arxiv: 2605.12705 · v1 · submitted 2026-05-12 · 💻 cs.LG

Recognition: unknown

Early Data Exposure Improves Robustness to Subsequent Fine-Tuning

Aditi Raghunathan, Gaurav R. Ghosal, Jacob Mitchell Springer, Lawrence Feng, Ziqian Zhong

Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:40 UTC · model grok-4.3

classification 💻 cs.LG

keywords early exposurerobustness to fine-tuningcatastrophic forgettingpretraining data allocationpost-training specializationlanguage model trainingdata mixing

0 comments

The pith

Mixing some target data into pretraining improves retention of that capability after later fine-tuning on new tasks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a three-stage training pipeline for language models: pretraining on broad data, post-training to acquire a specific capability, and then downstream fine-tuning on a different objective. It finds that immediate performance right after post-training does not reliably indicate how well the capability will survive the subsequent fine-tuning step. Instead, including some of the post-training data already during pretraining produces models that retain upstream performance better while still gaining on the downstream task. In settings where the total compute budget for the target data is fixed, splitting that data between the pretraining and post-training stages outperforms putting it all in one stage or the other.

Core claim

Early exposure, by mixing post-training data into pretraining, improves the frontier between retained upstream performance and downstream performance across 135M and 1B models, two post-training domains, and two downstream tasks. In compute-matched allocations of target data, the optimum lies neither at full pretraining exposure nor at full post-training specialization. Post-training drives immediate specialization while early exposure builds robustness to later forgetting; replay and dropout applied during post-training provide additional gains on top of early exposure.

What carries the argument

Early exposure: the inclusion of a portion of the post-training target data directly in the pretraining corpus, which alters how the capability is initially acquired and thereby affects its resistance to overwriting during later fine-tuning.

Load-bearing premise

The controlled three-stage pipeline and the specific model sizes and tasks used here reflect the forgetting dynamics that arise in larger-scale training with more complex and overlapping data distributions.

What would settle it

Train a larger model or a model on more naturalistic overlapping data mixtures and check whether the retention advantage of early exposure over pure post-training disappears or reverses after the downstream fine-tuning stage.

Figures

Figures reproduced from arXiv: 2605.12705 by Aditi Raghunathan, Gaurav R. Ghosal, Jacob Mitchell Springer, Lawrence Feng, Ziqian Zhong.

**Figure 1.** Figure 1: Overview of our three-stage experimental setup, in contrast to a typical two-stage setup. A first party pretrains then post-trains a model with the goal of achieving high performance on domain X. Subsequently, downstream users fine-tune θpost for a task Y , causing catastrophic forgetting of domain X. Previous work investigates interventions in the third stage: how can we fine-tune for Y while mitigating f… view at source ↗

**Figure 2.** Figure 2: Mixing during pretraining improves the frontier across four training pipelines (135M). Each panel corresponds to one 3-stage pipeline. Within each panel, the left plot shows retained post-training loss versus downstream fine-tuning loss, and the right plot shows retained pretraining loss versus retained post-training loss. Black denotes the frontier obtained from unmixed pretraining, and purple denotes the… view at source ↗

**Figure 3.** Figure 3: Left: As the mixture fraction λ increases, immediate MusicPile loss after post-training remains nearly constant, while retained MusicPile loss after downstream fine-tuning on ChemPile improves. This shows that the benefits of mixing can be latent: they may not be visible immediately after post-training, but emerge after subsequent fine-tuning. Right: In a compute-matched setting where total MusicPile expos… view at source ↗

**Figure 4.** Figure 4: Replay and dropout provide complementary gains on top of mixed pretraining. Each subfigure shows one 3-stage pipeline. Within each subfigure, the left panel compares unmixed pretraining, mixed pretraining, and mixed pretraining + dropout, while the right panel compares unmixed pretraining, mixed pretraining, and mixed pretraining + replay. Across both downstream settings, adding dropout or replay to mixed … view at source ↗

**Figure 5.** Figure 5: Dropout and replay preserve broader pretraining capability in addition to the posttraining capability (135M). Companion to [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

