Searching Parameterized AP Loss for Object Detection
Authors: Chenxin Tao
Zizhang Li
Xizhou Zhu
Gao Huang
Yong Liu
Jifeng Dai
Venue: NeurIPS 2021 Poster (2021)
Keywords: Loss Function Search
Metric Surrogate
Object Detection

Abstract: Loss functions play an important role in training deep-network-based object detectors. The most widely used evaluation metric for object detection is Average Precision (AP), which captures the performance of localization and classification sub-tasks simultaneously. However, due to the non-differentiable nature of the AP metric, traditional object detectors adopt separate differentiable losses for the two sub-tasks. Such a mis-alignment issue may well lead to performance degradation. To address this, existing works seek to design surrogate losses for the AP metric manually, which requires expertise and may still be sub-optimal. In this paper, we propose Parameterized AP Loss, where parameterized functions are introduced to substitute the non-differentiable components in the AP calculation. Different AP approximations are thus represented by a family of parameterized functions in a unified formula. Automatic parameter search algorithm is then employed to search for the optimal parameters. Extensive experiments on the COCO benchmark with three different object detectors (i.e., RetinaNet, Faster R-CNN, and Deformable DETR) demonstrate that the proposed Parameterized AP Loss consistently outperforms existing handcrafted losses. Code shall be released.

Searching Parameterized AP Loss for Object Detection
Chenxin Tao Zizhang Li Xizhou Zhu Gao Huang', Yong Liu', Jifeng Dai3.41
TTsinghua University. "Zhejiang University, "SenseTime Research, SShanghai Jiao Tong University
tcx200mails..siinggeee edu. en, zzliēzju.edu. cn,
(zhuwalter, daijifeng) com
gachuangttsinghua. edu, cn, yongliuGiipc. edu, cn
Abstract
Loss functions play an important role training deep-network- object de- tectors. The most widely used evaluation metric for object detection is Average Precision (AP), which captures the performance localization and classification sub-tasks simultancously. However, due to the non-diffferentiable nature of the AP metric, traditional object detectors adopt separate differentiable losses for the two sub-tasks. Such mis-alignment issue may well lead to performance degradation. To address this, existing works seek to design surrogate losses for the AP metric manually, which requires expertise and may still be sub-optimal. In this paper, we propose Parameterized AP Loss, where parameterized functions are introduced to substitute the non- diffeeferentiable components in the AP calculation. Different AP approximations are thus represented by family of parameterized functions in a uni- fied formula. Automatic parameter search algorithm is then employed search for the optimal parameters. Extensive experiments on the COCO henchmark with three different object detectors (i.e., RetinaNet, Faster R-CNN, and Deformable DETR) demonstrate that the proposed Parameterized AP Loss consistently outperforms existing handcrafted losses. Code shall be released.
Introduction
The past decade has witnessed the significant success of deep neural networks in object detection, in which loss functions play an indispensable role in training networks. To evaluate the object detection methods, the Average Precision (AP) metric usually used, which captures the performance of localization and classification simultancously. However, as in most object detectors [26]. the training of localization and classification sub-tasks are driven by two separate losses (see Figure [T). For
example, the Ll/smooth-LI 11 or GloU (38 losses are usually employed for localization, while the
cross-entropy or Focal 25 losses are usually used for classification. Such a mis-alignment between network training and evaluation may well lead to performance degradation.
To mitigate this mis-alignment issue, straight-forwart solution is approximate the AP metric in network training. Because the AP metric is non-differentiable: many works 3 5.
13
40
35
have explored hand-crafted losses based the mathematical formula of the AP metric. the pon-differentiable parts in the AP metric with hand-crafted differentiable approximations. The loss gradient is obtained by taking derivation with respect to the hand-crafted function. However, as both [5] and our experiments show, such a hand-crafted AP approximated loss produces lower
3 replaces
"Equal contribution. 'This work is done when Chenxin Tao and Zizhang Li are interns at SenseTime Research. 'Corresponding author.
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Existing Separate Classification/ Localization Losses
Our Parameterized AP Loss
fonward bockward
disesfication
clssslfication
cisssifiation brandh
branch
Hand-cratted:
od. Entropy, Focal
cla fiottion
2. Searched:
e. ASE-Autolosa, Autokoss-Zaro
classtfication spores
scorcs
3. AP optimization: eg. Eroo--Driven Psoudo-Gradiant
Parameterized
locallaation branch
localization
localization branch
trand--rafted:
egg. loU,,
features
gradient
coordinates
Searched:
ag CSE-Autokoss Autoioss-Zaro
box cocrcinates
Figure 1: Comparison between existing approaches and our Parameterized AP Loss.
performance than the commonly used cross-entropy and LI/smooth-LI losses. Another line of works
40 35 These works try to manually design the loss gradient. Therein, the AP approximated gradient only drives the training of the classification branch, while the training of the localization branch is still supervised by traditional regression losses. In practice, these methods still do not address the mis- miiiaaligment issue well.
try to estimate the gradient for the AP metric directly [ 13
The common issue of existing approaches is they do not optimize over the numerous approximations of the non-differentiable discrete AP metric. The AP metric itself piecewise constant function, whose differentiable approximations have infinite possibilities. The cenddfdd approximations / gradients may well be sub-optimal for driving network training. Instead of manually determining the surrogate loss form, we propose to approximate the -on-differentiable parts with family of continuous parameterized functions, which helps to represent the numerous AP approximations in a unified formula. Then, an efficient parameter search procedure is employed to find out the desired loss function, so as to optimize the trained object detector's performance on the evaluation set with the AP metric. Because the parameterized AP approximations constitute a compact search space, the search process would be very effective.
To this end, we propose the Parameterized AP Loss, which is built on top of the AP metric to mitigate the mis-alignment between network training and evaluation. utilizes parameterization construct the search space so that the optimal parameters can be searched automatically. Specifically, we first explicitly reformulate the AP metric for object detection function of classification scores and box coordinates. Then we replace the non-differentiable components in the reformulated function with parameterized functions. Finally, to obtain the optimal loss function for network training, we search the parameters through reinforeement-lamming based search process, which aims to maximize the AP score on the evaluation set.
We evaluate the searched Parameterized AP Loss on various object detectors, including RetinaNet 255 Faster R-CNN (37) and Deformable DETR 42 Extensive experiments the COCO benchmark 23 demonstrate that the proposed Parameterized AP Loss consistently outperforms existing delicately designed losses.
The main contributions of our work can be summarized as:
By reformulating the AP metric and introducing differentiable parameterized substitutions, Parameterized AP Loss represent numerous AP approximations in unified formula, which captures classification and localization sub-tasks simultaneously in single loss function.
Instead of hand-crafting AP losses gradient approximations, our approach automatically searches the optimal parameter, optimizing for the trained object detector performance.
Extensive experiments on different object detectors demonstrate that the searched Parameterized AP Loss consistently outperforms the existing losses.
2 Related work
Hand-crafted Loss Functions for Object Detection. Designing loss functions has been an active direction in the field of object detection for long. Cross Cross-entropy and smooth-L1 losses are widely used for the classification and localization sub-tasks, respectively (26). For the classification sub-
task, to mitigate the imbalance problem in one-stage detectors, Focal Loss 25 and GHM [17] propose to adjust the weight of loss or gradient for each predicted box. DR Loss 34 proposes to convert the classification problem into ranking problem. For the localization sub-task, a series of

