Learning Credit Assignment

Deep learning has achieved impressive prediction accuracies in a variety of scientific and industrial domains. However, the nested nonlinear feature of deep learning makes the learning highly nontransparent, i.e., it is still unknown how the learning coordinates a huge number of parameters to achieve decision-making. To explain this hierarchical credit assignment, we propose a mean-field learning model by assuming that an ensemble of subnetworks, rather than a single network, is trained for a classification task. Surprisingly, our model reveals that apart from some deterministic synaptic weights connecting two neurons at neighboring layers, there exists a large number of connections that can be absent, and other connections can allow for a broad distribution of their weight values. Therefore, synaptic connections can be classified into three categories: very important ones, unimportant ones, and those of variability that may partially encode nuisance factors. Therefore, our model learns the credit assignment leading to the decision and predicts an ensemble of sub-networks that can accomplish the same task, thereby providing insights toward understanding the macroscopic behavior of deep learning through the lens of distinct roles of synaptic weights.

Figure 1

Schematic illustration of a model of learning credit assignment. A deep neural network of four layers, including two hidden layers, is used to recognize a handwritten digit, say zero, with the softmax output indicating the probability of the categorization. Each connection is specified by a spike and slab distribution, where the spike indicates the probability of the absence of this connection and the slab is modeled by a Gaussian distribution of weight values, as pictorially shown only on strong connections with different means and variances. Other weak connections indicate nearly unit spike-probabilities, although they also carry a slab distribution (not shown in illustration for simplicity).

Figure 2

Properties of gBP in testing performances on unseen data. (a) Training trajectories of a network architecture of three or four layers. The architecture is defined as 784-100-100-10 (four layers) or 784-100-10 (three layers), where each number indicates the corresponding layer width. Networks are trained on the data size of $T={10}^4$ images and tested on another unseen-data size of $V={10}^4$ images. The fluctuation is computed from five independent runs. (b) Test errors of gBP versus training data size. The error bar characterizes the fluctuation across ten independently sampled network architectures from the SaS distribution. The same network of four layers as in (a) is used. The inset shows the sparsity per connection as a function of layers, for which networks of five layers are used and the hidden layer width is still 100 nodes. Each marker is an average over ten independent runs. The “k” in the inset means the training data size in the unit of ${10}^3$.

Figure 3

Statistical properties of trained four-layer networks with gBP. Unless stated otherwise, other training conditions are the same as in Fig. 2. Distribution of hyperparameters $(\mathit{\pi },\mathbf{m},\mathbf{\Xi })$ for a typical trained network. (b) Distribution of connection entropy $S_{\ell }$ for a typical trained network. (c) Entropy per connection versus layers ($\mathcal{B}=100$). Different training data sizes are considered, and the result is averaged over ten independent runs of five-layer networks.

Figure 4

Effects of targeted-weight perturbation on test performances. Training conditions are the same as in Fig. 2, and the result is averaged over ten independent runs. (a) VIP weights and all weights (without weight categorization) are stochastically turned off, keeping the same number, while UIP weights are stochastically turned on and sampled from a standard Gaussian distribution. The perturbation is carried out at the first hierarchical stage of a four-layer network. (b) The fraction of VIP and UIP weights between neighboring layers.