Semi-gradient Q-learning is widely applied across various fields; however, due to the absence of an explicit loss function, understanding its preference for critical points in the parameter space—namely, the implicit bias—remains challenging. This paper leverages the Fokker–Planck equation to construct and visualize the non-equilibrium loss landscape in a two-dimensional parameter space. Our visualization reveals that the semi-gradient method transforms stable critical points into unstable ones, resulting in training dynamics biased against unstable critical points. Furthermore, we show that this phenomenon extends to high-dimensional parameter spaces and neural network settings, such as DQN. This paper provides valuable insights into the implicit bias of semi-gradient Q-learning.

Although transformer-based models have shown exceptional empirical performance, the fundamental principles governing their training dynamics are inadequately characterized beyond configuration-specific studies. Inspired by empirical evidence showing improved reasoning capabilities under small initialization scales in language models, we employ the gradient flow analytical framework established in \cite{zhou2022towards} to systematically investigate linearized Transformer training dynamics. Our theoretical analysis dissects the dynamics of attention modules into two distinct stages. In the first stage, asymmetric weight perturbations from random initialization sustain non-degenerate gradient dynamics in parameter matrices, facilitating systematic escape from small initialization regimes. Subsequently, these matrices undergo condensation, progressively aligning toward the target orientation. In the second stage, the previously static key-query matrices actively participate in training, driving the normalized matrices toward asymptotic rank collapse. This two-stage framework generalizes classical directional convergence results.

Dropout is a widely utilized regularization technique in the training of neural networks, nevertheless, its underlying mechanism and impact on achieving good generalization abilities remain to be further understood. In this work, we start by undertaking a rigorous theoretical derivation of the stochastic modified equations, with the primary aim of providing an effective approximation for the discrete iterative process of dropout. Meanwhile, we experimentally verify SDE's ability to approximate dropout under a wider range of settings. Subsequently, we empirically delve into the intricate mechanisms by which dropout facilitates the identification of flatter minima. This exploration is conducted through intuitive approximations, exploiting the structural analogies inherent in the Hessian of loss landscape and the covariance of dropout. Our empirical findings substantiate the ubiquitous presence of the Hessian-variance alignment relation throughout the training process of dropout.

Substantial work indicates that the dynamics of neural networks (NNs) is closely related to their initialization of parameters. Inspired by the phase diagram for two-layer ReLU NNs with infinite width (Luo et al., 2021), we make a step towards drawing a phase diagram for three-layer ReLU NNs with infinite width. First, we derive a normalized gradient flow for three-layer ReLU NNs and obtain two key independent quantities to distinguish different dynamical regimes for common initialization methods. With carefully designed experiments and a large computation cost, for both synthetic datasets and real datasets, we find that the dynamics of each layer also could be divided into a linear regime and a condensed regime, separated by a critical regime. The criteria is the relative change of input weights (the input weight of a hidden neuron consists of the weight from its input layer to the hidden neuron and its bias term) as the width approaches infinity during the training, which tends to $0$, $+\infty$ and $O(1)$, respectively. In addition, we also demonstrate that different layers can lie in different dynamical regimes in a training process within a deep NN. In the condensed regime, we also observe the condensation of weights in isolated orientations with low complexity. Through experiments under three-layer condition, our phase diagram suggests a complicated dynamical regimes consisting of three possible regimes, together with their mixture, for deep NNs and provides a guidance for studying deep NNs in different initialization regimes, which reveals the possibility of completely different dynamics emerging within a deep NN for its different layers.

Empirical works show that for ReLU neural networks (NNs) with small initialization, input weights of hidden neurons (the input weight of a hidden neuron consists of the weight from its input layer to the hidden neuron and its bias term) condense onto isolated orientations. The condensation dynamics implies that the training implicitly regularizes a NN towards one with much smaller effective size. In this work, we illustrate the formation of the condensation in multi-layer fully connected NNs and show that the maximal number of condensed orientations in the initial training stage is twice the multiplicity of the activation function, where ``multiplicity'' indicates the multiple roots of activation function at origin. Our theoretical analysis confirms experiments for two cases, one is for the activation function of multiplicity one with arbitrary dimension input, which contains many common activation functions, and the other is for the layer with one-dimensional input and arbitrary multiplicity. This work makes a step towards understanding how small initialization leads NNs to condensation at the initial training stage.

Abstract It has been an important approach of using matrix completion to perform image restoration. Most previous works on matrix completion focus on the low‐rank property by imposing explicit constraints on the recovered matrix, such as the constraint of the nuclear norm or limiting the dimension of the matrix factorisation component. Recently, theoretical works suggest that deep linear neural network has an implicit bias towards low rank on matrix completion. In this work, a regularised deep matrix factorised (RDMF) model for image restoration is proposed, which utilises the implicit bias of the low rank of deep neural networks and the explicit bias of total variation. RDMF is a powerful and flexible framework for inverse problems in image processing while the combination of implicit and explicit regularisation represents the intrinsic characteristics of a natural image. The effectiveness of the RDMF model with extensive experiments are demonstrated, in which the method surpasses the recently proposed models in common examples, especially for the restoration from very few observations. This work sheds light on a more general framework for solving other inverse problems by combining the implicit bias of deep learning with explicit regularisation.

Understanding the structure of loss landscape of deep neural networks (DNNs) is obviously important. In this work, we prove an embedding principle that the loss landscape of a DNN "contains" all the critical points of all the narrower DNNs. More precisely, we propose a critical embedding such that any critical point, e.g., local or global minima, of a narrower DNN can be embedded to a critical point/affine subspace of the target DNN with higher degeneracy and preserving the DNN output function. Note that, given any training data, differentiable loss function and differentiable activation function, this embedding structure of critical points holds.This general structure of DNNs is starkly different from other nonconvex problems such as protein-folding.Empirically, we find that a wide DNN is often attracted by highly-degenerate critical points that are embedded from narrow DNNs. The embedding principle provides a new perspective to study the general easy optimization of wide DNNs and unravels a potential implicit low-complexity regularization during the training.Overall, our work provides a skeleton for the study of loss landscape of DNNs and its implication, by which a more exact and comprehensive understanding can be anticipated in the near future.

How neural network behaves during the training over different choices of hyperparameters is an important question in the study of neural networks. In this work, inspired by the phase diagram in statistical mechanics, we draw the phase diagram for the two-layer ReLU neural network at the infinite-width limit for a complete characterization of its dynamical regimes and their dependence on hyperparameters related to initialization. Through both experimental and theoretical approaches, we identify three regimes in the phase diagram, i.e., linear regime, critical regime and condensed regime, based on the relative change of input weights as the width approaches infinity, which tends to $0$, $O(1)$ and $+\infty$, respectively. In the linear regime, NN training dynamics is approximately linear similar to a random feature model with an exponential loss decay. In the condensed regime, we demonstrate through experiments that active neurons are condensed at several discrete orientations. The critical regime serves as the boundary between above two regimes, which exhibits an intermediate nonlinear behavior with the mean-field model as a typical example. Overall, our phase diagram for the two-layer ReLU NN serves as a map for the future studies and is a first step towards a more systematical investigation of the training behavior and the implicit regularization of NNs of different structures.