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- 1. Solving Large-Scale Machine Learning Problems in a Distributed Way Martin Tak´aˇc Cognitive Systems Institute Group Speaker Series June 09 2016 1 / 28
- 2. Outline 1 Machine Learning - Examples and Algorithm 2 Distributed Computing 3 Learning Large-Scale Deep Neural Network (DNN) 2 / 28
- 3. Examples of Machine Learning binary classiﬁcation classiﬁes person to have cancer or not decided for an input image to which class it belongs, e.g. car/person spam detection/credit card fraud detection multi-class classiﬁcation hand-written digits classiﬁcation speech understanding face detection product recommendation (collaborative ﬁltering) stock trading . . . and many many others. . . 3 / 28
- 4. Support Vector Machines (SVM) blue: healthy person green: e.g. patient with lung cancer Exhaled breath analysis for lung cancer: predict if patient has cancer or not 4 / 28
- 5. ImageNet - Large Scale Visual Recognition Challenge Two main chalanges Object detection - 200 categories Object localization - 1000 categories (over 1.2 million images for training) 5 / 28
- 6. ImageNet - Large Scale Visual Recognition Challenge Two main chalanges Object detection - 200 categories Object localization - 1000 categories (over 1.2 million images for training) The state-of-the-art solution method is Deep Neural Network (DNN) E.g. input layer has dimension of input image The output layer has dimension of e.g. 1000 (how many categories we have) 5 / 28
- 7. Deep Neural Network we have to learn the weights between neurons (blue arrows) the neural network is deﬁning a non-linear and non-convex function (of weights w) from input x to output y: y = f (w; x) 6 / 28
- 8. Example - MNIST handwritten digits recognition A good w could give us f w; = 0 0 0 0.991 ... f w; = 0 0 ... 0 0.999 7 / 28
- 9. Mathematical Formulation Expected Loss Minimization let (X, Y ) be the distribution of input samples and its labels we would like to ﬁnd w such that w∗ = arg min w E(x,y)∼(X,Y )[ (f (w; x), y)] is a loss function, i.e. (f (w; x), y) = f (w; x) − y 2 8 / 28
- 10. Mathematical Formulation Expected Loss Minimization let (X, Y ) be the distribution of input samples and its labels we would like to ﬁnd w such that w∗ = arg min w E(x,y)∼(X,Y )[ (f (w; x), y)] is a loss function, i.e. (f (w; x), y) = f (w; x) − y 2 Impossible, as we do not know the distribution (X, Y ) 8 / 28
- 11. Mathematical Formulation Expected Loss Minimization let (X, Y ) be the distribution of input samples and its labels we would like to ﬁnd w such that w∗ = arg min w E(x,y)∼(X,Y )[ (f (w; x), y)] is a loss function, i.e. (f (w; x), y) = f (w; x) − y 2 Impossible, as we do not know the distribution (X, Y ) Common approach: Empirical loss minimization: we sample n points from (X, Y ): {(xi , yi )}n i=1 we minimize regularized empirical loss w∗ = arg min w 1 n n i=1 (f (w; xi ), yi ) + λ 2 w 2 8 / 28
- 12. Stochastic Gradient Descent (SGD) Algorithm How can we solve min w F(w) := 1 n n i=1 (f (w; xi ); yi ) + λ 2 w 2 9 / 28
- 13. Stochastic Gradient Descent (SGD) Algorithm How can we solve min w F(w) := 1 n n i=1 (f (w; xi ); yi ) + λ 2 w 2 1 we can use an iterative algorithm 2 we start with some initial w 3 we compute g = F(w) 4 we get a new iterate w ← w − αg 5 if w is still not good enough go to step 3 if n is very large, computing g can take a while.... even few hours/days 9 / 28
- 14. Stochastic Gradient Descent (SGD) Algorithm How can we solve min w F(w) := 1 n n i=1 (f (w; xi ); yi ) + λ 2 w 2 1 we can use an iterative algorithm 2 we start with some initial w 3 we compute g = F(w) 4 we get a new iterate w ← w − αg 5 if w is still not good enough go to step 3 if n is very large, computing g can take a while.... even few hours/days Trick: choose i ∈ {1, . . . , n} randomly deﬁne gi = (f (w; wi ); yi ) + λ 2 w 2 use gi instead of g in the algorithm (step 4) 9 / 28
