learning meta face recognition in unseen...

Learning Meta Face Recognition in Unseen Domains

Jianzhu Guo1,2 Xiangyu Zhu1,2 Chenxu Zhao3 Dong Cao1,2 Zhen Lei1,2∗ Stan Z. Li4

1CBSR & NLPR, Institute of Automation, Chinese Academy of Sciences, Beijing, China2School of Artificial Intelligence, University of Chinese Academy of Sciences, Beijing 100049, China

3Mininglamp Academy of Sciences, Mininglamp Technology4School of Engineering, Westlake University, Hangzhou, China

{jianzhu.guo, xiangyu.zhu, dong.cao, zlei, szli}@nlpr.ia.ac.cn, [email protected]

Abstract

Face recognition systems are usually faced with unseen

domains in real-world applications and show unsatisfac-

tory performance due to their poor generalization. For ex-

ample, a well-trained model on webface data cannot deal

with the ID vs. Spot task in surveillance scenario. In this

paper, we aim to learn a generalized model that can directly

handle new unseen domains without any model updating.

To this end, we propose a novel face recognition method

via meta-learning named Meta Face Recognition (MFR).

MFR synthesizes the source/target domain shift with a meta-

optimization objective, which requires the model to learn

effective representations not only on synthesized source do-

mains but also on synthesized target domains. Specifi-

cally, we build domain-shift batches through a domain-level

sampling strategy and get back-propagated gradients/meta-

gradients on synthesized source/target domains by optimiz-

ing multi-domain distributions. The gradients and meta-

gradients are further combined to update the model to

improve generalization. Besides, we propose two bench-

marks for generalized face recognition evaluation. Experi-

ments on our benchmarks validate the generalization of our

method compared to several baselines and other state-of-

the-arts. The proposed benchmarks and code will be avail-

able at https://github.com/cleardusk/MFR.

1. Introduction

Face recognition is a long-standing topic in the research

community. Recent works [1, 2, 3, 4, 5, 6, 7, 8] have pushed

the performance to a very high level on several common

benchmarks, e.g. LFW [9], YTF [10] and MegaFace [11].

These methods are based on the assumption that the training

sets like CASIA-Webface [12], MS-Celeb [13] and testing

sets have similar distribution. However, in real-world ap-

∗Corresponding author

African

Asian

Caucasian

Indian

Source Domains Target Domains

Learning generalized

face recognition

NIR-VIS

Large Pose

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Cross Age

Eye-glass Occlusion

ID vs. Spot

Meta-train

Meta-test

Meta

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Figure 1: An illustration of our MFR for generalized face

recognition problem. The left column contains four source

domains of different races, the right includes five target

domains: cross-age (CACD-VS), NIR-VIS face matching

(CASIA NIR-VIS 2.0), large pose (Multi-PIE), eyeglass oc-

clusion (MeGlass) and ID vs. Spot (Public-IvS), which are

unseen in training. By meta-learning on the simulated meta-

train/meta-test shifts in source domains, our model learns

the transferable knowledge across domains to generalize

well on target unseen domains.

plications of face recognition, the model trained on source

domainsDS is usually deployed in another domainDT with

a different distribution. There are two kinds of scenarios: (i)

the target domain DT is known and the data is accessible.

(ii) the target domain is unseen. Approaches to the first sce-

nario are categorized into domain adaptation for face recog-

nition, where the common setting is that the source domain

DS contains a labelled face domain and the target domain

DT is with or without labels. Domain adaption methods try

to adapt the knowledge learned from DS to DT so that the

model generalizes well on DT . The second scenario can

be regarded as domain generalization for face recognition,

and we call it Generalized Face Recognition, which is more

common as the trained model is usually deployed in un-

known scenarios and faced with unseen data. As illustrated

in Fig. 1, the deployed model should be able to generalize

to unseen domains without any updating or fine-tuning.

6163

Compared with domain adaptation, generalized face

recognition is less studied and more challenging, since it

makes no assumptions about target domains. To the best of

our knowledge, there are no relative studies on generalized

face recognition problem. A related task is domain gener-

alization on visual recognition, it assumes that the source

and target domains share the same label space, and has a

small set, e.g., 7 categories [14]. However, generalized face

recognition is an open-set problem and has a much larger

scale of categories, making existing methods inapplicable.

In this paper, we aim to learn a model for general-

ized face recognition problem. Once trained on a set

of source domains, the model can be directly deployed

on an unseen domain without any model updating. In-

spired by [14, 15], we propose a novel face recognition

framework via meta-learning named Meta Face Recogni-

tion (MFR). MFR simulates the source/target domain shift

with a meta-optimization objective, which optimizes the

model to learn effective face representations not only on

synthesized source domains but also on synthesized target

domains. Specifically, a domain-level sampling strategy is

adopted to simulate the domain shift such that source do-

mains are divided into meta-train/meta-test domains. To op-

timize multi-domain distributions, we propose three com-

ponents: 1) the hard-pair attention loss optimizes the local

distribution with hard pairs, 2) soft-classification loss con-

siders the global relationship within a batch and 3) domain

alignment loss learns to reduce meta-train domains discrep-

ancy by aligning domain centers. These three losses are

combined to learn domain-invariant and discriminative face

representations. The gradients from meta-train domains and

meta-gradients from meta-test domains are finally aggre-

gated by meta-optimization, and are then used to update the

network to improve model generalization. Compared with

traditional meta-learning methods, our MFR does not need

model updating for target domains and can directly handle

unseen domains.

Our main contributions include: (i) For the first time, we

highlight the generalized face recognition problem, which

requires a trained model to generalize well on unseen do-

mains without any updating. (ii) We propose a novel Meta

Face Recognition (MFR) framework to solve generalized

face recognition, which meta-learns transferable knowledge

across domains to improve model generalization. (iii) Two

generalized face recognition benchmarks are designed for

evaluation. Extensive experiments on the proposed bench-

marks validate the efficacy of our method.

2. Related work

Domain Generalization. Domain generalization can be

traced back to [16, 17]. DICA [17] adopts the kernel-based

optimization to learn domain-invariant features. CCSA [18]

can handle both domains adaptation and domain generaliza-

tion problems by aligning a source domain distribution to a

target domain distribution. MLDG [14] firstly applies the

meta-learning method MAML [15] for domain generaliza-

tion. Compared with domain adaptation, domain general-

ization is a less investigated problem. Besides, the above

domain generalization works mainly focus on the closed-

set category-level recognition problems, where the source

and target domains share the same label space. In contrast,

our generalized face recognition problem is much more

challenging because the target classes are disjoint from the

source ones. It means that generalized face recognition is

an open-set problem rather than the closed-set problem like

MLDG [14], and we must handle the domain gap and the

disjoint label space simultaneously. One related work is

DIMN [19], but it differs from ours in both task and method.

Meta Learning. Recent meta-learning studies concen-

trate on: (i) learning a good weight initialization for fast

adaptation on a new task, such as the foundational work

MAML [15] and its variants Reptile [20], meta-transfer

learning [21], iMAML [22] and so on. (ii) learning an em-

bedding space with a well-designed classifier that can di-

rectly classify samples on a new task without fast adapta-

tion [23, 24, 25]. (iii) learning to predict the classification

parameters [26, 27] after pre-training a good feature extrac-

tor on the whole training set. These works focus on few-

shot learning, where the common setting is that the target

task has very few data points (1/5/20 shots per class). In

contrast, generalized face recognition should handle thou-

sands of classes, making it more challenging and generally

applicable. Our approach is most related to MAML [15]

that tries to learn a transferable weight initialization. How-

ever, MAML requires fast adaptation on a target task, while

our MFR does not require any model updating as target do-

mains are unseen.

3. Methology

This section describes the proposed MFR to address gen-

eralized face recognition problem. MFR consists of three

parts: (i) the domain-level sampling strategy. (ii) three

losses for optimizing multi-domain distributions to learn

domain-invariant and discriminative face representations.

(iii) the meta-optimization procedure to improve model

generalization, shown in Fig. 3. The overview is shown in

Fig. 2 and Algorithm 1.