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works [3] 41 introduce IoU variants for better localization. GFL [22 and GFLv2 [2] point out that the classification and localization branches are separately trained but compositely used in inference, so they merge the localization representation into the classification branch for joint optimization. But such combination still not consistent with the AP metric.
Recent works [S] 30] have noticed the mis-alignment issue between network training and evaluation, and try address it with specifically designed surrogate loss functions. AP Loss (5] replaces the classification task with a ranking task derived from the AP metric via a hand-crafted error- driven gradient. aLRP Loss 30 extends the framework of AP Loss, and unifies classification and localization losses under the LRP metric [29].
These loss functions all hand-crafted, and may well be sub-optimat in guiding network training. In contrast, our proposed Parameterized AP Loss is automatically searched, which can better align
the network training and evaluation.
Direct Optimization for the AP Metric. Pioneer works [13 40 35 have also tried to use the AP
metric training objective. Due to the non-differentiable nature of the AP metric, these works adopt different methods to estimate the back-propagated gradient. [13 uses finite difference and linear envelope to derive the useudo-gradients. [40] applies the loss-augmented inference for the gradient estimation. 35 resorts to reinforcement learning for policy gradient.
While these works take step towards direct optimization for the AP metric, they all focus on the gradient for the classification task, ignoring the localization task, which may well be sub-optimal for the optimization of AP metric. By contrast, our proposed Parameterized AP Loss deals with these two tasks simultaneously in a single loss function, and thus better align training and evaluation processes.
Another line of works try to approximate the AP metric via interpolation or neural network [27] 32 (8.
27 refactors the computation process and interpolates the loss value with differentiable functions, learns an embedding for prediction and target, so the Euclidean distance between them approximates the metric value. g trains a network for sorting operation. so as to construct a differentiable surrogate. Although these methods use differentiable functions to approximate AP metric, they ignore the training process. Our proposed Parameterized AP Loss, however, is searched to directly optimize the AP metric during evaluation, and thus consider the training behavior of loss function. This leads to a more effective loss function for network training.
32
Searching Loss Functions for Object Detection. Recent works [28] [9] have also tried to search suitable loss functions for object detection. In these methods, loss functions are formulated as computational graphs composed of basic mathematical operators. Since the combinations of operators are randomly chosen, the search space is very sparse with a large number of unpromising loss functions. Thus, they have to design specific techniques to accelerate the search process. (28 also relies on hand-crafted initialization. Nevertheless, their searched loss functions only gain marginal improvement on detection tasks. On the other hand, these methods search for separate losses for the localization and classification sub-tasks. thus the mis-alignment between network training and evaluation still exists.
Our proposed Parameterized AP Loss constitutes compact search space, where different loss functions are represented by different parameters. Since the parameterized search space is continuous, the search process would be more effective than the discrete combinatorial optimization in 28 19]. Moreover, Parameterized AP Loss drives the localization and classification training with a single unified loss, which better mitigates the mis-alignment issue.
Hyper-Parameter Optimization. Previously, grid search or random search [ are commonly used for typer-parameter tuning. As the search space becomes more complex, many efficient hyper- parameter optimization methods have been proposed. Bayesian optimization 22] 14) aims to build prediction model from history data and it to select the most promising hyper-parameters for evaluation. Bandit-based methods [5]
10] view it as a resource allocation problem and allocate most of the computational resource to the most promising syper-parametert Evolutionary algorithms [36] (33 evolve the optimal hyper-parameters from a population of models. Reinforcement learning techniques [3] have also been used to explore the search space via sampling candidates and adjusting the sampling policy.
20
The introduced parameters in our proposed Parameterized AP Loss indeed hyper-paramette parameters, which fit the hyper- parameerr optimization methods. In this work, we adopt reinforcement learning
3

