Tag
This paper addresses the open question of maximum step size for gradient descent convergence on non-L-smooth objectives, introducing adaptive methods that operate at the edge of stability and can minimize sharpness globally.
This paper shows that discrete Gradient Descent with large step sizes restores symmetry in multi-pathway Deep Linear Networks, countering the symmetry-breaking predicted by Gradient Flow, and leads to signal re-balancing across pathways. The authors theoretically prove that balanced solutions are flatter (less sharp) than sparse ones, and large learning rates drive the network toward stable, balanced configurations.
MIT researchers show that the edge of stability (EoS) in neural network training is not merely a global optimization phenomenon but selectively redistributes learning across subsets of the training distribution, amplifying progress on some data groups while suppressing others. They identify two key conditions governing this allocation: gradient alignment with the top Hessian eigenvector and sustained non-vanishing gradient magnitude.
This paper develops a sharp pseudospectral theory for block-triangular Jacobians in coupled gradient descent, proving Kreiss-constant bounds and establishing iteration complexity results. The work exposes non-asymptotic, instance-dependent transient amplification phenomena relevant to bilevel optimization, two-time-scale stochastic approximation, and GAN training.
This paper derives exact closed-form expressions for gradients and test loss after one and two steps of gradient descent in two-layer and three-layer linear neural networks, characterizing optimal learning rate selection and revealing a distinct early-training regime where unequal layer-wise learning rates are initially optimal.
This paper proposes Lie group embedded dynamical neural networks (LieEDNN) with learning algorithms based on gradient descent and metric projection on smooth manifolds, enabling stable dynamics on Lie groups like SO(3) and SE(3) for robotics and control applications.
This paper studies retrieval-augmented generation as an in-context optimization process, showing that linear self-attention can implement gradient descent on a unified RAG objective. It proposes a lightweight method for frozen RAG LLMs that predicts context-conditioned updates, improving performance across multiple QA benchmarks.
This blog post explains the math behind gradient descent, the fundamental optimization algorithm used to train machine learning models, with a step-by-step numeric example and intuition.
This paper studies how depth alone induces an implicit low-rank bias in deep unconstrained feature models trained without regularization, shifting the optimal solution from neural collapse to softmax codes, and provides the first asymptotic and dynamic characterization of this bias under gradient descent with cross-entropy loss.
An article explaining the concepts of strong convexity and L-smoothness in optimization, known as the quadratic sandwich, and their role in gradient descent performance.
This paper formally proves that training neural networks with asymmetric activation functions like ReLU, GELU, or SiLU causes weights to drift negative, leading to up to 90% activation sparsity. It also shows that squared activations like ReLU² improve performance but cause activation spikes, which can be fixed by clipping, with GELU² achieving the best validation loss.
This paper develops a local theory of gradient descent near bifurcations in dynamical models, showing that the state-space neural tangent kernel collapses to a rank-one operator that dominates learning dynamics, making optimization effectively low-dimensional and predictable from normal forms.