3.1. Overview

In the training stage, we have access to N source do-

mains DS = {DS1 , · · · ,D

SN | N > 1}, and each domain

DSi = {(xi

j , yij)} has its own label set. In the testing phase,

the trained model is evaluated on one or several unseen tar-

get domains, DT = {DT1 , · · · ,D

TM | M ≥ 1}, without any

model updating. Besides, the label sets YT of the target do-

mains are disjoint from the label sets YS of source domains,

6164

Source Domains

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Domain-level

Sampling







Meta-batch

meta-train

meta-test

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ProbeGallery

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Train

Next Step

Target Domains

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Test

…

ProbeGallery

…

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meta-train meta-test

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…

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Hard positive pairs

Hard negative pairs

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Lcls<latexit sha1_base64="7Je8Y6qRwGwCdmWjaFut5yi5XHM=">AAAB+nicdVDLSsNAFJ3UV62vVJduBovgKiQxtBVcFNy4cFHBPqANYTKdtEMnD2YmSon5FDcuFHHrl7jzb5y0FVT0wMDhnHu5Z46fMCqkaX5opZXVtfWN8mZla3tnd0+v7ndFnHJMOjhmMe/7SBBGI9KRVDLSTzhBoc9Iz59eFH7vlnBB4+hGzhLihmgc0YBiJJXk6dVhiOQEI5Zd5V6Gmcg9vWYaZ8267dShaZhmw7KtgtgN59SBllIK1MASbU9/H45inIYkkpghIQaWmUg3Q1xSzEheGaaCJAhP0ZgMFI1QSISbzaPn8FgpIxjEXL1Iwrn6fSNDoRCz0FeTRVDx2yvEv7xBKoOmm9EoSSWJ8OJQkDIoY1j0AEeUEyzZTBGEOVVZIZ4gjrBUbVVUCV8/hf+Trm1YpmFdO7XW+bKOMjgER+AEWKABWuAStEEHYHAHHsATeNbutUftRXtdjJa05c4B+AHt7RNGfJSk</latexit><latexit sha1_base64="7Je8Y6qRwGwCdmWjaFut5yi5XHM=">AAAB+nicdVDLSsNAFJ3UV62vVJduBovgKiQxtBVcFNy4cFHBPqANYTKdtEMnD2YmSon5FDcuFHHrl7jzb5y0FVT0wMDhnHu5Z46fMCqkaX5opZXVtfWN8mZla3tnd0+v7ndFnHJMOjhmMe/7SBBGI9KRVDLSTzhBoc9Iz59eFH7vlnBB4+hGzhLihmgc0YBiJJXk6dVhiOQEI5Zd5V6Gmcg9vWYaZ8267dShaZhmw7KtgtgN59SBllIK1MASbU9/H45inIYkkpghIQaWmUg3Q1xSzEheGaaCJAhP0ZgMFI1QSISbzaPn8FgpIxjEXL1Iwrn6fSNDoRCz0FeTRVDx2yvEv7xBKoOmm9EoSSWJ8OJQkDIoY1j0AEeUEyzZTBGEOVVZIZ4gjrBUbVVUCV8/hf+Trm1YpmFdO7XW+bKOMjgER+AEWKABWuAStEEHYHAHHsATeNbutUftRXtdjJa05c4B+AHt7RNGfJSk</latexit><latexit sha1_base64="7Je8Y6qRwGwCdmWjaFut5yi5XHM=">AAAB+nicdVDLSsNAFJ3UV62vVJduBovgKiQxtBVcFNy4cFHBPqANYTKdtEMnD2YmSon5FDcuFHHrl7jzb5y0FVT0wMDhnHu5Z46fMCqkaX5opZXVtfWN8mZla3tnd0+v7ndFnHJMOjhmMe/7SBBGI9KRVDLSTzhBoc9Iz59eFH7vlnBB4+hGzhLihmgc0YBiJJXk6dVhiOQEI5Zd5V6Gmcg9vWYaZ8267dShaZhmw7KtgtgN59SBllIK1MASbU9/H45inIYkkpghIQaWmUg3Q1xSzEheGaaCJAhP0ZgMFI1QSISbzaPn8FgpIxjEXL1Iwrn6fSNDoRCz0FeTRVDx2yvEv7xBKoOmm9EoSSWJ8OJQkDIoY1j0AEeUEyzZTBGEOVVZIZ4gjrBUbVVUCV8/hf+Trm1YpmFdO7XW+bKOMjgER+AEWKABWuAStEEHYHAHHsATeNbutUftRXtdjJa05c4B+AHt7RNGfJSk</latexit><latexit sha1_base64="7Je8Y6qRwGwCdmWjaFut5yi5XHM=">AAAB+nicdVDLSsNAFJ3UV62vVJduBovgKiQxtBVcFNy4cFHBPqANYTKdtEMnD2YmSon5FDcuFHHrl7jzb5y0FVT0wMDhnHu5Z46fMCqkaX5opZXVtfWN8mZla3tnd0+v7ndFnHJMOjhmMe/7SBBGI9KRVDLSTzhBoc9Iz59eFH7vlnBB4+hGzhLihmgc0YBiJJXk6dVhiOQEI5Zd5V6Gmcg9vWYaZ8267dShaZhmw7KtgtgN59SBllIK1MASbU9/H45inIYkkpghIQaWmUg3Q1xSzEheGaaCJAhP0ZgMFI1QSISbzaPn8FgpIxjEXL1Iwrn6fSNDoRCz0FeTRVDx2yvEv7xBKoOmm9EoSSWJ8OJQkDIoY1j0AEeUEyzZTBGEOVVZIZ4gjrBUbVVUCV8/hf+Trm1YpmFdO7XW+bKOMjgER+AEWKABWuAStEEHYHAHHsATeNbutUftRXtdjJa05c4B+AHt7RNGfJSk</latexit>

Optimizing Distributions

Optimizing Multi-

domain Distributions

Meta-optimization

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Meta-

optimization

gradient

meta-gradient

Figure 2: Overview of our proposed MFR. Three source domains are presented in this figure for a demonstration. Each

symbol represents a face image for convenience. MFR consists of three parts: domain-level sampling for simulating domain

shifts, multi-domain distributions optimization to learn face representations and meta-optimization procedure to improve

model generalization. Once trained on source domains, the model can be directly deployed on target unseen domains.

thus making ours problem open-set. During training, we de-

fine a single model represented by a parametrized function

f(θ) with parameters θ. Our proposed MFR aims to train θon source domains DS , such that it can generalize well on

target unseen domains DT , as illustrated in Fig. 1.

3.2. Domainlevel Sampling

To achieve domain generalization, we split source do-

mains into meta-train and meta-test domains during each

training iteration. Specifically, we split N source domains

DS into N−1 domainsDmtr for meta-train and 1 target do-

main Dmte for meta-test, simulating the domain shift prob-

lem existed when deployed in real-world scenarios. In this

way, the model is encouraged to learn transferable knowl-

edge about how to generalize well on the unseen domains

with different distributions. We further build a meta-batch

consisting of several batches as follows: (i) we iterate on

N source domains; (ii) in the i-th iteration, DSi is selected

as the meta-test domain Dmte; (iii) the rest ones as meta-

train domains Dmtr; (iv) we randomly choose B identities

in meta-train domains and B identities in meta-test domain,

and two face images are selected for each identity, in which

one as the gallery the other one as probe. Therefore, a meta-

batch of N batches is built. Our model is then updated by

the accumulated gradients of each meta-batch. The details

are illustrated in Algorithm 1. Different from MAML [15],

our sampling is domain-level for open-set face recognition.

MLDG [14] also performs a similar sampling, but their do-

mains are randomly divided in each training iteration and

no meta-batch is built.

3.3. Optimizing Multidomain Distributions

To aggregate back-propagated gradients within each

batch, we optimize multi-domain distributions such that the

same identities are mapped into nearby representation and

different identities are mapped apart from each other. Tradi-

tional metric losses like contrastive [28, 29] and triplet [3]

take randomly sampled pairs or triplets to build the train-

ing batches. These batches consist of lots of easy pairs or

triplets, leading to the slow convergence of training. To alle-

viate it, we propose to optimize and learn domain-invariant

and discriminative representations with three components.

The hard-pair attention loss optimizes the local distribu-

tion with hard pairs, the soft-classification loss considers the

global distribution within a batch and the domain alignment

loss learns to align domain centers.

Hard-pair Attention Loss. Hard-pair attention loss fo-

cuses on optimizing hard positive and negative pairs. A

batch of B identities are sampled and each identity con-

tains a gallery face and a probe face. We denote the in-

put as X , the gallery and probe embeddings are extracted:

Fg = f(Xg; θ) ∈ RB×C , Fp = f(Xp; θ) ∈ RB×C , where

C is the dimension length. After l2 normalization on Fg

and Fp, we can efficiently construct a similarity matrix by

computing M = FgFTp ∈ RB×B . Then we use a positive

threshold τp and negative threshold τn to filter out the hard

positive pairs and negative pairs: P = {i|Mi,i < τp} and

N = {(i, j)|Mi,j > τp, i 6= j}. This operation just needs

O(B2 log(B)) complexity and it can be formulated as:

Lhp =1

2|P|

∑

i∈P

‖Fgi − Fpi‖22−

1

2|N |

∑

(i,j)∈N

‖Fgi−Fpj‖22,

(1)

where P is indices of hard positive pairs determined by τp,

N is indices of hard negative pairs determined by τn.

Soft-classification Loss. Hard-pair attention loss only

concentrates on hard pairs and tends to converge to a lo-

cal optimum. To alleviate it, we introduce a specific soft-

classification loss to perform classification within a batch.

The loss is formulated as:

Lcls=1

2B

B∑

i=1

(

CE(yi, s · FgiWT ) + CE(yi, s · Fpi

WT ))

,

(2)

where yi = i indicates the i-th identity, FgiWT or Fpi

WT

is the logit of i-th identity and s is a fixed scaling factor. Wis initialized as (Fg + Fp)/2 ∈ RB×C and each row of Wis l2 normalized.

