DNNs as Layers of Cooperating Classifiers

We conduct our experiments in a relatively simple setup. Our aim is to understand trends, while retaining the key elements that are likely to be common to high-performance DNNs. Thus, we use only fully-connected feedforward networks with highly regular topologies, and investigate their behavior on two widely-used image-recognition tasks, namely MNIST [12] and FMNIST [13]. No data augmentation is employed. Refinements such as drop-out and batch normalization are also avoided in order to focus on the essential mechanisms of DNN learning. (Such refinements do not contribute much to test set accuracy in this setting, in contrast to data augmentation, which does [14, 15].)

All hidden nodes have Rectified Linear Unit (ReLU) activation functions, and a standard mean squared error (MSE) loss function is employed, unless stated otherwise. The popular Adam [16] optimizer is used to train the networks after normalized uniform initialization with three different training seeds [17], and the global learning rates are manually adjusted to ensure training set convergence. This is verified by ensuring that the performance obtained is comparable with prior results reported on both MNIST [12, 15] and FMNIST [18, 19], where similar topologies were employed. We implement early stopping by choosing networks with the smallest validation error.

The performance of the trained models are shown in Figure 2. Our first analysis investigates several networks of fixed width and increasing depth. “Depth” here refers to the number of hidden layers, without counting the input or output layers. For a width of 100 nodes per layer, both the MNIST and FMNIST systems initially achieve decreasing error rates as the number of hidden layers grows, but the performance quickly saturates. However, increasing the number of layers (and thus parameters) beyond this level does not degrade performance, even when as many as 20 hidden layers are employed. (Results here shown up to a a depth of 10.) In the second analysis, network depth is kept constant at 10 layers, and the width (number of nodes per layer) is adjusted. As with increased depth, increased width leads to a similar saturation in performance.

Using the trained networks from Figure 2, we now analyse the distinctiveness of their activation patterns. The per-layer mean perplexity of each network is shown in Figure 3, with mean values obtained by averaging over all classes. The perplexities are measured with regard to the test set samples: with the focus of our analyses being on generalization, we are interested in encodings that are applicable during categorization of unseen samples, and not only those created for optimization purposes.

There are several interesting observations that can be made from these graphs. Notice the relatively sharp drop in perplexity values for all networks, and take note that the drop is more gradual for the FMNIST models and sharper for wider networks. Additionally, for the networks with sufficient depth and width:

• •

  The perplexity in the later layers is very near 1. That is, all activation patterns have become fully class-specific.

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  The perplexity values in the first 2 layers are almost equal to the total number of samples for each class (approximately 1,000 for the test sets for both MNIST and FMNIST), which means that an individual encoding is created per sample.

• •

  The transition from high perplexity to low perplexity is very similar across networks.

Lastly, notice that if the network width is below some threshold, the perplexity values in the earlier layers reduce accordingly. This cannot be due to a lack of representational power seeing as even the smallest layer (20 nodes) can represent more patterns than required by the number of training samples. Taking into account their lower test error, this suggests that wider networks represent sample information in a way that is more conducive to good generalization. This phenomenon was recently explored by [20] where it is attributed to better weight exploration and a small number of observed prototype weight vectors.

Provided the network is large enough, there seems to be a range of earlier layers within which the nodes have high (virtually maximal) perplexity, and a corresponding range of later layers where the nodes have relatively low (virtually minimal) perplexity. Furthermore, the transition from the former behavior to the latter is consistent across all networks, irrespective of size, as long as they are deeper and wider than a task-specific threshold. After this transition, the class-specific discrete behavior in the excess layers is relatively trivial. (Perplexity is already at a minimum.)

The nodes in the earlier layers appear to perform most of the information processing required to produce a feature space that supports the ability to differentiate among samples relating to different classes. In this setup, the deeper layers effectively produce no new benefits and merely propagate the information forward through the network. The forward propagation of information, at this point, takes the form of a class-specific encoding, which is unique to each layer. By varying either width or depth, the same message emerges: a task-specific threshold exists with regard to both width and depth, beyond which network behavior is strikingly regular and similar, irrespective of network size.

