Order Regularization on Ordinal Loss for Head Pose, Age and Gaze Estimation

Ordinal loss is widely used in solving regression problems with deep learning technologies. Its basic idea is to convert regression to classification while preserving the natural order. However, the order constraint is enforced only by ordinal label implicitly, leading to the real output values not strictly in order. It causes the network to learn separable feature rather than discriminative feature, and possibly overfit on training set. In this paper, we propose order regularization on ordinal loss, which makes the outputs in order by explicitly constraining the ordinal classifiers in order. The proposed method contains two parts, i.e. similar-weights constraint, which reduces the ineffective space between classifiers, and differential-bias constraint, which enforces the decision planes in order and enhances the discrimination power of the classifiers. Experimental results show that our proposed method boosts the performance of original ordinal loss on various regression problems such as head pose, age, and gaze estimation, with significant error reduction of around 5%. Furthermore, our method outperforms the state of the art on all these tasks, with the performance gain of 14.4%, 2.2% and 6.5% on head pose, age and gaze estimation respectively. Introduction Benefiting from the strong ability of feature representation, convolution neural network (CNN) is widely used to solve regression problems, such as head pose (Yang et al. 2019; Ruiz, Chong, and Rehg 2018), age (Li et al. 2019; Chen et al. 2017; Zhang et al. 2017b), gaze (Park et al. 2019; Krafka et al. 2016; Cheng et al. 2020), and depth estimation (Fu et al. 2018). Most researchers prefer enhanced Softmax (Gao et al. 2017) or ordinal loss (Chen et al. 2017; Fu et al. 2018) to L2 loss, because such loss functions quantize the continuous value to discrete value, converting the regression problem to a classification problem, which is less sensitive to outliers compared with L2 loss. Among them, ordinal loss is outstanding, because it preserves the property of the regression problem, which means that the farther from the ground truth the prediction, the larger the punishment. In order to employ the ordinal loss, a continuous value gt is converted to an ordinal label y, which is a vector with the *Work was done when they were employed by SRC-B. Copyright © 2021, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. length of N , using the following formula: y = { 1, if (n+ 1) ·BinSize+Rmin ≤ gt 0, otherwise (1) Where y(0≤n < N) is the n-th component of y, and BinSize quantizes the regression range [Rmin, Rmax] into N + 1 intervals. Each y has a corresponding binary classifier, i.e. ordinal classifier, and the ordinal loss is defined as the cross-entropy loss to supervise all N binary classifiers with y. Now we focus on the ordinal classifier. The decision plane of the n-th classifier is denoted as g(wn, bn) := wnx + bn, for a feature x extracted by CNN. It judges whether the condition in Eq. 1 is satisfied. For greater regression values, more and more classifiers output 1 sequentially. Thus, intuitively there should be the following constraint: g(w0, b0) ≥ g(w1, b1) ≥ . . . ≥ g(wN−1, bN−1) We call it implicit order constraint in ordinal loss. However, this constraint may not be satisfied in real situations. We observed that the values computed with the decision planes are not strictly in order, as shown in Fig. 1(a). The invalid order problem may cause the classifiers easy to overfit, since the learned feature is separable rather than discriminative. Fig. 1-(b) shows the 2D geometric interpretation with a toy model consisting of three classifiers. The training samples (represented as black shapes) can be perfectly classified. However, the feature is separable rather than discriminative. Thus a test sample in star category (i.e. the red star) may be misclassified to the circle category, crossing several planes, which has larger error than misclassified to the neighbouring triangle category. In this paper, we propose an order regularization to constrain the order of the classifiers explicitly. The basic idea is that given a x, the output values, i.e. wnx + bn, n = 0, 1 · · ·N − 1 in order can be accomplished through constraining the decision planes in order. To achieve this goal, firstly, we make the weights of all decision planes be similar by introducing similar-weights constraint, which means w0 ≈ w1 ≈ · · · ≈ wN−1. Secondly, we make all the bias bn, n = 0, 1, · · ·N − 1 in order by introducing differentialbias constraint. The 2D geometric interpretation is shown in The Thirty-Fifth AAAI Conference on Artificial Intelligence (AAAI-21)