Simple and efficient online algorithms for real world applications


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1 Simple and efficient online algorithms for real world applications Università degli Studi di Milano Milano, Italy Centro de Visión por Computador
2 Something about me PhD in Robotics at LIRALab, University of Genova PostDoc in Machine Learning
3 Something about me PhD in Robotics at LIRALab, University of Genova PostDoc in Machine Learning I like theoretical motivated algorithms
4 Something about me PhD in Robotics at LIRALab, University of Genova PostDoc in Machine Learning I like theoretical motivated algorithms But they must work well too! ;)
5 Outline 1 2 OM2 Projectron BBQ
6 Outline 1 2 OM2 Projectron BBQ
7 What do they have in common? Spam filtering: minimize the number of wrongly classified mails, while training
8 What do they have in common? Spam filtering: minimize the number of wrongly classified mails, while training Shifting distributions: the Learner must track the best classifier
9 What do they have in common? Spam filtering: minimize the number of wrongly classified mails, while training Shifting distributions: the Learner must track the best classifier Learning with limited feedback: selecting publicity banners
10 What do they have in common? Spam filtering: minimize the number of wrongly classified mails, while training Shifting distributions: the Learner must track the best classifier Learning with limited feedback: selecting publicity banners Active learning: asking for specific samples to label
11 What do they have in common? Spam filtering: minimize the number of wrongly classified mails, while training Shifting distributions: the Learner must track the best classifier Learning with limited feedback: selecting publicity banners Active learning: asking for specific samples to label Interactive learning: the agent and the human are learning at the same time
12 What do they have in common? Spam filtering: minimize the number of wrongly classified mails, while training Shifting distributions: the Learner must track the best classifier Learning with limited feedback: selecting publicity banners Active learning: asking for specific samples to label Interactive learning: the agent and the human are learning at the same time These problems cannot be solved with standard batch learning!
13 What do they have in common? Spam filtering: minimize the number of wrongly classified mails, while training Shifting distributions: the Learner must track the best classifier Learning with limited feedback: selecting publicity banners Active learning: asking for specific samples to label Interactive learning: the agent and the human are learning at the same time These problems cannot be solved with standard batch learning! But they can in the online learning framework!
14 Learning in an artificial agent We have the Learner and the Teacher. The Learner observes examples in a sequence of rounds, and constructs the classification function incrementally.
15 Learning in an artificial agent Given an input, the Learner predicts its label.
16 Learning in an artificial agent Then the Teacher reveals the true label.
17 Learning in an artificial agent The Learner compares its prediction with the true label and update its knowledge. The aim of the learner is to minimize the number of mistakes.
18 Machine learning point of view on online learning The Teacher is a black box It can chose the examples arbitrarily  in the worst case the choice can be adversarial! There is no IID assumption on the data! Useful to model interaction between the user and the algorithm or nonstationary data Performance is measured while training : no separate testing set Update after each sample: efficiency of the update is important See [CesaBianchi & Lugosi, 2006] for a full introduction to the theory of online learning. universitylogo
19  Formal setting Learning goes on in a sequence of T rounds Instances: x t X Images, sounds, etc. Labels: y t Y Labels, numbers, structured output, etc. Prediction rule, f t (x) = ŷ for binary classification ŷ = sign( w, x ) Loss, l(ŷ, y) R +
20 Regret bounds The learner must minimize its loss on the T observed samples T l(ŷ t, y t ) t=1
21 Regret bounds The learner must minimize its loss on the T observed samples, compared to the loss of the best fixed predictor T l(ŷ t, y t ) min f t=1 T l (f (x t ), y t ) + R T t=1
22 Regret bounds The learner must minimize its loss on the T observed samples, compared to the loss of the best fixed predictor T l(ŷ t, y t ) min f t=1 T l (f (x t ), y t ) + R T t=1 We want the regret, R T, to be small: a small regret indicates that the performance of the learner is not too far from the one of the best fixed classifier
23 A loss for everyone Mistake, l 01 (w, x, y) := 1(y sign( w, x )) Hinge, l hinge (w, x, y) := max (0, 1 y w, x ) Logistic regression, l logreg (w, x, y) := log (1 + exp ( y w, x )) exponential loss, etc.
