Learning distinct features helps, provably^††thanks: Preprint

Neural networks are a powerful class of non-linear function approximators that have been successfully used to tackle a wide range of problems. They have enabled breakthroughs in many tasks, such as image classification [31], speech recognition [20], and anomaly detection [16]. However, neural networks are often over-parameterized, i.e., have more parameters than the data they are trained on. As a result, they tend to overfit to the training samples and not generalize well on unseen examples [18]. Avoiding overfitting has been extensively studied [45, 43, 47, 14, 15] and various approaches and strategies have been proposed, such as data augmentation [18, 64], regularization [32, 8, 1], and Dropout [21, 38, 39], to close the gap between the empirical loss and the expected loss.

Formally, the output of a neural network consisting of P layers can be defined as follows:

where $\rho^{i}(.)$ is the element-wise activation function, e.g., ReLU or Sigmoid, of the $i^{th}$ layer and ${\bm{\mathsfit{W}}}=\{{\bm{W}}^{1},\dots,{\bm{W}}^{P}\}$ are the weights of the network with the superscript denoting the layer. By defining $\Phi(\cdot)=\rho^{P-1}(\cdots\rho^{2}({\bm{W}}^{2}\rho^{1}({\bm{W}}^{1}\cdot)))$, the output of neural network becomes

where $\Phi({\bm{x}})=[\phi_{1}({\bm{x}}),\cdots,\phi_{M}({\bm{x}})]$ is the $M$-dimensional feature representation of the input ${\bm{x}}$. This way neural networks can be interpreted as a two-stage process, with the first stage being representation learning, i.e., learning $\Phi(\cdot)$, followed by the final prediction layer. Both parts are jointly optimized.

Learning a rich and diverse set of features, i.e., the first stage, is critical for achieving top performance [3, 34, 10]. Studying the different properties of the learned features is an active field of research [11, 29, 11, 13]. For example, [13] showed theoretically that learning a good feature representation can be helpful in few-shot learning. In this paper, we focus on the diversity of the features. This property has been empirically studied in [36, 35, 10] and has been shown to boost performance and reduce overfitting. However, no theoretical guarantees are provided. In this paper, we close this gap and we conduct a theoretical analysis of feature diversity. In particular, we propose to quantify the diversity of the feature set $\{\phi_{1}(\cdot),\cdots,\phi_{M}(\cdot)\}$ using the average pairwise $L_{2}$-distance between their outputs. Formally, given a dataset $\{{\bm{x}}_{i}\}_{i=1}^{i=N}$, we have

Intuitively, $diversity$ measures how distinct the learned features are. If the mappings learned by two different units are redundant, then, given the same input, both units would yield similar output. This yields in low $L_{2}$-distance and as a result a low diversity. In contrast, if the mapping learned by each unit is distinct, the corresponding average distances to the outputs of the other units within the layer are high. Thus, this yields a high global diversity.

To confirm this intuition and further motivate the analysis of this attribute, we conduct empirical simulations. We track the diversity of the representation of the last hidden layer, as defined in equation 3, during the training of three different ResNet [19] models on CIFAR10 [30]. The results are reported in Figure 1. Indeed, diversity consistently increases during the training for all the models. This shows that, in order to solve the task at hand, neural networks learn distinct features.

Our contributions can be summarized as follows:

• •

  We analyze the effect the feature diversity on the generalization error bound of a neural network. The analysis is presented in Section 3. In Theorem 3.1, we derive an upper bound for the generalization gap which is inversely proportional to the diversity factor. Thus, we provide theoretical evidence that learning distinct features can help reduce the generalization error.

• •

  We extend our analysis to different losses and general multi-layer networks. These results are presented in Theorems 4.1, 4.2, 4.3, 4.4, and 4.5.

Generalization theory [50, 28] focuses on the relation between the empirical loss defined as

and the expected risk, for any $f$ in the hypothesis class $\mathcal{F}$, defined as

where $\mathcal{Q}$ is the underlying distribution of the dataset and $y_{i}$ the corresponding label of $x_{i}$. Let $f^{*}=\operatorname*{arg\,min}_{f\in\mathcal{F}}L(f)$ be the expected risk minimizer and $\hat{f}=\operatorname*{arg\,min}_{f\in\mathcal{F}}\hat{L}(f)$ be the empirical risk minimizer. We are interested in the estimation error, i.e., $L(f^{*})-L(\hat{f})$, defined as the gap in the loss between both minimizers [6]. The estimation error represents how well an algorithm can learn. It usually depends on the complexity of the hypothesis class and the number of training samples [5, 63].