- 15. Stochastic Gradient Descent (SGD) Algorithm How can we solve min w F(w) := 1 n n i=1 (f (w; xi ); yi ) + λ 2 w 2 1 we can use an iterative algorithm 2 we start with some initial w 3 we compute g = F(w) 4 we get a new iterate w ← w − αg 5 if w is still not good enough go to step 3 if n is very large, computing g can take a while.... even few hours/days Trick: choose i ∈ {1, . . . , n} randomly deﬁne gi = (f (w; wi ); yi ) + λ 2 w 2 use gi instead of g in the algorithm (step 4) Note: E[gi ] = g, so in expectation, the ”direction” the algorithm is going is the same as if we use the true gradient, but we can compute it n times faster! 9 / 28
- 16. Outline 1 Machine Learning - Examples and Algorithm 2 Distributed Computing 3 Learning Large-Scale Deep Neural Network (DNN) 10 / 28
- 17. The Architecture What if the size of data {(xi , yi )} exceeds the memory of a single computing node? 11 / 28
- 18. The Architecture What if the size of data {(xi , yi )} exceeds the memory of a single computing node? each node can store portion of the data {(xi , yi )} each node is connected to the computer network they can communicate with any other node (over maybe 1 or more switches) Fact: every communication is much more expensive then accessing local data (can be even 100,000 times slower). 11 / 28
- 19. Outline 1 Machine Learning - Examples and Algorithm 2 Distributed Computing 3 Learning Large-Scale Deep Neural Network (DNN) 12 / 28
- 20. Using SGD for DNN in Distributed Way assume that the size of data or the size weights (or both) is so big, that we cannot store them on one machine . . . or we can store them but it takes too long to compute something . . . SGD: we need to compute w (f (w; xi ); yi ) The DNN has a nice structure w (f (w; xi ); yi ) can we computed by backpropagation procedure (this is nothing else just automated diﬀerentiation) 13 / 28
- 21. Why is SGD a Bad Distributed Algorithm it samples only 1 sample and computes gi (this is very fast) then w is updated each update of w requires a communication (cost c seconds) hence one iteration is suddenly much slower then if we would run SGD on one computer 14 / 28
- 22. Why is SGD a Bad Distributed Algorithm it samples only 1 sample and computes gi (this is very fast) then w is updated each update of w requires a communication (cost c seconds) hence one iteration is suddenly much slower then if we would run SGD on one computer The trick: Mini-batch SGD In each iteration 1 Choose randomly S ⊂ {1, 2, . . . , n} with |S| = b 2 Use gb = 1 b i∈S gi instead of just gi 14 / 28
- 23. Why is SGD a Bad Distributed Algorithm it samples only 1 sample and computes gi (this is very fast) then w is updated each update of w requires a communication (cost c seconds) hence one iteration is suddenly much slower then if we would run SGD on one computer The trick: Mini-batch SGD In each iteration 1 Choose randomly S ⊂ {1, 2, . . . , n} with |S| = b 2 Use gb = 1 b i∈S gi instead of just gi Cost of one epoch number of MPI calls / epoch n/b amount of data send over network n b × log(N) × sizeof (w) if we increase b → n we would minimize amount of data and number of number of communications per epoch! 14 / 28
- 24. Why is SGD a Bad Distributed Algorithm it samples only 1 sample and computes gi (this is very fast) then w is updated each update of w requires a communication (cost c seconds) hence one iteration is suddenly much slower then if we would run SGD on one computer The trick: Mini-batch SGD In each iteration 1 Choose randomly S ⊂ {1, 2, . . . , n} with |S| = b 2 Use gb = 1 b i∈S gi instead of just gi Cost of one epoch number of MPI calls / epoch n/b amount of data send over network n b × log(N) × sizeof (w) if we increase b → n we would minimize amount of data and number of number of communications per epoch! Caveat: there is no free lunch! Very large b means slower convergence! 14 / 28
