Organizations: IRMA, Universit´e de Strasbourg, France · Independent Researcher
Abstract
We study when a skip connection around a single-hidden-layer MLP can be absorbed into a residual-free MLP of the same width. We first show that for any architecture whose skip branch is an invertible linear map (including Hyper-Connections and their manifold-constrained variants), the problem reduces to the identity skip case. For homogeneous activations of degree k=1, such as ReLU2 and ReGLU, absorption is unconditionally impossible by a degree argument. For gated activations whose gate is differentiable at the origin with g(0)=0, including SwiGLU and GeGLU, a linearization argument gives the same conclusion. These impossibility results extend to arbitrary depth: a composition of L residual blocks using such activations cannot be replicated by any composition of L residual-free blocks of the same width. For ungated ReLU and GELU, the situation is richer. For generic weight matrices, absorption holds at the single-block level if and only if there exists an index set S of size at least d such that Wdown[:,S]Wup[S,:]=−Id. This condition is non-generic (it fails with probability one under continuous weight distributions), so skip-connected and residual-free MLPs of the same width represent generically disjoint function classes. Whether this disjointness persists for deep compositions of ReLU or GELU blocks remains open.
The analogy between deep neural network forward passes and renormalization group (RG) flows has been repeatedly noted in the literature, but existing treatments remain qualitative: depth is described as a coarse-graining scale, attention is likened to a partition function, and representations are said to flow toward fixed points. No existing work has defined a measurable RG order parameter, tested it under controlled variation of the input distribution, or made quantitative predictions that are empirically verified. We study the simplest architecture for which the analogy is tractable: a pure MLP residual stack trained on masked token prediction over synthetic Markov chain sequences with known spectral properties. We report three findings. (i) The effective rank of the residual stream decreases monotonically with depth after training, consistent with progressive integration of irrelevant degrees of freedom. (ii) This rank collapse is selective: it occurs for chains with short correlation length approximately 1 but is absent for chains with long correlation length approximately 7, measured at the position level to control for mean-pooling artifacts. The network preserves exactly the degrees of freedom relevant to the prediction task, the content of the RG relevance criterion. (iii) Inter-layer kernel drift is concentrated at one or two specific transitions, with the remainder of the network near a fixed point, consistent with a discrete fixed-point plateau. Together these findings constitute the first quantitative, position-level evidence that MLP residual networks implement a selective coarse-graining procedure governed by the spectral structure of the input distribution.
Cryptanalytic extraction has been demonstrated for ReLU networks, for networks using componentwise activations such as GELU or SiLU, and for a Transformer's final projection matrix. These methods do not recover the bias-free Gated Linear Unit (GLU) feed-forward blocks used in many modern language models. Such a block multiplies an activated linear projection by a second learned linear projection within each hidden unit, a two-branch structure absent from the network classes and final-layer setting addressed by those methods. We give a constructive, multi-stage forward-query recovery primitive for isolated bias-free GLU blocks. Finite-difference curvature supplies gate-direction candidates, and paired observations at x and -x separate gate magnitude, orientation, and value-branch coupling. Across high-precision targets, six Qwen layers, an 8,192-unit Llama subproblem, and a full-dimensional Gemma block all reach sub-percent median validation error. Four finite-precision configurations remain below 5 percent median error, but none reproduces every stored weight. These isolated-block experiments are not an end-to-end model-API attack: deriving the required internal block responses from final model outputs remains unsolved.
While Attention Residuals has shown some effectiveness in addressing the widespread issue of unbounded activation growth across deep residual layers, it inevitably incurs significant communication overhead. To circumvent this bottleneck, we propose Multi-Gate Residuals (MGR), which stabilizes activation scales without additional communication burden. It utilizes a straightforward scoring and gating mechanism to maintain multi-stream context, coupled with Attention Pooling to extract hidden states from the stream states. Empirical experiments demonstrate that MGR is practical for large-scale training and deployment, offering tangible performance improvements over existing architectures.