6165

Algi<latexit sha1_base64="8OpVrfKEGytyBAhBznvx3QQuMVM=">AAAB7nicbZC7SwNBEMbn4ivGV9TSZvEIWIU7Gy0jNikjmAckR9jb7CVL9vaO3TkhHAF7KxsLRWz9e+z8b9w8Ck38YOHH982wMxOmUhj0vG+nsLG5tb1T3C3t7R8cHpWPT1omyTTjTZbIRHdCargUijdRoOSdVHMah5K3w/HtLG8/cG1Eou5xkvIgpkMlIsEoWqt9I4f9XEz7ZderenORdfCX4NbcytMjADT65a/eIGFZzBUySY3p+l6KQU41Cib5tNTLDE8pG9Mh71pUNOYmyOfjTknFOgMSJdo+hWTu/u7IaWzMJA5tZUxxZFazmflf1s0wug5yodIMuWKLj6JMEkzIbHcyEJozlBMLlGlhZyVsRDVlaC9UskfwV1deh9Zl1feq/p3v1uqwUBHO4BwuwIcrqEEdGtAEBmN4hld4c1LnxXl3PhalBWfZcwp/5Hz+AH/mkSo=</latexit><latexit sha1_base64="hUpiIcWb+eAey+hbSZuqHTpzuUE=">AAAB7nicbZC7SwNBEMbnfMb4ilraLB4Bq3Bno2XEJmUE84AkhL3NXLJkb+/Y3RPCEQRbKxsLRSSdf4+d/42bR6GJHyz8+L4ZdmaCRHBtPO/bWVvf2Nzazu3kd/f2Dw4LR8d1HaeKYY3FIlbNgGoUXGLNcCOwmSikUSCwEQxvpnnjHpXmsbwzowQ7Ee1LHnJGjbUa16Lfzfi4W3C9kjcTWQV/AW7ZLT49Pkwm1W7hq92LWRqhNExQrVu+l5hORpXhTOA43041JpQNaR9bFiWNUHey2bhjUrROj4Sxsk8aMnN/d2Q00noUBbYyomagl7Op+V/WSk141cm4TFKDks0/ClNBTEymu5MeV8iMGFmgTHE7K2EDqigz9kJ5ewR/eeVVqF+UfK/k3/puuQJz5eAUzuAcfLiEMlSgCjVgMIRneIU3J3FenHfnY1665ix6TuCPnM8f2miS7Q==</latexit><latexit sha1_base64="hUpiIcWb+eAey+hbSZuqHTpzuUE=">AAAB7nicbZC7SwNBEMbnfMb4ilraLB4Bq3Bno2XEJmUE84AkhL3NXLJkb+/Y3RPCEQRbKxsLRSSdf4+d/42bR6GJHyz8+L4ZdmaCRHBtPO/bWVvf2Nzazu3kd/f2Dw4LR8d1HaeKYY3FIlbNgGoUXGLNcCOwmSikUSCwEQxvpnnjHpXmsbwzowQ7Ee1LHnJGjbUa16Lfzfi4W3C9kjcTWQV/AW7ZLT49Pkwm1W7hq92LWRqhNExQrVu+l5hORpXhTOA43041JpQNaR9bFiWNUHey2bhjUrROj4Sxsk8aMnN/d2Q00noUBbYyomagl7Op+V/WSk141cm4TFKDks0/ClNBTEymu5MeV8iMGFmgTHE7K2EDqigz9kJ5ewR/eeVVqF+UfK/k3/puuQJz5eAUzuAcfLiEMlSgCjVgMIRneIU3J3FenHfnY1665ix6TuCPnM8f2miS7Q==</latexit><latexit sha1_base64="GNG5nWMsAW1xZ33xhPWnTHo3/4w=">AAAB7nicbVA9SwNBEJ3zM8avqKXNYhCswp2NlhGblBHMByRH2NvMJUt2947dPSEc+RE2ForY+nvs/Ddukis08cHA470ZZuZFqeDG+v63t7G5tb2zW9or7x8cHh1XTk7bJsk0wxZLRKK7ETUouMKW5VZgN9VIZSSwE03u537nCbXhiXq00xRDSUeKx5xR66TOnRgNcj4bVKp+zV+ArJOgIFUo0BxUvvrDhGUSlWWCGtML/NSGOdWWM4Gzcj8zmFI2oSPsOaqoRBPmi3Nn5NIpQxIn2pWyZKH+nsipNGYqI9cpqR2bVW8u/uf1MhvfhjlXaWZRseWiOBPEJmT+OxlyjcyKqSOUae5uJWxMNWXWJVR2IQSrL6+T9nUt8GvBQ1CtN4o4SnAOF3AFAdxAHRrQhBYwmMAzvMKbl3ov3rv3sWzd8IqZM/gD7/MHYAKPlw==</latexit>

gradient flow

meta-gradient flow

rθLS(θ;D2,3)<latexit sha1_base64="GijdPUeU2XVPAligCQCSyGXL8Yk=">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</latexit><latexit sha1_base64="w2oFYrKnqLPecNmPkFfYJ3HqZco=">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</latexit><latexit sha1_base64="w2oFYrKnqLPecNmPkFfYJ3HqZco=">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</latexit><latexit sha1_base64="smArexpJQpTIyQIuPu73Ckey/Y4=">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</latexit>

rθLT (θ01;D1)

<latexit sha1_base64="BvuWkMuUSXqvAZARZXHPaF+RzxE=">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</latexit><latexit sha1_base64="rNjdZR77Kk9FNlUPQG8rNxeCiaw=">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</latexit><latexit sha1_base64="rNjdZR77Kk9FNlUPQG8rNxeCiaw=">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</latexit><latexit sha1_base64="kLncvGFPvsf5xDdYuSM6k56ImEA=">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</latexit>

rθLS(θ;D1,3)<latexit sha1_base64="Vcj2UNKKA+vze5iZP8XJc0t1wTg=">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</latexit><latexit sha1_base64="mrgVU7yVbNYTYOb3mQZ9l8izAdk=">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</latexit><latexit sha1_base64="mrgVU7yVbNYTYOb3mQZ9l8izAdk=">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</latexit><latexit sha1_base64="MBazpNiiAT/mf4YmQL7kb14o3qY=">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</latexit>

rθLS(θ;D1,2)<latexit sha1_base64="hzMe5Z/XMk0cJZW6/egWFORpPCg=">AAACInicdZBNaxRBEIZr4ldcP7JJjrk0rkIEGXpW3WTJZUly8OAhopsEdpahprc326SnZ+iuCSzD/BYv+Q05e/HiIWI8Cf4N7/buKn6gLzS8PFVFV71poZUjzr8ES9eu37h5a/l2487de/dXmqtrhy4vrZB9kevcHqfopFZG9kmRlseFlZilWh6lp3uz+tGZtE7l5g1NCznM8MSosRJIHiXNbmww1ZhUMU0kYc3iDGkiUFcv6+Q121zgnV94v06q6Em7fpw0WzzsdvmzqMN4+JzzdqfrDX/a3u50WBTyuVq9h98u3gHAQdK8ike5KDNpSGh0bhDxgoYVWlJCy7oRl04WKE7xRA68NZhJN6zmJ9bskScjNs6tf4bYnP4+UWHm3DRLfedsU/d3bQb/VRuUNN4eVsoUJUkjFh+NS80oZ7O82EhZKUhPvUFhld+ViQlaFORTbfgQfl7K/m8O22HEw+hV1OrtwkLLsAEPYBMi2IIevIAD6IOAt/AeLuFjcB58CD4FnxetS8GPmXX4Q8HX70KIpwI=</latexit><latexit sha1_base64="PucDsbXej5fD8u7AKrdgMA2sT9Y=">AAACInicdZBBSxtBFMdnrW1ttDWtx14GY8FCWWZjmxq8hLaHHjwoGhWyYXk7mZjB2dll5q0Qlv0svfQz9AuIlx4qtifBr6FnJ4miFf3DwJ/fe4957x9nSlpk7MybejL99NnzmReV2bmXr+arr9/s2DQ3XLR5qlKzF4MVSmrRRolK7GVGQBIrsRsffB3Vdw+FsTLV2zjMRDeBfS37kgM6FFWboYZYQVSEOBAIJQ0TwAEHVayX0RZdnuC1W/ytjIrgQ718H1VrzG822cegQZn/ibF6o+kMW6mvNho08NlYtdbSxa+jw9nLjaj6N+ylPE+ERq7A2k7AMuwWYFByJcpKmFuRAT+AfdFxVkMibLcYn1jSd470aD817mmkY3p3ooDE2mESu87RpvZ+bQQfqnVy7K92C6mzHIXmk4/6uaKY0lFetCeN4KiGzgA30u1K+QAMcHSpVlwIN5fSx81O3Q+YH2wGtdYXMtEMeUsWyTIJyGfSIt/JBmkTTn6QY/KHnHg/vd/eqfdv0jrlXc8skP/knV8BOyiofA==</latexit><latexit sha1_base64="PucDsbXej5fD8u7AKrdgMA2sT9Y=">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</latexit><latexit sha1_base64="S7GnL7iFnV7nhIYiokGSozAKy+I=">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</latexit>

rθLT (θ02;D2)

