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To sample the uncertain data examples, the logistic classifier which is offline, prob- abilistic, and linear in the parameters is applied. To meet the online requirement of our approach, this classifier will be adapted as will be shown below.

Logistic regression corresponds to the following binary classification model:

y|x∼Bern(µ0) (2.11)

y|x has the Bernoulli distribution with parameter µ0 given as:

µ0 =p(y= 1|x,θ0) =sigm(θ0Tx) (2.12) where sigm(a) refers to the sigmoid function, sigm(a) = _1+exp(1 _−a).

Note that in this work, binary classification is considered. However, it is easy to extend logistic regression to multi-class classification. In the following, the adaptation of logistic regression to online classification is discussed and then our method for handling concept drift using online logistic regression is introduced.

2.2.2.1 Bayesian view of online logistic regression

Logistic regression can be trained in an online mode either by using stochastic optimization or by adopting a Bayesian view which has the obvious advantage of returning posterior instead of just point estimate. In this work, we use the Bayesian view because we want to capture the non-deterministic nature (uncertainty) of the querying process that itself exploits the label uncertainty criterion. The Bayesian approach of the logistic regression is expressed through:

p(θ0_t|DLt)∝p((xt, yt)|θ 0

t)p(θ

0

t|DLt−1) (2.13) where DLt is the labeled data seen up to timet. Suppose that at time t−1 our knowledge about the parameters θ0_t−1 is summarized by the posterior distribution

p(θ0_t−1|DLt−1). After receiving an observationxt, we consider the probability:

p(yt|xt, DLt−1) =

Z

θ0

t

where p(θ_t0|DLt−1) is the predicted posterior which can be expressed as: p(θ_t0|DLt−1) = Z θ0 t−1 p(θ0_t|θ0_t−1)p(θ_t0₋₁|DLt−1)dθt0−1 (2.15)

In order to calculate the expressionp(θ0_t|θ_t0₋₁), we must specify how the parameters change over time. Following [28], we assume no knowledge of the drifting distribu- tion p(θ_t0|θ_t0₋₁). Thus, Eq. (2.15) can be eliminated by estimating p(yt|xt, DLt−1) which is done as follows:

p(yt|xt, DLt−1) = Z θ0_t−1 p(yt|xt,θ_t0−1)p(θ 0 t−1|DLt−1)dθ0_t−1 (2.16)

Here, two challenges need to be dealt with:

• predicting the class of the new data example xt by computing the posterior

predictive distribution (Eq. (2.16))

• updating the posterior distribution p(θ_t0|DLt) as soon as the label of xt is obtained by computing Eq. (2.13)

Both challenges require to approximate the prior distribution over the weight θ0_t as a Gaussian distribution N(µt,Σt). Two methods have been proposed in the

literature to approximate the integral of Eq. (2.16): Monte Carlo approximation and probit approximation [82]. We use the probit approximation as it does not require sampling, thus it takes less computational time. However, it has been found that probit approximation gives very similar results to the Monte Carlo approximation [82]. The result of the approximation is as follows:

p(ˆyt= 1|xt, DLt−1)≈∇ˆt=sigm(K(st)¯at) (2.17) s2_t =xtTΣt−1xt (2.18) ¯ at=µt−1Txt (2.19) K(st) = (1 + πs2 t 8 ) −1 2 (2.20)

For more details about the approximation steps, the interested reader is referred to [82].

In the following, we show how the parameters of the proposed online logistic regression classifier are updated. We use Newton’s method as formulated in [83] to sequentially updateθ0_t = (Σt,µt) of Eq. (2.13) using Eq. (2.17). Therefore, the

mean µt and covariance Σt can be updated as follows:

Σt =Σt−1− ˆ ∇t(1−∇ˆt) 1 + ˆ∇t(1−∇ˆt)s2t (Σt−1xt)((Σt−1xt))T µt =µt−1+Σtxt(yt−∇ˆt) (2.21)

These recursive equations reflect on the current and the past data. Nevertheless, the effect of data is implicitly decreasing as more data is processed due to the variance shrinkage.

2.2.2.2 Handling of concept drift

In non stationary setting, a variant version of (2.21) proposed in [28] is used. The situation is exactly the same as for the stationary case, except that the prior distribution is nowN(µt−1,Σt−1+vtA). HereA=cI withcis a constant andI

is the identity matrix. vtAassumes that the weight vectorθ_t0changes are of similar

magnitude. Alternatively, one could use a separate forgetting matrix parameter for every weight coordinate as discussed in [84]. In this work, we consider the first assumption: Σt =(Σt−1+vtA)− ˆ ∇t(1−∇ˆt) 1 + ˆ∇t(1−∇ˆt)s0t2 [(Σt−1+vtA)xt][(Σt−1+vtA)xt]T µt =µt−1 +Σtxt(yt−∇ˆt) (2.22)

wheres0_t2 =xT_t(Σt−1+vtA)xt. Here vt can be thought of as the Bayesian version

of the window size in batch learning for data stream. In order to adjust the model as soon as it becomes unable to estimate the true changing θ0 distribution, we compute the discrepancy between the predictive class uncertainty after and before observing the true class label:

where ˆU(xt) is the uncertainty of the label predicted from xt and ˜U(xt) is the

uncertainty remaining after incorporating the true class label.

ˆ

U(xt) = E[(ˆyt−yt)2|xt]

˜

U(xt) = E[(˜yt−yt)2|xt] (2.24)

ˆ

yt is the predicted label and ˆyt = 1( ˆ∇t > 0.5). ˜yt is the predicted label after

updating the classification model with the true label. ˜yt=1( ˜∇t >0.5), where ˜∇

can be computed as follows:

p(˜yt= 1|xt, DLt)≈∇˜t=sigm(K(˜st)˜at) (2.25) ˜ s2_t =xtTΣtxt (2.26) ˜ at=µtTxt (2.27) K(˜st) = (1 + πs˜2_t 8 ) −1 2 (2.28)

The discrepancy Gt of the model uncertainty is monitored using a forgetting

technique[85]: vt = αvt−1+ max(Gt,0) N Lt (2.29) N Lt=αN Lt−1+l(xt) (2.30) N Lt is the prequential number of labeled data. l(xt) is 1 if xt is labeled and 0

otherwise. α is a constant fading factor empirically set to 0.9. Equation (2.29) will be used to update Eq. (2.22).

Finally, the online logistic regression classifier is incrementally updated using (2.22) that addresses concept drift.