24 Let s start from the Perceptron for t = 1, 2,..., T do Receive new instance x t Predict ŷ t = sign( w, x t ) Receive label y t if y t ŷ t then w = w + y t x t end if end for
25 The mistake bound of the Perceptron Let (x 1, y 1 ),, (x T, y T ) be any sequence of instancelabel pairs, y t { 1, +1}, and x t R. The number of mistakes of the Perceptron is bounded by where L = T i l hinge (u, x i, y i ) min L + u 2 R 2 + u R L u } {{ } R T
26 The mistake bound of the Perceptron Let (x 1, y 1 ),, (x T, y T ) be any sequence of instancelabel pairs, y t { 1, +1}, and x t R. The number of mistakes of the Perceptron is bounded by where L = T i l hinge (u, x i, y i ) min L + u 2 R 2 + u R L u } {{ } R T If the problem is linearly separable the maximum number of mistakes is R 2 u 2, regardless of the ordering of the samples!
27 The mistake bound of the Perceptron Let (x 1, y 1 ),, (x T, y T ) be any sequence of instancelabel pairs, y t { 1, +1}, and x t R. The number of mistakes of the Perceptron is bounded by where L = T i l hinge (u, x i, y i ) min L + u 2 R 2 + u R L u } {{ } R T If the problem is linearly separable the maximum number of mistakes is R 2 u 2, regardless of the ordering of the samples! Video universitylogo
28 Pros and Cons of the Perceptron Pros :) Very efficient! It can be easily trained with huge datasets Theoretical guarantee on the maximum number of mistakes
29 Pros and Cons of the Perceptron Pros :) Cons :( Very efficient! It can be easily trained with huge datasets Theoretical guarantee on the maximum number of mistakes Linear hyperplane only Binary classification only
30 Pros and Cons of the Perceptron Pros :) Cons :( Very efficient! It can be easily trained with huge datasets Theoretical guarantee on the maximum number of mistakes Linear hyperplane only Binary classification only Let s generalize the Perceptron!
31 Nonlinear classifiers using Kernels! Suppose to transform the inputs through a nonlinear transform φ(x), to the feature space A linear classifier in the feature space will result in a nonlinear classifier in the input space We can do even better: the algorithms just need to access to φ(x 1 ), φ(x 2 ), so if we have such a function, K (x 1, x 2 ), we do not need φ()!
32 Kernel Perceptron for t = 1, 2,..., T do Receive new instance ( x t ) Predict ŷ t = sign x i S y ik (x i, x t ) Receive label y t if y t ŷ t then Add x t to the support set S end if end for
33 Follow The Regularized Leader algorithm is just a particular case of a more general algorithm: the follow the regularized leader (FTRL) algorithm In FTRL, at each time step we predict with the approximate batch solution using a linearization of the loss, instead of the true loss functions Regret bounds will come almost for free! See [Kakade et al., 2009] for FTRL and [Orabona&Crammer, 2010] for an even more general algorithm
34 The general algorithm for t = 1, 2,..., T do Receive new instance x t Predict ŷ t Receive label y t θ = θ η t l(w, x t, y t ) w = g t (θ) end for F. Orabona, and K. Crammer. New Adaptive for Online Classification. Accepted in Neural Information Processing Systems (NIPS) 2010
35 The general algorithm for t = 1, 2,..., T do Receive new instance x t Predict ŷ t Receive label y t θ = θ η t l(w, x t, y t ) w = g t (θ) end for Suppose g t (θ) = 1 2 θ 2 g t (θ) = θ Let s use the hinge loss l(w, x t, y t ) = y t x t η t = 1 on mistakes, 0 otherwise We recovered the Perceptron algorithm! F. Orabona, and K. Crammer. New Adaptive for Online Classification. Accepted in Neural Information Processing Systems (NIPS) 2010
36 The Recipe Create the online algorithm that best suits your needs! Define a task this will define a (convex) loss function. Define which charateristic you would like your solution to have this will define a (strongly convex) regularizer. Compute the gradient of the loss. Compute the gradient of the fenchel dual of the regularizer. Just code it!
37 Some examples Multiclassmultilabel classification, M classes, M different classifiers w i. l( w, x, Y) = max(1 + max y / Y w y, x min y Y w y, x, 0) Do you prefer sparse classifiers? Use the regularizer w 2 p with 1 < p 2. K different kernels? Use the regularizer [ w 1 2,..., w K 2 ] 2 with 1 < p 2. p
38 Batch Solutions with Online What most of the persons do when they want a batch solution: they stop the online algorithm and use the last solution found. This is wrong: the last solution can be arbitrarly bad! If the data are IID you can simply use the averaged solution [CesaBianchi et al., 2004]. Alternative way: several epochs of training, until convergence.