Gradient-based optimization has many variations but is essentially a straightforward process. In its basic form, each weight update is accumulated over a batch of random samples, each sample contributing a $\Delta w_{i,j,k}$. Each sample-specific update is proportional to the derivative of the error function $E$ with regard to this weight, and the learning rate $\eta$ (which could potentially be adaptive, as with Adam). In practice, the derivative of the error function with regard to each parameter is calculated using backpropagation

$\displaystyle\Delta w_{i,j,k}=-\eta\frac{\partial{E}}{\partial{w_{i,j,k}}}=-\eta\beta_{i,j}a_{i-1,k}$

(3)

with $a_{i-1,k}$ the activation result at layer $i-1$ for node $k$ and $\beta_{i,j}$ as defined below. Using $z_{i,j}$ to describe the sum of the input to node $j$ in layer $i$, and defining the symbols

$\displaystyle\alpha_{i,j}=\frac{\partial a_{i,j}}{\partial z_{i,j}}\hskip 14.22636pt\lambda_{j}=\frac{\partial{E}}{\partial{a_{N,j}}}$

(4)

$\beta_{i,j}$ is calculated by counting through all $n$ forward connections from node $j$ to the next layer, working backwards from the last layer (also counting the output layer) $N$:

$\displaystyle\beta_{i,j}=\begin{cases}\alpha_{i,j}\sum\limits_{n}w_{i+1,n,j}\beta_{i+1,n}&\text{if }i\neq N\\ \alpha_{i,j}\lambda_{j}&\text{if }i=N\end{cases}$

(5)

This recursive update rule is important for computational efficiency but, while not commonly done, the derivative can also be written as an iterative expression¹¹1See Appendix A.:

	$\displaystyle\beta_{i,j}$	$\displaystyle=$	$\displaystyle\sum\limits_{b=0}^{B_{i}-1}\lambda_{I_{i,j}(N,b)}\prod\limits_{g=i}^{N}\alpha_{g,I_{i,j}(g,b)}\prod\limits_{r=i+1}^{N}w_{r,I_{i.j}(r,b),I_{i,j}(r-1,b)}$		(6)
	$\displaystyle\textrm{where }B_{L}$	$\displaystyle=$	$\displaystyle\begin{cases}\prod\limits_{m=L+1}^{N}s_{m}&ifL\neq N\\ 1&ifL=N\end{cases}$
	$\displaystyle I_{i,j}(r,b)$	$\displaystyle=$	$\displaystyle\begin{cases}(b\div B_{r})\mod s_{r}&\textrm{ if }r\neq i\\ j&\textrm{ if }r=i\end{cases}$

with $s_{i}$ the number of nodes in layer $i$, and each $I_{i,j}(r,b)$ indexing function specific to the layer and node position of the $\beta_{i,j}$ required. When inner node activations are ReLUs, this equation simplifies further. Noting that

	$\displaystyle Relu(x)$	$\displaystyle=$	$\displaystyle xT(x)$		(7)
	$\displaystyle\textrm{where }T(x)$	$\displaystyle=$	$\displaystyle\begin{cases}1\text{ if }x>0\\ 0\text{ if }x\leq 0\\ \end{cases}$		(8)

the weight update becomes

	$\displaystyle\Delta w_{i,j,k}=-\eta\hskip 2.84526pta_{i-1,k}\sum\limits_{b=0}^{B_{i}-1}$	$\displaystyle\lambda_{I_{i,j}(N,b)}\prod\limits_{g=i}^{N-1}T(z_{g,I_{i,j}(g,b)})$			(9)
		$\displaystyle\prod\limits_{r=i+1}^{N}w_{r,I_{i,j}(r,b),I_{i,j}(r-1,b)}$