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to search for the optimal parameters because of its simplicity and efficiency. Other efficient hyper- parameter optimization algorithms can also be employed.
Method
In this section, we present how Parameterized AP Loss is constructed to better align the training target with the evaluation metric. The overview of our method illustrated in Figure
3.1 Revisiting AP Metric
AP metric is the most widely used evaluation metric for object detection. Given an image, we assume an object detector outputs N detected bounding boxes for each category as {(b, ) Here, b, E I denotes the box coordinates of the i-th prediction, and 8; E R is its corresponding classification score. In the AP metric, these predictions will be matched with set of ground truth bounding boxes G. Each prediction will be assigned with zero or one ground-truth bounding box. Those predictions assigned to ground-truth bounding boxes constitute the positive set P, while other predictions form the negative set N. Then, the AP metric score is defined as the area under the precision-recall curve. Following the above notations, can be written as
1
Σ p(i). REP
AP
(1)
P
where p(i) denotes the precision of the predictions, whose classification scores are higher than ① can be explicitly calculated as the function of
that of the i-th prediction. Following [5 31 Eq. classification scores
rank (s:) rank(s;)
EjeN H(s - si) 1+ ΣjeB, H(s - 8.)
AP(()))
Σ EEP
Σ iEP
P
PP]
where rank (s;) and (an denote the ranks of the classification score 8, among the negative prediction N and the whole prediction set B, respectively. H(.) is the Heaviside step function.
1, if >
H(x)
0. otherwise.
Eq. ( explicitly includes the classification scores into the AP calculation, which helps previous works 5 [ to explore surrogate losses for the classification task. However, the localization task is also very important for object detection, which is just implicitly contained in Eq. (
3.2 Parameterized AP Loss
Instead of hand-crafting AP approximations for training networks. we propose to represent the numerous potential AP approximations in a unified parameterized formula, which is denoted as the Parameterized AP Loss. Specifically, we first explicitly reformulate the AP metric as function of classification scores and box coordinates {(b, &)),, output by the detector. Then, the non- differentiable components in the reformulated function are substituted with parameterized functions, so as to extend the AP metric to differentiable parameterized approximations.
Reformulating AP metric as Function of Classification Scores and Box Coordinates. Since Eq. (2) is only related to classification scores explicitly, it needs be further extended as the function of box coordinates. In Eq. (2). the only part related to the localization task for object detection the summation range (i.e., the positive set and the negative set N)) which motivates us to replace the summation range by explicitly including the localization results.
To this end, define the localization score for the i-th prediction
ToU by-
i P, EE
1(b.)=
(4)
0,
where denotes the assigned ground-truth bounding box the i-th prediction, and ToU(b,, b..) calculates the area overlap ratio between these two boxes. Here, box IoU is used to measure the

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Non-Differeniabll Components
Non www.Diiffrrnbbbbccc. AP Metric
Evaluation Motric
AP
SubsltWuuted
Approximated
Parameterized Functions
Porameterizid AP Loss
Trained Detectors
Rewards
f(:68)
(BE)
R(
Parameter Sampling Pollcy
Sampling f(x;8))
Traiiing
Evalustion
or (BS)
R(
Morona (3)
(()
1
R(O
Updeling Mess argmaxy 3 Ei (..
Figure 2: Overview of searching Parameterized AP Loss for object detection.
localization results, which follows the AP metric. Other measurements, such as GloU 38 and L1 are also applicable. Results Section 4.3 demonstrate that GIoU actually delivers the best results.
By further replacing the summation range in Eq. 2 using Eq. (4) the AP metric can be explicitly formulated the function of classification scores and box coordinates:
1 IPI
(((bb))) EjeB,44 H(85 8:) (1+H((()))) H(((b)). (5) IEB
AP([(b,, s;))
1+ ++EEEB.41 H(ss-b)
Here, H (I(b)) is the Heaviside step function with the localization score as input, which indicates whether the i-th prediction belongs the positive We multiply each summation term with the appropriate indicator function to replace the summation range.
Extending AP metric to Differentiable Parameterized Approximations. Designing approxi- mated loss functions for the AP metric may well mitigate the mis mis-alignment between network training and evaluation. We need to replace the non-differentiable Heaviside step function H(-) in Eq. (5) with its differentiable substitutions. Previous works [5] 30 have tried to address this problem with specially designed crror-driven update technique. Nevertheless, these hand-crafted methods all based on heuristics and expertise, which may not be optimal to guide the training process.
In contrast to hand-crafting differentiable surrogates for the AP metric, we substitute the non- differentiable components with parameterized functions, which helps to represent different AP approximations in unified formula. To achieve this, we replace the non-differentiable Heaviside step function (-) )) in Eq. (5) with parameterized functions f(;00 parameterized by 0. Then, the proposed Parameterized AP Loss can be obtained
Σ (((()):0))) EjeB,jti f(s, - 3, 0;102) - f(((b)):(a)) P
1
f(((b::;;ss)
1+ EJE [83 8;; 04)
Note that we normalize the input of f(x; 0) to æ e [0, for unified domain (see Section 4.1 for implementation details). The output range is also restricted to 0) E [0, 1], which follows the value range H (-). We adopt different parameters [0, in Eq. (6 to substitute different H()) functions in Eq. (5). which makes the Parameterized AP Loss more flexible. Our experiments in Section 4.3 also demonstrate that such parameterization delivers better performance than that of employing shared parameters for different H-)) functions. We also block the gradient of f(s; s;; f.), on the denominator, and empirically find such modification brings more stable training (see Appendix IA.I).
Piecewise I Linear Function.
The parameterized function f(x; 0) can be of any family of differen- tiable functions. Here, we adopt the piecewise linear function for simplicity. The piecewise linear function is composed of linear segments over different input ranges, where the first point of one
5