Several techniques have been proposed to quantify the generalization error, such as Probably Approximately Correctly (PAC) learning [50, 53], VC dimension [52], and the Rademacher complexity [50]. The Rademacher complexity has been widely used as it usually leads to a tighter generalization error bound than the other metrics [51, 45, 17]. The formal definition of the empirical Rademacher complexity is given as follows:

In this work, we rely on the Rademacher complexity to study diversity. We recall the following three lemmas related to the Rademacher complexity and the generalization error:

Lemma 3 upper-bounds the generalization error using the Rademacher complexity defined over the loss set and $\sup_{x,y,f}|l(f(x),y)|$. Our analysis aims at expressing this bound in terms of diversity, in order to understand how it affects the generalization.

In order to study the effect of diversity on the generalization, given a layer with M units $\{\phi_{1}(\cdot),\cdots,\phi_{M}(\cdot)\}$, we make the following assumption:

$d_{min}$ lower-bounds the average $L_{2}$-distance between the different units’ activations within the same representation layer. Intuitively, if several neuron pairs $i$ and $j$ have similar outputs, the corresponding $L_{2}$ distance is small. Thus, the lower bound $d_{min}$ is also small and the units within this layer are considered redundant and “not diverse”. Otherwise, if the average distance between the different pairs is large, their corresponding $d_{min}$ is large and they are considered “diverse”. By studying how the lower bound $d_{min}$ affects the generalization of the model, we can analyze how the diversity theoretically affects the performance of neural networks. In the rest of the paper, we derive generalization bounds for neural networks using $d_{min}$.

In this section, we derive generalization bounds for neural networks depending on their diversity. Here, we consider a simple tow-layer neural network with a hidden layer composed of $M$ neurons and one-dimensional output trained for a regression task. The full characterization of the setup can be summarized as follows:

• •

  The activation function of the hidden layer, $\rho(\cdot)$, is a positive $L_{\rho}$-Lipschitz continuous function.

• •

  The input vector ${\bm{x}}\in\mathbb{R}^{D}$ satisfies $||{\bm{x}}||_{2}\leq C_{1}$ and the output scalar $y\in\mathbb{R}$ satisfies $|y|\leq C_{2}$.

• •

  The weight matrix ${\bm{W}}=[{\bm{w}}_{1},{\bm{w}}_{2},\cdots,{\bm{w}}_{M}]\in\mathcal{R}^{D\times M}$ connecting the input to the hidden layer satisfies $||{\bm{w}}_{m}||_{2}\leq C_{3}$.

• •

  The weight vector ${\bm{v}}\in\mathbb{R}^{M}$ connecting the hidden-layer to the output satisfies $||{\bm{v}}||_{\infty}\leq C_{4}$.

• •

  The hypothesis class is $\mathcal{F}=\left\{f|f({\bm{x}})=\sum_{m=1}^{M}v_{m}\phi_{m}({\bm{x}})=\sum_{m=1}^{M}v_{m}\rho({\bm{w}}_{m}^{T}{\bm{x}})\right\}$.

• •

  Loss function set is $\mathcal{A}=\left\{l|l(f({\bm{x}}),y)=\frac{1}{2}|f({\bm{x}})-y|^{2}\right\}$.

• •

  Given an input ${\bm{x}}$, $\frac{1}{2M(M-1)}\sum_{n\neq m}^{M}(\phi_{n}({\bm{x}})-\phi_{m}({\bm{x}}))^{2}\geq d_{min}^{2}$.

Our main goal is to analyze the generalization error bound of the neural network and to see how its upper-bound is linked to the diversity of the different units, expressed by $d_{min}$. The main result of the paper is presented in Theorem 3.1. Our proof consists of three steps: At first, we derive a novel bound for the hypothesis class $\mathcal{F}$ depending on $d_{min}$. Then, we use this bound to derive bounds for the loss class $\mathcal{A}$ and its Rademacher complexity $\mathcal{R}_{N}(\mathcal{A})$. Finally, we plug all the derived bounds in Lemma 3 to complete the proof of Theorem 3.1.