**Figure 6.** Figure 6: Dropout and replay applied without pretraining-time mixing (135M). To isolate the effect of post-training interventions from pretraining-time mixing, each panel applies dropout or replay on top of unmixed pretraining (λ = 0), with the mixed-pretraining frontier shown for reference. Within each pipeline, the left panel adds dropout during Stage 2 post-training; the right panel adds a small fraction (1%) of … view at source ↗

**Figure 7.** Figure 7: FFT vs LoRA fine-tuning–retention frontiers (135M). Each panel shows four frontiers obtained by sweeping Stage 2 post-training hyperparameters and Stage 3 fine-tuning learning rates: FFT with unmixed pretraining (black circles, solid), FFT with mixed pretraining (purple circles, solid), LoRA with unmixed pretraining (black squares, dashed), and LoRA with mixed pretraining (purple squares, dashed). Mixed pr… view at source ↗

**Figure 8.** Figure 8: Mixing frontiers at 1B, MusicPile post-training pipelines. Companion to [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗

**Figure 9.** Figure 9: Mixing frontiers at 1B, FLAN post-training pipeline. Left: retained post-training loss vs fine-tuning loss. Right: retained pretraining loss (C4) vs retained post-training loss. In this pipeline, mixed pretraining does not noticeably improve the retained post-training vs fine-tuning frontier (left), but it does improve the retained pretraining vs retained post-training frontier (right), indicating that the… view at source ↗

**Figure 10.** Figure 10: Replay and dropout provide complementary gains on top of mixed pretraining at 1B. Companion to [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗

**Figure 11.** Figure 11: Dropout and replay on top of mixed pretraining, broader pretraining retention (1B). Companion to [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗

**Figure 12.** Figure 12: Dropout and replay applied without pretraining-time mixing (1B). Companion to [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗

read the original abstract

How can we train models whose post-trained capabilities survive subsequent fine-tuning? Rather than focusing on downstream interventions to mitigate forgetting of upstream capabilities, we study how upstream training choices - that is, the manner in which a capability is acquired - shape how robustly that capability is retained. We investigate this question in a controlled three-stage language-model pipeline: pretraining, post-training to acquire a target capability, and downstream fine-tuning on a new objective. Across 135M and 1B models, two post-training domains, and two downstream fine-tuning tasks, we find that immediate post-training performance does not reliably predict retention after subsequent fine-tuning: training recipes that look equivalent immediately after post-training can retain the target capability very differently after subsequent fine-tuning. In particular, early exposure - mixing post-training data into pretraining - consistently improves the frontier between retained upstream performance and downstream performance. In compute-matched experiments, where the target data must be allocated between pretraining and post-training, we find that the optimum lies at neither extreme. Together with our other empirical and theoretical findings, this supports the view that post-training drives immediate specialization while early exposure improves robustness to later forgetting. Replay and dropout, typically used to mitigate forgetting as it occurs during fine-tuning, provide complementary gains to early exposure when applied during post-training. Our findings suggest that robustness to subsequent fine-tuning should be treated as a first-class objective of upstream training, addressed preventatively through choices like early exposure rather than reactively during fine-tuning itself.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Early mixing of target data into pretraining improves retention after fine-tuning in small models, with best results at intermediate allocations.

read the letter

Hi, the one thing to know is that this paper finds early exposure to target data during pretraining helps models keep that capability better after later fine-tuning, and in compute-matched runs the sweet spot is splitting the data rather than all pre or all post. They do this cleanly across 135M and 1B models, two domains, and two tasks. The key observation is that post-training performance doesn't forecast retention well, and early mixing improves the trade-off frontier. It also stacks with replay and dropout. The empirical results look consistent and the setup isolates the effect without obvious circularity. The soft spot is the scale. Everything stops at 1B, and forgetting dynamics likely shift at frontier sizes where models have more capacity. No larger tests or scaling analysis means we can't be sure the benefit carries over. The controlled pipeline is a strength for the claim they make, but real-world training has blurrier stages. This is useful for anyone thinking about how to allocate data in pretraining to support later adaptation. Readers working on forgetting or continual learning in LLMs would get something practical out of it. I'd say bring it to reading group as maybe. The result is new enough and the experiments back it. I'd cite the finding if it fit my work on training recipes. It should go to peer review; the evidence is there for the tested regime and the question is worth referee time even if more scale work is needed. Cheers