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segment coincides with the last point of the previous segment. Assuming a piecewise linear function f(; has M segments, the k-th segment is defined
9k+1 yk (I (x) + Ih S < Ik+l. k= M
fa; 0)
(7)
Tk+1
2
The k-th segment is controlled by two points (Ik. yke) and (IK+1) 9k+1). The coordinates these control points build up the set of parameters 0.
Following [8], we apply the end-point constraint and monotonicity constraint to regularize the search space. The end-point constraint requires that the end points of the parameterized function should take the same value with the original function, while the monotonicity constraint requires the parameterized function to be monotonically increasing. These constraints can be easily applied via
30 = =0, y/0 = 0,
J = 1, yM 1;
(end-point constraint)
2k-1 1, 0 ys 1
0
1, k=
1 M 1. (montonicity constraint)
IM -k-1
3M yk-1
To apply the above restrictions in optimization, the specific form of the parameters is defined as
Fk 2k-1
yk yk
θ =
k
M
(8)
MM Tk yM
3/-1
Such parameterization also makes each parameter independent, which simplifies the search.
Gradient Scale for Localization Branch. In object detection, to balance the training of the localization and classification branches, traditional detectors 3I usually add loss weight to the localization branch. While the Parameterized AP Loss unified loss, the network itself still preserves independent branches for these two tasks. In order to balance the training weights, multiply the gradient pack-propagated through the localization branch with a gradient scale À > 0 (see Figure D. which serves similar effect with the traditional loss weight.
In practice, the actual searched parameter 0₂ 1/ (log10 + 1). Such search parameterization form simplifies the search for small and large gradient scale À values. We also empirically restrict the search range for 0x as (0, 1), i.e., (0.1, 10).
The collection of the searched parameter for the gradient scale 0x and other parameters in the Parameterized AP Loss {6 are denoted θ, which could be optimized by efficient parameter search methods.
3.3 Searching for Optimal Parameters
The optimal parameter set θ for the Parameterized AP Loss is found by optimizing the network performance on the validation set with the AP metric. The massive parameter space of θ makes impractical determine the desired parameters manually. Efficient automatic search thus necessary to find the optimal parameters.
Our search process is described in Algorithm 1 Specifically, divide the training set into two subsets in the search process, Dirain for training and Deval for evaluation, respectively. The whole search process is treated as bi-level optimization problem, which can be formulated as
maxe R(O) AP Deval) s.t. w (+) min, Le(w; Durii))
where Le is the Parameterized AP Loss with the loss parameter set θ, stands for the network weights, and w (9) indicates the trained network weights with the given loss parameterized with e. (w; D) evaluates the AP metric for the network with weights on the given dataset D.
To optimize Eq. 9 we optimize the inner level and outer level problem iteratively. At the inner level, we train detector network on Durain with the certain loss parameter set θ for one epoch. At the outer level, following [43 [8 we adopt reinforcement learning, commonly used hyper- hyper-parameter optimization algorithm, to search for the optimal parameters. Other efficient search methods like evolutionary algorithm may also be applied. Specifically, we adopt the PPO2 (39) algorithm. Here, consider search process consisting of I'' rounds. In the t-th round, we sample S parameter set

