Grounding latent algorithm routing in transformer reasoning
Authors: Xiangbo Zhang, Xiaoxu Ma
Organizations: Georgia Institute of Technology
Abstract
A central question in the in-context learning literature is whether transformers can organize episode-level adaptation around different inductive-bias families. We study this question in a controlled setting through latent algorithm routing: route-like behavior in which the solver-family preference changes with the latent data-generating regime while prompt form is held fixed, remains stable under nuisance perturbations, and is selectively influenced by targeted activation interventions without large losses in answer quality. We introduce ROUTEBENCH, a diagnostic benchmark whose regimes differentially favor global shrinkage, sparsity, robustness, and locality, operationalized by ridge-like, lasso-like, Huber-like, and kNN-like family representatives. Across dense decoder-only transformers trained from scratch at 44M-612M parameters, a 306M model closes 80.9 percent of the oracle-routing gap and achieves route F1 of 84.1. The effect remains substantial under natural-language renderings, shuffled supports, lexical paraphrases, and a unified four-way routing setting. Stronger adaptive alternatives, including an input-conditioned soft mixture and an unsupervised Gumbel router, narrow the gap but remain below the 306M and 612M models on route F1 and OOD performance. Probe controls and matched activation-patching controls further show that route-relevant internal directions are decodable and functionally involved in solver-family-consistent output behavior. These results provide controlled evidence that dense transformers trained on ROUTEBENCH can develop route-like internal variables, but they do not establish universal routing in pretrained language models or unrestricted natural-language reasoning.
Standard Large Language Models (LLMs) execute layers sequentially. Dynamic layer routing, i.e. search for a different execution path through layers involving layer repetitions, skips and other moves, can improve performance. Existing routing approaches often require updating model weights, running expensive search loops per test instance, or demand ground-truth labels during inference. In this work, we propose Markov Chain Routing of Transformer Layers (MACRO), a framework that learns task-specific routes over LLM architectures without modifying underlying parameters. MACRO models layer routing as a context-dependent Markov policy conditioned on layer indices, computation budget phases, directional displacements, and operator context, supporting skip, repeat, and residual hidden-state addition operations. The Markov route distribution is updated via feedback on training data and decoded using a top-k Viterbi algorithm to isolate high-probability candidate programs. We evaluate MACRO across diverse reasoning and knowledge benchmarks on multiple open-weight LLMs. MACRO achieves a +5.0% average accuracy improvement over the unrouted baselines, with largest gains on small models. We outperform the best dynamic routing approach Dr. LLM by +7.2%, while reducing route-search time 9.4x (from 14.8 to 1.6 hours). Our code is publicly available at https://github.com/Batorskq/MACRO.
Paweł Batorski, Abtin Pourhadi, Akylgali Aitaza +2
Complex structured reasoning tasks often require additional computation, yet current language models obtain it mainly by increasing parameter scale or by serializing intermediate steps as chain-of-thought (CoT) tokens. The former raises training and deployment costs, while the latter ties reasoning computation to autoregressive output length. We introduce Penelope, an efficient latent-reasoning framework for pretrained decoder-only Transformers that localizes recurrent computation to a selected decoder interval. The lower decoder prefix is evaluated once to construct a problem-conditioned boundary memory, which is then iteratively refined through time-modulated GRU dynamics and recurrent readout states before answer generation. A progressive CoT-to-latent curriculum transfers visible reasoning into this internal recurrent path, allowing additional computation to be allocated in latent space without repeatedly executing the complete decoder or generating a long intermediate trace. Experiments on open-source structured-reasoning benchmarks show that, at validation-selected latent budgets, Penelope attains competitive accuracy relative to established latent-reasoning models while reducing measured inference latency. These results show that latent refinement can be localized to a narrow decoder interval, reducing repeated full-decoder execution without generating a long visible reasoning trace and providing a practical accuracy-efficiency tradeoff for decoder-only Transformer models.
We present a theoretical framework to explain the emergence of inductive reasoning abilities in Transformer language models. While previous works on Transformer learning dynamics have so far been mostly tied to specific tasks, we study a generalized class of inductive tasks that unifies several synthetic tasks known in the literature, including in-context n-grams and multi-hop reasoning. In this class, we theoretically prove that the training dynamics of attention models can be confined to a highly interpretable, low-dimensional invariant manifold. On this manifold, the learning dynamics are captured by a handful of interpretable coordinates rather than millions of parameters, making both theoretical and empirical analysis more tractable. Using this framework, we characterize how data statistics govern the competition between in-context and in-weights learning, we study how random initializations determine the `winning' circuit when multiple solutions are possible, and we demonstrate that the coordinate frame associated with the manifold can be used to automatically detect which circuits have been learned in trained models. By casting circuit formation as a low-dimensional dynamical phenomenon, we take a step toward a predictive theory of how Transformers learn.