# Data Selection

For targeted instruction-tuning data selection, QSGS organizes candidate examples by gradient similarity into a graph and applies a submodular graph cut that pulls the subset toward the target task while penalizing internal redundancy, then adds gradient-guided synthesis to fill directions the pool cannot cover. On MMLU, BBH and TydiQA across Llama-2, Llama-3 and Mistral, QSGS at a 0.5% budget (1,350 of 270,000 examples) matches or exceeds full-data performance on Mistral-7B and beats the strongest baseline on every benchmark.

For targeted instruction-tuning data selection, QSGS organizes candidate examples by gradient similarity into a graph and applies a submodular graph cut that pulls the subset toward the target task while penalizing internal redundancy, then adds gradient-guided synthesis to fill directions the pool cannot cover. On MMLU, BBH and TydiQA across Llama-2, Llama-3 and Mistral, QSGS at a 0.5% budget (1,350 of 270,000 examples) matches or exceeds full-data performance on Mistral-7B and beats the strongest baseline on every benchmark.

Reinforcement learning for formal theorem proving is bottlenecked by scarce hard tasks: existing Lean problems are either already solved by current provers or too uninformative. ProofFrontier starts from verified Lean theorem–proof pairs, keeps each theorem statement unchanged, and hides proof-structural guidance around key local nodes so the prover must rediscover intermediate reasoning; base-prover pass rates then select tasks that are solvable in principle but not easily solved. GRPO training on 500 synthesized tasks improves DeepSeek-Prover-V2-7B and Goedel-Prover-V2-8B on miniF2F, ProofNet and PutnamBench over each base model.

Reinforcement learning for formal theorem proving is bottlenecked by scarce hard tasks: existing Lean problems are either already solved by current provers or too uninformative. ProofFrontier starts from verified Lean theorem–proof pairs, keeps each theorem statement unchanged, and hides proof-structural guidance around key local nodes so the prover must rediscover intermediate reasoning; base-prover pass rates then select tasks that are solvable in principle but not easily solved. GRPO training on 500 synthesized tasks improves DeepSeek-Prover-V2-7B and Goedel-Prover-V2-8B on miniF2F, ProofNet and PutnamBench over each base model.

Reasoning post-training no longer feeds every teacher chain-of-thought to the student at equal weight. Scaling the corpus had raised training cost without proportional gains. Here Monte Carlo reasoning trees estimate process uncertainty at intermediate states to contract the corpus into a teacher-informed pool, and online routing during supervised fine-tuning concentrates updates on what the student has not yet learned, cutting the optimization budget to 0.22–1.91% of full-data SFT across reasoning benchmarks and model families while matching or improving performance.

Reasoning post-training no longer feeds every teacher chain-of-thought to the student at equal weight. Scaling the corpus had raised training cost without proportional gains. Here Monte Carlo reasoning trees estimate process uncertainty at intermediate states to contract the corpus into a teacher-informed pool, and online routing during supervised fine-tuning concentrates updates on what the student has not yet learned, cutting the optimization budget to 0.22–1.91% of full-data SFT across reasoning benchmarks and model families while matching or improving performance.

The work shows that loss-based data selection drives the post-continual-pre-training Fisher diagonal downward on exactly the high-Fisher coordinates the pretrained model had committed to while leaving low-Fisher coordinates largely untouched, and proposes a Fisher-aware selector that decomposes each candidate's gradient into an anchor component and a frontier component and aggregates them with a log-determinant submodular objective optimized in a single streaming pass; on TinyLlama-1.1B and Llama-3.1-8B continual pre-training over medical data it improves target-domain quality while bounding forgetting on held-out pretraining benchmarks, and 1B selected tokens outperform a replay strategy trained with 10B tokens on both adaptation and forgetting, a roughly 10x token-efficiency advantage.

The work shows that loss-based data selection drives the post-continual-pre-training Fisher diagonal downward on exactly the high-Fisher coordinates the pretrained model had committed to while leaving low-Fisher coordinates largely untouched, and proposes a Fisher-aware selector that decomposes each candidate's gradient into an anchor component and a frontier component and aggregates them with a log-determinant submodular objective optimized in a single streaming pass; on TinyLlama-1.1B and Llama-3.1-8B continual pre-training over medical data it improves target-domain quality while bounding forgetting on held-out pretraining benchmarks, and 1B selected tokens outperform a replay strategy trained with 10B tokens on both adaptation and forgetting, a roughly 10x token-efficiency advantage.