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Algorithm 1: Parameterized AP Loss Search Process
Input: Network initial weights we, initial parameter set distribution (#1, o²1), training dataset Drrain and evaluation dataset Peval for the proxy task Output: Searched optimal parameter set @*
for ł <- 1 to T do
for 2 < 1 to do
Sample parameter set O N (Pt. of1);
Inner-level network training with the initial network weights wo.
(*O min Le (w: Perain;;
Evaluate the AP metric for the trained network as the corresponding reward,
(())) AP( (
end
Outer-level distribution update, μ++1 = argmax
3 Eis R (H, Me: O.):
end
return θ" argmaxe R(O)), AA = 1, T,
S
samples [0] independently from truncated normal distribution Nr (Me, I), where Me and o² are the mean and variance values, respectively. The truncated range of the distribution is set to [0, 1] so as to satisfy the monotonicity constraint of independent parameters Eq. 8 For the i-th sample, the AP metric score of the network trained in the inner level is regarded its reward R = R) In the PPO2 algorithm, the mean value of the truncated normal distribution for the next (t 1)-th round is updated as
14+1 argmaa /
Σ
( 14, 0.)
(10)
S
1=1
Here (R, H4, O2) is calculated from the original reward Rd of each sample as
(r,, pe, θ₁) min (,,0): pp (e, 0²1) R CLIP
p(0%:u,, o21) p(0) 14 0²1)
1, - R
(11)
where min(:-)) outputs the smaller value between two inputs, p(0); μr, T² denotes the PDF of given truncated normal distribution, and the CLIP function clips input value within 1 - and + as stated in PPO2 (39) for more stable and effective search. Note that the mean reward the t-th round is subtracted from each R2 for better convergence. More implementation details are described
Section 4.1
Experiments
We evaluate our approach on the COCO 2017 object detection benchmark 23 with various object detectors, including RetinaNet (25). Faster R-CNN [37] and Deformable DETR 42). Note that for Faster R-CNN, we search the losses for both the RPN and R-CNN heads simultancously.
4.1
Implementation Details
Loss Function Calculation. As described in Section 3.2
the inputs the parameterized functions f(x; 0) are normalized to range [0, 1] for a unified domain. For classification score differences
sy - St because their original values are unbounded, we first clip them within the range of [-1, 11 1] following [5] 30 The clipped values are further mapped to the range of [o, 1] via min-max re-scaling, For the localization scores l(b;), GloU 38 is used the measurement by default due its good performance. The localization scores are also mapped to the range of [o, 1] via min-max re-scaling. The default value of number of segments M fixed as 5. In experiments, the whole prediction in Eq. 66) consists of all predicted boxes in the current mini batch for all categories. The positive
"coCO 2017 publicly available under the Creative Commons Altribution 4.0 License. As far as we know, it does not contain any personally identifiable information offensive coutent.
7

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Table 1: Performance of different losses the COCO benchmark.
Model
Loss
AP
Ao s0
AP
APs
APM
APL
Focal Loss 25
L1
37.5
57.3
39.5
20.4
41.9
52.2
Focal Loss
25
GloU
38
39.2
58.1
41.4
22.0
43.6
53.3
RetinaNet
25
AP Loss 5
+ 1.
35.4
58.1
37.0
19.2
39.7
49.2
ResNet-50
12
FPN
24
aLRP Loss
30
39.0
58.7
40.7
22.0
43.5
54.0
Parameterized AP Loss (ours)
40.5
59.0
43.4
23.9
44,9
56.1
Cross Entropy + L1
39.0
59.6
42.2
22.4
43,5
53.2
Cross Entropy + GloU (8
39.1
59.3
42.4
23.2
43.4
53.5
Faster R-CNN
37
al.RP Loss 50
40.7
60.7
43.3
23.2
45.2
56.6
ResNet-50
12
FPN
24
AutoLoss Zero 19
39.3
59.0
42.4
21.4
44.0
54.0
CSE-AutoLoss-A
28
40.4
60.6
43.8
23.8
45.0
55.7
Parameterized AP Loss (ours)
42.0
60.7
45.0
25.3
46.6
57.7
Deformahle DETR
42
Focal Loss 25
L1 GloU
B8
43.8
62.6
47.7
26.4
47.1
58.0
Re 12
Parameterized AP Loss (ours)
45.3
63.1
49.6
27.9
49.3
60.2
prediction set P and the negative prediction set / are determined by the original pre-defined training target assignment of each object detector.
Search Settings. In COCO 2017 [23]] there are 1 118k images in the train subset. As described in
we randomly divide the original train subset into Diaain and Deval, which constitute I 113k training images and 5k evaluation images for the proxy task, respectively. In the inner-level network training of Algorithm|] we train the object detectors for one epoch. For Deformable DETR 422, we also set the number of feature levels to 1 to further accelerate the training. In the outer-level PPO2 [39] updating of Algorithm[] [] we sample S samples each round, and search for T = 40 rounds in total. The mean vector (41 of the truncated normal distribution is initialized to make fa; 0) z. The standard deviation a is initialized as 0.2, which decays linearly to 0 with respect to the search round. The clip operation in Eq. (1T) is applied on each component value in θ independently. The clip range e is set to 0.1 following PPO2 [39]. To solve Eq. (1O), we employ the Adam optimizer [16] for 100 iterations with base learning rate of 0.01. In the first 30 iterations, linear warm-up from learning rate of is utilized.
Section 3.3
Training Settings. After the parameter search, we re-train the object detectors with the searched Parameterized AP Loss on the COCO 2017 train subset, and evaluate them on the val subset, which
consists of 5k images. For RetinaNet (25] and Faster R-CNN (37], ImageNet (7] pre- pre-trained ResNet-
50 I2 with FPN [24] is utilized as the backbone, and the training settings strictly follow [5] 30]. For Faster R-CNN, For RetinaNet, the base learning rate is set to 0.016 and the batch size is 64 (8 images on each GPU). For Deformable DETR, ImageNet 7 pre-trained ResNet-50 [2] is utilized as the backbone, and the training settings strictly follow [4). where the learning rate is set to 0.0002 and the batch size is 32 (4 images on each GPU). All the experiments are conducted on 8 NVIDIA V100 GPUs. Our method is implemented based on the open-sourced MMDetection codebase
base learning rate of 0.024 and batch size of 64 (8 images
each GPU).
4.2 Main Results
Table I summarizes the performance of our approach and the existing losses on COCO 2017 23
Among these existing losses, Cross Entropy and Focal Loss (25) are commonly used for classification, while L. and GloU (38) are commonly used for localization. AP Loss [5] is a classification loss handcrafted for error-driven optimization of the AP metric, while aLRP Loss (00 is an error-driven method specifically designed for the LRP metric (29]. Compared with these handcrafted losses, Parameterized AP Loss can yield 1.5 3.0 AP score gain. We also compared with AutoLoss- Zero 19 and CSE AutoLoss (28). which are two AutoML-based losses. Our approach can obtain over 1.5 AP score improvement over them. We also seek to compare with the direct optimization methods B3 40 35), but their codes are not released and their original training settings differ too much from ours, which makes them difficult to be compared with. The searched parameterized functions / (ar; 0) of our approach are demonstrated in Appendix A.1
"MDetection open-soureed codebase under the Apache-2.0 Liceuse.

