Quantifying Error Tolerance in Synthetic Data: An Atomic-level Operand vs. Operator Perturbation Study

Synthetic data are increasingly used to scale LLM training, yet more synthetic data do not necessarily produce better models. Useful synthetic data must add information relevant to the target task without introducing errors that offset their benefit, and the value of an example can change as the training set grows. We develop a linear theory that characterizes this tradeoff and determines where synthetic data are useful, how much should be added, and the marginal value of adding one example to an existing set. The analysis shows the conditions when input coverage alone is sufficient and when synthetic errors must also be considered. Guided by these results, we introduce Training-Aware Target Coverage (TATC), a synthetic data selection method for LLM fine-tuning. TATC identifies candidates whose training effects are beneficial to the target task and selects among them to expand coverage of target-relevant directions not already represented by the available data. Experiments on text and image data verify the linear theory. With mathematical reasoning tasks, TATC selects synthetic solutions for fine-tuning Qwen2.5-Math-1.5B-Instruct and outperforms alternative synthetic-data selection methods on GSM8K across selection budgets. In summary, we provide a principled approach to synthetic data selection by quantifying and maximizing its value to the target task.

Large language models trained recursively on their own or other models' outputs undergo model collapse, in which distributional tails and factual accuracy deteriorate while fluency survives. Prior work diagnoses collapse after training; the actionable problem is screening a corpus of unknown provenance before training. We introduce SynthSentry, a corpus-level, model-agnostic contamination signal requiring no access to the generating model, no generation history, and no synthetic labels. The score is a distributional divergence over three statistics: lexical diversity collapse, n-gram tail truncation, and perplexity variance across reference models. We evaluate on corpora contaminated by small open-weight generators and an instruction-tuned open-weight model under a leave-one-generator-out protocol. A domain-stratified study measures false positives on naturally repetitive human text (legal, clinical, source code). The score ranks corpora by severity with little loss when whole generator families are held out. Per-domain calibration holds near its nominal false-positive budget once covariance shrinkage and a bootstrap threshold replace a naive quantile, which runs four times over budget. A downstream fine-tuning check showed no contamination-driven accuracy deficit at our scale, so whether pruning recovers one remains open; the same run shows over-pruning risk once pruning exceeds the true contamination fraction. We frame screening as a data-curation defense rather than a post-hoc diagnosis and release the scoring toolkit. All results are small-scale; scope is English-language, batch-mode corpus screening. Contamination sources are single-generation or hand-authored rather than recursively generated, so results speak to synthetic contamination generally and not to recursion depth.

While synthetic data generation with large language models (LLMs) is widely used in post-training pipelines, existing approaches typically generate full outputs before applying quality filters, leading to substantial token waste on samples that are ultimately discarded. To address this, we propose Multi-Stage In-Flight Rejection (MSIFR), a lightweight, training-free framework that detects and terminates low-quality generation trajectories at intermediate checkpoints before they reach full completion. MSIFR decomposes the generation process into sequential stages and applies fast rule-based validators to identify arithmetic inconsistencies, hallucination patterns, and formatting violations, enabling early rejection of faulty samples. We formalize in-flight rejection as a sequential decision process and show that any non-trivial discard policy reduces expected token consumption, with stage-wise savings increasing when rejection occurs earlier in the generation pipeline. We further demonstrate that conditional utility estimates form a martingale, ensuring that early, in-flight rejection does not bias the expected utility of retained samples. Across five instruction-tuned models and seven reasoning benchmarks, MSIFR reduces token consumption by 11%-77% as a standalone method, and up to 78.2% when combined with early-exit methods, while preserving or improving evaluation accuracy. These results confirm that MSIFR provides a practical mechanism for improving the efficiency of LLM-based synthetic data generation without additional training or architectural changes.