Referee Report

2 major / 2 minor

Summary. The paper claims that in a controlled three-stage language model pipeline (pretraining, post-training to acquire a target capability, and downstream fine-tuning), mixing post-training data into pretraining ('early exposure') improves robustness to subsequent fine-tuning. This is shown empirically across 135M and 1B models, two post-training domains, and two fine-tuning tasks: immediate post-training performance does not predict retention after fine-tuning, early exposure improves the retained-upstream vs. downstream performance frontier, and in compute-matched data allocation the optimum lies at neither extreme (all data in pretraining or all in post-training). Replay and dropout during post-training provide complementary gains.

Significance. If the result holds, the work is significant for shifting emphasis from reactive forgetting mitigation during fine-tuning to preventative upstream training choices such as data mixing. It offers consistent empirical patterns across model scales, domains, and tasks in a controlled setting, with credit for the compute-matched allocation experiments that directly compare allocation strategies. This could influence practical training pipelines by treating robustness as a first-class upstream objective.

major comments (2)

[Experiments] Experiments section: The robustness benefit of early exposure is demonstrated only for 135M and 1B models; no scaling experiments, scaling-law analysis, or discussion of how forgetting dynamics or loss landscape curvature might change at larger scales (e.g., 7B+) is provided, which is load-bearing for the claim that early exposure 'consistently improves' the frontier in general.
[Methods and Results] Methods and Results: Exact data mixing proportions, precise definitions of upstream/downstream metrics, and statistical controls (number of runs, variance reporting, significance tests) are not fully explicit, which limits assessment of the strength and reproducibility of the reported patterns across the three-stage pipeline.

minor comments (2)

[Abstract] Abstract: The reference to 'other empirical and theoretical findings' should be clarified with a brief pointer to the specific sections or results that constitute the theoretical component.
[Figures] Figures: Ensure legends and axis labels explicitly indicate mixing ratios and the two performance axes for all frontier plots to improve readability.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their positive assessment of the work and recommendation for minor revision. We address each major comment below and will update the manuscript accordingly to improve clarity and acknowledge limitations.

read point-by-point responses

Referee: [Experiments] Experiments section: The robustness benefit of early exposure is demonstrated only for 135M and 1B models; no scaling experiments, scaling-law analysis, or discussion of how forgetting dynamics or loss landscape curvature might change at larger scales (e.g., 7B+) is provided, which is load-bearing for the claim that early exposure 'consistently improves' the frontier in general.

Authors: We thank the referee for this important observation. Our experiments are deliberately limited to 135M and 1B models to enable a fully controlled three-stage pipeline with compute-matched allocations. We agree that the absence of scaling experiments and analysis limits the strength of general claims. In the revised manuscript we will add a paragraph in the Discussion section that (i) explicitly qualifies the scope of our results to the studied scales, (ii) references existing literature on how forgetting and loss-landscape properties evolve with model size, and (iii) outlines why we expect the qualitative benefit of early exposure to persist while noting that quantitative scaling behavior remains an open question. We do not have the resources to run new 7B+ experiments in this revision. revision: partial
Referee: [Methods and Results] Methods and Results: Exact data mixing proportions, precise definitions of upstream/downstream metrics, and statistical controls (number of runs, variance reporting, significance tests) are not fully explicit, which limits assessment of the strength and reproducibility of the reported patterns across the three-stage pipeline.

Authors: We agree that greater explicitness is required. The revised manuscript will (i) state the exact mixing proportions used in every experiment (e.g., the percentage of post-training data introduced during pretraining), (ii) provide formal definitions of the upstream retention metric and downstream performance metric, and (iii) report the number of independent runs (typically three random seeds), standard deviations, and any statistical significance tests performed. These details will be placed in a new subsection of Methods and referenced in Results. revision: yes

standing simulated objections not resolved

Absence of scaling experiments and scaling-law analysis for models larger than 1B parameters due to computational constraints.

