The data collection process is often biased (Torralba and Efros, 2011). One of the most prominent factors of such dataset bias is the background, such that some aspects of the background systematically co-occur with certain objects, e.g., airplanes may tend to always appear in the sky. This co-occurrence is a confounding factor for the network, and the network may learn to associate the background with the object, e.g., the sky may be regarded as part of the airplane. Previous works have shown that DNNs for image recognition fail to classify objects in novel and uncommon backgrounds (Choi et al., 2012; Volokitin et al., 2017; Beery et al., 2018; Tian et al., 2018). Remarkably, popular object recognition datasets are biased to such an extent that DNNs can predict the object category even when the objects are removed from the image (Zhu et al., 2017; Tian et al., 2018; Xiao et al., 2021). Barbu et al. (2019) introduced a new benchmark which addresses the biased co-occurrence of objects and background, among other types of bias. DNNs exhibit large performance drops in this benchmark compared to ImageNet (Deng et al., 2009). Recently, Borji has shown that a large portion of the performance drop comes from the bias in object's background, as classifying the object in isolation substantially alleviates the performance drop (Borji, 2021).

In contrast to previous works, we analyse the impact of the object's background to the DNN's generalization performance when the dataset is unbiased, i.e., there is no significant correlation between the objects and backgrounds and the statistics of the object's background are the same between training and testing times. To the best of our knowledge, our work is the first to investigate the effects of object's background on DNNs when these are unbiased. We show that just the presence of background, even if it is unbiased, can degrade the data efficiency of the DNN.

A remarkable characteristic of DNNs is that the test error follows a double-descend when the DNN's width is increased by adding more hidden units. Thus, the test error decreases as the network's width is increased in both the underparameterized and overparameterized regimes, except in a critical region between these two where a substantial error increase can take place (Belkin et al., 2019; Advani et al., 2020; Nakkiran et al., 2020). The overparameterized regime has received a lot of attention because DNNs with many more parameters than training examples can achieve high test accuracy, and a theoretical understanding of this phenomenon is an active area of research. Robustness to overparametrization relates to unnecessary input dimensions because unnecessary input dimensions also increase the number of parameters of the network, albeit in the input layer rather than in the intermediate layers. As we show in the sequel, increasing the number of unnecessary input dimensions can have the opposite effect of increasing the number of hidden units in the test error.

A theoretical understanding of this phenomenon using mathematical tools is an open question. The PAC Bayes theory appears as a promising approach to describe the generalization capacity of DNNs [e.g., (Dziugaite and Roy, 2017; De Palma et al., 2019; Bernstein and Yue, 2021)]. While these theoretical results provide insights about the trends of the behaviour of the DNN, an empirical, quantitative assessment of the effect of unnecessary dimensions to the DNN's data efficiency is missing. Our analysis derives from theoretical insights of the exact solution of a linear network in a regression task. In this way, we can relate and compare empirical results for DNNs with cases that are well understood theoretically.

Another strand of research relates the structure of the dataset with the generalization ability of the network. Several works in statistical learning theory for kernel machines relate the spectrum of the dataset with the generalization performance (Zhang, 2005). For neural networks, Ansuini et al. (2019); Recanatesi et al. (2019) define the intrinsic dimensionality based on the dimension of the data manifold. These works analyze how the network reduces the intrinsic dimension across layers. Yet, these metrics based on manifolds do not provide insights about how specific aspects of the dataset, e.g., unnecessary dimensions, contribute to the intrinsic dimensionality.

We use a linearly separable dataset for binary classification, as it facilitates relating results of classic machine learning and DNNs. We generate a binary classification dataset of 30 input dimensions, which follow a Gaussian distribution with (μ = 0, σ = 1). The ground-truth label is the output of a linear classifier, such that the dataset is linearly separable with a hyperplane randomly chosen. Unnecessary input dimensions are appended to the data samples. Task-unrelated dimensions follow a Gaussian distribution with (μ = 0, σ = 0.1). The task-related dimensions are linear combinations of the dimensions of the original dataset samples.

We evaluate the following linear and Multi-Layer Perceptron (MLP) networks: linear network trained with square loss (pseudo-inverse solution), MLP with linear activation functions trained with either square loss or cross entropy loss, and MLP with ReLU trained with cross entropy loss.