<latexit sha1_base64="AZJjUxbBHnPgaJ+EcTsRNHK/O9A=">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</latexit><latexit sha1_base64="dOd4FPKqLiJzPVqsNyPLRxZz1Xs=">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</latexit><latexit sha1_base64="dOd4FPKqLiJzPVqsNyPLRxZz1Xs=">AAACI3icdZDPbtNAEMbXgbYh0DaFI5dVI0SrSpHtRk2BSwQceuihlfKnUhxZ4806WWW9tnbHSJHlt+ABuPAqXDiAIi498C6sGyooKp+00qffN6OdmSiTwqDrXju1Bw83NrfqjxqPn2zv7Db3ng5NmmvGByyVqb6KwHApFB+gQMmvMs0hiSQfRYt3VT76wLURqerjMuOTBGZKxIIBWhQ2XwcKIglhEeCcI5Q0SADnDGRxXoZ9erDGof/yzZ/kfRkWfnkYNltu+9Xpid85oW7bdbue71XG73aOO9SzpFKrtx8cfbzuLS/C5iqYpixPuEImwZix52Y4KUCjYJKXjSA3PAO2gBkfW6sg4WZS3OxY0heWTGmcavsU0hv6d0cBiTHLJLKV1Zzm36yC92XjHOPTSSFUliNXbP1RnEuKKa0ORqdCc4ZyaQ0wLeyslM1BA0N71oY9wu2m9P9m6Lc9t+1deq3eW7JWnTwn++SAeKRLeuSMXJABYeQT+UK+ke/OZ+ers3J+rEtrzu+eZ+SOnJ+/AEaJp54=</latexit><latexit sha1_base64="1e0+Qql3CB58TEEmHa4hXilhOmQ=">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</latexit>

rθLT (θ03;D3)

<latexit sha1_base64="EZq1g/a3SO7fqSwAptm7ckRIseE=">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</latexit><latexit sha1_base64="yZsC8lNXay3QknrftX4UEuXTvJk=">AAACI3icdZDPattAEMZXaf7VbROnPfayxJQmBIxkmzhtL6bJIYccHLBjg2XEaL2Ol6xWYndUEEJv0Qfopa+SSw4NIZcc+i5dRQltSvvBwsfvm2FnJkykMOi6d87Ss+WV1bX157UXL19tbNa3Xp+ZONWMD1ksYz0OwXApFB+iQMnHieYQhZKPwovDMh994dqIWA0wS/g0gnMl5oIBWhTUP/oKQglB7uOCIxTUjwAXDGR+UgQDulPhoP3+0+/kqAjydrEb1Btu88PBfquzT92m63a9lleaVrfT7lDPklKN3ra/9/Wul/WD+o0/i1kacYVMgjETz01wmoNGwSQvan5qeALsAs75xFoFETfT/H7Hgr6zZEbnsbZPIb2nf3bkEBmTRaGtLOc0f2cl/Fc2SXF+MM2FSlLkilUfzVNJMablwehMaM5QZtYA08LOStkCNDC0Z63ZIzxuSv9vzlpNz216p16j95lUWidvyTbZIR7pkh45Jn0yJIx8I5fkB7l2vjtXzo1zW5UuOQ89b8gTOT9/AUmmp6A=</latexit><latexit sha1_base64="yZsC8lNXay3QknrftX4UEuXTvJk=">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</latexit><latexit sha1_base64="YgQ6XZTXo8XaGP84rjwb8PdvDHQ=">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</latexit>

final update direction

θ<latexit sha1_base64="1c2PZxTZwdV/YiIlRJ5+hHqbPg8=">AAAB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMeAF49RzAOSJcxOOsmY2ZllplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHNzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDDbH7tlJ45pU8H2rhSSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MMBRThxh3Ah3K+UjZhhHF1DJhRAsv7xKmhfVwK8Gd5eV2n0eR5GckFNyTgJyRWrkltRJg3DySJ7JK3nztPfivXsfi9aCl88ckz/wPn8AqMePOA==</latexit><latexit sha1_base64="1c2PZxTZwdV/YiIlRJ5+hHqbPg8=">AAAB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMeAF49RzAOSJcxOOsmY2ZllplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHNzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDDbH7tlJ45pU8H2rhSSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MMBRThxh3Ah3K+UjZhhHF1DJhRAsv7xKmhfVwK8Gd5eV2n0eR5GckFNyTgJyRWrkltRJg3DySJ7JK3nztPfivXsfi9aCl88ckz/wPn8AqMePOA==</latexit><latexit sha1_base64="1c2PZxTZwdV/YiIlRJ5+hHqbPg8=">AAAB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMeAF49RzAOSJcxOOsmY2ZllplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHNzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDDbH7tlJ45pU8H2rhSSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MMBRThxh3Ah3K+UjZhhHF1DJhRAsv7xKmhfVwK8Gd5eV2n0eR5GckFNyTgJyRWrkltRJg3DySJ7JK3nztPfivXsfi9aCl88ckz/wPn8AqMePOA==</latexit><latexit sha1_base64="1c2PZxTZwdV/YiIlRJ5+hHqbPg8=">AAAB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMeAF49RzAOSJcxOOsmY2ZllplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHNzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDDbH7tlJ45pU8H2rhSSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MMBRThxh3Ah3K+UjZhhHF1DJhRAsv7xKmhfVwK8Gd5eV2n0eR5GckFNyTgJyRWrkltRJg3DySJ7JK3nztPfivXsfi9aCl88ckz/wPn8AqMePOA==</latexit>

θ01<latexit sha1_base64="lbis7ko2Po32t7FqutnSfgKvC+A=">AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPoKeyKoMeAF8FLBBODSQizk95kyOzsMtMrhJC/8OJBEa/+jTf/xsnjoIkFDUVVN91dYaqkJd//9nIrq2vrG/nNwtb2zu5ecf+gbpPMCKyJRCWmEXKLSmqskSSFjdQgj0OFD+HgeuI/PKGxMtH3NEyxHfOelpEUnJz02KI+Eu8E7LRTLPllfwq2TII5KcEc1U7xq9VNRBajJqG4tc3AT6k94oakUDgutDKLKRcD3sOmo5rHaNuj6cVjduKULosS40oTm6q/J0Y8tnYYh64z5tS3i95E/M9rZhRdtUdSpxmhFrNFUaYYJWzyPutKg4LU0BEujHS3MtHnhgtyIRVcCMHiy8ukfl4O/HJwd1Gq3M7jyMMRHMMZBHAJFbiBKtRAgIZneIU3z3ov3rv3MWvNefOZQ/gD7/MHi9GQMA==</latexit><latexit sha1_base64="lbis7ko2Po32t7FqutnSfgKvC+A=">AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPoKeyKoMeAF8FLBBODSQizk95kyOzsMtMrhJC/8OJBEa/+jTf/xsnjoIkFDUVVN91dYaqkJd//9nIrq2vrG/nNwtb2zu5ecf+gbpPMCKyJRCWmEXKLSmqskSSFjdQgj0OFD+HgeuI/PKGxMtH3NEyxHfOelpEUnJz02KI+Eu8E7LRTLPllfwq2TII5KcEc1U7xq9VNRBajJqG4tc3AT6k94oakUDgutDKLKRcD3sOmo5rHaNuj6cVjduKULosS40oTm6q/J0Y8tnYYh64z5tS3i95E/M9rZhRdtUdSpxmhFrNFUaYYJWzyPutKg4LU0BEujHS3MtHnhgtyIRVcCMHiy8ukfl4O/HJwd1Gq3M7jyMMRHMMZBHAJFbiBKtRAgIZneIU3z3ov3rv3MWvNefOZQ/gD7/MHi9GQMA==</latexit><latexit sha1_base64="lbis7ko2Po32t7FqutnSfgKvC+A=">AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPoKeyKoMeAF8FLBBODSQizk95kyOzsMtMrhJC/8OJBEa/+jTf/xsnjoIkFDUVVN91dYaqkJd//9nIrq2vrG/nNwtb2zu5ecf+gbpPMCKyJRCWmEXKLSmqskSSFjdQgj0OFD+HgeuI/PKGxMtH3NEyxHfOelpEUnJz02KI+Eu8E7LRTLPllfwq2TII5KcEc1U7xq9VNRBajJqG4tc3AT6k94oakUDgutDKLKRcD3sOmo5rHaNuj6cVjduKULosS40oTm6q/J0Y8tnYYh64z5tS3i95E/M9rZhRdtUdSpxmhFrNFUaYYJWzyPutKg4LU0BEujHS3MtHnhgtyIRVcCMHiy8ukfl4O/HJwd1Gq3M7jyMMRHMMZBHAJFbiBKtRAgIZneIU3z3ov3rv3MWvNefOZQ/gD7/MHi9GQMA==</latexit><latexit sha1_base64="lbis7ko2Po32t7FqutnSfgKvC+A=">AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPoKeyKoMeAF8FLBBODSQizk95kyOzsMtMrhJC/8OJBEa/+jTf/xsnjoIkFDUVVN91dYaqkJd//9nIrq2vrG/nNwtb2zu5ecf+gbpPMCKyJRCWmEXKLSmqskSSFjdQgj0OFD+HgeuI/PKGxMtH3NEyxHfOelpEUnJz02KI+Eu8E7LRTLPllfwq2TII5KcEc1U7xq9VNRBajJqG4tc3AT6k94oakUDgutDKLKRcD3sOmo5rHaNuj6cVjduKULosS40oTm6q/J0Y8tnYYh64z5tS3i95E/M9rZhRdtUdSpxmhFrNFUaYYJWzyPutKg4LU0BEujHS3MtHnhgtyIRVcCMHiy8ukfl4O/HJwd1Gq3M7jyMMRHMMZBHAJFbiBKtRAgIZneIU3z3ov3rv3MWvNefOZQ/gD7/MHi9GQMA==</latexit>

