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Explainer

Articles

  • The Edge of Stability: How Progressive Sharpening and Hessian Dynamics Govern Deep Learning Optimization

    In classical convex optimization, the behavior of gradient descent is dictated by the Lipschitz smoothness constant of the objective function. If a function $f(\theta)$ has an $L$-smooth gradient—meaning the largest eigenvalue of its Hessian matrix is bounded by $\lambda_{\max}(\nabla^2 f(\theta)) \le L$—gradient descent with learning rate $\eta$ is guaranteed to monotonically reduce the loss if and only if $\eta < 2/L$. When the step size exceeds this threshold ($\eta > 2/\lambda_{\max}$), stan

    1 min

  • Sparse Attention and BigBird: How Window, Global, and Random Graphs Preserve Turing Completeness in Linear Time

    Standard self-attention in transformer architectures scales quadratically with sequence length. Computing full pairwise interactions between n tokens requires evaluating an n x n attention matrix, yielding O(n^2) computational complexity and memory consumption. While hardware accelerators and IO-aware tiling algorithms like FlashAttention optimize memory traffic, the quadratic compute and KV footprint remains a barrier for processing long contexts, document-level summarization, and genomic seque

    1 min

  • Modern Hopfield Networks: How Continuous Energy Landscapes Explain Transformer Attention and Exponential Memory

    When Vaswani et al. introduced the Transformer architecture in 2017, scaled dot-product self-attention was presented primarily as a pragmatic computational mechanism: an efficient, highly parallelizable alternative to recurrence and convolutions. By computing pairwise inner products between queries and keys, normalizing via softmax, and taking a weighted sum of values, attention allowed models to route information dynamically across arbitrarily distant tokens. For several years, self-attention

    1 min