Reasoning as Energy Minimization over Structured Latent Trajectories

David K. Johansson

cs.AIMar 30, 2026

Reasoning as Energy Minimization over Structured Latent Trajectories

Authors: David K. Johansson

Organizations: Polished Snow Inc.

Abstract

Single-shot neural decoders commit to answers without iterative refinement, while chain-of-thought methods introduce discrete intermediate steps but lack a scalar measure of reasoning progress. We propose Energy-Based Reasoning via Structured Latent Planning (EBRM), which models reasoning as gradient-based optimization of a multi-step latent trajectory $z_{1:T}$ under a learned energy function $E(h_x, z)$ . The energy decomposes into per-step compatibility, transition consistency, and trajectory smoothness terms. Training combines supervised encoder-decoder learning with contrastive energy shaping using hard negatives, while inference performs gradient descent or Langevin dynamics over $z$ and decodes from $z_T$ . We identify a critical failure mode: on CNF logic satisfaction, latent planning reduces accuracy from $\approx 95\%$ to $\approx 56\%$ . This degradation arises from a distribution mismatch, where the decoder is trained on encoder outputs $h_x$ but evaluated on planner outputs $z_T$ that drift into unseen latent regions. We analyze this behavior through per-step decoding, latent drift tracking, and gradient decomposition. To address it, we propose dual-path decoder training and latent anchoring. We further introduce a six-part ablation protocol covering component contributions, trajectory length, planner dynamics, initialization, decoder training distribution, and anchor weight. Experiments on three synthetic tasks show that energy decreases monotonically and induces structured latent trajectories on graph and logic tasks, while remaining flat on arithmetic ( $r = 0.073$ ), indicating a negative result. Code is available at https://github.com/dkjo8/ebr-via-structured-latent-planning.

Figures & tables

Panel of Figure 1 — Figure 1: EBRM overview. Encode problem $x$ to context $h_{x}$ ; minimize $E(h_{x},z)$ over latent trajectory $z_{1:T}$ via gradient descent or Langevin dynamics; decode $z_{T}$ to answer $\hat{y}$ .

Figure 2: Direct vs planner endpoint performance across tasks. Planning degrades logic accuracy from ${\approx}95\%$ to ${\approx}56\%$ , motivating the failure analysis in Section 6 .

Panel of Figure 3 — Figure 3: Energy during latent planning. Left: Graph — energy decreases consistently. Center: Logic — monotonic descent across formulas. Right: Arithmetic — energy is flat, indicating limited optimization progress.

Panel of Figure 4 — Figure 4: Latent trajectories in PCA space. Left: Graph — trajectories diverge from a shared start to instance-specific endpoints. Right: Logic — trajectories from diverse starts converge to a shared terminal cluster.

Panel of Figure 5 — Figure 5: Energy landscapes around $z_{T}$ . Left: Graph — smooth directional gradients. Center: Logic — structured surface with clear low-energy basin. Right: Arithmetic — nearly flat surface with negligible gradient signal.

Figure 6: Logic: per-step SAT% during planning. Rows are test problems, columns are planning steps. SAT% is highest at step 1 and degrades, confirming the spurious attractor hypothesis.

Figure 7: Logic: energy vs clause satisfaction during planning. Energy (solid) decreases while SAT% (dashed) remains flat.

Figure 8: Arithmetic: final energy vs prediction error ( $r=0.073$ ). Energy does not reliably predict answer quality.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Latent space
Parameter	Value
Latent dimension $d$	64
Trajectory length $T$	8
Training
Epochs	100
Batch size	32

Appendix

Table 1: Default hyperparameters. See config.toml in the repository for the complete specification.

Latent-Space Planning

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Jul 28, 2026

Penelope: Localized Latent Recurrence for Efficient Structured Reasoning

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.

Yutong Chen, Shouqian Shi, Xinran Liu +5

Structured Reasoning in Language Models Decoder-Only Language Models Recurrent Transformers Structured Reasoning Efficient Language Model Inference Large Reasoning Models+1

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Jan 29, 2026

Latent Chain-of-Thought as Planning: Decoupling Reasoning from Verbalization

Chain-of-Thought (CoT) empowers Large Language Models (LLMs) to tackle complex problems, but remains constrained by the computational cost and early token commitments in discrete reasoning traces. Recent latent reasoning approaches attempt to optimize efficiency by performing reasoning within continuous hidden states. However, many such methods optimize latent states end to end without a trained interface for intermediate textual readout, and several representative configurations use a pre-defined number of latent steps during inference. In this work, we introduce PLaT (Planning with Latent Thoughts), a framework that decouples latent planning from verbalization. The Planner deterministically evolves latent planning states, while an independent Decoder provides textual readouts when needed. Answer-aware textual stopping allows the latent rollout to use a problem-dependent number of groups rather than a pre-specified chain length. PLaT achieves competitive coverage at larger

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in several mathematical settings, with lower Pass@1: on Llama-1B GSM8K, it reaches 80.59% Pass@128 versus CODI's 72.37%. These results support PLaT as a candidate-generation interface supplying multiple textual readouts for downstream verification or reranking.

Jiecong Wang, Hao Peng, Zhanyi Wang +2

Beihang University · Didi Chuxing

LLM Planning Efficient Language Model Reasoning Implicit Reasoning in Language Models Continuous Latent Reasoning Efficient Language Model Inference LLM Reasoning+1

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Jun 25, 2026

Retrieval-Warmed Energy-Based Reasoning: A Five-Arm Ablation Methodology for Diffusion-as-Inference on Structured Reasoning Tasks

Warm-started diffusion samplers accelerate iterative inference, but it is rarely clear which part of the pipeline carries the gain. We study retrieval-warmed energy-based reasoning (RW-EBR) -- an IRED energy-based diffusion model \cite{du2024ired} augmented with a Modern Hopfield trajectory memory -- and contribute a five-arm ablation methodology (oracle, best-constant, per-query-random, shuffled, aligned) that separates three confounded effects: class-prior bias shift, stochastic warm-starting, and graph-aligned value reuse. The diagnostic decomposition is adapted from LLM-RAG evaluation \cite{ru2024ragchecker}. On connectivity-2 (Erdős--Rényi all-pairs reachability), the aligned-vs-shuffled-oracle swing reaches \textbf{

+35

,pp} balanced accuracy on a fixed 1{,}000-graph validation-set diagnostic, with value distribution and retrieval mechanics fixed, only per-graph alignment destroyed, while per-query random initialisation falls below cold -- per-graph alignment, not bias shift or stochasticity, dominates. Yet the deployable cold-prediction pipeline misses the acceptance gate at stored-value quality. The same diagnostic logic, stopped at the key-quality screen, applied to Sudoku with a task-specific key encoder produces a clean negative at a different component -- key quality, under the current setup. The decomposition names the first blocking component on each task. The setting -- graph reachability refined by an iterative diffusion sampler, with explainability of failure modes as the lens -- places the work within structured and spatio-temporal reasoning.

Libo Sun, Po-Wei Harn, Zewei Zhang +2

Department of Computer Science and Software Engineering, Auburn University, Auburn, AL 36830, USA · Department of Information Management, National Central University, Taoyuan 320317, Taiwan

Diffusion Sampling Energy-Based Models Structured Reasoning Diffusion Model Inference Acceleration

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Abstract

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