39 Batch Solutions with Online What most of the persons do when they want a batch solution: they stop the online algorithm and use the last solution found. This is wrong: the last solution can be arbitrarly bad! If the data are IID you can simply use the averaged solution [CesaBianchi et al., 2004]. Alternative way: several epochs of training, until convergence. Video
40 Outline OM2 Projectron BBQ 1 2 OM2 Projectron BBQ
41 Multi Kernel Learning online? OM2 Projectron BBQ Cue 1 w 1 Cue 2 w 2... Σ Prediction Cue F w F We have F different features, e.g. color, shape, etc.
42 Multi Kernel Learning online? OM2 Projectron BBQ Cue 1 w 1 Cue 2 w 2... Σ Prediction Cue F w F We have F different features, e.g. color, shape, etc. Solution: Online MultiClass MultiKernel learning algorithm L. Jie, F. Orabona, M. Fornoni, B. Caputo, and N. CesaBianchi. OM2: An Online Multiclass Multikernel Learning Algorithm. In Proc. of the 4th IEEE for Computer Vision Workshop (in CVPR10) universitylogo
43 OM2: Pseudocode OM2 Projectron BBQ Input: q Initialize: θ 1 = 0, w 1 = 0 for t = 1, 2,..., T do Receive new instance x t Predict ŷ t = argmax y=1,...,m Receive label y t z t = φ(x t, y t ) φ(x t, ŷ t ) if l( w t, x t, y t ) > 0 then η t = min { 2 w t zt 1, 1 z t 2 2,q else η t = 0 θ t+1 = θ t + η t z ( t w j t+1 = 1 q end for w t φ(x t, y) } ) q 2 θ j t+1 2 θ j θ t+1 2,q t+1, j = 1,, F Code available! universitylogo
44 Experiments OM2 Projectron BBQ We compared OM2 to OMCL [Jie et al. ACCV09], and to PAI [Crammer et al. JMLR06] using the best feature and the sum of the kernels. We also used SILP [Sonnenburg et al. JMLR06], a stateoftheart MKL batch solver. We used the Caltech101 with 39 different kernels, as in [Gehler and Nowozin ICCV09].
45 OM2 Projectron BBQ Caltech101: online performance 100 Caltech 101 (5 splits, 30 training examples per category) Online average training error rate OM 2, p=1.01 OMCL PA I (average kernel) PA I (best kernel) number of training samples Best results with p = OM2 achieves the best performance among the online algorithms.
46 OM2 Projectron BBQ Caltech101: batch performance classification rate on test data Caltech 101 (5 splits, 30 training examples per category) OM 2, p=1.01 OMCL PA I (average kernel) PA I (best kernel) SILP number of epochs Matlab implementation of OM2 takes 45 mins, SILP more than 2 hours. The performance advantage of OM2 over SILP is due the fact that OM2 is based on a native multiclass formulation. universitylogo
47 OM2 Projectron BBQ Learning with bounded complexity Perceptronlike algorithms will never stop updating the solution if the problem is not linearly separable.
48 OM2 Projectron BBQ Learning with bounded complexity Perceptronlike algorithms will never stop updating the solution if the problem is not linearly separable. On the other hand, if we use kernels, sooner or later the memory of the computer will finish...
49 OM2 Projectron BBQ Learning with bounded complexity Perceptronlike algorithms will never stop updating the solution if the problem is not linearly separable. On the other hand, if we use kernels, sooner or later the memory of the computer will finish... Is it possible to bound the complexity of the learner?
