Sharpness-A ware Minimization (SAM) is a recently proposed gradient-based optimizer (Foret et al., ICLR 2021) that greatly improves the prediction performance of deep neural networks.

Machine learning has matured to the point where we often take key design decisions for granted.

This work finds the analytical expression of the global minima of a deep linear network with weight decay and stochastic neurons, a fundamental model for understanding the landscape of neural networks. Our result implies that zero is a special point in deep neural network architecture. We show that weight decay strongly interacts with the model architecture and can create bad minima at zero in a network with more than $1$ hidden layer, qualitatively different from a network with only $1$ hidden layer. Practically, our result implies that common deep learning initialization methods are insufficient to ease the optimization of neural networks in general.

The Rashomon effect -- the existence of multiple, distinct models that achieve nearly equivalent predictive performance -- has emerged as a fundamental phenomenon in modern machine learning and statistics. In this paper, we explore the causes underlying the Rashomon effect, organizing them into three categories: statistical sources arising from finite samples and noise in the data-generating process; structural sources arising from non-convexity of optimization objectives and unobserved variables that create fundamental non-identifiability; and procedural sources arising from limitations of optimization algorithms and deliberate restrictions to suboptimal model classes. We synthesize insights from machine learning, statistics, and optimization literature to provide a unified framework for understanding why the multiplicity of good models arises. A key distinction emerges: statistical multiplicity diminishes with more data, structural multiplicity persists asymptotically and cannot be resolved without different data or additional assumptions, and procedural multiplicity reflects choices made by practitioners. Beyond characterizing causes, we discuss both the challenges and opportunities presented by the Rashomon effect, including implications for inference, interpretability, fairness, and decision-making under uncertainty.

Low-rank matrix factorization (LRMF) is a canonical problem in non-convex optimization, the objective function to be minimized is non-convex and even non-smooth, which makes the global convergence guarantee of gradient-based algorithm quite challenging.

It is generally accepted that starting neural networks training with large learning rates (LRs) improves generalization. Following a line of research devoted to understanding this effect, we conduct an empirical study in a controlled setting focusing on two questions: 1) how large an initial LR is required for obtaining optimal quality, and 2) what are the key differences between models trained with different LRs? We discover that only a narrow range of initial LRs slightly above the convergence threshold lead to optimal results after fine-tuning with a small LR or weight averaging. By studying the local geometry of reached minima, we observe that using LRs from this optimal range allows for the optimization to locate a basin that only contains high-quality minima. Additionally, we show that these initial LRs result in a sparse set of learned features, with a clear focus on those most relevant for the task. In contrast, starting training with too small LRs leads to unstable minima and attempts to learn all features simultaneously, resulting in poor generalization. Conversely, using initial LRs that are too large fails to detect a basin with good solutions and extract meaningful patterns from the data.

Gradient-based algorithms are effective for many machine learning tasks, but despite ample recent effort and some progress, it often remains unclear why they work in practice in optimising high-dimensional non-convex functions and why they find good minima instead of being trapped in spurious ones.Here we present a quantitative theory explaining this behaviour in a spiked matrix-tensor model.Our framework is based on the Kac-Rice analysis of stationary points and a closed-form analysis of gradient-flow originating from statistical physics. We show that there is a well defined region of parameters where the gradient-flow algorithm finds a good global minimum despite the presence of exponentially many spurious local minima. We show that this is achieved by surfing on saddles that have strong negative direction towards the global minima, a phenomenon that is connected to a BBP-type threshold in the Hessian describing the critical points of the landscapes.