θ02<latexit sha1_base64="K5PijTDQd0UyQkGinQk39pv+ihk=">AAAB8XicbVBNS8NAEN3Ur1q/qh69LBbRU0mKoMeCF8FLBfuBbSib7aRdutmE3YlQQv+FFw+KePXfePPfuG1z0NYHA4/3ZpiZFyRSGHTdb6ewtr6xuVXcLu3s7u0flA+PWiZONYcmj2WsOwEzIIWCJgqU0Ek0sCiQ0A7GNzO//QTaiFg94CQBP2JDJULBGVrpsYcjQNav0fN+ueJW3TnoKvFyUiE5Gv3yV28Q8zQChVwyY7qem6CfMY2CS5iWeqmBhPExG0LXUsUiMH42v3hKz6wyoGGsbSmkc/X3RMYiYyZRYDsjhiOz7M3E/7xuiuG1nwmVpAiKLxaFqaQY09n7dCA0cJQTSxjXwt5K+YhpxtGGVLIheMsvr5JWreq5Ve/+slK/y+MokhNySi6IR65IndySBmkSThR5Jq/kzTHOi/PufCxaC04+c0z+wPn8AY1XkDE=</latexit><latexit sha1_base64="K5PijTDQd0UyQkGinQk39pv+ihk=">AAAB8XicbVBNS8NAEN3Ur1q/qh69LBbRU0mKoMeCF8FLBfuBbSib7aRdutmE3YlQQv+FFw+KePXfePPfuG1z0NYHA4/3ZpiZFyRSGHTdb6ewtr6xuVXcLu3s7u0flA+PWiZONYcmj2WsOwEzIIWCJgqU0Ek0sCiQ0A7GNzO//QTaiFg94CQBP2JDJULBGVrpsYcjQNav0fN+ueJW3TnoKvFyUiE5Gv3yV28Q8zQChVwyY7qem6CfMY2CS5iWeqmBhPExG0LXUsUiMH42v3hKz6wyoGGsbSmkc/X3RMYiYyZRYDsjhiOz7M3E/7xuiuG1nwmVpAiKLxaFqaQY09n7dCA0cJQTSxjXwt5K+YhpxtGGVLIheMsvr5JWreq5Ve/+slK/y+MokhNySi6IR65IndySBmkSThR5Jq/kzTHOi/PufCxaC04+c0z+wPn8AY1XkDE=</latexit><latexit sha1_base64="K5PijTDQd0UyQkGinQk39pv+ihk=">AAAB8XicbVBNS8NAEN3Ur1q/qh69LBbRU0mKoMeCF8FLBfuBbSib7aRdutmE3YlQQv+FFw+KePXfePPfuG1z0NYHA4/3ZpiZFyRSGHTdb6ewtr6xuVXcLu3s7u0flA+PWiZONYcmj2WsOwEzIIWCJgqU0Ek0sCiQ0A7GNzO//QTaiFg94CQBP2JDJULBGVrpsYcjQNav0fN+ueJW3TnoKvFyUiE5Gv3yV28Q8zQChVwyY7qem6CfMY2CS5iWeqmBhPExG0LXUsUiMH42v3hKz6wyoGGsbSmkc/X3RMYiYyZRYDsjhiOz7M3E/7xuiuG1nwmVpAiKLxaFqaQY09n7dCA0cJQTSxjXwt5K+YhpxtGGVLIheMsvr5JWreq5Ve/+slK/y+MokhNySi6IR65IndySBmkSThR5Jq/kzTHOi/PufCxaC04+c0z+wPn8AY1XkDE=</latexit><latexit sha1_base64="K5PijTDQd0UyQkGinQk39pv+ihk=">AAAB8XicbVBNS8NAEN3Ur1q/qh69LBbRU0mKoMeCF8FLBfuBbSib7aRdutmE3YlQQv+FFw+KePXfePPfuG1z0NYHA4/3ZpiZFyRSGHTdb6ewtr6xuVXcLu3s7u0flA+PWiZONYcmj2WsOwEzIIWCJgqU0Ek0sCiQ0A7GNzO//QTaiFg94CQBP2JDJULBGVrpsYcjQNav0fN+ueJW3TnoKvFyUiE5Gv3yV28Q8zQChVwyY7qem6CfMY2CS5iWeqmBhPExG0LXUsUiMH42v3hKz6wyoGGsbSmkc/X3RMYiYyZRYDsjhiOz7M3E/7xuiuG1nwmVpAiKLxaFqaQY09n7dCA0cJQTSxjXwt5K+YhpxtGGVLIheMsvr5JWreq5Ve/+slK/y+MokhNySi6IR65IndySBmkSThR5Jq/kzTHOi/PufCxaC04+c0z+wPn8AY1XkDE=</latexit>

θ03<latexit sha1_base64="4bR5vdGg29J2hSPCPHBQtrjt5eg=">AAAB8XicbVBNS8NAEN3Ur1q/qh69LBbRU0lU0GPBi+Clgv3ANpTNdtIu3WzC7kQoof/CiwdFvPpvvPlv3LY5aOuDgcd7M8zMCxIpDLrut1NYWV1b3yhulra2d3b3yvsHTROnmkODxzLW7YAZkEJBAwVKaCcaWBRIaAWjm6nfegJtRKwecJyAH7GBEqHgDK302MUhIOtd0NNeueJW3RnoMvFyUiE56r3yV7cf8zQChVwyYzqem6CfMY2CS5iUuqmBhPERG0DHUsUiMH42u3hCT6zSp2GsbSmkM/X3RMYiY8ZRYDsjhkOz6E3F/7xOiuG1nwmVpAiKzxeFqaQY0+n7tC80cJRjSxjXwt5K+ZBpxtGGVLIheIsvL5PmedVzq979ZaV2l8dRJEfkmJwRj1yRGrklddIgnCjyTF7Jm2OcF+fd+Zi3Fpx85pD8gfP5A47dkDI=</latexit><latexit sha1_base64="4bR5vdGg29J2hSPCPHBQtrjt5eg=">AAAB8XicbVBNS8NAEN3Ur1q/qh69LBbRU0lU0GPBi+Clgv3ANpTNdtIu3WzC7kQoof/CiwdFvPpvvPlv3LY5aOuDgcd7M8zMCxIpDLrut1NYWV1b3yhulra2d3b3yvsHTROnmkODxzLW7YAZkEJBAwVKaCcaWBRIaAWjm6nfegJtRKwecJyAH7GBEqHgDK302MUhIOtd0NNeueJW3RnoMvFyUiE56r3yV7cf8zQChVwyYzqem6CfMY2CS5iUuqmBhPERG0DHUsUiMH42u3hCT6zSp2GsbSmkM/X3RMYiY8ZRYDsjhkOz6E3F/7xOiuG1nwmVpAiKzxeFqaQY0+n7tC80cJRjSxjXwt5K+ZBpxtGGVLIheIsvL5PmedVzq979ZaV2l8dRJEfkmJwRj1yRGrklddIgnCjyTF7Jm2OcF+fd+Zi3Fpx85pD8gfP5A47dkDI=</latexit><latexit sha1_base64="4bR5vdGg29J2hSPCPHBQtrjt5eg=">AAAB8XicbVBNS8NAEN3Ur1q/qh69LBbRU0lU0GPBi+Clgv3ANpTNdtIu3WzC7kQoof/CiwdFvPpvvPlv3LY5aOuDgcd7M8zMCxIpDLrut1NYWV1b3yhulra2d3b3yvsHTROnmkODxzLW7YAZkEJBAwVKaCcaWBRIaAWjm6nfegJtRKwecJyAH7GBEqHgDK302MUhIOtd0NNeueJW3RnoMvFyUiE56r3yV7cf8zQChVwyYzqem6CfMY2CS5iUuqmBhPERG0DHUsUiMH42u3hCT6zSp2GsbSmkM/X3RMYiY8ZRYDsjhkOz6E3F/7xOiuG1nwmVpAiKzxeFqaQY0+n7tC80cJRjSxjXwt5K+ZBpxtGGVLIheIsvL5PmedVzq979ZaV2l8dRJEfkmJwRj1yRGrklddIgnCjyTF7Jm2OcF+fd+Zi3Fpx85pD8gfP5A47dkDI=</latexit><latexit sha1_base64="4bR5vdGg29J2hSPCPHBQtrjt5eg=">AAAB8XicbVBNS8NAEN3Ur1q/qh69LBbRU0lU0GPBi+Clgv3ANpTNdtIu3WzC7kQoof/CiwdFvPpvvPlv3LY5aOuDgcd7M8zMCxIpDLrut1NYWV1b3yhulra2d3b3yvsHTROnmkODxzLW7YAZkEJBAwVKaCcaWBRIaAWjm6nfegJtRKwecJyAH7GBEqHgDK302MUhIOtd0NNeueJW3RnoMvFyUiE56r3yV7cf8zQChVwyYzqem6CfMY2CS5iUuqmBhPERG0DHUsUiMH42u3hCT6zSp2GsbSmkM/X3RMYiY8ZRYDsjhkOz6E3F/7xOiuG1nwmVpAiKzxeFqaQY0+n7tC80cJRjSxjXwt5K+ZBpxtGGVLIheIsvL5PmedVzq979ZaV2l8dRJEfkmJwRj1yRGrklddIgnCjyTF7Jm2OcF+fd+Zi3Fpx85pD8gfP5A47dkDI=</latexit>