In effect, the $b$ index runs through all possible paths from node $j$ in layer $i$ to each of the nodes in layer $N$, the $g$ index runs through all the activation values of a single path, and the $r$ index multiplies the weights along the same path. Using MSE as loss function and linear activation functions in the outer layer results in $\lambda_{i,j}=z_{N,I_{i,j}(N,b)}-y_{I_{i,j}(N,b)}$ where $y_{j}$ the true target value at the outer node $j$, that is, the classification gap. Note that $\lambda_{i,j}$ has the same form when using a cross entropy loss function with softmax activation functions in the outer layer, as long as one-hot encodings are used for classification targets.

Per sample, each weight update then only takes into account the activation strength at node $k$ feeding into the weight, and all the active paths – where all the $T(.)$ values are 1 – supported from node $j$ onward. Each path contributes a single product of all the weights along the active path, multiplied by the classification gap at the path end point. The $T(.)$ values can therefore by viewed as switches, selecting which samples contribute to a weight update at each point in the network, and the weight update rewritten as:

$$\Delta w_{i,j,k}=\eta\hskip 2.84526pt\sum\limits_{s\in S}\sum\limits_{p\in P_{s}}(a^{s}_{i-1,k})(\prod\limits_{g=1}^{N-i}w^{\prime}_{p_{g}})(y^{s}-z^{s}_{N,p_{N-i}})$$

(10)

where $S$ consists of all the samples active at both nodes $j$ and $k$, $P_{s}$ is the set of active paths that starts at node $j$ (generated specifically by $s$) and $w^{\prime}_{p_{g}}$ runs through the $g$ weights along the active path $p=p_{1},p_{2},\ldots p_{N-i}$. The $s$ superscript emphasizes that these are sample-specific values.

The update process of Equation 10 can be viewed as two interacting systems: one continuous and one discrete, both utilizing the same underlying network architecture and parameters. Each node plays a role in both systems:

1. 1.

   The discrete system associates an “on/off” value with every single sample-node pair, depending on whether the node is active or not for that sample. This system is fully specified by the $T(.)$ values of Equation 9. Nodes can therefore be considered as switched either on or off, giving rise to a discrete information processing system that creates a discrete set of samples at each node.

2. 2.

   The continuous system associates a continuous value with each sample-node pair (the pre-activation value of the sample at the given node) and updates the continuous values of the weight vector feeding into this node during gradient descent.

The training process utilizes both systems to optimize the network, but the relative importance of the two systems with regard to eventual classification ability changes, both during the training process and through the layers of the network. Each node in effect acts as a local feature transformation, combining multiple features from an earlier level to form a single new feature, made available to the next level. The node only optimizes its weights (weights feeding into the node) with regard to the set of samples it is sensitive to: with regard to these, it determines the relative importance of the features available at the previous layer in closing the classification gap it is aware of. The training process uses the two systems interactively: (1) During the forward pass, the discrete systems determines whether a sample should be included or excluded from the set of similar samples at that node. (2) During the backward pass, only the selected samples are used by the continuous system to update the relative weighting of the input features: creating a new feature more attuned to these specific samples, and these only.

This also means that the optimization process is simultaneously taking into account both global and local information. Globally, the extent to which all the collaborating nodes have already “solved” the task posed by a specific sample determines the influence of that sample, while locally, each node that is active for an unsolved sample adjusts its parameters according to its own set of active samples only. Locally, nodes solve subsets of the class differentiation task; globally, nodes in a layer cooperate.

We now interpret each node as a classifier, implicitly estimating $P(z|y_{n})$, where $z$ is the pre-activation value and $y_{n}$ a class. A discrete, continuous and combined estimate of this value is created at each node:

• •

  discrete: if $z>0$, $P(z|y_{n})$ is estimated as the ratio of class $n$ training samples with positive activation values with regard to all class $n$ training samples; 1 minus this value otherwise.