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4.3 Ablation Study
In this section, we ablate different design choices and the search effectiveness. RetinaNet [25 is employed as the baseline model in the following experiments.
Comparing Searched and Hand-crafted Substitutions for H(()) Here, we compare searched parameterized function (() with several handcrafted differentiable substitutions, including sig- moid, sqrt, linear, and square functions. Table shows that the searched parameterized function demonstrates significant improvement. The low performance of hand-crafted substitutions indicates that it is non-trivial hand-craft appropriate differentiable approximations, As analyzed in [5], the sigmoid substitution for H.) actually may lead to non-convergence of the network training. Note that the linear function is also the initialization of searched parameterized function, which indicates the effectiveness of the search process.
Comparing Separate and Shared Parameters for Different
(1)... As stated in Section 3.2
use separate parameters for different (-)) functions. We have also tried using shared parameters for all the (-) functions. Table shows that using separate parameters can yield nearly 3.0 AP improvement. That is because separate parameters can enhance the flexibility, so that different components in Eq. (6) can search for its own shape to better adjust the loss function.
Comparing with and without Gradient Scale X. As described in Section 3.2 the searched gradient scale is adopted to serve similar effect as the traditional loss weight. We have also tried not using the gradient scale, i.e., is fixed as 1. Table shows that the automatically searched gradient scale can indeed improve the performance.
Comparing Different Measurements for Localization Score l(b.). Table shows the results of using L1, IoU and GloU (38) for the localization score. GloU and LI produce sumilar results, both of which outperform the performance IoU. We argue that this is because IoU will back propaatee meaningful gradients only if two boxes are overlapped, while GIoU and L1 do not have such problem.
Comparing Different Number of Segments in (=; Tshle the results. Too less segments will limit the expressiveness [(; 0), which leads to significant drop in performance. However, too many segments may increase the search difficulty. In practice, 5 segments is enough.
Comparing PPO2 [3] and Random Search. Figure that PPO2 can find better parameters more efficiently than random search, which suggests that loss function search is non-trivial and reinforcement learning helps to accelerate the search process.
Table 2: Comparison of the searched parameterized
function and the hand-crafted substitutions for H()).
soare hoot reward
Differentiable
AP
AP30
APTs
APs
APM M
APr.
Substitution
Sigmoid
3.2
5.9
2.8
2.7
4.0
4.9
random
Sqrt
2.3
2.2
1.8
2.9
0-
Linear
22.9
39.3
23.2
16,9
27.3
29.2
sampling round
Square
36.4
56.5
39.9
20.3
41.7
53.0
Figure 3: Comparison of PPO2 and random search.
Searched
40.5
59.0
43.4
23.9
44.9
56.1
The search is repeated four times. Each curve presents the highest searched reward up to the 1-th round.
Table 3: Comparison shared and separate parame- ters for different H()).
Table 4: Comparison our proposed loss with and without the gradient scale A.
Parameters AP
AP30
AP7o
APs
APM
APL.
Gradient Scale
AP
AP7s
APs
APM
APL
Shared 37.8
58.3
39.9
21.6
42.2
52.1
w/o
39.4
58.9
42.0
22.8
43.9
54.2
Separate 40.5
43.4
23.9
44.9
w/
40.5
59.0
43.4
23.9
44.9
56.1
Table 5: Comparison of different measurements for the localization score l(b.).
Table Comparison different number of segments in the piecewise linear functions (=:0).
Measurement
AP
APso
APrs
APs
APM
APt
Segments
AP
APso
AP1s
APs
APM
APt
1.1
40.0
58.8
43.2
22.3
44.5
55.3
3
34.2
48.3
36.8
9.2
43.8
56.9
loU
38.1
56.2
40.8
21.8
43.6
53.4
5
40.5
59.0
43.4
23.9
44.9
56.1
GIoU
40.5
59.0
43.4
23.9
44,9
56.1
7
40.3
58.7
43.2
23.8
44.6
55.9

