Modern deep neural networks (DNNs) are typically trained with a global cross-entropy loss in a supervised end-to-end manner: neurons need to store their outgoing weights; training alternates between a forward pass (computation) and a top-down backward pass (learning) which is biologically implausible. Alternatively, greedy layer-wise training eliminates the need for cross-entropy loss and backpropagation. By avoiding the computation of intermediate gradients and the storage of intermediate outputs, it reduces memory usage and helps mitigate issues such as vanishing or exploding gradients. However, most existing layer-wise training approaches have been evaluated only on relatively small datasets with simple deep architectures. In this paper, we first systematically analyze the training dynamics of popular convolutional neural networks (CNNs) trained by stochastic gradient descent (SGD) through an information-theoretic lens. Our findings reveal that networks converge layer-by-layer from bottom to top and that the flow of information adheres to a Markov information bottleneck principle. Building on these observations, we propose a novel layer-wise training approach based on the recently developed deterministic information bottleneck (DIB) and the matrix-based Rényi's $α$-order entropy functional. Specifically, each layer is trained jointly with an auxiliary classifier that connects directly to the output layer, enabling the learning of minimal sufficient task-relevant representations. We empirically validate the effectiveness of our training procedure on CIFAR-10 and CIFAR-100 using modern deep CNNs and further demonstrate its applicability to a practical task involving traffic sign recognition. Our approach not only outperforms existing layer-wise training baselines but also achieves performance comparable to SGD.

End-to-end (E2E) training, optimizing the entire model through error backpropagation, fundamentally supports the advancements of deep learning. Despite its high performance, E2E training faces the problems of memory consumption, parallel computing, and discrepancy with the functionalities of the actual brain. Various alternative methods have been proposed to overcome these difficulties; however, no one can yet match the performance of E2E training, thereby falling short in practicality. Furthermore, there is no deep understanding regarding differences in the trained model properties beyond the performance gap. In this paper, we reconsider why E2E training demonstrates a superior performance through a comparison with layer-wise training, a non-E2E method that locally sets errors. On the basis of the observation that E2E training has an advantage in propagating input information, we analyze the information plane dynamics of intermediate representations based on the Hilbert-Schmidt independence criterion (HSIC). The results of our normalized HSIC value analysis reveal the E2E training ability to exhibit different information dynamics across layers, in addition to efficient information propagation. Furthermore, we show that this layer-role differentiation leads to the final representation following the information bottleneck principle. It suggests the need to consider the cooperative interactions between layers, not just the final layer when analyzing the information bottleneck of deep learning.

Many recent studies integrate federated learning (FL) with self-supervised learning (SSL) to take advantage of raw training data distributed across edge devices. However, edge devices often struggle with high computation and communication costs imposed by SSL and FL algorithms. To tackle this hindrance, we propose LW-FedSSL, a layer-wise federated self-supervised learning approach that allows edge devices to incrementally train one layer of the model at a time. LW-FedSSL comprises server-side calibration and representation alignment mechanisms to maintain comparable performance with end-to-end FedSSL while significantly lowering clients' resource requirements. The server-side calibration mechanism takes advantage of the resource-rich server in an FL environment to assist in global model training. Meanwhile, the representation alignment mechanism encourages closeness between representations of FL local models and those of the global model. Our experiments show that LW-FedSSL has a $3.3 \times$ lower memory requirement and a $3.2 \times$ cheaper communication cost than its end-to-end counterpart. We also explore a progressive training strategy called Prog-FedSSL that outperforms end-to-end training with a similar memory requirement and a $1.8 \times$ cheaper communication cost.

End-to-end training of graph neural networks (GNN) on large graphs presents several memory and computational challenges, and limits the application to shallow architectures as depth exponentially increases the memory and space complexities. In this manuscript, we propose Layer-wise Regularized Graph Infomax, an algorithm to train GNNs layer by layer in a self-supervised manner. We decouple the feature propagation and feature transformation carried out by GNNs to learn node representations in order to derive a loss function based on the prediction of future inputs. We evaluate the algorithm in inductive large graphs and show similar performance to other end to end methods and a substantially increased efficiency, which enables the training of more sophisticated models in one single device. We also show that our algorithm avoids the oversmoothing of the representations, another common challenge of deep GNNs.

