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Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group Mario Lezcano-Casado David Martnez-Rubio Mathematical Institute Department of Computer Science June 12, 2019 Cheap Orthogonal

  1. Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group Mario Lezcano-Casado David Martínez-Rubio Mathematical Institute Department of Computer Science June 12, 2019

  2. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints We study the optimization of neural networks with orthogonal constraints B ∈ R n × n , B ⊺ B = I Visit our poster (#27 on Wednesday) 1 4

  3. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints We study the optimization of neural networks with orthogonal constraints B ∈ R n × n , B ⊺ B = I Motivation: Visit our poster (#27 on Wednesday) 1 4

  4. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints We study the optimization of neural networks with orthogonal constraints B ∈ R n × n , B ⊺ B = I Motivation: ◮ Orthogonal matrices have eigenvalues with norm 1 . Visit our poster (#27 on Wednesday) 1 4

  5. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints We study the optimization of neural networks with orthogonal constraints B ∈ R n × n , B ⊺ B = I Motivation: ◮ Orthogonal matrices have eigenvalues with norm 1 . ◮ Convenient for exploding and vanishing gradient problems within RNN s. ◮ They constitute a implicit regularization method. Visit our poster (#27 on Wednesday) 1 4

  6. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints We study the optimization of neural networks with orthogonal constraints B ∈ R n × n , B ⊺ B = I Motivation: ◮ Orthogonal matrices have eigenvalues with norm 1 . ◮ Convenient for exploding and vanishing gradient problems within RNN s. ◮ They constitute a implicit regularization method. ◮ They are the basic building block for matrix factorizations like SVD or QR . Visit our poster (#27 on Wednesday) 1 4

  7. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints We study the optimization of neural networks with orthogonal constraints B ∈ R n × n , B ⊺ B = I Motivation: ◮ Orthogonal matrices have eigenvalues with norm 1 . ◮ Convenient for exploding and vanishing gradient problems within RNN s. ◮ They constitute a implicit regularization method. ◮ They are the basic building block for matrix factorizations like SVD or QR . ◮ They allow for the implementation of factorized linear layers. Visit our poster (#27 on Wednesday) 1 4

  8. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints B ∈ SO( n ) f ( B ) min is equivalent to solving A ∈ Skew( n ) f (exp ( A )) min Visit our poster (#27 on Wednesday) 2 4

  9. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints B ∈ SO( n ) f ( B ) min is equivalent to solving A ∈ Skew( n ) f (exp ( A )) min � �� � � �� � constrained problem. unconstrained problem. Visit our poster (#27 on Wednesday) 2 4

  10. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints B ∈ SO( n ) f ( B ) min is equivalent to solving A ∈ Skew( n ) f (exp ( A )) min � �� � � �� � constrained problem. unconstrained problem. ◮ The matrix exponential maps skew-symmetric matrices to orthogonal matrices . Visit our poster (#27 on Wednesday) 2 4

  11. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints B ∈ SO( n ) f ( B ) min is equivalent to solving A ∈ Skew( n ) f (exp ( A )) min � �� � � �� � constrained problem. unconstrained problem. ◮ The matrix exponential maps skew-symmetric matrices to orthogonal matrices . ◮ Compute the exponential to optimize over the unconstrained space of skew symmetric matrices. Visit our poster (#27 on Wednesday) 2 4

  12. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints B ∈ SO( n ) f ( B ) min is equivalent to solving A ∈ Skew( n ) f (exp ( A )) min � �� � � �� � constrained problem. unconstrained problem. ◮ The matrix exponential maps skew-symmetric matrices to orthogonal matrices . ◮ Compute the exponential to optimize over the unconstrained space of skew symmetric matrices. ◮ No orthogonality needs to be enforced. Visit our poster (#27 on Wednesday) 2 4

  13. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints B ∈ SO( n ) f ( B ) min is equivalent to solving A ∈ Skew( n ) f (exp ( A )) min � �� � � �� � constrained problem. unconstrained problem. ◮ The matrix exponential maps skew-symmetric matrices to orthogonal matrices . ◮ Compute the exponential to optimize over the unconstrained space of skew symmetric matrices. ◮ No orthogonality needs to be enforced. ◮ It has negligible overhead in your neural network. Visit our poster (#27 on Wednesday) 2 4

  14. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints B ∈ SO( n ) f ( B ) min is equivalent to solving A ∈ Skew( n ) f (exp ( A )) min � �� � � �� � constrained problem. unconstrained problem. ◮ The matrix exponential maps skew-symmetric matrices to orthogonal matrices . ◮ Compute the exponential to optimize over the unconstrained space of skew symmetric matrices. ◮ No orthogonality needs to be enforced. ◮ It has negligible overhead in your neural network. ◮ General purpose optimizers can be used ( SGD , ADAM , ADAGRAD , . . . ). Visit our poster (#27 on Wednesday) 2 4

  15. Cheap Orthogonal Constraints in Neural Networks Optimization with orthogonal constraints B ∈ SO( n ) f ( B ) min is equivalent to solving A ∈ Skew( n ) f (exp ( A )) min � �� � � �� � constrained problem. unconstrained problem. ◮ The matrix exponential maps skew-symmetric matrices to orthogonal matrices . ◮ Compute the exponential to optimize over the unconstrained space of skew symmetric matrices. ◮ No orthogonality needs to be enforced. ◮ It has negligible overhead in your neural network. ◮ General purpose optimizers can be used ( SGD , ADAM , ADAGRAD , . . . ). ◮ No new extremal points are created in the main parametrization region. Visit our poster (#27 on Wednesday) 2 4

  16. Cheap Orthogonal Constraints in Neural Networks 0 . 020 Baseline EURNN LSTM scoRNN expRNN 0 . 015 Cross entropy 0 . 010 0 . 005 0 . 000 0 500 1000 1500 2000 2500 3000 3500 4000 Iterations Cross entropy in the copying problem for L = 2000 . The copying problem uses synthetic data of the form: Random numbers Wait for L steps Recall Input: 14221 ------ :---- Output: ----- ------ 14221 Visit our poster (#27 on Wednesday) 3 4

  17. Cheap Orthogonal Constraints in Neural Networks M ODEL # PARAM V ALID . T EST N 224 ≈ 83 K 5 . 34 5 . 30 EXPRNN 322 ≈ 135 K 4 . 42 4 . 38 EXPRNN 425 ≈ 200 K 5 . 52 5 . 48 EXPRNN 224 ≈ 83 K 9 . 26 8 . 50 SCORNN 322 ≈ 135 K 8 . 48 7 . 82 SCORNN 425 ≈ 200 K 7 . 97 7 . 36 SCORNN ≈ 83 K 15 . 42 14 . 30 LSTM 84 120 ≈ 135 K 13 . 93 12 . 95 LSTM 158 ≈ 200 K 13 . 66 12 . 62 LSTM 158 ≈ 83 K 15 . 57 18 . 51 EURNN ≈ 135 K 15 . 90 15 . 31 EURNN 256 378 ≈ 200 K 16 . 00 15 . 15 EURNN RGD 128 ≈ 83 K 15 . 07 14 . 58 192 ≈ 135 K 15 . 10 14 . 50 RGD ≈ 200 K 14 . 96 14 . 69 RGD 256 RNN s trained on a speech prediction task on the TIMIT dataset. It shows the best validation MSE accuracy. Visit our poster (#27 on Wednesday) 4 4

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