Your Model Diversity, Not Method, Determines Reasoning Strategy

Compute scaling for LLM reasoning requires allocating budget between exploring solution approaches ($breadth$) and refining promising solutions ($depth$). Most methods implicitly trade off one for the other, yet why a given trade-off works remains unclear, and validation on a single model obscures the role of the model itself. We argue that $\textbf{the optimal strategy depends on the model's diversity profile, the spread of probability mass across solution approaches, and that this must be characterized before any exploration strategy is adopted.}$ We formalize this through a theoretical framework decomposing reasoning uncertainty and derive conditions under which tree-style depth refinement outperforms parallel sampling. We validate it on Qwen-3 4B and Olmo-3 7B families, showing that lightweight signals suffice for depth-based refinement on low-diversity aligned models while yielding limited utility for high-diversity base models, which we hypothesize require stronger compensation for lower exploration coverage.

Decomposing the Delta: What Do Models Actually Learn from Preference Pairs?

Preference optimization methods such as DPO and KTO are widely used for aligning language models, yet little is understood about what properties of preference data drive downstream reasoning gains. We ask: what aspects of a preference pair improve a reasoning model's performance on general reasoning tasks? We investigate two distinct notions of quality delta in preference data: generator-level delta, arising from the differences in capability between models that generate chosen and rejected reasoning traces, and sample-level delta, arising from differences in judged quality differences within an individual preference pair. To study generator-level delta, we vary the generator's scale and model family, and to study sample-level delta, we employ an LLM-as-a-judge to rate the quality of generated traces along multiple reasoning-quality dimensions. We find that increasing generator-level delta steadily improves performance on out-of-domain reasoning tasks and filtering data by sample-level delta can enable more data-efficient training. Our results suggest a twofold recipe for improving reasoning performance through preference optimization: maximize generator-level delta when constructing preference pairs and exploit sample-level delta to select the most informative training examples.

Routing with Generated Data: Annotation-Free LLM Skill Estimation and Expert Selection

Large Language Model (LLM) routers dynamically select optimal models for given inputs. Existing approaches typically assume access to ground-truth labeled data, which is often unavailable in practice, especially when user request distributions are heterogeneous and unknown. We introduce Routing with Generated Data (RGD), a challenging setting in which routers are trained exclusively on generated queries and answers produced from high-level task descriptions by generator LLMs. We evaluate query-answer routers (using both queries and labels) and query-only routers across four diverse benchmarks and 12 models, finding that query-answer routers degrade faster than query-only routers as generator quality decreases. Our analysis reveals two crucial characteristics of effective generators: they must accurately respond to their own questions, and their questions must produce sufficient performance differentiation among the model pool. We then show how filtering for these characteristics can improve the quality of generated data. We further propose CASCAL, a novel query-only router that estimates model correctness through consensus voting and identifies model-specific skill niches via hierarchical clustering. CASCAL is substantially more robust to generator quality, outperforming the best query-answer router by 4.6% absolute accuracy when trained on weak generator data.

Leveraging Parameter Space Symmetries for Reasoning Skill Transfer in LLMs

Task arithmetic is a powerful technique for transferring skills between Large Language Models (LLMs), but it often suffers from negative interference when models have diverged during training. We address this limitation by first aligning the models' parameter spaces, leveraging the inherent permutation, rotation, and scaling symmetries of Transformer architectures. We adapt parameter space alignment for modern Grouped-Query Attention (GQA) and SwiGLU layers, exploring both weight-based and activation-based approaches. Using this alignment-first strategy, we successfully transfer advanced reasoning skills to a non-reasoning model. Experiments on challenging reasoning benchmarks show that our method consistently outperforms standard task arithmetic. This work provides an effective approach for merging and transferring specialized skills across evolving LLM families, reducing redundant fine-tuning and enhancing model adaptability.

SPEAR-MM: Selective Parameter Evaluation and Restoration via Model Merging for Efficient Financial LLM Adaptation

Large language models (LLMs) adapted to financial domains often suffer from catastrophic forgetting of general reasoning capabilities essential for customer interactions and complex financial analysis. We introduce Selective Parameter Evaluation and Restoration via Model Merging (SPEAR-MM), a practical framework that preserves critical capabilities while enabling domain adaptation. Our method approximates layer-wise impact on external benchmarks through post-hoc analysis, then selectively freezes or restores transformer layers via spherical interpolation merging. Applied to LLaMA-3.1-8B for financial tasks, SPEAR-MM achieves 91.2% retention of general capabilities versus 69.7% for standard continual pretraining, while maintaining 94% of domain adaptation gains. The approach provides interpretable trade-off control and reduces computational costs by 90% crucial for resource-constrained financial institutions.

