semiboost : boosting for semi-supervised learning

SemiBoost : Boosting for Semi-supervised Learning

Pavan Kumar Mallapragada, Student Member, IEEE, Rong Jin, Member, IEEE,

Anil K. Jain, Fellow, IEEE, and Yi Liu, Student Member, IEEE

Presented by Yueng-Tien ,LoReference:SemiBoost: Boosting for Semi-supervised Learning P. K. Mallapragada, R. Jin, A. K. Jain, and Y. Liu, IEEE Transaction on Pattern Analysis and Machine Intelligence(PAMI), 31(11):2000-2014, 2009

outline

• Introduction• Related Work• Semi-Supervised Boosting• Results and Discussion• Conclusions and Future Work• Gaussian Fields and Harmonic Functions

Introduction

• The key idea of semi-supervised learning, specifically semi-supervised classification, is to exploit both labeled and unlabeled data to learn a classification model.

• There is an immense need for algorithms that can utilize the small amount of labeled data, combined with the large amount of unlabeled data to build efficient classification systems.

Introduction

• Existing semi-supervised classification algorithms may be classified into two categories based on their underlying assumptions.– manifold assumption – cluster assumption

Introduction

• manifold assumption : the data lie on a low dimensional manifold in the input space

• cluster assumption : the data samples with high similarity between them must share the same label.

Introduction

• Most semi-supervised learning approaches design specialized learning algorithms to effectively utilize both labeled and unlabeled data.

• We refer to this problem of improving the performance of any supervised learning algorithm using unlabeled data as Semi-supervise Improvement, to distinguish our work from the standard semi-supervised learning problems.

Introduction

• The key difficulties in designing SemiBoost are: – how to sample the unlabeled examples for training a new

classification model at each iteration – what class labels should be assigned to the selected

unlabeled examples.

Introduction

• One way to address the above questions is to exploit both the clustering assumption and the large margin criterion

• Selecting the unlabeled examples with the highest classification confidence and assign them the class labels that are predicted by the current classifier.

• A problem with this strategy is that the introduction of examples with predicted class labels may only help to increase the classification margin, without actually providing any novel information to the classifier.

Introduction

• To overcome the above problem, we propose using the pairwise similarity measurements to guide the selection of unlabeled examples at each iteration, as well as for assigning class labels to them.

Related Work

Related Work

• An inductive algorithm can be used to predict the labels of samples that are unseen during training (irrespective of it being labeled or unlabeled).

• Transductive algorithms are limited to predicting only the labels of the unlabeled samples seen during training.

Related Work

• A popular way to define the inconsistency between the labels of the samples and the pairwise similarities Sij is the quadratic criterion:

where L is the combinatorial graph Laplacian.• The task is to assign values to the unknown labels in

such a way that the overall inconsistency is minimized.

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Semi-Supervised Improvement

• denote the entire data set, including both the labeled and the unlabeled examples.– the first nl examples are labeled, given by – the imputed class labels of unlabeled examples

• Let denote the symmetric similarity matrix, where represents the similarity between xi and xj.– denote the submatrix of the similarity matrix– denote the submatrix of the similarity matrix

• ‘A’ denote the given supervised learning algorithm

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Semi-Supervised Improvement

• The goal of semi-supervised improvement is to improve the performance of A iteratively by treating A like a black box, using the unlabeled examples and the pairwise similarity S.

• In the semi-supervised improvement problem, we aim to build an ensemble classifier which utilizes the unlabeled samples in the way a graph-based approach would utilize.

An outline of the SemiBoost algorithm for semi-supervised improvement

• Start with an empty ensemble.• At each iteration,– Compute the peusdo label (and its confidence)

for each unlabeled example (using existing ensemble,and the pairwise similarity).

– Sample most confident pseudo labeled examples;combine them with the labeled samples andtrain a component classifier using the supervised learning algorithm A.

– Update the ensemble by including the componentclassifier with an appropriate weight.

SemiBoost

• The unlabeled samples must be assigned labels following two main criteria: – The points with high similarity among unlabeled samples

must share the same label – Those unlabeled samples which are highly similar to a

labeled sample must share its label.