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  continuous: the estimate provided by a kernel density estimator trained using all class $n$ training data activation values observed at this node.

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  combined: using the discrete estimate if $z\leq 0$, the continuous estimate otherwise.

This estimate is combined with the prior probability $P(y_{n})$ of a class being observed to estimate the posterior $P(y_{n}|z)$:

$\displaystyle P(y_{n}|z)$

$\displaystyle=$

$\displaystyle\frac{P(z|y_{n})P(y_{n})}{\sum_{m}{P(z|y_{m})P(y_{m})}}$

(11)

We view the nodes as independent classifiers (we ignore possible dependence) and multiply the probability estimates per class over all the nodes in a layer, to obtain a layer-specific probability estimate for each of the three systems. (In practice, the log probabilities are summed.) These probability estimates can then be used directly to classify samples based on maximum probability, creating three layer-specific classifiers for each layer in the network: a continuous, a discrete and a combined classifier. While neither the nodes nor the layers use these probabilities directly, they provide insight into the information available locally at each point in the network. By evaluating layer-specific classification ability at different layers and at different stages in the training process, we can better demonstrate the interaction between the discrete and continuous systems.

Using the nodes as individual classifiers, we evaluate the performance of the discrete, continuous and combined systems generated from the trained models in Figure 2, during the training process. In Figure 4, we demonstrate the performance of an MLP with $6$ hidden layers of $100$ nodes each, trained on the FMNIST classification dataset; the behavior of this model during training expresses the overall tendencies for all the analysed models very well.

The most striking observation is that, at the later hidden layers, the accuracies of the three systems are virtually identical. In the first layer, the accuracy of the combined system is higher than both the discrete and continuous systems. This difference in classification accuracy among the three systems becomes smaller at later layers in the network, until it disappears. While it is to be expected that the combined system would outperform the other two (since its probability estimates have access to information pertaining to both the continuous and discrete subsystems) this is not what happens: at later layers, the other two systems are able to perform at levels comparable to the system subsuming them.

Additionally, it can be seen that the accuracies in the later layers improve visibly over iterations of learning while the performance of the earlier layers improves less. This reinforces the idea that the function of earlier layers is not to classify samples into the classes involved in the global classification problem, but instead act as general sample differentiators (that is, earlier layers attempt to group and solve subsets of the main task, which may not necessarily be class-specific); later layers use these elements to more efficiently perform the classification task. During training, the overall accuracy of each system in later layers increases on the train and on the test set until it reaches the same, or slightly better accuracy as the network itself. At the end of the first epoch, significant training has already occurred. We therefore also investigate how the performance of these systems changes during mini-batch updates in the first epoch, as shown in Figure 5. Note how poorly the continuous system performs initially (relative to the discrete system), until the training process stabilises and the previously discussed trends emerge.

Similar trends²²2Also see Appendix B. are observed when changing either network width or depth. Figure 6 depicts the classification accuracy of the three systems for a set of FMNIST networks with fixed width (100 nodes) and increasing depth (1 to 9 layers). It is striking to note that the three systems start overlapping when sufficient depth becomes available, but struggle to beforehand. Similarly, when the network layers lack width, the earlier layers underperform significantly. This is especially true for the discrete system. As expected, there is a clear increase in accuracy (across all systems) in the later layers with an increase in width. Curiously, the continuous performance appears to reduce with an increase in width in the first layers.

While not shown here, trends for FMNIST and MNIST are similar, except that for MNIST (1) the depth at which the three systems converge is earlier; (2) higher accuracies are observed overall; and (3) there is an anomalously low performance measurement for the discrete system at one of the layers of the model with a width of 20. (We know that the discrete subsystem tends to underperform significantly at low widths.) Finally, it is clear that the the nodes at each layer have the ability to solve the classification task when applied in collaboration. It is worth noting that, in the earlier layers, nodes are formed that range from very general (active for many samples) to very specific (active for only one or two samples).