Circularity Check

0 steps flagged

No circularity: purely empirical comparisons with no derivation chain

full rationale

The paper is an empirical study reporting direct experimental outcomes from controlled three-stage training pipelines (pretraining, post-training, downstream fine-tuning) on 135M and 1B models. No equations, derivations, fitted parameters renamed as predictions, or self-citation chains are present in the provided text or abstract. Claims rest on observed performance differences across data-mixing recipes and compute-matched allocations, which are measured independently rather than reduced to inputs by construction. This is the standard non-circular outcome for purely experimental work.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

The central claim rests on empirical observations from controlled experiments rather than theoretical derivations; no new entities are postulated and free parameters are limited to experimental design choices such as mixing ratios that are varied rather than fitted to produce the result.

free parameters (1)

mixing proportion of post-training data into pretraining
Varied across experiments to identify the optimum allocation; not a fitted constant but an experimental variable.

pith-pipeline@v0.9.0 · 5584 in / 1112 out tokens · 30107 ms · 2026-05-14T20:40:28.418904+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 22 canonical work pages · 6 internal anchors

[1]

URL https://arxiv.org/abs/2502.02737. Christina Baek, Ricardo Pio Monti, David Schwab, Amro Abbas, Rishabh Adiga, Cody Blakeney, Maximilian Böther, Paul Burstein, Aldo Gael Carranza, Alvin Deng, Parth Doshi, Vineeth Dorna, Alex Fang, Tony Jiang, Siddharth Joshi, Brett W. Larsen, Jason Chan Lee, Katherine L. Mentzer, Luke Merrick, Haakon Mongstad, Fan Pan,...

work page internal anchor Pith review Pith/arXiv arXiv
[2]

Louis Bethune, David Grangier, Dan Busbridge, Eleonora Gualdoni, Marco Cuturi, and Pierre Ablin

URL https://arxiv.org/abs/2603.16177. Louis Bethune, David Grangier, Dan Busbridge, Eleonora Gualdoni, Marco Cuturi, and Pierre Ablin. Scaling laws for forgetting during finetuning with pretraining data injection,

work page arXiv
[3]

URL https://arxiv.org/ abs/2502.06042. Dan Biderman, Jacob Portes, Jose Javier Gonzalez Ortiz, Mansheej Paul, Philip Greengard, Connor Jennings, Daniel King, Sam Havens, Vitaliy Chiley, Jonathan Frankle, Cody Blakeney, and John P. Cunningham. Lora learns less and forgets less,

work page arXiv
[4]

Yangyi Chen, Binxuan Huang, Yifan Gao, Zhengyang Wang, Jingfeng Yang, and Heng Ji

URLhttps://arxiv.org/abs/2405.09673. Yangyi Chen, Binxuan Huang, Yifan Gao, Zhengyang Wang, Jingfeng Yang, and Heng Ji. Scaling laws for predicting downstream performance in llms,

work page arXiv
[5]

Zhengxiao Du, Aohan Zeng, Yuxiao Dong, and Jie Tang

URLhttps://arxiv.org/abs/2410.08527. Zhengxiao Du, Aohan Zeng, Yuxiao Dong, and Jie Tang. Understanding emergent abilities of language models from the loss perspective. InThe Thirty-eighth Annual Conference on Neural Information Processing Systems,

work page arXiv
[6]

Edward J

URLhttps://arxiv.org/abs/1904.13262. Edward J. Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. Lora: Low-rank adaptation of large language models,

work page arXiv 1904
[7]

LoRA: Low-Rank Adaptation of Large Language Models

URL https://arxiv.org/ abs/2106.09685. James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei A. Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Grabska-Barwinska, Demis Hassabis, Claudia Clopath, Dharshan Kumaran, and Raia Hadsell. Overcoming catastrophic forgetting in neural networks. Proceedings of the Nat...

work page internal anchor Pith review Pith/arXiv arXiv
[8]

Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Grabska-Barwinska, Demis Hassabis, Claudia Clopath, Dharshan Kumaran, and Raia Hadsell