End-to-end backpropagation has a few shortcomings: it requires loading the entire model during training, which can be impossible in constrained settings, and suffers from three locking problems (forward locking, update locking and backward locking), which prohibit training the layers in parallel. Solving layer-wise optimization problems can address these problems and has been used in on-device training of neural networks. We develop a layer-wise training method, particularly welladapted to ResNets, inspired by the minimizing movement scheme for gradient flows in distribution space. The method amounts to a kinetic energy regularization of each block that makes the blocks optimal transport maps and endows them with regularity. It works by alleviating the stagnation problem observed in layer-wise training, whereby greedily-trained early layers overfit and deeper layers stop increasing test accuracy after a certain depth. We show on classification tasks that the test accuracy of block-wise trained ResNets is improved when using our method, whether the blocks are trained sequentially or in parallel.

Deep neural networks have long training and processing times. Early exits added to neural networks allow the network to make early predictions using intermediate activations in the network in time-sensitive applications. However, early exits increase the training time of the neural networks. We introduce QuickNets: a novel cascaded training algorithm for faster training of neural networks. QuickNets are trained in a layer-wise manner such that each successive layer is only trained on samples that could not be correctly classified by the previous layers. We demonstrate that QuickNets can dynamically distribute learning and have a reduced training cost and inference cost compared to standard Backpropagation. Additionally, we introduce commitment layers that significantly improve the early exits by identifying for over-confident predictions and demonstrate its success.

Graph convolution networks (GCN) are increasingly popular in many applications, yet remain notoriously hard to train over large graph datasets. They need to compute node representations recursively from their neighbors. Current GCN training algorithms suffer from either high computational costs that grow exponentially with the number of layers, or high memory usage for loading the entire graph and node embeddings. In this paper, we propose a novel efficient layer-wise training framework for GCN (L-GCN), that disentangles feature aggregation and feature transformation during training, hence greatly reducing time and memory complexities. We present theoretical analysis for L-GCN under the graph isomorphism framework, that L-GCN leads to as powerful GCNs as the more costly conventional training algorithm does, under mild conditions. We further propose L^2-GCN, which learns a controller for each layer that can automatically adjust the training epochs per layer in L-GCN. Experiments show that L-GCN is faster than state-of-the-arts by at least an order of magnitude, with a consistent of memory usage not dependent on dataset size, while maintaining comparable prediction performance. With the learned controller, L^2-GCN can further cut the training time in half. Our codes are available at https://github.com/Shen-Lab/L2-GCN.

Deep neural networks have been used in various machine learning applications and achieved tremendous empirical successes. However, training deep neural networks is a challenging task. Many alternatives have been proposed in place of end-to-end back-propagation. Layer-wise training is one of them, which trains a single layer at a time, rather than trains the whole layers simultaneously. In this paper, we study a layer-wise training using a block coordinate gradient descent (BCGD) for deep linear networks. We establish a general convergence analysis of BCGD and found the optimal learning rate, which results in the fastest decrease in the loss. More importantly, the optimal learning rate can directly be applied in practice, as it does not require any prior knowledge. Thus, tuning the learning rate is not needed at all. Also, we identify the effects of depth, width, and initialization in the training process. We show that when the orthogonal-like initialization is employed, the width of intermediate layers plays no role in gradient-based training, as long as the width is greater than or equal to both the input and output dimensions. We show that under some conditions, the deeper the network is, the faster the convergence is guaranteed. This implies that in an extreme case, the global optimum is achieved after updating each weight matrix only once. Besides, we found that the use of deep networks could drastically accelerate convergence when it is compared to those of a depth 1 network, even when the computational cost is considered. Numerical examples are provided to justify our theoretical findings and demonstrate the performance of layer-wise training by BCGD.