1 Introduction
Synthetic images rendered by graphics engines have emerged as a promising source of training data for deep networks, especially for vision and robotics tasks that involve perceiving 3D structures from RGB pixels Butler et al. (2012); Yeh et al. (2012); Varol et al. (2017); Ros et al. (2016); McCormac et al. (2017); Xia et al. (2018); Chang et al. (2017); Kolve et al. (2017); Song et al. (2017); Richter et al. (2016, 2017); Zhang et al. (2017); Li and Snavely (2018). A major appeal of generating training images from computer graphics is that they have a virtually unlimited supply and come with highquality 3D ground truth for free. Despite its great promise, however, using synthetic training images from graphics poses its own challenges. One of them is ensuring that the synthetic training images are useful for real world tasks, in the sense that they help train a network to perform well on real images. Ensuring this is challenging because a graphicsbased generation pipeline requires numerous design decisions including the selection of 3D shapes, the composition of scene layout, the application of texture, the configuration of lighting, and the placement of the camera. These design decisions can profoundly impact the usefulness of the generated training data, but have largely been made manually by researchers in prior work, potentially leading to suboptimal results. In this paper we address the problem of automatically optimizing a generation pipeline of synthetic 3D training data, with the explicit objective of improving the generalization performance of a trained deep network on real images. One idea is blackbox optimization: we try a particular configuration of the pipeline, use the pipeline to generate training images, train a deep network on these images, and evaluate the network on a validation set of real images. We can treat the performance of the trained network as a blackbox function of the configuration of the generation pipeline, and apply blackbox optimization techniques. In fact, recent work by Yang and Deng (2018)
has explored this exact direction. They use genetic algorithms to optimize the 3D shapes used in the generation pipeline. In particular, they start with a collection of simple primitive shapes such as cubes and spheres, and evolve them through mutation and combination into complex shapes, whose fitness is determined by the generalization performance of a trained network. They show that the 3D shapes evolved from scratch can provide more useful training data than manually created 3D CAD models. The advantage of blackbox optimization is that it makes no assumption about the function being optimized as long as it can be evaluated. As a result, it can be applied to any existing function including advanced photorealistic renderers. On the other hand, blackbox optimization is computationally expensive—knowing nothing else about the function, it needs many trials to find a good update to the current solution. In contrast, gradientbased optimization can be much more efficient by assuming the availability of analytical gradients, which can be efficiently computed and directly correspond to good updates to the current solution, but the downside is that analytical gradients are often unavailable, especially for many advanced photorealistic renderers. In this work, we propose a new method that optimizes the generation of 3D training data based on what we call “hybrid gradient”. The basic idea is to make use of analytical gradients where they are available, and combine them with blackbox optimization for the rest of the function. Our hypothesis is that hybrid gradient will lead to more efficient optimization than blackbox methods because it makes use of the partially available analytical gradient. Concretely, if we parametrize the design decisions as a real vector
, the function mapping to network performance can be decomposed into two parts: (1) from design parameters to the generated training images , and (2) from the training images to the network performance . The first part often does not have analytical gradients, due to the use of advanced photorealistic renderers. We instead compute an approximate gradient by averaging finite difference approximations along random directions Mania et al. (2018). For the second part, we compute analytical gradients through backpropagation—with SGD training unrolled, the performance of the network is a differentiable function of the training images. Then we combine the approximate gradient and the analytical gradient to obtain the hybrid gradient of the network performance
with respect to parameters , as illustrated in Fig. 1.A key ingredient of our approach is representing design decisions as real vectors of fixed dimensions, including the selection and composition of shapes. Yang and Deng (2018) represent 3D shapes as a finite set of graphs, one for each shape. This representation is suitable for a genetic algorithm but is incompatible with our method. Instead, we propose to represent 3D shapes as random samples generated by a Probabilistic ContextFree Grammar (PCFG) Harrison (1978)
. To sample a 3D shape, we start with an initial shape, and repeatedly sample a production rule in the grammar to modify it. The (conditional) probabilities of applying the production rules are parametrized as a real vector of a fixed dimension. Our approach is novel in multiple aspects. First, to the best our knowledge, we are the first to propose the idea of hybrid gradient, i.e. combining approximate gradients and analytical gradients, especially in the context of optimizing the generation of 3D training data. Second, the integration of PCFGbased shape generation and hybrid gradient is also novel. We evaluate our approach on the task of estimating surface normals from a single image. Experiments on standard benchmarks show that our approach can outperform the prior state of the art on optimizing the generation of 3D training data, particularly in terms of computational efficiency.