50 OM2 Projectron BBQ Learning with bounded complexity Perceptronlike algorithms will never stop updating the solution if the problem is not linearly separable. On the other hand, if we use kernels, sooner or later the memory of the computer will finish... Is it possible to bound the complexity of the learner? Yes! Just update with a noisy version of the gradient If the new vector can be well approximated with the old ones, just update the coefficients of the old ones. F. Orabona, J. Keshet, and B. Caputo. Bounded KernelBased. Journal of Machine Learning Research, 2009 universitylogo
51 The Projectron Algorithm OM2 Projectron BBQ for t = 1, 2,..., T do Receive new instance x t Predict ŷ t = sign( w, x t ) Receive label y t if y t ŷ t then w = w + y t x t w = w + y t P(x t ) if δ t = w w η then w = w else w = w end if end if end for It is possible to calculate the projection even using Kernels. The algorithm has a mistake bound and a bounded memory growth. Code available! universitylogo
52 OM2 Projectron BBQ Average Online Error vs Budget Size Adult9, samples, 123 features, Gaussian Kernel Number of Mistakes (%) A9A Gaussian Kernel RBP Forgetron Plain Perceptron PA I Projectron Size of the support set = (36.9) Size of the Support Set universitylogo
53 OM2 Projectron BBQ Robot Navigation & Phoneme Recognition Place Recognition IDOL2 database 5 rooms CRFH features Phoneme recognition Subset of the TIMIT corpus 55 phonemes MFCC + + features
54 Place recognition OM2 Projectron BBQ 1 IDOL2 exp χ 2 kernel 0.9 Accuracy on Test Set (%) Perceptron Projectron++ η 1 Projectron++ η 2 Projectron++ η Training step
55 Place recognition OM2 Projectron BBQ Size of the Support Set Perceptron Projectron++ η 1 Projectron++ η 2 Projectron++ η 3 IDOL2 exp χ 2 kernel Training step
56 Phoneme recognition OM2 Projectron BBQ Phoneme recognition Gaussian kernel Projectron++ PA I 0.31 Number of Mistakes (%) Size of the Support Set x 10 4
57 Phoneme recognition OM2 Projectron BBQ Phoneme recognition Gaussian kernel Projectron++ PA I Error on Test Set (%) Size of the Support Set x 10 4
58 OM2 Projectron BBQ Semisupervised online learning We are given N training samples and a regression problem, {x i, y i } N i=1, y i [ 1, 1] Obtaining the output for a given sample can be expensive, so we want to ask for as few as possible labels. We assume the outputs y t are realizations of random variables Y t such that E Y t = u x t for all t, where u R d is a fixed and unknown vector such that u = 1. The order of the data is still adversarial, but the outputs are generated by a stochastic source
59 OM2 Projectron BBQ Semisupervised online learning We are given N training samples and a regression problem, {x i, y i } N i=1, y i [ 1, 1] Obtaining the output for a given sample can be expensive, so we want to ask for as few as possible labels. We assume the outputs y t are realizations of random variables Y t such that E Y t = u x t for all t, where u R d is a fixed and unknown vector such that u = 1. The order of the data is still adversarial, but the outputs are generated by a stochastic source Solution: Bound on Bias Query algorithm (BBQ) N. CesaBianchi, C. Gentile and F. Orabona. Robust Bounds for Classification via Selective Sampling. In Proc. of the International Conference on Machine Learning (ICML), 2009 universitylogo
60 OM2 Projectron BBQ The Parametric BBQ Algorithm Parameters: 0 < ε, δ < 1 for t = 1, 2,..., T do Receive new instance x t Predict ŷ t = w x ( t ) r = x t I + S t 1 St 1 + x t x 1 t xt ( ) q = St 1 I + S t 1 St 1 + x t x 1 t xt ( ) s = I + S t 1 St 1 + x t x 1 t xt if [ ε r s ] + < q t(t + 1) 2 ln then 2δ Query label y t Update w with a Regularized Least Square S t = [S t, x t ] end if end for The algorithm follows the general framework Every time a query is not issued the predicted output is far from the correct one at most by ε
61 OM2 Projectron BBQ Regret Bound of Parametric BBQ Theorem If Parametric BBQ is run with input ε, δ (0, 1) then: with probability at least 1 δ, t t ε holds on all time steps t when no query is issued; the number N T of queries issued after any number T of steps is bounded as ( ( d N T = O ε 2 ln T ) ) ln(t /δ) ln. δ ε
62 Synthetic Experiment OM2 Projectron BBQ Maximum error Theoretical maximum error Measured maximum error Fraction of queried labels ε Fraction of queried labels We tested Parametric BBQ. 10,000 random examples on the unit circle in R 2. The labels were generated according to our noise model using a randomly selected hyperplane u with unit norm. universitylogo
63 Real World Experiments OM2 Projectron BBQ F measure SOLE Random Parametric BBQ Fraction of queried labels F measure SOLE 0.48 Random Parametric BBQ Fraction of queried labels Fmeasure and fraction of queried labels for different algorithms on Adult9 dataset (left)(gaussian Kernel) and RCV1 (right)(linear kernel). universitylogo
64 Summary OM2 Projectron BBQ Many real world problem cannot be solved using the standard batch framework The online learning framework offers a useful tool in these cases A general algorithm that covers many of the previous known online learning algorithms has been presented The framework allows to easily design algorithms for specific problems
65 OM2 Projectron BBQ Thanks for your attention Code: My website:
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