Affine-Scaled Attention: Towards Flexible and Stable Transformer Attention

Transformers are a dominant architecture in modern machine learning, powering applications across vision, language, and beyond. At the core of their success lies the attention layer, where the query, key, and value matrices determine how token dependencies are captured. While considerable work has focused on scaling and optimizing Transformers, comparatively little attention has been paid to how the weights of the queries, keys and values are initialized. Common practice relies on random initialization or alternatives such as mimetic initialization, which imitates weight patterns from converged models, and weight selection, which transfers weights from a teacher model. In this paper, we argue that initialization can introduce an optimization bias that fundamentally shapes training dynamics. We propose conditioned initialization, a principled scheme that initializes attention weights to improve the spectral properties of the attention layer. Theoretically, we show that conditioned initialization can potentially reduce the condition number of the attention Jacobian, leading to more stable optimization. Empirically, it accelerates convergence and improves generalization across diverse applications, highlighting conditioning as a critical yet underexplored area for advancing Transformer performance. Importantly, conditioned initialization is simple to apply and integrates seamlessly into a wide range of Transformer architectures.

We develop a sensitivity analysis for transformer attention in a geometry aligned with tokenwise computation. Our main result is the exact identity $\|J_τ(u)\|_{\infty\to1}=θ(p)/τ$ for the Jacobian $J_τ(u)$ of the tempered softmax $u\mapsto\mathrm{softmax}(u/τ)$, where $θ(p)=4\max_{S\subseteq[L]}p(S)(1-p(S))$ measures how evenly the attention distribution can be bisected rather than how concentrated it is. We combine this identity with a block-$\infty$/RMS norm under which row-stochastic attention mixing is nonexpansive. This yields a distribution-aware local Jacobian bound for multi-head attention and a sequence-length-independent Lipschitz bound on bounded input sets, with explicit dependence on width, input magnitude, temperature, and projection norms. We also identify a structural distinction between normalization placements: a pre-LN residual-sublayer Jacobian contains an additive identity term, whereas a post-LN residual-sublayer Jacobian does not. A LayerNorm projection lemma gives a sufficient condition under which the LayerNorm-only term in the post-LN expansion contracts geometrically; the condition is not tested by our experiments. Across three Pre-LN early-training runs of $774$M-parameter models, attention becomes substantially more concentrated while the median lower-bound certificate for $θ(p)$ remains near one at every sampled layer and checkpoint. This certifies near-maximal exact sensitivity for at least half of the sampled rows within each layer. A minority of rows enters a dominant-atom regime with lower exact sensitivity, consistent with the deterministic relationship between $p_{\max}$ and $θ(p)$.

The Transformer model architecture has become one of the most widely used in deep learning and the attention mechanism is at its core. The standard attention formulation uses a softmax operation applied to a scaled dot product between query and key vectors. We explore the role played by norms of the queries and keys, which can cause training instabilities when they arbitrarily increase. We demonstrate how this can happen even in simple Transformer models, in the presence of easy-to-learn spurious patterns in the data. We propose a new attention formulation, QUEry-modulated Spherical aTtention (QUEST), that constrains the keys to a hyperspherical latent space, while still allowing individual tokens to flexibly control the sharpness of the attention distribution. QUEST can be easily used as a drop-in replacement for standard attention. We focus on vision applications while also exploring other domains to highlight the method's generality. We show that (1) QUEST trains without instabilities and (2) produces models with improved performance (3) that are robust to data corruptions and adversarial attacks.