The first step of our analysis is presented in Lemma 4:

Note that in Lemma 4, we have expressed the upper-bound of $\sup_{{\bm{x}},f}|f({\bm{x}})|$ in terms of $d_{min}$. Using this bound, we can now find an upper-bound for $\sup_{{\bm{x}},f,y}|l(f({\bm{x}}),y)|$ in the following lemma:

Next, using the result of lemmas 1, 2, and 5, we can derive a bound for the Rademacher complexity of $\mathcal{A}$. We have, thus, expressed all the elements of Lemma 3 using the diversity term $d_{min}$. By plugging in the derived bounds in Lemmas 4, 5, we obtain Theorem 3.1.

Theorem 3.1 provides an upper-bound for the generalization gap. We note that it is a decreasing function of $d_{min}$. Thus, this suggests that higher $d_{min}$, i.e., more diverse activations, yields a lower generalization error bound. This shows that learning distinct features helps in neural network context.

We note that the bound in Theorem 3.1 is non-vacuous in the sense that it converges to zero when the number of training samples $N$ goes to infinity. Moreover, we note that in this paper we do not claim to reach a tighter generalization bound for neural networks in general [48, 24, 44, 14]. Our main claim is that we derive a generalization bound which depends on the diversity of learned features, as measured by $d_{min}$. To the best of our knowledge, this is the first work that performs such theoretical analysis based on the average $L_{2}$-distance between the units within the hidden layer.

In this section, we show how to extend our theoretical analysis for classification, for general multi-layer networks, and for different losses.

In this paper, we showed how the diversity of the features learned by a two-layer neural network trained with the least-squares loss affects generalization. We quantified the diversity by the average $L_{2}$-distance between the hidden-layer features and we derived novel diversity-dependant generalization bounds based on Rademacher complexity for such models. The derived bounds are inversely-proportional to the diversity term, thus demonstrating that more distinct features within the hidden layer can lead to better generalization. We also showed how to extend our results to deeper networks and different losses.

The bound found in Theorem 3.1 suggests that the generalization gap, with respect to diversity, is inversely proportional to $d_{min}$ and scales as $\sim(C_{5}^{2}-d^{2}_{min})/\sqrt{N}$. We validate this finding empirically in Figure 2. We train a two-layer neural network on the MNIST dataset for 100 epochs using SGD with a learning rate of $0.1$ and batch size of 256. We show the generalization gap, i.e., test error - train error, and the theoretical bound, i.e., $(C_{5}^{2}-d^{2}_{min})/\sqrt{N}$, for different training set sizes. $d_{min}$ is the lower bound of diversity. Empirically, it can be estimated as the minimum feature diversity over the training data $S$: $d_{min}=\min_{x\in S}\frac{1}{2M(M-1)}\sum_{n\neq m}^{M}(\phi_{n}({\bm{x}})-\phi_{m}({\bm{x}}))^{2}$. We experiment with different sizes of the hidden layer, namely 128, 256, and 512. The average results using 5 random seeds are reported for different training sizes in Figure 2 showing that the theoretical bound correlates consistently well (correlation $>$ 0.9939) with the generalization error.

As shown in Figure 1, diversity increases for neural networks along the training phase. To further investigate this observation, we conduct additional experiments on ImageNet [49] dataset using 4 different state-of-the-art models: ResNet50 and ResNet101, i.e., the standard ResNet model [19] with 50 layers and 101 layers, ResNext50 [60], and WideResNet50 [62] with 50 layers. All models are trained with SGD using standard training protocol [64, 22, 10]. We track the diversity, as defined in equation 3, of the features of the last intermediate layer. The results are shown in Figure 3 (a) and (b). As it can be seen, SGD without any explicit regularization implicitly optimizes diversity and converges toward regions with high features’ distinctness. These observations suggest the following conjecture:

Studying the fundamental properties of SGD is extremely important to understand generalization in deep learning [27, 25, 54, 65, 23]. Conjecture 1 suggests a new implicit bias for SGD, showing that it favors regions with high feature diversity.