−rθ0

iLT (θ

0i;Dmte)

<latexit sha1_base64="6omrB83RLg06CGRSRzL08T95BVQ=">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</latexit><latexit sha1_base64="6omrB83RLg06CGRSRzL08T95BVQ=">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</latexit><latexit sha1_base64="6omrB83RLg06CGRSRzL08T95BVQ=">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</latexit><latexit sha1_base64="6omrB83RLg06CGRSRzL08T95BVQ=">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</latexit>

θ0 = θ −rθLS(θ;Dmtr)<latexit sha1_base64="T8+WkpizbtovzkxMuR7LDKfI2v0=">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</latexit><latexit sha1_base64="T8+WkpizbtovzkxMuR7LDKfI2v0=">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</latexit><latexit sha1_base64="T8+WkpizbtovzkxMuR7LDKfI2v0=">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</latexit><latexit sha1_base64="T8+WkpizbtovzkxMuR7LDKfI2v0=">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</latexit>

Figure 3: Overview of the meta-optimization procedure in

a meta-batch. Given three source domains D1,2,3, a meta

batch contains three meta-train/meta-test divisions: D2,3 /

D1, D1,3 / D2, D1,2 / D3. Each division contributes a gra-

dient from meta-train and a meta-gradient from meta-test.

The model is finally updated towards a direction that per-

forms well on both meta-train and meta-test domains by ac-

cumulating all the gradients and meta-gradients.

Domain Alignment Loss. We find negative pairs across

meta-train domains tend to be easier than within domains.

By adding a domain alignment regularization to make the

embeddings domain-invariant, we can reduce domain gap

of different meta-train domains. Besides, negative pairs

across meta-train domains become harder, which is bene-

ficial to learn more discriminative representations. To per-

form domain alignment, we make the mean embeddings of

multiple meta-train domains close to each other. Specifi-

cally, we first calculate the embedding center of all mean

embeddings of meta-train domains, then optimize the dis-

crepancies between all mean embeddings and this embed-

ding center. The domain alignment loss is only applied on

meta-train domains, as meta-test has only one domain. The

loss is formulated as:

cj =1

B

B∑

i=1

(

(FDj

gi+ FDj

pi)/2

)

,

cmtr =1

n

n∑

j=1

cj ,

Lda =1

n

n∑

j=1

‖s · (cj − cmtr)‖22,

(3)

where Fgi , Fpiare normalized embeddings, cj is the mean

embedding within a batch sampled from domain Dj , cmtr is

the embedding center of all mean embeddings of meta-train

domains, n is the number of meta-train domains and s is

the scaling factor. In meta-optimization, we will adaptively

utilize the back-propagated signals from these three losses

to improve the model generalization.

3.4. Metaoptimization

This section describes how the model is optimized to im-

prove model generalization. The whole meta-optimization

procedure is summarized in Algorithm 1 and illustrated in

Fig. 3.

Meta-train. Based on domain-level sampling, during

each batch within a meta-batch, we sample N − 1 source

domains Dmtr and B pairs of images XS from Dmtr. Then

we conduct the proposed losses in each batch as follows:

LS = Lhp(XS ; θ) + Lcls(XS ; θ) + Lda(XS ; θ), (4)

where θ represents the model parameters. The model is next

updated by gradient ∇θ as: θ′ = θ − α∇θLS(θ). This

update step is similar to the conventional metric learning.

Meta-test. In each batch, the model is also tested on the

meta-test domain Dmte. This testing procedure simulates

the evaluating on an unseen domain with a different dis-

tribution, so as to make the the model to learn to generalize

across domains. We also sample B pairs of images XT from

the meta-test domain Dmte. Then the loss is conducted on

the updated parameters θ′ as below:

LT = Lhp(XT ; θ′) + Lcls(XT ; θ

′). (5)

Summary. To optimize the meta-train and meta-test si-

multaneously, the final MFR objective is:

argminθ

γLS(θ) + (1− γ)LT (θ − αL′S(θ)), (6)

where α is the meta-train step-size and γ balances meta-

train and meta-test. This objective can be understood as:

optimize the model parameters, such that after updating

on the meta-train domains, the model also performs well

on the meta-test domain. From another perspective, the

second term of Eqn. 6 serves as an extra regularization to

update the model with high order gradients, and we call

it meta-gradients. For example, given three source do-

mains DS = {D1,D2,D3}, a meta-batch consists of three

meta-train/meta-test divisions: D2,D3/D1, D1,D3/D2 and

D1,D2/D3. For each division or batch, a gradient and a

meta-gradient are back-propagated on meta-train and meta-

test, respectively. By accumulating all the gradients and

meta-gradients in the meta-batch, the model is finally op-

timized to perform well on both meta-train and meta-test

domains. Fig. 3 illustrates how the gradients and meta-

gradients flow on the computation graph.

4. Experiments

To evaluate our proposed MFR for generalized face

recognition problem, we conduct several experiments on

two proposed benchmarks.

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Algorithm 1: MFR for generalized face recogni-

tion problem.

Input: Source (training) domains

DS = {D1,D2, · · · ,DN}.Init: A pre-trained model f(θ) parametrized by θ,

hyperparameters α, β, γ and batch-size of B.

1 for ite in max iterations do

2 Init the gradient gθ as 0;

// For a meta-batch

3 for each Dmte in DS do

// For a batch

4 Sampling remaining domains as Dmtr;

5 Meta-train:

6 Sampling B paired images XS from Bidentities of meta-train domains Dmtr;

7 LS=Lhp(XS ; θ)+Lcls(XS ; θ)+Lda(XS ; θ);8 Update model parameters by:

θ′ = θ − α∇θLS(θ);9 Meta-test:

10 Sampling B paired images XT from Bidentities of the meta-test domain Dmte;

11 LT = Lhp(XT ; θ′) + Lcls(XT ; θ

′);12 Gradient aggregation:

13 gθ ← gθ + γ∇θLS + (1− γ)∇θLT ;

14 end

15 Meta-optimization:

16 Update θ ← θ − βNgθ by SGD;

17 end

4.1. GFR Benchmark and Protocols

Generalized face recognition has not attracted much at-

tention and we do not have a common protocol for eval-

uation, thus we introduce two well-designed benchmarks

to evaluate the generalization of a model. One benchmark

is for crossing race evaluation named GFR-R and another

one is crossing facial variety named GFR-V. We use variety

here to emphasize that there is a large gap between source

domains and target unseen domains on GFR-V.

In a real-world scenario, a large-scale base dataset like

MS-Celeb [13] is usually available for pre-training, but

the model may generalize poorly on a new domain with

a different distribution. To simulate it, we use MS-Celeb

as the base dataset. RFW [35] is originally proposed to

study the racial bias in face recognition and it labels four

racial datasets (Caucasian, Asian, African, Indian) from

MS-Celeb. We choose to select these four datasets as our

four racial domains. Note that RFW [35] overlaps MS-

Celeb [13], we remove all the overlapped identities from

MS-Celeb according to the identity keyword, thus building

our base dataset named MS-Celeb-NR1, which means MS-

Celeb without RFW. MS-Celeb-NR can be regarded as an

independent base dataset of four racial ones.

GFR-R. Each race has about 2K∼3K identities. We

randomly choose 1K identities for testing and the remain-

ing 1K∼2K identities for training. The dataset details are

shown in Table 1. In our experiment setting, each race is

regarded as one domain. We randomly select three domains

in four as source domains and and the rest one as the testing

domain, which is not accessible in training. Therefore, we

build four sub-protocols for GFR-R, shown in Table 2.

GFR-V. The GFR-V benchmark is for crossing facial va-

riety evaluation, which is a harder setting and can better re-

flect the generalization ability of a model. As is shown in

Table 2, four racial datasets (Caucasian, Asian, African, In-

dian) are treated as source domains, and five datasets are

as target domains. Specifically, the target datasets include

CACD-VS [30], CASIA NIR-VIS 2.0 [31], MultiPIE [32],

MeGlass [33], Public-IvS [34]. For CASIA NIR-VIS 2.0,

we follow the standard protocol in View 2 evaluation [31]

and we report the average value of 10 folds. For Me-

Glass and Public-IvS, we follow the standard testing pro-

tocols [34, 33]. For CACD-VS, in addition to the standard

protocol [30], we use the provided 2,000 positive cross-age

image pairs and split them into gallery and probe for our

ROC/Rank-1 evaluation. For Multi-PIE, we select 337 iden-

tities and each identity contains about 3∼4 frontal gallery

images and 3∼4 probe images with the 45◦ view.

