Exact Attention Sensitivity and the Geometry of Transformer Stability

We study signal propagation at initialization in transformers through the averaged partial Jacobian norm (APJN), a measure of gradient amplification across layers. We extend APJN analysis to transformers with bidirectional attention and permutation-symmetric input token configurations by deriving recurrence relations for activation statistics and APJNs across layers. Our theory predicts how attention modifies the asymptotic behavior of the APJN at large depth and matches APJNs measured in deep vision transformers. The criticality picture known from residual networks carries over to transformers: the pre-LayerNorm architecture exhibits power-law APJN growth, whereas transformers with LayerNorm replaced by elementwise $\tanh$-like nonlinearities have stretched-exponential APJN growth, indicating that the latter are subcritical. Applied to Dynamic Tanh (DyT) and Dynamic erf (Derf) transformers, the theory explains why these architectures can be more sensitive to initialization and optimization choices and require careful tuning for stable training.

Transformers are the state-of-the-art architecture for large language models, and a key to their scalability is the strategic usage of low-precision arithmetic. We develop a mixed-precision analysis of transformer inference, deriving bounds for the condition numbers and forward error of the architecture's constituent parts. Notably, we compare the numerical stability of LayerNorm and RMSNorm in the massive-outlier regime, tighten the error bound of softmax in the presence of attention sinks, and quantify the impact of its shifted evaluation on the sensitivity to perturbations. Furthermore, we derive novel sequence-length-independent bounds on the local Lipschitz constant of self-attention. Our worst-case error bound for transformer inference suggests that its numerical stability is determined by the interplay between weight magnitude and the growth of the residual stream. Crucially, and as validated by experiments with GPT-2, our analysis establishes that the scaling of residual-projection weights preserves the propagation of the relative rounding error unless it forces a qualitative transition in the dynamics of the residual stream.

Counting is a simple task on which transformers can nevertheless fail at long sequence lengths. We study the cost of counting in a single attention layer, up to a maximum length $L$, focusing on how softmax attention and layer normalization shape the representation from which the count is recovered. We mainly focus on regression read-out but also consider classification read-out. For regression, we show unnormalized linear attention counts exactly at parameter norms independent of $L$. For softmax attention, regression cannot count exactly with finite parameters; it recovers the count after rounding, but only when the vector parameter norms grow with $L$. We measure this cost by $B_{\min}$, the smallest bound that can be placed on every trainable vector to ensure exact counting after rounding, and derive tight rates. At fixed model dimension, $B_{\min}$ grows as $L$ without normalization, as $L^{3/2}$ under bare post-normalization, and as $L^{3/4}$ with learned post-normalization. The last rate is unexpected: post-normalization discards the magnitude that carries the count and might be expected to make counting harder, yet with a learnable gain it counts at a smaller norm than softmax without normalization. Bare pre-normalization is different: it bounds the attention scores, so beyond a finite length no parameter norm suffices. A classification read-out keeps every norm bounded under every placement, but a dense head stores one output vector per count and costs more in storage. Experiments on the post-normalized model reproduce this split between the two read-outs. Together, our theoretical and empirical findings characterize how self-attention, layer-normalization placement, and read-out choice shape the representation geometry and parameter cost of counting in the minimal single-head architecture studied here.