A Novel Manifold Regularized Online Semi-supervised Learning Model

Abstract

In the process of human learning, training samples are often obtained successively. Therefore, many human learning tasks exhibit online and semi-supervision characteristics, that is, the observations arrive in sequence and the corresponding labels are presented very sporadically. In this paper, we propose a novel manifold regularized model in a reproducing kernel Hilbert space (RKHS) to solve the online semi-supervised learning (OS²L) problems. The proposed algorithm, named Model-Based Online Manifold Regularization (MOMR), is derived by solving a constrained optimization problem. Different from the stochastic gradient algorithm used for solving the online version of the primal problem of Laplacian support vector machine (LapSVM), the proposed algorithm can obtain an exact solution iteratively by solving its Lagrange dual problem. Meanwhile, to improve the computational efficiency, a fast algorithm is presented by introducing an approximate technique to compute the derivative of the manifold term in the proposed model. Furthermore, several buffering strategies are introduced to improve the scalability of the proposed algorithms and theoretical results show the reliability of the proposed algorithms. Finally, the proposed algorithms are experimentally shown to have a comparable performance to the standard batch manifold regularization algorithm.

Appendix

In this Appendix, we give out the derivation process of Eq. 13.

For simplicity, we define D and W as

$$\begin{array}{@{}rcl@{}} D_{ij} = \left\{ \begin{array}{lcl} {w_{ij}} &\text{if} &0<i=j<t+1 \\ {{\sum}_{i=1}^{t}w_{it+1}} &\text{if} &i=j=t+1\\ {0} &\text{otherwise} \end{array} \right. \end{array} $$

(23)

$$\begin{array}{@{}rcl@{}} W_{ij} = \left\{ \begin{array}{lcl} {w_{ij}} &\text{if} &0<i<t+1,j=t+1 \\ {w_{ij}} &\text{if} &i=t+1,0<j<t+1 \\ {0} &\text{otherwise} \end{array} \right. \end{array} $$

(24)

Substituting (10), (23), (24) into (12) and letting L = D − W, we have

$$\begin{array}{@{}rcl@{}} \begin{array}{llll} {L(\alpha,\xi_{t+1},\gamma_{t+1},\beta_{t+1})}& =& \frac{1}{2}\alpha^T(K+\lambda_1K+\lambda_2KLK)\alpha \\ &&-\gamma_{t+1}(y_{t+1}\alpha^TJ-1+\xi_{t+1})\\ &&-\alpha^TK\tilde{\alpha}^t-\beta_{t+1}\xi_{t+1}+C\xi_{t+1}+c_0 \end{array}\\ \end{array} $$

(25)

where α = [α ₁,..., α _t+1]^T, $\tilde {\alpha }^{t} = [{\alpha ^{t}_{1}},...,{\alpha ^{t}_{t}},0]^{T}$, K is a (t + 1) × (t + 1) Gram Matrix with K _{i j} = K(x _i , x _j ), J = K e, e = [0,..., 0, 1]^T is a (t + 1)-dimensional vector and c ₀ is a constant.

Note that L(α, ξ _t+1, γ _t+1, β _t+1) attains its minimum with respect to α and ξ _t+1, if and only if the following conditions are satisfied:

$$\begin{array}{@{}rcl@{}} \nabla_{\alpha}{L(\alpha,\xi_{t+1},\gamma_{t+1},\beta_{t+1})} = 0, \end{array} $$

(26)

$$\begin{array}{@{}rcl@{}} \nabla_{\xi_{t+1}}{L(\alpha,\xi_{t+1},\gamma_{t+1},\beta_{t+1})} = 0. \end{array} $$

(27)

Therefore, we have

$$\begin{array}{@{}rcl@{}} \begin{array}{llll} \frac{\partial L}{\partial \xi_{t+1}} &= -\gamma_{t+1}-\beta_{t+1}+C=0\\ &\Longrightarrow \quad 0\leq\gamma_{t+1}\leq C. \end{array} \end{array} $$

