Sparse Growing Transformer: Training-Time Sparse Depth Allocation via Progressive Attention Looping

Looped Transformers scale sequential computation by applying a compact stack of physical blocks for multiple rounds, increasing unrolled depth without increasing stored parameters. This reuse changes the residual-scaling problem: in an untied Transformer, each residual branch receives and applies its own parameter update, whereas in a looped Transformer one shared update aggregates gradients from repeated visits and is read back by those same visits in the next linearized forward pass. We formalize this tied-depth effect through a first-order perturbation bound controlled by a visit-alignment coefficient $κ_R$. The bound recovers the DeepNorm exponent when visits decorrelate, but in the conservative aligned regime it requires the exponent to increase from $1/4$ to $1/2$ as loop count grows at fixed physical depth. The resulting method, DeepLoop, keeps the Post-LN DeepNorm architecture and sets $α=(2N)^{1/2}$ and $β=(8N)^{-1/2}$ for unrolled depth $N$. On GPT-style looped language models at GPT-2 small and GPT-2 medium scale, DeepLoop is neutral when no physical block is revisited and improves validation loss and downstream accuracy once recurrent depth is activated. These results show that stable recurrent depth requires residual scaling rules that account for parameter visits, not only nominal layer count.

Multi-head attention conventionally partitions the hidden dimension equally across all heads at every layer, enforcing an identical representational subspace dimension (dh = dmodel/h) throughout the models depth. In this work, we identify this uniform allocation as a fundamental structural bottleneck: due to their restricted dimensional space, early-layer heads are unable to faithfully capture complex, high-dimensional contextual patterns. To resolve this, we introduce the Prism Transformer, a novel architectural paradigm that replaces the static, uniform head configuration with a progressive head schedule. By monotonically increasing the head count across layers, the Prism Transformer naturally establishes a local-to-global representational hierarchy: early layers leverage fewer, exceptionally wide heads to capture complex, local compositional patterns, while deep layers deploy many, narrow heads to decompose these patterns into specialized linguistic features. Crucially, this structural shift is parameter-neutral, compute-neutral, and introduces zero training or inference overhead, preserving identical weight matrices and FLOP budgets as the standard Transformer. Across three model scales (124M, 354M, and 757M), the Prism Transformer consistently outperforms uniform baselines, achieving consistent reductions in validation loss alongside consistent gains on downstream zero-shot benchmarks (including PIQA, HellaSwag, ARC-Easy, and WinoGrande). Our findings demonstrate that non-uniform subspace allocation unlocks latent capacity within the standard Transformer budget, enabling more effective use of model capacity.

Deep Transformers are composed of uniformly stacked residual blocks, yet their deepest layers often add little value. We present two efficiency methods that exploit this asymmetry. CascadeFormer tapers width with depth to match the uneven information flow across layers, achieving comparable perplexity to a uniform baseline at the same training budget while reducing latency by 8.6% and increasing throughput by 9.4%. CascadeFlow Pruning removes layers using accumulated training gradients, with no post hoc analysis. It outperforms standard heuristics on perplexity and rank-stability and stays competitive on downstream accuracy. To motivate these methods, we propose Gradient Fan-in Asymmetry (GFA) as a structural account of why deeper layers contribute less. In Pre-LayerNorm residual stacks, the gradient at a layer is the sum of an identity path and all downstream functional paths, producing a gradient fan-in that decays linearly with depth (and quadratically under deep supervision), yielding richer gradients for early layers and sparser ones for later layers. We provide correlational and interventional evidence for GFA on models trained from scratch up to 1.2B parameters. Across Transformers and ResNets, accumulated training gradients follow the theoretical fan-in and are associated with post hoc layer importance. Two interventions point to structure rather than magnitude as the bottleneck: equalizing per-layer gradient norms does not restore late-layer value, while increasing downstream path counts via parameter-shared repetition restores and elevates it. Whether gradient magnitude proxies fan-in beyond high-rank regimes, and how these dynamics behave at the 100B+ scale, remain open questions.