First-order test-time optimization (TTO) updates continuous variables such as logits or latent states for a single input. Successful methods usually rely on external reward models or verifiers to supply a semantic gradient. We ask whether reward-free signals from model distributions or prompted self-verification can supply comparable update directions for reasoning tasks. In logit-space TTO, we compare each reward-free gradient with an operational reward-model reference direction, instantiated as the reward-model gradient on the same logits. Across Qwen2.5-7B-Instruct, Qwen3-14B, and Llama-3.1-8B-Instruct, confidence gradients remain nearly orthogonal to the reference, entropy-slope gradients have vanishing scale near greedy rollouts, and terminal answer certainty is sparse and reference-sensitive. Prompted-verifier gradients are more reference-dependent: in a tokenizer-matched Llama diagnostic, four variants attain moderate RM-relative alignment and measurable local RM-score gains. Nevertheless, in end-to-end TTO on Qwen2.5-7B-Instruct and Llama-3.1-8B-Instruct, the selected reward-free signals produce no wrong-to-correct flips on MATH-500, AIME-2024, AIME-2025, or GSM8K, while a separate reward-model-guided TTO reproduction provides an external positive control with measurable gains.

In parameter-efficient fine-tuning, mixture-of-experts (MoE), which involves specializing functionalities into different experts and sparsely activating them appropriately, has been widely adopted as a promising approach to trade-off between model capacity and computation overhead. However, current MoE variants fall short on heterogeneous datasets, ignoring the fact that experts may learn similar knowledge, resulting in the underutilization of MoE’s capacity. In this paper, we propose Contrastive Representation for MoE (CoMoE), a novel method to promote modularization and specialization in MoE, where the experts are trained along with a contrastive objective by sampling from activated and inactivated experts in top-k routing. We demonstrate that such a contrastive objective recovers the mutual-information gap between inputs and the two types of experts. Experiments on several benchmarks and in multi-task settings demonstrate that CoMoE can consistently enhance MoE’s capacity and promote modularization among the experts.