Benchmark Protocols. For each image, the features

from both the original image and the flipped one are ex-

tracted then concatenated as the final representation. The

score is measured by the cosine distance of two representa-

tions. For performance evaluation, we use the receiver oper-

ating characteristic (ROC) curve and Rank-1 accuracy. For

ROC, we report the verification rate (VR) at low false ac-

ceptance rate (FAR) like 1%, 0.1% and 0.01%. For Rank-1

evaluation, each probe image is matched to all gallery im-

ages, if the top-1 result is within the same identity, it is cor-

rect.

4.2. Implementation Details

Our experiments are based on PyTorch [37]. The ran-

dom seed is set to a fixed value 2019 in comparative exper-

iments for fair comparisons. We use a 28-layer ResNet as

our backbone, but with a channel-number multiplier of 0.5.

Our backbone has only 128.7M FLOPs and 4.64M param-

eters, which is relatively light-weighted. The dimension of

the output embedding is 256. The model is pre-trained on

MS-Celeb-NR with CosFace [38]. During training, all faces

are cropped and resized to 120×120. The inputs are then

normalized by subtracting 127.5 and being divided by 128.

The meta-train step-size α, the meta optimization step-size

1We will release the list of MS-Celeb-NR.

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Facial Variety Dataset #Train IDs #Train images#Test IDs #Test images

#Gallery IDs #Probe IDs #Gallery images #Probe images

Race Caucasian 1,957 6,757 1,000 1,000 1,000 1,000

Race Asian 1,492 5,784 1,000 1,000 1,000 1,000

Race African 1,995 6,938 1,000 1,000 1,000 1,000

Race Indian 1,984 6,857 1,000 1,000 1,000 1,000

Age CACD-VS [30] - - 2,000 2,000 2,000 2,000

Illumination CASIA NIR-VIS 2.0 [31] - - 358 363 358 6,208

Pose MultiPIE [32] - - 337 337 1,184 1,181

Occlusion MeGlass [33] - - 1,710 1,710 3,420 3,420

Heterogeneity Public-IvS [34] - - 1,262 1,262 1,262 4,241

Table 1: The statistics of all involved datasets. CASIA NIR-VIS 2.0 has 10 folds and the first fold is shown. The other folds

own similar statistics.

Protocol Source Domains Target Domain(s)

GFR-R

I

Caucasian

IndianAsian

African

II

Caucasian

AfricanAsian

Indian

III

Caucasian

AsianAfrican

Indian

IV

Asian

CaucasianAfrican

Indian

GFR-V

Caucasian CACD-VS

Asian CASIA NIR-VIS 2.0

African MultiPIE

Indian MeGlass

Public-IvS

Table 2: The GFR-R and GFR-V benchmarks. Source do-

mains are for training, target domains are for evaluation and

are unseen during training.

Protocol MethodVR (%)

Rank-1 (%)FAR=1% FAR=0.1% FAR=0.01%

GFR-R I

(Indian)

Base 94 82.2 64.65 80.3

Base-Agg 94.1 80.9 65.3 81

Base-FT rnd. 62.5 39 21.05 39.3

Base-FT imp. [36] 87 69.9 51.2 69.6

MLDG [14] 94.2 83 66.3 80.5

MFR (Ours) 95.4 86.1 71.4 83.1

GFR-R II

(African)

Base 91.6 74.5 55.4 73.1

Base-Agg 90.5 74.8 56.3 74

Base-FT rnd. 26.2 10.9 3.5 21

Base-FT imp. [36] 78.7 56.6 36.45 57.9

MLDG [14] 91.9 74.8 55.7 73.8

MFR (Ours) 92.3 79.4 60.8 75.2

GFR-R III

(Asian)

Base 91.89 77.98 60.86 75.98

Base-Agg 91.49 78.08 59.41 76.28

Base-FT rnd. 40.44 17.32 7.67 27.53

Base-FT imp. [36] 80.58 57.56 39.79 61.86

MLDG [14] 92.29 78.28 60.3 76.68

MFR (Ours) 93.49 80.7 62.56 78.68

GFR-R IV

(Caucasian)

Base 96.6 89.6 78.6 86.6

Base-Agg 97 88.1 79.1 86.8

Base-FT rnd. 61.1 36.2 18.9 36.7

Base-FT imp. [36] 91.5 78.2 63.4 76.8

MLDG [14] 96.8 89.6 79.15 86.3

MFR (Ours) 98.2 92.9 81.1 88.9

Table 3: Comparative results of the GFR-R benchmark.

rnd. means random initializing the classification weight

template, imp. is weight-imprinted.

β, the weight γ balancing meta-train and meta-test loss are

initialized to 0.0004, 0.0004, 0.5, respectively. Batch-size

B is set to 128 and the scaling factor s of both the soft-

classification loss and domain alignment loss are set to 64.

The step-size α and β are decayed with every 1K steps and

the decay rate is 0.5. The positive threshold τp and nega-

tive threshold τn are initialized to 0.3, 0.04 and are updated

as τp = 0.3 + 0.1n and τn = 0.04/0.5n, where n is the

decayed number. For meta-optimization, we use SGD to

optimize the network with the weight decay of 0.0005 and

momentum of 0.9.

4.3. GFRR Comparisons

Settings. We compare our model with several baselines,

including the base model and several domain aggregation

baselines. To further compare our method with other do-

main generalization methods, we adapt MLDG [14] to an

open-set setting, so that it can be applied in our proto-

cols. The results are shown in Table 3. For four proto-

cols in GFR-R, we report the VRs at low FAR 1%, 0.1%,

0.01%, and the Rank-1 accuracy. Specifically, our com-

parisons include: (i) Base: the model pre-trained only on

MS-Celeb-NR using CosFace [38]. Note that MS-Celeb-

NR has no overlapped identities with four racial datasets

(Caucasian, Asian, African and Indian) and can be consid-

ered as an independent dataset. (ii) Base-Agg: the model

trained on MS-Celeb-NR and the aggregation of source do-

mains using CosFace [38]. Take GFR-R-I as an example,

Base-Agg is trained on MS-Celeb-NR and three source do-

mains Caucasian, Asian, African jointly. This is for the

fair comparison with our MFR, where the same training

datasets are involved. (iii) Base-FT rnd.: the base model

further fine-tuned on the aggregation of source domains.

The classification template of the last FC-layer is randomly

initialized. (iv) Base-FT imp.: the base model further fine-

tuned on the aggregation of source domains, but the classi-

fication template is initialized as the mean of embeddings

of the corresponding identities. It is refined from weight-

imprinted [36]. (v) MLDG: MLDG [14] adapted for gener-

alized face recognition problem.

Results. From the results in Table 3, the following obser-

vations can be made: (i) Overall, our method achieves the

6168

best result on four GFR-R protocols among all compared

settings and methods. (ii) The base model pre-trained on

MS-Celeb-NR is strong, but not generalizes well for target

domains, especially for Indian, African, Asian. The rea-

son may be that MS-Celeb-NR is occupied by Caucasian

people. (iii) Jointly training on MS-Celeb-NR and source

domains performs slightly better than the base model, but

is still not comparable to our MFR method. (iv) The per-

formance of Base-FT rnd. declines dramatically and we

attribute it to over-fitting on source domains. Weight-

imprinted (Base-FT imp.) can reduce such over-fitting to

some degree, but its performance is still lower than the base

model. (v) MLDG [14], which is originally designed for

closed-set and category-level recognition problems, fail to

compete with our method on the open-set generalized face

recognition problem.

GFR-V

(CACD-VS)

VR (%)Rank-1 (%) Acc. AUC.

FAR=0.01% FAR=0.001%

Base 96.55 92.55 96.85 99.35 99.53

Base-Agg 96.75 92.98 97.15 99.42 99.6

MLDG [14] 96.75 92.9 97.25 99.45 99.57

LF-CNNs [39] - - - 98.5 99.3

Human, Voting [40] - - - 94.2 99

OE-CNNs [41] - - - 99.2 99.5

AIM+CAFR [42] - - - 99.76 -

MFR (Ours) 97.25 94.05 97.8 99.78 99.81

Table 4: Comparative results on CACD-VS.

GFR-V

CASIA NIR-VIS 2.0

VR (%)Rank-1 (%)

FAR=1% FAR=0.1% FAR=0.01%

Base 97.8 89.89 69.27 93.18

Base-Agg 98.31 90.47 71.35 94.29

MLDG [14] 98.28 90.44 69.32 93.56

IDR [43] 98.9 95.7 - 97.3

WCNN [44] 99.4 97.6 - 98.4

MFR (Ours) 99.32 95.97 81.92 96.92

Table 5: Comparative results on CASIA NIR-VIS 2.0. The

highest two results are highlighted.

GFR-V

(Multi-PIE)

VR (%)Rank-1 (%)

FAR=0.1% FAR=0.01% FAR=0.001%

Base 99.92 98.83 61.54 99.75

Base-Agg 99.92 98.96 68.49 99.82

MLDG [14] 99.84 98.87 62.95 99.83

MFR (Ours) 100 99.96 74.54 99.92

Table 6: Comparative results on MultiPIE.

