How Much Rank Does LoRA Need? Rank-Error Bounds for Transformer Attention

Choosing the rank of a low-rank adaptation (LoRA) update is usually an empirical task. In this paper, we provide a task-dependent theory of the approximation error achievable at each LoRA rank for Transformer attention. We fix a pretrained attention head, a target attention function, and a distribution over inputs from the downstream task, and bound the smallest expected Kullback--Leibler (KL) error achievable by a rank-$r$ query LoRA update. When target attention probabilities are bounded away from zero, we prove a lower bound of the error proportional to $ψ(\|d\|_2)$, where $d$ is the difference between candidate and target attention scores and $ψ(t)=\min\{t^2,t\}$. We also prove an unconditional upper bound $\min\{\|d\|_2^2/4,\sqrt2\|d\|_2\}$. Under explicit realizability, geometry, and moment conditions, we then bound the best rank-$r$ error between an explicit multiple of $ψ(\sqrt{T_r})$ and $\min\{T_r/4,\sqrt{2T_r}\}$, where $T_r$ is the downstream-weighted tail energy of the target update. We also provide target-Fisher bounds when candidate scores remain within a fixed range of the target scores, and an unrestricted lower bound when a subset of tokens carries most of the probability mass. These spectral bounds describe finite-score approximation. We then construct explicit families in which softmax saturation makes the rank required to match the attention function strictly smaller than the rank required to match the finite logits. Finally, we extend the analysis to fused multi-head LoRA and joint query/key updates, exposing the effects of rank sharing and query/key factorization constraints.

Training-Free Universal Approximation by Prompting Random Transformers

How expressive is prompting a transformer? Answering this question is important for separating the roles of prompting, architecture, and pretraining in transformer models, and for determining whether task-specific behavior must be stored in model weights or can instead be induced at inference time through the prompt. We show, in an approximation-theoretic sense, that pretraining is optional: a single-layer softmax attention network with random, untrained weights can approximate any Hölder function on a compact manifold when steered by an appropriate soft prompt. Guided by the connection between softmax attention and kernel methods, we construct explicit soft prompts (a prompt per target function, independent of the query) as solutions to linear systems matching attention logits to Gaussian kernel exponents, under which the frozen transformer emulates the classical Nadaraya-Watson kernel estimator. The construction requires only a mild rank condition on the weights, which we show holds almost surely under Gaussian initialization. The prompted network inherits the theoretical guarantees of kernel regression, leading to universal approximation theorems with minimax-optimal rates that depend on the intrinsic dimension. We further quantify the cost of prompting, exposing a tradeoff between the norm of the constructed soft prompt tokens, prompt length, and hidden dimension. Numerical experiments corroborate the constructions and predicted rates.

The Entropic Bound for Transformers: Why Static Rank Fails and Attention-Native Rank Recovers

Neural scaling laws describe how loss decreases as models, data, and compute grow, but they do not answer a prior question: for a fixed task, what is the minimum model capacity required to solve it? We study this through the Entropic Bound, a spectral notion of task-intrinsic capacity for Transformers. We first prove that, in a linear attention surrogate, the intrinsic rank $r^*$ of the token-mixing operator is a tight lower bound: any rank-deficient model incurs unavoidable excess risk, and the bound is achievable at $r^*$. We further show that gradient descent recovers this rank under standard low-rank implicit-bias assumptions, confirm all three properties empirically, and show $r^*$ is recoverable from data before training. We then ask whether this transfers to real attention. A naive transfer fails, and a controlled interpolation ladder localizes the cause precisely: it is not softmax and not a rank constraint, but the input-conditioned nature of attention's mixing operator, which a static weight kernel cannot summarize. Motivated by this, we introduce an attention-native intrinsic rank -- the minimum query-key kernel rank realizing the task within the attention class -- and show that under this definition the full Entropic Bound structure (deficiency, achievability, recovery) is restored for both linear and softmax attention, with the energy effective rank as the estimator robust to softmax distortion. Finally, we map the boundary of data-only predictability: $r^*$ is exactly recoverable for linear QK attention, even without the value map at scale, while softmax attention admits only partial pre-training recovery due to nonlinear inversion and kernel-value identifiability effects. Our results reframe the Entropic Bound from a post-hoc descriptor into an attention-native capacity measure with a precisely characterized predictability frontier.

The Capability Convergence Hypothesis: Capability from Access Structure, Not Scale

The Platonic Representation Hypothesis (PRH) holds that as models scale, representations of heterogeneous networks converge toward a shared model of reality. We propose its sequel and boundary, the Capability Convergence Hypothesis (CCH): under a fixed per-token inference budget, representational convergence does not entail capability convergence. Capability instead converges toward a class, the access-complete hybrid: any architecture holding both a compressive O(1)-state channel and a scalable verbatim-index channel. We anchor it on a witness task, the Newton's-apple problem in an infinite stream, and name three resource walls: a Shannon wall barring any o(Nb)-state architecture, a horizon wall barring any fixed window, and a circuit wall barring fixed-depth attention-only composition (conditional on TC0 != NC1). Under an explicit separability assumption a hybrid crosses all three by paying each wall's price, so capability is strictly super-additive under composition. We separate what we prove from what we conjecture: the access-completeness principle rests on information-theoretic lower bounds and pre-registered experiments, while the field-level convergence trend is an economics-motivated conjecture. We report the first pre-registered small-scale tests under criteria frozen before the data: the predicted scissors gap is measured (exact-retrieval error 0.994 vs. 0.000 once a 64-scalar state gains one global-attention layer), the state-tracking bifurcation lands at the registered boundary, and a conjunction witness shows an irreducibly two-channel solution; one prediction failed with its direction reversed and is reported as such. Representational convergence is given freely by scale; capability convergence must be purchased by access structure.