Transformers empirically perform precise probabilistic reasoning in carefully constructed ``Bayesian wind tunnels'' and in large-scale language models, yet the mechanisms by which gradient-based learning creates the required internal geometry remain opaque. We provide a complete first-order analysis of how cross-entropy training reshapes attention scores and value vectors in a transformer attention head. Our core result is an \emph{advantage-based routing law} for attention scores, \[ \frac{\partial L}{\partial s_{ij}} = α_{ij}\bigl(b_{ij}-\mathbb{E}_{α_i}[b]\bigr), \qquad b_{ij} := u_i^\top v_j, \] coupled with a \emph{responsibility-weighted update} for values, \[ Δv_j = -η\sum_i α_{ij} u_i, \] where $u_i$ is the upstream gradient at position $i$ and $α_{ij}$ are attention weights. These equations induce a positive feedback loop in which routing and content specialize together: queries route more strongly to values that are above-average for their error signal, and those values are pulled toward the queries that use them. We show that this coupled specialization behaves like a two-timescale EM procedure: attention weights implement an E-step (soft responsibilities), while values implement an M-step (responsibility-weighted prototype updates), with queries and keys adjusting the hypothesis frame. Through controlled simulations, including a sticky Markov-chain task where we compare a closed-form EM-style update to standard SGD, we demonstrate that the same gradient dynamics that minimize cross-entropy also sculpt the low-dimensional manifolds identified in our companion work as implementing Bayesian inference. This yields a unified picture in which optimization (gradient flow) gives rise to geometry (Bayesian manifolds), which in turn supports function (in-context probabilistic reasoning).

Gradient dynamics play a central role in determining the stability and generalization of deep neural networks. In this work, we provide an empirical analysis of how variance and standard deviation of gradients evolve during training, showing consistent changes across layers and at the global scale in convolutional networks. Motivated by these observations, we propose a hyperparameter-free gradient normalization method that aligns gradient scaling with their natural evolution. This approach prevents unintended amplification, stabilizes optimization, and preserves convergence guarantees. Experiments on the challenging CIFAR-100 benchmark with ResNet-20, ResNet-56, and VGG-16-BN demonstrate that our method maintains or improves test accuracy even under strong generalization. Beyond practical performance, our study highlights the importance of directly tracking gradient dynamics, aiming to bridge the gap between theoretical expectations and empirical behaviors, and to provide insights for future optimization research.

In this work, we use the same setting as Saxe et al. [2018]: a regression problem with least-squares

In this paper, we study the implicit bias of gradient descent for sparse regression.

The Kullback-Leibler (KL) divergence plays a central role in probabilistic machine learning, where it commonly serves as the canonical loss function. Optimization in such settings is often performed over the probability simplex, where the choice of parameterization significantly impacts convergence. In this work, we study the problem of minimizing the KL divergence and analyze the behavior of gradient-based optimization algorithms under two dual coordinate systems within the framework of information geometry$-$ the exponential family ($θ$ coordinates) and the mixture family ($η$ coordinates). We compare Euclidean gradient descent (GD) in these coordinates with the coordinate-invariant natural gradient descent (NGD), where the natural gradient is a Riemannian gradient that incorporates the intrinsic geometry of the underlying statistical model. In continuous time, we prove that the convergence rates of GD in the $θ$ and $η$ coordinates provide lower and upper bounds, respectively, on the convergence rate of NGD. Moreover, under affine reparameterizations of the dual coordinates, the convergence rates of GD in $η$ and $θ$ coordinates can be scaled to $2c$ and $\frac{2}{c}$, respectively, for any $c>0$, while NGD maintains a fixed convergence rate of $2$, remaining invariant to such transformations and sandwiched between them. Although this suggests that NGD may not exhibit uniformly superior convergence in continuous time, we demonstrate that its advantages become pronounced in discrete time, where it achieves faster convergence and greater robustness to noise, outperforming GD. Our analysis hinges on bounding the spectrum and condition number of the Hessian of the KL divergence at the optimum, which coincides with the Fisher information matrix.

In this paper, the authors are using a neural network with one hidden layer and one output layer. The activation function is ReLU, and each hidden unit also has a bias term that can be trained. The input is 1-dimensional, i.e., a scalar, so the task is equivalent to interpolation. The loss function is square loss, and the authors use gradient descent to learn the network weights. The authors use an over-parametrization scheme where the number of nodes in the hidden layer tends to infinity, which means the network is infinitely wide. There are two main learning schemes mentioned in this paper: Kernel and adaptive.

We study the implicit regularization of gradient descent towards structured sparsity via a novel neural reparameterization, which we call a diagonally grouped linear neural network. We show the following intriguing property of our reparameterization: gradient descent over the squared regression loss, without any explicit regularization, biases towards solutions with a group sparsity structure. In contrast to many existing works in understanding implicit regularization, we prove that our training trajectory cannot be simulated by mirror descent. We analyze the gradient dynamics of the corresponding regression problem in the general noise setting and obtain minimax-optimal error rates. Compared to existing bounds for implicit sparse regularization using diagonal linear networks, our analysis with the new reparameterization shows improved sample complexity. In the degenerate case of size-one groups, our approach gives rise to a new algorithm for sparse linear regression. Finally, we demonstrate the efficacy of our approach with several numerical experiments.

We describe an implicit sparsity-inducing mechanism based on minimization over a family of kernels: \begin{equation*} \min_{\beta, f}~\widehat{\mathbb{E}}[L(Y, f(\beta^{1/q} \odot X)] + \lambda_n \|f\|_{\mathcal{H}_q}^2~~\text{subject to}~~\beta \ge 0, \end{equation*} where $L$ is the loss, $\odot$ is coordinate-wise multiplication and $\mathcal{H}_q$ is the reproducing kernel Hilbert space based on the kernel $k_q(x, x') = h(\|x-x'\|_q^q)$, where $\|\cdot\|_q$ is the $\ell_q$ norm. Using gradient descent to optimize this objective with respect to $\beta$ leads to exactly sparse stationary points with high probability. The sparsity is achieved without using any of the well-known explicit sparsification techniques such as penalization (e.g., $\ell_1$), early stopping or post-processing (e.g., clipping). As an application, we use this sparsity-inducing mechanism to build algorithms consistent for feature selection.

A deep equilibrium model uses implicit layers, which are implicitly defined through an equilibrium point of an infinite sequence of computation. It avoids any explicit computation of the infinite sequence by finding an equilibrium point directly via root-finding and by computing gradients via implicit differentiation. In this paper, we analyze the gradient dynamics of deep equilibrium models with nonlinearity only on weight matrices and non-convex objective functions of weights for regression and classification. Despite non-convexity, convergence to global optimum at a linear rate is guaranteed without any assumption on the width of the models, allowing the width to be smaller than the output dimension and the number of data points. Moreover, we prove a relation between the gradient dynamics of the deep implicit layer and the dynamics of trust region Newton method of a shallow explicit layer. This mathematically proven relation along with our numerical observation suggests the importance of understanding implicit bias of implicit layers and an open problem on the topic. Our proofs deal with implicit layers, weight tying and nonlinearity on weights, and differ from those in the related literature. A feedforward deep neural network consists of a stack of H layers, where H is the depth of the network. The value for the depth H is typically a hyperparameter and is chosen by network designers (e.g., ResNet-101 in He et al. 2016).