2 Related Work
Generating 3D training data
Synthetic images generated by computer graphics have been extensively used for training deep networks for numerous tasks, including single image 3D reconstruction Song et al. (2015); Hua et al. (2016); McCormac et al. (2017); Janoch et al. (2011); Yang and Deng (2018); Chang et al. (2015), optical flow estimation Mayer et al. (2018); Butler et al. (2012); Gaidon et al. (2016), human pose estimation Varol et al. (2017); Chen et al. (2016), action recognition Roberto de Souza et al. (2017), natural language modeling Johnson et al. (2017), and many others Weichao Qiu (2017); MartinezGonzalez et al. (2018); Xia et al. (2018); Tobin et al. (2017); Richter et al. (2017, 2016); Wu et al. (2018). The success of these works has demonstrated the effectiveness of synthetic images. To ensure the relevance of the generated the training data to real world tasks, a large amount of manual effort has been necessary, particularly in acquiring 3D assets such as shapes and scenes Chang et al. (2015); Janoch et al. (2011); Choi et al. (2016); Xiang et al. (2016); Hua et al. (2016); McCormac et al. (2017); Song et al. (2017)
. To reduce manual labor, some heuristics have been proposed to automatically generate 3D configurations. For example,
Zhang et al. (2017) design an approach to use entropy of object masks and color distribution of the rendered image to select sampled camera poses. McCormac et al. (2017) simulate gravity for physically plausible object configurations inside a room. Prior work has also performed explicit optimization of 3D configurations. For example, Yeh et al. (2012) synthesizes layouts with the target of satisfying constraints such as nonoverlapping and occupation. Jiang et al. (2018) learns a probabilistic grammar model for indoor scene generation, with parameters learned using maximum likelihood estimation on the existing 3D configurations in SUNCG Song et al. (2017). Similarly, Veeravasarapu et al. (2017) tunes the parameters for stochastic scene generation using generative adversarial networks, with the goal of making synthetic images indistinguishable from real images. Qi et al. (2018) synthesize 3D room layouts based on humancentric relations among furniture, to achieve visual realism, functionality and naturalness of the scenes. However, these optimization objectives are different from ours, which is the generalization performance of a trained network on real images. The closest prior work to ours is that of Yang and Deng (2018), who use a genetic algorithm to optimize the 3D shapes used for rendering synthetic training images. Their optimization objective is the same as ours, but their optimization method is different in that they do not use any gradient information.Unrolling and backpropogating through network training
One component of our approach is unrolling and backpropagating through the training iterations of a deep network. This is a technique that has often been used by existing work in other contexts, including hyperparameter optimization
Maclaurin et al. (2015) and metalearning Andrychowicz et al. (2016); Ha et al. (2017); Munkhdalai and Yu (2017); Li and Malik (2017); Finn et al. (2018). Our work is different in that we apply this technique in a novel context: it is used to optimize the generation of 3D training data and it is integrated with approximate gradients to form hybrid gradients.Hyperparameter optimization
Our method is connected to hyperparameter optimization in the sense that we can treat the design decisions of the 3D generation pipeline as hyperparameters of the training procedure. Hyperparameter optimization is typically approached as blackbox optimization Bergstra and Bengio (2012); Bergstra et al. (2011); Lacoste et al. (2914); Brochu et al. (2010). Since blackbox optimization does not assume knowledge about the function being optimized, it requires repeated evaluation of the function, which is expensive in this case because it contains the process of training and evaluating a deep network. In contrast, we combine analytical gradients from backpropagation and approximate gradient from generalized finite difference for more efficient optimization.
3 Problem Setup
Suppose we have a probabilistic generative pipeline that takes a real vector and a random number as input. After 3D composition and rendering, an image and its 3D ground truth are computed through a function . By randomly sampling for times, we obtain a dataset of size for training:
(1) 
Then, a deep neural network with initialized weights
is trained on the training data , with the function representing the optimization process and generating the weights of the trained network. The network is then evaluated on real data with a validation loss to obtain a generalization performance :(2) 
Combining the above two functions, is a function of , and the task is to optimize this value with respect to the parameters . As we mentioned in the previous section, blackbox algorithms typically need repeated evaluation of this function, which is expensive.
4 Approach
4.1 Generative Modeling of Synthetic Training Data
We decompose the function into two parts: 3D composition and rendering.