---

Conclusion
In this paper, we proposed Parameterized AP Loss, which represents numerous AP approximations in unified formula, and automatically searches for the optimal loss function. Parameterized AP Loss is a single unified loss function capturing the classification and localization sub-tasks simultaneously, which consistently outperforms existing loss functions various object detectors. Although we have verified the effectiveness parameterization there are still open questions about whether such technique can be extended to other non- non-differentiable metrics in different tasks, such as the widely used BLEU metric in machine translation, where the calculation of n-gram might be non-trivial to be parameterized.
searching for optimal loss functions for the AP metric,
Potential Negative Societal Impacts. Our searched losses share the same societal issues with other hand-crafted ones, that the trained object detectors may have inexplicable detection failures and suffer from data bias. The automatic search process in our method is laborsaving, which may also have a negative impact on social employment opportunities.
Acknowledgments and Disclosure of Funding
The work is supported by the National Key R&D Program of China (2020AAA0105200) and Beijing Academy of Artificial Intelligence.
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes] See Section S
(c) Did you discuss any potential negative societal impacts of your work? [Yes] We provide discussion in Section
(d) Have you read the ethics review guidelines and ensured that your paper conforms to
them? [Yes] We have discussed potential negative societal impacts. The used data follows the general ethical conduct.
2. If you are including theoretical results...
(a) Did you state the full set of assumptions all theoretical results? [N/A]
(b) Did you include complete proofs of all theoretical results? [N/A]
3. If you ran experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or URL)? [No] Our code will
be released once the paper is accepted, and the dataset we use are publicly available.
We have provided detailed implementation details in Section 4.1 and the pseudo-code algorithm in Algorithm!] |
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 4.1
(c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)? [No] However, we have ran the search experiments multiple times, as in Figure [
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 4.1
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes] We cite the open dataset that we use, and the codebase that we build our work on.
(b) Did you mention the license of the assets? [Yes] See Section
(c) Did you include any new assets either in the supplemental material or as a URL? [No] Our code will be released once the paper is accepted.
(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [Yes] See Section 4
(c) Did you discuss whether the data you using/curating contains personally identifiable information or offensive content? [Yes] See Section
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
12

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(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent participant compensation? [N/A]
13