To further evaluate the generality of results on data distributions that are not linearly separable, we use a mixture of Gaussians to generate non-linearly separable datasets for binary classification. Each class consists of three multivariate Gaussians of dimensions p = 30. We generate a sample by randomly selecting with the same probability one of the three distribution. To give a more comprehensive evaluation on the effect of different types of noise, we generate unnecessary dimensions using Gaussian distributions with different variance, and we also evaluate two other noise distributions, namely, Gaussian noise with Σ_ii = 1, ∀i, with Σ_ij = 0.5, ∀i≠j, and salt and pepper noise, where each vector component can assume value (0, or u), based on a Bernoullian distribution on {−1, 1}.

We consider the MLP with ReLU and soft-max with cross-entropy loss because among the different variants it is the only well suited to fit non-linearly separable data.

We evaluate object recognition datasets based on extensions of the MNIST dataset (LeCun et al., 1998) and the Stanford Dogs dataset (Khosla et al., 2011).

Synthetic and Natural MNIST. We generate two datasets based on MNIST: the synthetic MNIST and the natural MNIST, which have synthetic and natural background, respectively. In both datasets, the MNIST digit is always at the center of the image and normalized between 0 and 1.

In the synthetic MNIST dataset, the task-unrelated dimensions are sampled from a Gaussian distribution with (μ = 0, σ = 0.2) and the task-related dimensions are the result of upscaling the MNIST digit. We also combine task-related and unrelated dimensions by fixing the size of the image and changing the ratio of task-related and unrelated dimensions by upscaling the MNIST digit.

In the natural MNIST dataset, the background is taken from the Places dataset (Zhou et al., 2014), as in Volokitin et al. (2017). The size of the image is constant across experiments (256 × 256 pixels), and the size of the MNIST digits determines the amount of task-related and unrelated dimensions.

We use the MLP with ReLU and cross entropy loss, and also Convolutional Neural Networks (CNNs). The architecture of the CNN consists of three convolutional layers each with max-pooling, followed by two fully connected layers. Since the receptive field size of the CNN neurons may have an impact on the data efficiency, we evaluate different receptive field sizes. We use a factor r to scale the receptive field size, such that the convolution filter size is (r·3) × (r·3) and the pooling region size is (r·2) × (r·2). We experiment by either fixing r to a constant value or adapting r to the scale of the MNIST digit, such that the receptive fields of the neurons capture the same object region independently of the scale of the digit.

Stanford Dogs. Recall our analysis focuses on unnecessary input dimensions that are unbiased. We use the Stanford Dogs dataset (Khosla et al., 2011) as it is reasonable to assume that the bias between breeds of dogs and background is negligible. This dataset contains natural images (227 × 227 pixels) of dogs at different image positions. The amount of task-unrelated dimensions is determined by the dog size, which is different for each image. To evaluate the effect of unnecessary input dimensions, we introduce the following five versions of the dataset. Case 1 corresponds to the original image. In case 2, we multiply by zero the pixels of the background, which reduces the variability of the task-unrelated dimensions. In case 3, the dog is centered in the image. In case 4, we fix the ratio of task-related/unrelated dimensions by centering the dog and scaling it to half of the image size. In case 5, we remove the background by cropping and scaling the dog.

We use a ResNet-18 (He et al., 2016), following the standard pre-processing of the image used in ImageNet.

Figure 1A shows the test accuracy of the pseudo-inverse solution for different number of training examples and unnecessary input dimensions. Figure 1B reports the data efficiency of all networks tested for different number of unnecessary input dimensions. Recall that the data efficiency is measured with the AUTC and summarizes the test accuracy as a function of the amount of training examples, e.g., for the pseudo-inverse solution, the curves in Figure 1A are summarized by the AUTC in Figure 1B. We observe that increasing the amount of task-unrelated dimensions harms the data efficiency, i.e., the AUTC drops. Also, the task-related dimensions alone do not harm data efficiency, and they alleviate the effect of the task-unrelated dimensions.