3D composition
Contextfree grammars have been used in scene generation Jiang et al. (2018); Qi et al. (2018) and in parsing of Constructive Solid Geometry (CSG) shapes (Sharma et al., 2018). Here, we design a probabilistic contextfree grammar (PCFG) Foley et al. (1990)
to control the random generation of unlimited shapes. In a PCFG, a tree is randomly sampled given a set of probabilities. Starting from a root node, the leaf nodes of the tree keeps expanding according to a set of rules. The process is stopped until all leaf nodes cannot expand. Since multiple rules may apply, the parameters in a PCFG define the probability distribution of applying different rules. In our PCFG, a shape can be constructed by composing two other shapes through union and difference, and this construction can be recursively applied until all leaf nodes are a predefined set of concrete primitive shapes (terminals). The parameters can be the probability of either expanding the node or replacing it with a terminal. Given our PCFG model with the probability parameters
, a 3D shape can be composed:(3) 
Rendering training images
we use a graphics renderer to render the composed shape . The rendering configurations (e.g. camera poses), are also sampled from a distribution controlled by a set of parameters :
(4) 
Now that we have Eq. 3 and 4, The full function for training data generation can be represented as follows:
(5) 
where and . By drawing the random number
from a uniform distribution, we obtain a set of training images and their 3D ground truth
.4.2 Hybrid Gradient
After a deep network is trained on synthetic training data , it is evaluated on a set of validation images to obtain the generalization loss . Recall that to compute the hybrid gradient to optimize , we multiply two gradients: the gradient of network training and the gradient of image generation , as is shown in Fig. 2.
Analytical gradient from backpropagation
We assume the network is trained on a a set of previously generated training images
. Without loss of generality, we assume minibatch stochastic gradient descent (SGD) with a batch size of 1 is used for weight update. Let function
denote the SGD step and let denote the training loss:(6) 
Note that the SGD step is differentiable with respect to the network weights as well as the training batch , if our training loss is twice (sub)differentiable. This requirement is satisfied in most practical cases. To simplify the equation, we assume the training loss and the learning rate do not change during one update step of , so the variables can be safely discarded in the equation. Therefore, the gradient of the generalization loss for each sample can be computed through backpropagation:
(7)  
(8) 
with the boundary condition computed from the evaluation function :
(9) 
Aproximate gradient from finite difference
For the formulation in Eq. 5, the graphics renderer can be a general black box and nondifferentiable. We can approximate the gradient of each rendered image with ground truth with respect to the generation parameters , with Basic Random Search, a generalized finite difference method described in (Mania et al., 2018)
. First, we sample a set of noise from an uncorrelated multivariate Gaussian distribution
Mania et al. (2018):(10) 
Next, we approximate the Jacobian for each sample ( denotes cross product) Mania et al. (2018):
(11) 
Incremental training
Following Yang and Deng (2018), we incrementally update parameters and network weights . At timestamp , we update with the hybrid gradient; for network weights, we simply use the latest trained network for initialization in timestamp :
(12) 
5 Experimental Setup
We experiment on the task of surface normal estimation, a standard task for singleimage 3D. The input is a RGB image and the output is pixelwise surface normals. We evaluate on two datasets of real images: MITBerkeley Intrinsic Images Dataset (MBII) Barron and Malik (2015), which focuses on images of single objects, and NYU Depth Silberman et al. (2012), which focuses on indoor scenes. For MBII, we use pure synthetic shapes Yang and Deng (2018) to render training images. We first compare our method with ablation baselines, then show that our algorithm is better than the previous state of the art. For NYU Depth, we base our generative model on SUNCG Song et al. (2017) and augment the original 3D configurations in Zhang et al. (2017). We report the performance of surface normal directions with the metrics commonly used in previous works, including mean angle error (MAE), median angle error, mean squared error (MSE), and the proportion of pixels that normals fall in an error range ().
5.1 MITBerkeley Intrinsic Images
Following the work of Yang and DengYang and Deng (2018), we recover the surface normals of an object from a single image.
Synthetic shape generation
In Yang and Deng (2018), a population of primitive shapes such as cylinders, spheres and cubes are evolved and rendered to train deep networks. The evolution operators are defined as transformations of individual shapes, as well as boolean operations of shapes in Constructive Solid Geometry (CSG) Foley et al. (1990). In our algorithm, we also use the CSG grammar for our PCFG:
S => E; E => C(E, T(E))  P; C => union  subtract; P => sphere  cube  truncated_cone  tetrahedron; T => attach * rand_translate * rand_rotate * rand_scale;
In this PCFG, the parameter vector
consists of three parts: (1) The probability of the different rules; (2) The means and variations of lognormal distributions controlling shape primitives (
P
), such as
the radius of the sphere;
(3) The means and variations of lognormal distributions controlling transformation parameters (T
),
such as scale values.