Reviews:
[{"review_id": "zjizazBJPL9", "reviewer": "Reviewer_sg2b", "summary": "This paper proposes a Parameterized AP Loss to better align the network training and evaluation in object detection. The main technical contributions include:\n1) It proposes a parameterized loss function by reformulating the AP metric and replacing the non-differentiable component with parameterized functions.\n2) It automatically searches for the optimal parameters of the proposed loss function through a reinforcement-learning-based search process.\n3) Finally, it demonstrates competitive results on COCO.\n", "questions": "", "limitations": "", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Pros:\n1. The idea of introducing localization score into AP loss and parameterizing the reformulated AP loss is novel and interesting.\n2. The paper is clearly written and easy to follow with detailed equations.\n3. The authors conducted adequate experiments and ablation studies, verifying the efficacy of their method.\nCons:\nThere is no big flaw in this paper, here are some small issues:\n1. In lines 285-292 and Table 2, what is the value of the gradient scale $\\lambda$ used in the experiments with hand-crafted substitutions? The experiments with hand-crafted substitutions share H(.), what is the performance of our method when using the same $lambda$ and shared parameters of H(.) as the hand-crafted ones?\n2. What is the difference of the shape of the five parameterized functions $f(x; \\theta_{i})$ searched by our method. Will they change dramatically in different object detectors (e.g., RetinaNet, Faster R-CNN, and Deformable DETR)? It would be better to visualize the shape of the parameterized functions.\n3. How is the generalization of the Parameterized AP Loss. Does it also achieve a good performance when using the hyper-parameters searched on different datasets? For example, searching the hyper-parameters on COCO and using the same hyper-parameters to train the object detector on VOC.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Yes.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1.5"}, "extra_text": {}}, {"review_id": "xELgG7rFWE", "reviewer": "Reviewer_KBgc", "summary": "This paper describes a technique for estimating a loss for object detection that is well-aligned with the AP loss used at evaluation time. The work builds on a prior formulation of the AP loss as a function on classification scores, but converts the formulation to account for per-bounding box results, and also replaces the Heaviside function with parametrized, searchable functions. Experiments show the gain from the proposed technique using three backbone detectors, and compares to different prior losses.\n\nPost-rebuttal: The rebuttal addressed my concerns.", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "1. The work is well-written and sound. All steps are well-explained and justified. The results are strong and extensive, including detailed ablations that verify the contribution of each method component. \n\n2. The choice of baseline losses seems reasonable, and techniques from each family described in Sec. 2 are included in the experiments, except for the \"Direct Optimization for the AP Metric\" family-- why are results from one of those methods not included? \n\n3. Some statements are subjective, e.g. L91-92. Others are a bit unclear, i.e. L97-98, 104-105. Some text on the other hand is repetitive, e.g. L168-170 have been previously stated multiple times.\n\n4. The authors explain why they use different theta parameters for different parts of Eq. 6, but one choice especially stands out, e.g. the use of theta_5 as opposed to theta_1 for different occurrences of the exact same l(b_i). How important is this experimentally?\n\n5. A few places in the text indicate the approach relies a bit too much on experimental settings, e.g. L179. In general, using a searchable, tunable metric is great, but is there a risk of overfitting? \n\n6. In general, the proposed approach works well, but is there any cost to pay? Does some part of the method's workings get slower or less stable compared to baseline losses?\n\n7. Minor questions:\n- Eq. 7, what is the value of M, i.e. how computed?\n- L215, which exact equation is used? ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Discussed", "review_text": "", "extra_scores": {"ethical_concerns": "N/A", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "1.5"}, "extra_text": {}}, {"review_id": "XNQYQxwQxo", "reviewer": "Reviewer_L4vJ", "summary": "This paper proposes a parameterized AP loss. Parameterizd functions are introduced to substitute the non-differentiable components in the AP calculation. Then automatic parameter search algorithms can be employed to search the optimal parameters.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The method proposed by this paper is novel to some extent and technically sound. The written is also clear.\n\nStrong points:\n\nThe parameterized AP Loss can produce better performance, while hand-crafted AP approximated loss produce lower performance than the commonly used cross-entropy and L1/smooth-L1 losses.\n\nWeak points:\n\n1.Parameterization makes the optimal parameters of the proposed AP Loss different on different datasets, so it is necessary to search for the optimal parameters for each dataset. However, the process of finding optimal parameters is computationally expensive.\n\n2.The AP loss proposed in this paper uses a series of parameterized functions to approximate the calculation of AP, but this paper does not show whether the finally searched functions are really close to the Heaviside step function. If the searched function curve can be drawn, it will give readers a more intuitive understanding.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The searched losses in this paper share the same societal issues with other hand-crafted ones, that the trained object detectors may have inexplicable detection failures and suffer from data bias. This problem mainly needs to be solved from the perspective of dataset.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "6hr_-07mgeJ", "reviewer": "Reviewer_8qFe", "summary": "Authors propose a loss for object detection, that is a relaxation of mAP suitable for use in gradient ascent. Where AP includes Heaviside (step) functions, the authors replace these by learnt piecewise-linear functions. These functions are trained to maximise AP on a held-out set achieved by a detection model that was itself trained using them as the loss (so they do not aim to approximate AP closely, but to derive a surrogate loss that is 'structurally similar' to AP and achieves good training performance). Experiments on COCO 2017 show accuracy improvements over baselines (trained with focal/XE + regression losses) and other techniques approximating AP.", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths:\n\nThe proposed method is reasonably well-motivated, in terms of limitations of existing methods that do not optimise the surrogate itself to be 'good' as a training loss.\n\nThe precise structure of the proposed surrogate loss is novel; also I don't know of any existing methods for object detection that aim to tune the surrogate.\n\nEmpirical results indicate some improvement over earlier methods; these gains are consistent across various different detection architectures. There is also a detailed ablation study. There is plenty of detail on the experimental details and training protocols.\n\n\nConcerns / suggestions:\n\nThere are no error bars / standard deviations on the main experiments. This makes it difficult to judge how much better the proposed method really is.\n\nBy replacing the Heaviside functions with an arbitrary monotonic function, the loss ceases to be AP; rather, it becomes a proxy loss chosen to yield high test-set AP. But given this, there is not really any justification for retaining the full structure of AP in the loss -- larger chunks of eq. 2 than just the H(...) terms could be replaced. What is the justification for not doing so?\n\n\"A Unified Framework of Surrogate Loss by Refactoring and Interpolation\" [Liu, ECCV 20] is very relevant, and should be cited (and either evaluated against, or discussed why it is not applicable).\n\n\"Learning Surrogates via Deep Embedding\" [Patel, ECCV 20] is relevant and should be cited.\n\n\"SoDeep: a Sorting Deep net to learn ranking loss surrogates\" [Engilberge, CVPR 19] is relevant and should be cited.\n\nIt would be useful to add some insights on why the various learnt piecewise-linear functions look so different.\n\nWhat happens if the Heaviside functions are instead replaced by sigmoids of varying slope/offset? This results in many fewer parameters in the loss, but gives more flexibility than just sigmoid.\n\nThere is no discussion of how long the PPO training of the surrogate loss takes (e.g. does it dominate the main detector training time)?\n\nThe same training process used for the loss parameters (i.e. PPO2) could, theoretically, also be used to train the main detection model. There should be some discussion of why this is not done (or even better, an evaluation).\n\n42: \"helps to represent the numerous AP approximations\" -- clarify -- is this supposed to refer back to the AP approximations discuss in the previous bit of text?\n\n46: \"more effective\" -- than what?\n\n---\n\n## Post-rebuttal\n\nThe authors' comment clarified several concerns raised in my review (e.g. the search does not need to be repeated for each dataset; the standard deviations are small). I therefore still broadly favor acceptance. However, the rebuttal simply says that UniLoss [Liu, ECCV '20] -- which is one of the most relevant recent works -- will be discussed in the final version; there is no attempt to justify why it was ignored so far, and no promise to include an evaluation against it. Therefore, I cannot raise my score higher. Concerns raised by the other reviewers seem to have been adequately addressed by the authors' comments.\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "There is only very minimal discussion of limitations of the method. Societal impact is discussed sufficiently.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2.5"}, "extra_text": {}}]

Author rebuttal:

Meta-review: All the reviewers have supported the acceptance of this paper, so I recommend accepting as well.
Decision: Accept (Poster)