Pushpak Bhattacharyya Computer Science and Engineering Department IIT Bombay. To learn the identity function The setting is probabilistic, x = 1 or x = -1, with uniform probability, i.e. , P(x=1) = 0.5, P(x=-1) = 0.5

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CS623: Introduction to Computing with Neural Nets (lecture-19)

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  1. CS623: Introduction to Computing with Neural Nets(lecture-19) Pushpak Bhattacharyya Computer Science and Engineering Department IIT Bombay

  2. To learn the identity function The setting is probabilistic, x = 1 or x = -1, with uniform probability, i.e., P(x=1) = 0.5, P(x=-1) = 0.5 For, x=1, y=1 with P=0.9 For, x=-1, y=-1 with P=0.9 Illustration of the basic idea of Boltzmann Machine y x 2 1 w12 y x 1 1 -1 -1

  3. Illustration of the basic idea of Boltzmann Machine (contd.) • Let α = output neuron states β = input neuron states Pα|β = observed probability distribution Qα|β = desired probability distribution Qβ = probability distribution on input states β

  4. Illustration of the basic idea of Boltzmann Machine (contd.) • The divergence D is given as: D = ∑α∑β Qα|β Qβln Qα|β / Pα|β called KL divergence formula D = ∑α∑β Qα|β Qβln Qα|β / Pα|β >= ∑α∑β Qα|β Qβ( 1 - Pα|β /Qα|β) >= ∑α∑βQα|β Qβ- ∑α∑βPα|β Qβ >= ∑α∑βQαβ - ∑α∑βPαβ {Qαβ and Pαβare joint distributions} >= 1 – 1 = 0

  5. Gradient descent for finding the weight change rule P(Sα) αexp(-E(Sα)/T) P(Sα) = (exp(-E(Sα)/T)) / (∑βє all statesexp(-E(Sβ)/T) ln(P(Sα))= (-E(Sα)/T)-ln Z D= ∑α∑βQα|β Qβln (Qα|β / Pα|β) Δwij= η (δD/δwij); gradient descent

  6. Calculating gradient: 1/2 δD / δwij = δ/δwij [∑α∑β Qα|β Qβln (Qα|β / Pα|β)] = δ/δwij [∑α∑β Qα|β Qβln Qα|β - ∑α∑β Qα|β Qβln Pα|β] δ(ln Pα|β)/δwij = δ/δwij [-E(Sα)/T– lnZ] Z = ∑βexp(-E(Sβ))/T Constant With respect To wij

  7. Calculating gradient: 2/2 δ [-E(Sα)/T]/δwij = (-1/T) δ/δwij [ - ∑i∑j>iwij si sj ] = (-1/T)[-sisj|α] = (1/T)[sisj|α] δ (ln Z)/δwij=(1/Z)(δZ/δwij) Z= ∑βexp(-E(Sβ)/T) δZ/δwij= ∑β[exp(-E(Sβ)/T)(δ(-E(Sβ/T)/δwij )] = (1/T) ∑βexp(-E(Sβ)/T).sisj|β

  8. Final formula for Δwij Δwij= [1/T/ [sisj|α– (1/Z) ∑βexp(-E(Sβ)/T).sisj|β = [1/T][sisj|α– ∑βP(Sβ).sisj|β] Expectation of ith and jth Neurons being on together

  9. Issue of Hidden Neurons • Boltzmann machines • can come with hidden neurons • are equivalent to a Markov Random field • with hidden neurons are like a Hidden Markov Machines • Training a Boltzmann machine is equivalent to running the Expectation Maximization Algorithm

  10. Use of Boltzmann machine • Computer Vision • Understanding scene involves what is called “Relaxation Search” which gradually minimizes a cost function with progressive relaxation on constraints • Boltzmann machine has been found to be slow in the training • Boltzmann training is NP-hard.

  11. Questions • Does the Boltzmann machine reach the global minimum? What ensures it? • Why is simulated annealing applied to Boltzmann machine? • local minimum  increase T  n/w runs gradually reduce T  reach global minimum. • Understand the effect of varying T • Higher T  small difference in energy states ignored, convergence to local minimum fast.

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