Examples of sampled shapes are shown in Fig. 3.
We compose our shape in mesh representations, slightly different from the implicit functions in (Yang and Deng, 2018). Therefore, we reimplemented their algorithm with mesh representations for fair comparison.
For network training and evaluation, we follow (Yang and Deng, 2018) and train the Stacked Hourglass Network Newell et al. (2016) on the images, and use the standard split of the MBII dataset for the optimization of and testing.
5.2 NYU Depth
Scene perturbation
We design our scene generation grammar as an augmentation of collected SUNCG scenes Song et al. (2015) with the cameras from Zhang et al. (2017):
S => E,P; E => T_shapes * R_shapes * E0; P => T_camera * R_camera * P0; T_shapes => translate(rand_x, rand_y, rand_z); R_shapes => rotate_euler(rand_yaw, rand_pitch, rand_roll);
For each 3D scene S
, we perturb the positions and poses of the original cameras (P0
) and shapes (E0
).
The position perturbations follow a mixture of uncorrelated Gaussians, and the perturbations
for pose angles (yaw, pitch & roll) follow a mixture of von Mises, i.e. wrapped Gaussians.
The vector consists of the parameters of the above distributions.
Our networks are only trained on synthetic images, and evaluated on NYU Depth V2 Silberman et al. (2012) with the same setup as in Zhang et al. (2017).
For real images in our optimization pipeline, we sample a subset of images from the standard validation images in NYU Depth V2.
6 Experiment Results
Summary Stats  Errors  

MAE  Median  MSE  
Fixed  
Hybrid gradient  
6.1 MITBerkeley Intrinsic Images
Ablation study
We first sample 10 random values of in advance, then for each we train a network, with the exact same training and evaluation configurations as in our hybrid gradient. We then report the best, median and worst performance of those 10 networks, and label the corresponding as , and . In hybrid gradient, we initialize from these three values and report the performance on test images also in Table 1. From the table we can observe that training with a fixed can hardly match the performance of our method, even with multiple trials. Instead, our hybrid gradient approach can optimize to a reasonable performance regardless of different initialization. This simple diagnostic experiment demonstrates that our algorithm is working properly.
Comparison with previous work
In this experiment, we compare with blackbox algorithms including Basic Random Search Mania et al. (2018) and Shape Evolution Yang and Deng (2018). Because we use mesh implementation instead of implicit computation graph in Yang and Deng (2018) for CSG, we reimplemented Shape Evolution with the same setting for fair comparison. We follow (Yang and Deng, 2018) for the initialization of , train the networks and update for the same number of steps. We then report the test performance of the network which has the best validation performance. The results are shown in Table 2.
Summary Stats  Errors  
MAE  Median  MSE  
SIRFS(Barron and Malik, 2015)  —  
Evolution (Yang and Deng, 2018)(Reported)  —  
Evolution (Yang and Deng, 2018)(Our Impl.)  
Basic Random Search Mania et al. (2018)  
Hybrid gradient 
We also run the experiments on the same set of CPUs and GPUs, and plot the test mean angle error with respect to the CPU time, GPU time and total computation time (Fig. 4). We see that our algorithm is more efficient than the above baselines. Shapes sampled from our optimized PCFG are shown in Fig. 3.
6.2 NYU Depth
We initialize our network using the original model in (Zhang et al., 2017) and initialize using a small value. To compare with random , we construct a dataset of k images with a small random for each image. We then load the same pretrained network and train for the same number of iterations as in hybrid gradient. We then evaluate the networks on the test set of NYU Depth V2 Silberman et al. (2012), following the same protocol. The results are reported in Table 3. Note that none of these networks has seen a single real image except for validation.
Summary Stats  Errors  

Mean  Median  
Original (Zhang et al., 2017)  
Training with random  
Hybrid gradient 
The numbers indicate that our parametrized generation of SUNCG augmentation exceeds the original baseline performance. Note that the network trained with random is worse than original performance. This means without proper optimization of perturbation parameters, such random augmentation may hurt generalization, demonstrating that good choices of these parameters are crucial for generalization to real images.
7 Conclusion
In this paper, we have proposed hybrid gradient, a novel approach to the problem of automatically optimizing a generation pipeline of synthetic 3D training data. We evaluate our approach on the task of estimating surface normals from a single image. Our experiments show that our algorithm can outperform the prior state of the art on optimizing the generation of 3D training data, particularly in terms of computational efficiency. Acknowledgments This work is partially supported by the National Science Foundation under Grant No. 1617767.
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