(28)

According to the above identity, we formulate a reduced Lagrangian:

$$\begin{array}{@{}rcl@{}} \begin{array}{llll} L^{R}(\alpha,\gamma_{t+1}) =& \frac{1}{2}\alpha^T(K+\lambda_1K+\lambda_2KLK)\alpha \\ &-\gamma_{t+1}(y_{t+1}\alpha^TJ-1)\\ &-\alpha^TK\tilde{\alpha}^t+c_0. \end{array} \end{array} $$

(29)

Taking derivative of Eq. 29 with respect to α, we have:

$$\begin{array}{@{}rcl@{}} \begin{array}{llll} \frac{\partial L^R}{\partial \alpha} = &(K+\lambda_1K+\lambda_2KLK)\alpha\\ &-K\tilde{\alpha}^t-Jy_{t+1}\gamma_{t+1}. \end{array} \end{array} $$

(30)

Note that ∂ L ^R/∂ α = 0. Therefore, we have:

$$\begin{array}{@{}rcl@{}} \begin{array}{lllll} \alpha = &(K+\lambda_1K+\lambda_2KLK)^{-1}\times\\ &(K\tilde{\alpha}^t+Jy_{t+1}\gamma_{t+1}). \end{array} \end{array} $$

(31)

Substituting (31) back into the reduced Lagrangian (29), we get:

$$\begin{array}{@{}rcl@{}} \begin{array}{lllll} \underset{\gamma_{t+1}}{\max} &\!-\frac{1}{2}(K\tilde{\alpha}^t\,+\,Jy_{t+1}\gamma_{t+1})^TA^{-1}(K\tilde{\alpha}^t\,+\,Jy_{t+1}\gamma_{t+1})\\ &+\gamma_{t+1}\\ \text{s.t.} &\quad \quad \quad \quad \quad \quad 0\leq\gamma_{t+1}\leq C, \end{array}\\ \end{array} $$

(32)

where A = K + λ ₁ K + λ ₂ K L K.

Let $\overline {\gamma }_{t+1}$ be the stationary point of the object function of Eq. 32.

Therefore,

$$\begin{array}{@{}rcl@{}} \begin{array}{lllll} \overline{\gamma}_{t+1} = \frac{1-y_{t+1}J^TA^{-1}K\tilde{\alpha}^t}{J^TA^{-1}J}. \end{array} \end{array} $$

(33)

Assume that the optimal solution of Eq. 32 is γ t+1∗ . Note that the object function (32) is quadratic, so the optimal solution γ t+1∗ in the interval [0, C] is at either 0, C or $\overline {\gamma }_{t+1}$. Hence

$$\begin{array}{@{}rcl@{}} {\gamma}_{t+1}^{*} = \left\{ \begin{array}{lcl} 0,\quad \text{if} \quad \overline{\gamma}_{t+1}\leq 0 \\ C,\quad \text{if} \quad \overline{\gamma}_{t+1}\geq 0 \\ \overline{\gamma}_{t+1},\quad \text{otherwise} \end{array} \right. \end{array} $$

(34)

Furthermore, if δ _t+1 = 0, we can obtain the solution of the proposed model by the similar process as above. Thus, the classifier obtained at time t + 1 is:

$$\begin{array}{@{}rcl@{}} \begin{array}{llllll} &f_{t+1}(x) = \sum\limits_{i=1}^{t+1}\alpha_i^{t+1}K(x_{t+1},x), \\ &h_{t+1} = \text{sign}({f_{t+1}(x)}), \end{array} \end{array} $$

(35)

where

$$\begin{array}{@{}rcl@{}} \begin{array}{lllll} \alpha^{t+1} =A^{-1}(K\tilde{\alpha}^{t}+\delta_{t+1} y_{t+1}\gamma_{t+1}^{*}J). \end{array} \end{array} $$