Optimizing Reasoning Efficiency through Prompt Difficulty Prediction

Reasoning language models perform well on complex tasks but are costly to deploy due to their size and long reasoning traces. We propose a routing approach that assigns each problem to the smallest model likely to solve it, reducing compute without sacrificing accuracy. Using intermediate representations from s1.1-32B, we train lightweight predictors of problem difficulty or model correctness to guide routing across a pool of reasoning models. On diverse math benchmarks, routing improves efficiency over random assignment and matches s1.1-32B's performance while using significantly less compute. Our results demonstrate that difficulty-aware routing is effective for cost-efficient deployment of reasoning models.

On Understanding of the Dynamics of Model Capacity in Continual Learning

The stability-plasticity dilemma, closely related to a neural network's (NN) capacity-its ability to represent tasks-is a fundamental challenge in continual learning (CL). Within this context, we introduce CL's effective model capacity (CLEMC) that characterizes the dynamic behavior of the stability-plasticity balance point. We develop a difference equation to model the evolution of the interplay between the NN, task data, and optimization procedure. We then leverage CLEMC to demonstrate that the effective capacity-and, by extension, the stability-plasticity balance point is inherently non-stationary. We show that regardless of the NN architecture or optimization method, a NN's ability to represent new tasks diminishes when incoming task distributions differ from previous ones. We conduct extensive experiments to support our theoretical findings, spanning a range of architectures-from small feedforward network and convolutional networks to medium-sized graph neural networks and transformer-based large language models with millions of parameters.

Training Dynamics Underlying Language Model Scaling Laws: Loss Deceleration and Zero-Sum Learning

This work aims to understand how scaling improves language models, specifically in terms of training dynamics. We find that language models undergo loss deceleration early in training; an abrupt slowdown in the rate of loss improvement, resulting in piecewise linear behaviour of the loss curve in log-log space. Scaling up the model mitigates this transition by (1) decreasing the loss at which deceleration occurs, and (2) improving the log-log rate of loss improvement after deceleration. We attribute loss deceleration to a type of degenerate training dynamics we term zero-sum learning (ZSL). In ZSL, per-example gradients become systematically opposed, leading to destructive interference in per-example changes in loss. As a result, improving loss on one subset of examples degrades it on another, bottlenecking overall progress. Loss deceleration and ZSL provide new insights into the training dynamics underlying language model scaling laws, and could potentially be targeted directly to improve language models independent of scale. We make our code and artefacts available at: https://github.com/mirandrom/zsl

Critique-Guided Distillation: Improving Supervised Fine-tuning via Better Distillation

Supervised fine-tuning (SFT) using expert demonstrations often suffer from the imitation problem, where the model learns to reproduce the correct responses without \emph{understanding} the underlying rationale. To address this limitation, we propose \textsc{Critique-Guided Distillation (CGD)}, a novel multi-stage framework that integrates teacher model generated \emph{explanatory critiques} and \emph{refined responses} into the SFT process. A student model is then trained to map the triplet of prompt, teacher critique, and its own initial response to the corresponding refined teacher response, thereby learning both \emph{what} to imitate and \emph{why}. Using entropy-based analysis, we show that \textsc{CGD} reduces refinement uncertainty and can be interpreted as a Bayesian posterior update. We perform extensive empirical evaluation of \textsc{CGD}, on variety of benchmark tasks, and demonstrate significant gains on both math (AMC23 +17.5%) and language understanding tasks (MMLU-Pro +6.3%), while successfully mitigating the format drift issues observed in previous critique fine-tuning (CFT) techniques.

Dense Backpropagation Improves Training for Sparse Mixture-of-Experts

Mixture of Experts (MoE) pretraining is more scalable than dense Transformer pretraining, because MoEs learn to route inputs to a sparse set of their feedforward parameters. However, this means that MoEs only receive a sparse backward update, leading to training instability and suboptimal performance. We present a lightweight approximation method that gives the MoE router a dense gradient update while continuing to sparsely activate its parameters. Our method, which we refer to as Default MoE, substitutes missing expert activations with default outputs consisting of an exponential moving average of expert outputs previously seen over the course of training. This allows the router to receive signals from every expert for each token, leading to significant improvements in training performance. Our Default MoE outperforms standard TopK routing in a variety of settings without requiring significant computational overhead. Code: https://github.com/vatsal0/default-moe.