Deep equilibrium models (DEQs) have recently emerged as a powerful paradigm for training infinitely deep weight-tied neural networks that achieve state of the art performance across many modern machine learning tasks. Despite their practical success, theoretically understanding the gradient descent dynamics for training DEQs remains an area of active research. In this work, we rigorously study the gradient descent dynamics for DEQs in the simple setting of linear models and single-index models, filling several gaps in the literature. We prove a conservation law for linear DEQs which implies that the parameters remain trapped on spheres during training and use this property to show that gradient flow remains well-conditioned for all time. We then prove linear convergence of gradient descent to a global minimizer for linear DEQs and deep equilibrium single-index models under appropriate initialization and with a sufficiently small step size. Finally, we validate our theoretical findings through experiments.

Reply to Reviewer 2: Thank you for the insightful and comprehensive summary of our work. We will add such information in revision. We can still learn short-run MCMC successfully. We shall also try to implement Burda et al. (thanks for the reference). Following your advice, we did 1D and 2D experiments.

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.

While the Transformer architecture has achieved remarkable success across various domains, a thorough theoretical foundation explaining its optimization dynamics is yet to be fully developed. In this study, we aim to bridge this understanding gap by answering the following two core questions: (1) Which types of Transformer architectures allow Gradient Descent (GD) to achieve guaranteed convergence? By analyzing the loss landscape of a single Transformer layer using Softmax and Gaussian attention kernels, our work provides concrete answers to these questions. Our findings demonstrate that, with appropriate weight initialization, GD can train a Transformer model (with either kernel type) to achieve a global optimal solution, especially when the input embedding dimension is large. Nonetheless, certain scenarios highlight potential pitfalls: training a Transformer using the Softmax attention kernel may sometimes lead to suboptimal local solutions.

To understand the complexity of the dynamic of learning in differential games, we decompose the game into components where the dynamic is well understood. One of the possible tools is Helmholtz's theorem, which can decompose a vector field into a potential and a harmonic component. This has been shown to be effective in finite and normal-form games. However, applying Helmholtz's theorem by connecting it with the Hodge theorem on $\mathbb{R}^n$ (which is the strategy space of differential game) is non-trivial due to the non-compactness of $\mathbb{R}^n$. Bridging the dynamic-strategic disconnect through Hodge/Helmoltz's theorem in differential games is then left as an open problem \cite{letcher2019differentiable}. In this work, we provide two decompositions of differential games to answer this question: the first as an exact scalar potential part, a near vector potential part, and a non-strategic part; the second as a near scalar potential part, an exact vector potential part, and a non-strategic part. We show that scalar potential games coincide with potential games proposed by \cite{monderer1996potential}, where the gradient descent dynamic can successfully find the Nash equilibrium. For the vector potential game, we show that the individual gradient field is divergence-free, in which case the gradient descent dynamic may either be divergent or recurrent.

This field has seen notable advancements through its marriage with contrastive learning, exemplified by Cluster-Contrastive Federated Clustering (CCFC). However, CCFC suffers from heterogeneous data across clients, leading to poor and unrobust performance. Our study conducts both empirical and theoretical analyses to understand the impact of heterogeneous data on CCFC. Findings indicate that increased data heterogeneity exacerbates dimensional collapse in CCFC, evidenced by increased correlations across multiple dimensions of the learned representations. To address this, we introduce a decorrelation regularizer to CCFC. Benefiting from the regularizer, the improved method effectively mitigates the detrimental effects of data heterogeneity, and achieves superior performance, as evidenced by a marked increase in NMI scores, with the gain reaching as high as 0.32 in the most pronounced case.

The choice of initial learning rate can have a profound effect on the performance of deep networks. We present a class of neural networks with solvable training dynamics, and confirm their predictions empirically in practical deep learning settings. The networks exhibit sharply distinct behaviors at small and large learning rates. The two regimes are separated by a phase transition. In the small learning rate phase, training can be understood using the existing theory of infinitely wide neural networks. At large learning rates the model captures qualitatively distinct phenomena, including the convergence of gradient descent dynamics to flatter minima. One key prediction of our model is a narrow range of large, stable learning rates. We find good agreement between our model's predictions and training dynamics in realistic deep learning settings. Furthermore, we find that the optimal performance in such settings is often found in the large learning rate phase. We believe our results shed light on characteristics of models trained at different learning rates. In particular, they fill a gap between existing wide neural network theory, and the nonlinear, large learning rate, training dynamics relevant to practice.

As a consequence of the universal approximation theorems, sufficiently wide single layer neural networks are expressive enough to accurately represent a broad class of functions [Cyb89, Bar93, PS91]. The existence of a neural network function arbitrarily close to a given target function, however, is not a guarantee that any particular optimization procedure can identify the optimal parameters. Recently, using mathematical tools from optimal transport theory and interacting particle systems, it was shown that gradient descent [RVE18b, MMN18, SS18, CB18b] and stochastic gradient descent converge asymptotically to the target function in the large data limit. This analysis relies on taking a "mean-field" limit in which the number of parameters n tends to infinity. In this setting, gradient descent optimization dynamics is described by a partial differential equation (PDE), corresponding to a Wasserstein gradient flow on a convex energy functional. While this PDE provides a powerful conceptual framework for analyzing the properties of neural networks evolving under gradient descent dynamics, the formula confers few immediate practical advantages.

While optimization methodology has provided much of the underlying algorithmic machinery that has driven the theory and practice of machine learning in recent years, sampling-based methodology, in particular Markov chain Monte Carlo (MCMC), remains of critical importance, given its role in linking algorithms to statistical inference and, in particular, its ability to provide notions of confidence that are lacking in optimization-based methodology. However, the classical theory of MCMC is largely asymptotic and the theory has not developed as rapidly in recent years as the theory of optimization. Recently, however, a literature has emerged that derives nonasymptotic rates for MCMC algorithms [see, e.g., 9, 12, 10, 8, 6, 14, 21, 22, 2, 5]. This work has explicitly aimed at making use of ideas from optimization; in particular, whereas the classical literature on MCMC focused on reversible Markov chains, the recent literature has focused on nonreversible stochastic processes that are built on gradients [see, e.g., 18, 20, 3, 1]. In particular, the gradient-based Langevin algorithm [33, 32, 13] has been shown to be a form of gradient descent on the space of probabilities [see, e.g., 36]. What has not yet emerged is an analog of acceleration. Recall that the notion of acceleration has played a key role in gradient-based optimization methods [26]. In particular, the Nesterov accelerated gradient descent (AGD) method, an instance of the general family of "momentum methods," provably achieves faster convergence rate than gradient descent (GD) in a variety of settings [25]. Moreover, it achieves the optimal convergence rate under an oracle model of optimization complexity in the convex setting [24].