GFR-V

(MeGlass)

VR (%)Rank-1 (%)

FAR=0.01% FAR=0.001% FAR=0.0001%

Base 85.92 71.96 53.5 97.6

Base-Agg 86.77 73.5 54.96 97.69

MLDG [14] 85.54 69.23 49.32 97.81

Face Syn. [33] 90.14 80.32 66.92 96.73

MFR (Ours) 90.79 80.86 66.15 98.57

Table 7: Comparative results on MeGlass. The highest two

results are highlighted.

GFR-V

(Public-IvS)

VR (%)Rank-1 (%)

FAR=0.1% FAR=0.01% FAR=0.001%

Base 94.38 86.71 74.83 92.74

Base-Agg 94.24 87.1 74.5 92.85

MLDG [14] 94.96 87.35 75.54 93.3

Contrastive [29] 96.52 91.71 84.54 -

LBL [34] 98.83 97.21 93.62 -

MFR (Ours) 96.66 92.96 85.28 95.82

Table 8: Comparative results on Public-IvS. The highest

two results are highlighted.

Method Base Base-Agg MLDG [14] MFR (Ours)

LFW 99.57 99.60 99.43 99.77

Table 9: Comparative results on LFW.

4.4. GFRV Comparisons

The GFR-V benchmark is for crossing facial variety

evaluation, which can better reflect model generalization.

Settings. We compare our model with two strong base-

lines Base, Base-Agg, an adapted MLDG [14] and other

competitors if existed. Since the standard protocols differ

among five target domains, we show them separately in Ta-

ble 4, 5, 6, 7, 8.

CACD-VS. CACD-VS [30] is for cross-age evaluation,

where each pair of images contain a young face and an old

one. We report ROC/Rank-1 as well as the standard proto-

col provided. Other competitors are only evaluated on the

standard protocol. The results in Table 4 show that our MFR

not only beats the baselines but also the competitors, which

use cross-age datasets for training.

CASIA NIR-VIS 2.0. In CASIA NIR-VIS 2.0 [31],

gallery images are collected under visible lighting, while

the probe one is under near infrared lighting, thus the

modality gap is huge. Table 5 shows: (i) we achieve great

performance improvements from 89.89% (69.27%) of Base

to 95.97% (81.92%) when FAR=0.1% (0.01%). (ii) even

with such a huge modality gap, our performance is com-

parable to several CNN-based methods [43, 44], which use

MS-Celeb for pre-training and the target domain NIR-VIS

dataset for fine-tuning. In comparison, our model has not

seen any NIR samples during training.

Multi-PIE. We compare our model with two baselines

and MLDG for cross-pose evaluation using Multi-PIE. Ta-

ble 6 validates the improvements of our MFR over baselines

and MLDG.

MeGlass. MeGlass [33] focuses on the effect of eyeglass

occlusion for face recognition. We select the hardest IV

protocol for evaluation. As shown in Table 7, our method

promotes the performance from 71.96% (53.5%) on Base

to 80.86% (66.15%) when at a low FAR 0.001%, which is

even slightly better than [33], which synthesizes wearing-

eyeglass image for the whole MS-Celeb for training.

Public-IvS. Public-IvS [34] is a testbed for ID vs. Spot

(IvS) verification. Compared to Base and Base-Agg, our

method greatly improves the generalization performance.

6169

The other two competitors are all pre-trained on MS-Celeb

and fine-tuned on CASIA-IvS, which has more than 2 mil-

lion identities and each identity has one ID and Spot face.

Even so, our method still performs slightly better than Con-

trastive [29].

LFW. We perform an extensive evaluation on LFW [9],

shown in Table. 9. The results demonstrate that our method

also generalizes better than baselines on a similar target do-

main.

The above results show that our method achieves great

improvement than baselines, and the performance is com-

petitive to the best supervised / non-generalization methods.

For a real-world face recognition application, our method is

the first choice because it generalizes well on all target do-

mains with competitive performances.

Protocol MethodVR (%)

Rank-1 (%)FAR=1% FAR=0.1% FAR=0.01%

GFR-R I

(Indian)

w/o hp. 95.1 84.5 69.2 82.2

w/o cls. 95.3 84.3 69 82.3

w/o da. 95.2 84.9 70.8 82.7

w/o meta (α = 0) 94.8 84.3 68.35 81.1

first order 95.3 85.7 70.9 82.6

Ours-full 95.4 86.1 71.4 83.1

GFR-R II

(African)

w/o hp. 92 77.9 59.2 74.6

w/o cls. 92.1 78.9 59.3 74.6

w/o da. 92.1 78.6 59.4 74.8

w/o meta (α = 0) 91.9 77.6 57.75 74.8

first order 92 78.05 59.5 75.2

Ours-full 92.3 79.4 60.8 75.2

GFR-R III

(Asian)

w/o hp. 93.39 79.9 61.56 77.78

w/o cls. 93.29 80.4 62.1 78.08

w/o da. 93.49 79.68 61.76 77.78

w/o meta (α = 0) 92.89 79.2 60.7 77.28

first order 93.49 79.9 61.7 77.88

Ours-full 93.49 80.7 62.56 78.68

GFR-R IV

(Caucasian)

w/o hp. 98.2 91.3 80.4 87.8

w/o cls. 98.3 92.6 80.4 88.4

w/o da. 98.4 92.4 80.5 87.5

w/o meta (α = 0) 97.3 91.1 79.6 87.3

first order 97.9 91.8 80.1 87.7

Ours-full 98.2 92.9 81.1 88.9

Table 10: Ablative results of the GFR-R benchmark. hp. is

the hard-pair attention loss, cls. is the soft-classification loss

and da. is the domain alignment loss on meta-train domains.

4.5. Ablation Study and Analysis

Contribution of Different Components. To evaluate

the contributions of different components, we compare our

full MFR with four degraded versions. The first three com-

ponents are the hard-pair attention loss, soft-classification

loss and domain alignment loss, which are designed for

learning domain-invariant and discriminative representa-

tions. The fourth component is the meta-gradient. If α is

set to 0 in Eqn. 6, the objective is degraded to the sum of

meta-train and meta-test and there is no meta-gradient com-

putation. Table 10 shows that each component contributes

to the performance. Among three components, the meta-

gradient is the most important one. For example, in GFR-

R I, the performance drops from 71.4% to 68.35% when

FAR=0.01% without the meta-gradient.

First Order Approximation. The meta-gradient needs

high order derivatives and is computationally expensive.

Therefore, we compare it with the first order approximation.

0.2 0.4 0.6 0.8

70.0

70.5

71.0

71.5

VR (%

) (FA

R=0.

01%

)

0.2 0.4 0.6 0.8

82.25

82.50

82.75

83.00

83.25

Rank

-1 (%

)

S1T1 S1T2 rand. S2T1Domain-level sampling

69.0

69.5

70.0

70.5

71.0

VR (%

) (FA

R=0.

01%

)

S1T1 S1T2 rand. S2T1Domain-level sampling

81.0

81.5

82.0

82.5

Rank

-1 (%

)

Figure 4: Ablation results on GFR-R I (Indian) protocol,

with different γ and domain-level sampling strategies.

To achieve the first order approximation, we only need to

change (1−γ)∇θLT in Algorithm 1 to (1−γ)∇θ′LT in the

gradient aggregation step. From Table 10, we can see that

the performance of the first order approximation is close to

high order. Considering that the first order approximation

takes only about 82% GPU memory and 63% time (in our

setting) of the high order, the first order approximation is a

practical substitute for the high order implementation.

Impact of γ. In Eqn. 6, γ is a hyperparameter weight-

ing the meta-train and meta-test losses. The ablative results

are shown in Fig. 4. A proper value 0.5 gives the best re-

sult, which indicates the meta-train and meta-test domains

should be equally learned.

Domains-level Sampling. Since domain alignment loss

cannot be applied when there is only one domain in meta-

train, we remove it for fair comparisons. For each batch,

SmTn ( m,n ∈ {(1, 1), (1, 2), (2, 1)} ) means sampling

m domain as meta-train and another n as meta-test. rand.

means randomly choosing m domains as meta-train (m is

a random number) and remaining one as meta-test. Fig. 4

shows that the setting m = 2 and n = 1 performs best.

5. Conclusion

In this paper, we highlight generalized face recogni-

tion problem and propose a Meta Face Recognition (MFR)

method to address it. Once trained on a set of source do-

mains, the model can be directly deployed on target do-

mains without any model update. Extensive experiments on

two newly defined generalized face recognition benchmarks

validate the effectiveness of our proposed MFR. We believe

generalized face recognition problem is of great importance

for practical applications, and our work is an important av-

enue for future works.

Acknowledgments

This work has been partially supported by the Chi-

nese National Natural Science Foundation Projects

#61876178, #61806196, #61976229, #61872367 and

Science and Technology Development Fund of Macau

(No. 0008/2018/A1, 0025/2019/A1, 0019/2018/ASC,

0010/2019/AFJ, 0025/2019/AKP).

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