- 25. Model Parallelism Model parallelism: we partition weights w across many nodes; every node has all data points (but maybe just few features of them) Hidden Layer 1 Hidden Layer 2 Output Input ForwardPropagation Hidden Layer 1 Hidden Layer 2 Output BackwardPropagation Node1 Node1 Node2 Node2 AllSamples AllSamples AllSamples AllSamples Exchange Activation Exchange Activation Exchange Deltas Exchange Deltas 15 / 28
- 26. Data Parallelism Data parallelism: we partition data-samples across many nodes, each node has a fresh copy of w Hidden Layer 1 Hidden Layer 2 Output Input ForwardPropagation Hidden Layer 1 Hidden Layer 2 Output BackwardPropagation Node1 Node1 Node2 Node2 PartialSamples PartialSamples PartialSamples PartialSamples Hidden Layer 1 Hidden Layer 2 Hidden Layer 1 Hidden Layer 2 Exchange Gradient 16 / 28
- 27. Large-Scale Deep Neural Network1 1Dipankar Das, Sasikanth Avancha, Dheevatsa Mudigere, Karthikeyan Vaidynathan, Srinivas Sridharan, Dhiraj Kalamkar, Bharat Kaul, Pradeep Dubey: Distributed Deep Learning Using Synchronous Stochastic Gradient Descent, arXiv:1602.06709 17 / 28
- 28. There is almost no speedup for large b 18 / 28
- 29. The Dilemma large b allows algorithm to be eﬃciently run on large computer cluster (more nodes) very large b doesn’t reduce number of iterations, but each iteration is more expensive! The Trick: Do not use just gradient, but use also Hessian (Martens 2010) 19 / 28
- 30. The Dilemma large b allows algorithm to be eﬃciently run on large computer cluster (more nodes) very large b doesn’t reduce number of iterations, but each iteration is more expensive! The Trick: Do not use just gradient, but use also Hessian (Martens 2010) Caveat: Hessian matrix can be very large, e.g. the dimension of weights for TIMIT datasets is almost 1.5M, hence to store Hessian we would need almost 10TB. 19 / 28
- 31. The Dilemma large b allows algorithm to be eﬃciently run on large computer cluster (more nodes) very large b doesn’t reduce number of iterations, but each iteration is more expensive! The Trick: Do not use just gradient, but use also Hessian (Martens 2010) Caveat: Hessian matrix can be very large, e.g. the dimension of weights for TIMIT datasets is almost 1.5M, hence to store Hessian we would need almost 10TB. The Trick: We can use Hessian Free approach (we need to be able to compute just Hessian-vector products) 19 / 28
- 32. The Dilemma large b allows algorithm to be eﬃciently run on large computer cluster (more nodes) very large b doesn’t reduce number of iterations, but each iteration is more expensive! The Trick: Do not use just gradient, but use also Hessian (Martens 2010) Caveat: Hessian matrix can be very large, e.g. the dimension of weights for TIMIT datasets is almost 1.5M, hence to store Hessian we would need almost 10TB. The Trick: We can use Hessian Free approach (we need to be able to compute just Hessian-vector products) Algorithm: w ← w − α[ 2 F(w)]−1 F(w) 19 / 28
- 33. Non-convexity We want to minimize min w F(w) 2 F(w) is NOT positive semi-deﬁnite at any w! 20 / 28
- 34. Computing Step recall the algorithm w ← w − α[ 2 F(w)]−1 F(w) we need to compute p = [ 2 F(w)]−1 F(w), i.e. to solve 2 F(w)p = F(w) (1) we can use few iterations of CG method to solve it (CG assumes that 2 F(w) 0) In our case it may not be true, hence, it is suggested to stop CG sooner, if it is detected during CG that 2 F(w) is indeﬁnite We can use a Bi-CG algorithm to solve (1) and modify the algorithm2 as follows w ← w − α p, if pT F(x) > 0, −p, otherwise PS: we use just b samples to estimate 2 F(w) 2Xi He, Dheevatsa Mudigere, Mikhail Smelyanskiy and Martin Tak´aˇc: Large Scale Distributed Hessian-Free Optimization for Deep Neural Network, arXiv:1606.00511, 2016. 21 / 28
- 35. Saddle Point Gradient descent slows down around saddle point. Second order methods can help a lot to prevent that. 22 / 28
