Searching Parameterized AP Loss for Object Detection

1 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 is usually used , which captures the performance of localization and classification simultaneously . However , as in most object detectors [ 26 ] , the training of localization and classification sub-tasks are driven by two separate losses ( see Figure 1 ) . For example , the L1/smooth-L1 [ 11 ] or GIoU [ 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 , a straight-forward solution is to 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 on the mathematical formula of the AP metric . [ 3 ] replaces the non-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 ∗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 ( NeurIPS 2021 ) . performance than the commonly used cross-entropy and L1/smooth-L1 losses . Another line of works try to estimate the gradient for the AP metric directly [ 5 , 13 , 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-alignment issue well . 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 is a piecewise constant function , whose differentiable approximations have infinite possibilities . The hand-crafted approximations / gradients may well be sub-optimal for driving network training . Instead of manually determining the surrogate loss form , we propose to approximate the non-differentiable parts with a 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 . It utilizes parameterization to construct the search space so that the optimal parameters can be searched automatically . Specifically , we first explicitly reformulate the AP metric for object detection as a 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 a reinforcement-learning-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 [ 25 ] , Faster R-CNN [ 37 ] and Deformable DETR [ 42 ] . Extensive experiments on 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 can represent numerous AP approximations in a unified formula , which captures classification and localization sub-tasks simultaneously in a single loss function . • Instead of hand-crafting AP losses or gradient approximations , our approach automatically searches for 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-entropy and smooth-L1 losses are widely used for the classification and localization sub-tasks , respectively [ 26 ] . For the classification subtask , 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 a ranking problem . For the localization sub-task , a series of works [ 38 , 41 ] introduce IoU variants for better localization . GFL [ 22 ] and GFLv2 [ 21 ] 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 a joint optimization . But such combination is still not consistent with the AP metric . Recent works [ 5 , 30 ] have noticed the mis-alignment issue between network training and evaluation , and try to 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 errordriven gradient . aLRP Loss [ 30 ] extends the framework of AP Loss , and unifies classification and localization losses under the LRP metric [ 29 ] . These loss functions are all hand-crafted , and may well be sub-optimal 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 as 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 pseudo-gradients . [ 40 ] applies the loss-augmented inference for the gradient estimation . [ 35 ] resorts to reinforcement learning for policy gradient . While these works take a 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 . [ 32 ] learns an embedding for prediction and target , so the Euclidean distance between them approximates the metric value . [ 8 ] 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 . Searching Loss Functions for Object Detection . Recent works [ 28 , 19 ] 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 a 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 [ 1 ] are commonly used for hyper-parameter tuning . As the search space becomes more complex , many efficient hyperparameter optimization methods have been proposed . Bayesian optimization [ 2 , 14 ] aims to build a prediction model from history data and use it to select the most promising hyper-parameters for evaluation . Bandit-based methods [ 15 , 20 , 10 ] view it as a resource allocation problem and allocate most of the computational resource to the most promising hyper-parameters . Evolutionary algorithms [ 36 , 33 ] evolve the optimal hyper-parameters from a population of models . Reinforcement learning techniques [ 43 ] have also been used to explore the search space via sampling candidates and adjusting the sampling policy . The introduced parameters in our proposed Parameterized AP Loss are indeed hyper-parameters , which fit the hyper-parameter optimization methods . In this work , we adopt reinforcement learning to search for the optimal parameters because of its simplicity and efficiency . Other efficient hyperparameter optimization algorithms can also be employed . 3 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 is illustrated in Figure 2 . 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 = { ( bi , si ) } Ni=1 . Here , bi ∈ R4 denotes the box coordinates of the i-th prediction , and si ∈ R is its corresponding classification score . In the AP metric , these predictions will be matched with a 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 , it can be written as AP = 1 |P| ∑ i∈P p ( i ) , ( 1 ) where p ( i ) denotes the precision of the predictions , whose classification scores are higher than that of the i-th prediction . Following [ 5 , 3 ] , Eq . ( 1 ) can be explicitly calculated as the function of classification scores AP ( { si } Ni=1 ) = 1 |P| ∑ i∈P 1− rank – ( si ) rank ( si ) = 1 |P| ∑ i∈P 1− ∑ j∈N H ( sj − si ) 1 + ∑ j∈B , j 6=iH ( sj − si ) , ( 2 ) where rank– ( si ) and rank ( si ) denote the ranks of the classification score si among the negative prediction set N and the whole prediction set B , respectively . H ( · ) is the Heaviside step function , H ( x ) = { 1 , if x > 0 , 0 , otherwise . ( 3 ) Eq . ( 2 ) explicitly includes the classification scores into the AP calculation , which helps previous works [ 5 , 3 ] 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 . ( 2 ) .

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.  Post-rebuttal: The rebuttal addressed my concerns.