These results clarify the difference between robustness to overparameterization in intermediate layers and unnecessary input dimensions. Note that the effect on the test accuracy of increasing the number of hidden units is the opposite of increasing the number of task-unrelated input dimensions, i.e., DNNs are not robust to all kinds of overparameterization.

Analytical results for linear regression using the square loss predicts an analogous effect of task-unrelated dimensions on the solution. For sake of clarity, we retrace these results in the Appendix 3.1, where we outline the effect of additional task-unrelated dimensions that are Gaussian-distributed on the pseudo-inverse solution. There, we show that task-unrelated dimensions lead to the pseudo-inverse, Tihkonov-regularized solution calculated in the dataset without task-unrelated dimensions. Since in this case the regularization can not be tuned or switched off as it is fixed by the number of task-unrelated dimensions, it is likely to harm the test accuracy, as we have observed.

The regularizer is beneficial in some specific cases. Following from regularization theory (Hastie et al., 2009), Appendix 3.2 highlights a noisy regression problem in which certain amounts of task-unrelated dimensions help to improve generalization. In a classification problem, Tiknhonov regularization may also be beneficial in some cases. This can be seen in Figure 1A, where we observe that for a given number of training examples, increasing the number of task-unrelated dimensions improves the test accuracy in some cases. This specific trend relates to the aforementioned double descend of DNNs (Belkin et al., 2019; Advani et al., 2020). As shown in Nakkiran et al. (2020), the location of the critical region is affected by the number of training examples and the complexity of the model. Here, the complexity of the model is affected by the number of task-unrelated dimensions due to its regularization effect.

In Figure 1C, we show the data efficiency of MLP with ReLU trained with cross entropy loss on non-linearly separable datasets. On the left of the quadrant, we report different distributions of the task-unrelated dimensions: Gaussian noise with different σ (corresponding to multiplicative factor applied on the identical covariance matrix), Gaussian noise with non-diagonal covariance matrix and salt and pepper noise. The amount of task-unrelated dimensions is reported through the colored bar (indicated with a gradient). We observe that, similarly to previous results in the linearly separable dataset (Figure 1B), task-unrelated dimensions harm data efficiency. Also, as expected, data efficiency deteriorates as the variance of Gaussian noise increases. The combination of task-related/unrelated dimensions alleviates the detrimental effect of task-unrelated dimensions. These empirical results on MLPs show a similar trend to the one predicted for linear networks, with exception of a less pronounced effect of the double descent behavior.

Figure 2A shows the log-AUTC for the MLP and the CNN for different amount of unnecessary dimensions (an increasing amount as we move left to right), for the synthetic MNIST dataset. In Appendix 4.1, we report the test accuracy for different number of unnecessary dimensions, which further strengthens the results of Figure 1. Conclusions are consistent with the previous results in the linearly separable dataset. Also, we observe that CNNs are overall much more data efficient than MLPs, which is expected because of their more adequate inductive bias given by the weight sharing of the convolutions.

Figure 2B shows results in natural MNIST dataset for different ratios of task-related/unrelated dimensions. The plots compare CNNs with different receptive field sizes, represented by the factor r (see Section 4.3). Since the CNN achieves high accuracy with few examples, the mean and standard deviation of the log-AUTC (left plot) hardly show any variation when computed on more than 20 training examples per class. Yet, the gap of the testing accuracy is considerable for 20 training examples per class (right plot). These results confirm that task-unrelated dimensions degrade data efficiency independently of the receptive field sizes (see Appendix 4.2 for additional results further supporting these conclusions).

Figure 2C shows results on the Stanford Dogs dataset, namely the AUTC score across the five cases of unnecessary dimensions that we evaluate. This dataset serves to assess a more realistic scenario, where the objects can appear at different positions and scales. We observe that the task-unrelated dimensions, which come from the background, harm the data efficiency (cases 1 to 4 versus case 5). Putting to zero the unnecessary dimensions improves the data efficiency of models trained on the original dataset (cases 2 to 4 vs. case 1). This is because the task-unrelated dimensions become redundant as they all take the same value in all images. We also observe that removing the variability of the position and scale of the object hardly affects the data efficiency (case 2 to 4). Thus, learning to discard the background requires more training examples than learning to handle the variability in scale and position of the object.