- 36. 50 100 150 200 250 300 350 400 10 −3 10 −2 10 −1 10 0 MNIST, 4 layers Number of iteration TrainError SGD, b=64 SGD, b=128 ggn−cg, b=512 hess−bicgstab, b=512 hess−cg, b=512 hybrid−cg, b=512 50 100 150 200 250 300 350 400 10 −3 10 −2 10 −1 10 0 MNIST, 4 layers Number of iteration TrainError SGD, b=64 SGD, b=128 ggn−cg, b=1024 hess−bicgstab, b=1024 hess−cg, b=1024 hybrid−cg, b=1024 50 100 150 200 250 300 350 400 10 −3 10 −2 10 −1 10 0 MNIST, 4 layers Number of iteration TrainError SGD, b=64 SGD, b=128 ggn−cg, b=2048 hess−bicgstab, b=2048 hess−cg, b=2048 hybrid−cg, b=2048 10 1 10 2 10 3 10 2 10 3 MNIST, 4 layers Size of Mini−batch NumberofIterations ggn−cg hess−bicgstab hess−cg hybrid−cg 23 / 28
- 37. 1 1.5 2 2.5 3 3.5 4 4.5 5 10 0 10 1 10 2 10 3 TIMIT, T=18, b=512 log2 (Number of Nodes) RunTimeperIteration Gradient CG Linesearch 1 1.5 2 2.5 3 3.5 4 4.5 5 10 0 10 1 10 2 10 3 TIMIT, T=18, b=1024 log2 (Number of Nodes) RunTimeperIteration Gradient CG Linesearch 1 1.5 2 2.5 3 3.5 4 4.5 5 10 0 10 1 10 2 10 3 TIMIT, T=18, b=4096 log2 (Number of Nodes) RunTimeperIteration Gradient CG Linesearch 1 1.5 2 2.5 3 3.5 4 4.5 5 10 1 10 2 10 3 TIMIT, T=18, b=8192 log2 (Number of Nodes) RunTimeperIteration Gradient CG Linesearch 24 / 28
- 38. 1 1.5 2 2.5 3 3.5 4 4.5 5 10 0 10 1 TIMIT, T=18 log2 (Number of Nodes) RunTimeperOneLineSearch b=512 b=1024 b=4096 b=8192 25 / 28
- 39. Learning Artistic Style by Deep Neural Network3 3Joint work with Jiawei Zhang, based on Leon A. Gatys, Alexander S. Ecker, Matthias Bethge, A Neural Algorithm of Artistic Style, arxiv 1508.06576 26 / 28
- 40. Learning Artistic Style by Deep Neural Network3 3Joint work with Jiawei Zhang, based on Leon A. Gatys, Alexander S. Ecker, Matthias Bethge, A Neural Algorithm of Artistic Style, arxiv 1508.06576 26 / 28
- 41. Learning Artistic Style by Deep Neural Network4 4Joint work with Jiawei Zhang, based on Leon A. Gatys, Alexander S. Ecker, Matthias Bethge, A Neural Algorithm of Artistic Style, arxiv 1508.0657627 / 28
- 42. Learning Artistic Style by Deep Neural Network4 4Joint work with Jiawei Zhang, based on Leon A. Gatys, Alexander S. Ecker, Matthias Bethge, A Neural Algorithm of Artistic Style, arxiv 1508.0657627 / 28
- 43. References 1 Albert Berahas, Jorge Nocedal and Martin Tak´aˇc: A Multi-Batch L-BFGS Method for Machine Learning, arXiv:1605.06049, 2016. 2 Xi He, Dheevatsa Mudigere, Mikhail Smelyanskiy and Martin Tak´aˇc: Large Scale Distributed Hessian-Free Optimization for Deep Neural Network, arXiv:1606.00511, 2016. 3 Chenxin Ma and Martin Tak´aˇc: Partitioning Data on Features or Samples in Communication-Eﬃcient Distributed Optimization?, OptML@NIPS 2015. 4 Chenxin Ma, Virginia Smith, Martin Jaggi, Michael I. Jordan, Peter Richt´arik and Martin Tak´aˇc: Adding vs. Averaging in Distributed Primal-Dual Optimization, ICML 2015. 5 Martin Jaggi, Virginia Smith, Martin Tak´aˇc, Jonathan Terhorst, Thomas Hofmann and Michael I. Jordan: Communication-Eﬃcient Distributed Dual Coordinate Ascent, NIPS 2014. 6 Richt´arik, P. and Tak´aˇc, M.: Distributed coordinate descent method for learning with big data, Journal Paper Journal of Machine Learning Research (to appear), 2016 7 Richt´arik, P. and Tak´aˇc, M.: On optimal probabilities in stochastic coordinate descent methods, Optimization Letters, 2015. 8 Richt´arik, P. and Tak´aˇc, M.: Parallel coordinate descent methods for big data optimization, Mathematical Programming, 2015. 9 Richt´arik, P. and Tak´aˇc, M.: Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function, Mathematical Programming, 2012. 10 Tak´aˇc, M., Bijral, A., Richt´arik, P. and Srebro, N.: Mini-batch primal and dual methods for SVMs, In ICML, 2013. 11 Qu, Z., Richt´arik, P. and Zhang, T.: Randomized dual coordinate ascent with arbitrary sampling, arXiv:1411.5873, 2014. 12 Qu, Z., Richt´arik, P., Tak´aˇc, M. and Fercoq, O.: SDNA: Stochastic Dual Newton Ascent for Empirical Risk Minimization, arXiv:1502.02268, 2015. 13 Tappenden, R., Tak´aˇc, M. and Richt´arik, P., On the Complexity of Parallel Coordinate Descent, arXiv: 1503.03033, 2015. 28 / 28

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