• Our objective function is a combination of two terms: – one measuring the inconsistency between labeled and

unlabeled examples and the other measuring the inconsistency among the unlabeled examples

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SemiBoost

• Inspired by the harmonic function approach, we define , the inconsistency between class labels y and the similarity measurement S, as

• Note that (1) can be expanded as

and due to the symmetry of S

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SemiBoost

• We have

where is the hyperbolic cosine function.

• Rewriting (1) using the function reveals the connection between the quadratic penalty used in the graph-Laplacian-based approaches, and the exponential penalty used in the current approach

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SemiBoost

• Using a penalty not only facilitates the derivation of boosting-based algorithms but also increases the classification margin.

• The inconsistency between labeled and unlabeled examples is defined as

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SemiBoost

• Combining (1) and (3) leads to the objective function,

• The constant C is introduced to weight the importance between the labeled and the unlabeled data.

• Given the objective function in (4), the optimal class label yu is found by minimizing F.

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SemiBoost

• The problem can now be formally expressed as

• The following procedure is adopted to derive the boosting algorithm:– The labels for the unlabeled samples yi

u are replaced by the ensemble predictions over the corresponding data sample.

– A bound-optimization-based approach is then used to find the ensemble classifier minimizing the objective function.

– The bounds are simplified further to obtain the sampling scheme, and other required parameters.

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The SemiBoost algorithm

• Compute the pairwise similarity Si,j between any two examples.

• Initialize H(x) = 0• For t = 1, 2, . . . , T

– Compute pi and qi for every example using Equations (9)and(10)

– Compute the class label zi = sign(pi − qi) for each example

– Sample example xi by the weight |pi − qi|– Apply the algorithm A to train a binary classifier ht(x) using the sampled examples and their class

labels zi– Compute αt using Equation (11)– Update the classification function as H(x) ← H(x) + αtht(x)

SemiBoost-Algorithm

• Let denote the two-class classifi-cation model that is learned at the t-th iteration by the algorithm A.

• Let denote the combined classification model learned after the first T iterations.

where is the combination weight.

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SemiBoost-Algorithm

• This leads to the following optimization problem:

• This expression involves products of variables and making it nonlinear and, hence , difficult to optimize.

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Appendix

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SemiBoost-Algorithm (prop.1)

• Minimizing (7) is equivalent to minimizing the function

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Appendix

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• The expression in (8) is difficult to optimize since the weight and the classifier h(x) are coupled together.

• Minimizing (8) is equivalent to minimizing

• We denote the upper bound in the above equation by F2.

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• To minimize F2, the optimal class label zi for the example xi is zi = sign(pi – qi) and the weight for sampling example xi is |pi-qi|. The optimal that minimizes F1 is

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SemiBoost-Algorithm

• At each relaxation, the “touch-point” is maintained between the objective function and the upper bound.

• As a result, the procedure guarantees: – The objective function always decreases through iterations– The final solution converges to a local minimum.

The SemiBoost algorithm

• Compute the pairwise similarity Si,j between any two examples.

• Initialize H(x) = 0• For t = 1, 2, . . . , T– Compute pi and qi for every example

using Equations (9)and(10)– Compute the class label zi = sign(pi − qi) for each example– Sample example xi by the weight |pi − qi|– Apply the algorithm A to train a binary classifier ht(x) using the sampled examples and their class labels zi– Compute αt using Equation (11)– Update the classification function as H(x) ← H(x) + αtht(x)

SemiBoost-Algorithm

• Let be the weighted error made by the classifier, where

• As in the case of AdaBoost, can be expressed as

which is very similar to the weighting factor of AdaBoost, differing only by a constant factor of 1/2 .

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SemiBoost-Algorithm

• Theorem 1. Let be the combination weights that are computed by running the SemiBoost algorithm (Fig. 1). Then, the objective function at the et t 1Tst iteration, i.e., Ftt1, is bounded as follows:

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Results and Discussion

• X. Zhu, Z. Ghahramani, and J. Lafferty, “Semi-supervised learning using gaussian fields and harmonic functions,” in Proc. 20t International Conference on Machine Learning, pp. 912–919, 2003