ISSN 1091-6490. doi: 10.1073/pnas.1611835114. URLhttp://dx.doi.org/10.1073/pnas.1611835114. Suhas Kotha and Percy Liang. Replaying pre-training data improves fine-tuning,

work page doi:10.1073/pnas.1611835114
[9]

Pratyush Maini, Zhili Feng, Avi Schwarzschild, Zachary C

URL https: //arxiv.org/abs/2603.04964. Pratyush Maini, Zhili Feng, Avi Schwarzschild, Zachary C. Lipton, and J. Zico Kolter. Tofu: A task of fictitious unlearning for llms,

work page arXiv
[10]

Lipton and J

URLhttps://arxiv.org/abs/2401.06121. 12 Preprint. Pratyush Maini, Sachin Goyal, Dylan Sam, Alex Robey, Yash Savani, Yiding Jiang, Andy Zou, Matt Fredrikson, Zacharcy C. Lipton, and J. Zico Kolter. Safety pretraining: Toward the next generation of safe ai,

work page arXiv
[11]

Michael McCloskey and Neal J Cohen

URLhttps://arxiv.org/abs/2504.16980. Michael McCloskey and Neal J Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. InPsychology of learning and motivation, volume 24, pp. 109–165. Elsevier,

work page arXiv
[12]

Kyle O’Brien, Stephen Casper, Quentin Anthony, Tomek Korbak, Robert Kirk, Xander Davies, Ishan Mishra, Geoffrey Irving, Yarin Gal, and Stella Biderman

URL https://arxiv.org/abs/2410.17194. Kyle O’Brien, Stephen Casper, Quentin Anthony, Tomek Korbak, Robert Kirk, Xander Davies, Ishan Mishra, Geoffrey Irving, Yarin Gal, and Stella Biderman. Deep ignorance: Filtering pretraining data builds tamper-resistant safeguards into open-weight llms,

work page arXiv
[13]

URLhttps://arxiv.org/abs/2508.06601. Team Olmo, :, Allyson Ettinger, Amanda Bertsch, Bailey Kuehl, David Graham, David Heineman, Dirk Groeneveld, Faeze Brahman, Finbarr Timbers, Hamish Ivison, Jacob Morrison, Jake Poznanski, Kyle Lo, Luca Soldaini, Matt Jordan, Mayee Chen, Michael Noukhovitch, Nathan Lambert, Pete Walsh, Pradeep Dasigi, Robert Berry, Saum...

work page arXiv
[14]

Olmo 3

URLhttps://arxiv.org/abs/2512.13961. Long Ouyang, Jeff Wu, Xu Jiang, Diogo Almeida, Carroll L. Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, John Schulman, Jacob Hilton, Fraser Kelton, Luke Miller, Maddie Simens, Amanda Askell, Peter Welinder, Paul Christiano, Jan Leike, and Ryan Lowe. Training language models to fol...

work page internal anchor Pith review Pith/arXiv arXiv
[15]

Training language models to follow instructions with human feedback

URL https://arxiv.org/abs/ 2203.02155. Xiangyu Qi, Yi Zeng, Tinghao Xie, Pin-Yu Chen, Ruoxi Jia, Prateek Mittal, and Peter Henderson. Fine- tuning aligned language models compromises safety, even when users do not intend to!,

work page internal anchor Pith review Pith/arXiv arXiv
[16]

Fine-tuning Aligned Language Models Compromises Safety, Even When Users Do Not Intend To!

URL https://arxiv.org/abs/2310.03693. David Rolnick, Arun Ahuja, Jonathan Schwarz, Timothy P. Lillicrap, and Greg Wayne. Experience replay for continual learning,

work page internal anchor Pith review Pith/arXiv arXiv
[17]

Dylan Sam, Sachin Goyal, Pratyush Maini, Alexander Robey, and J

URLhttps://arxiv.org/abs/1811.11682. Dylan Sam, Sachin Goyal, Pratyush Maini, Alexander Robey, and J. Zico Kolter. When should we introduce safety interventions during pretraining?,

work page arXiv
[18]

Jacob Mitchell Springer, Sachin Goyal, Kaiyue Wen, Tanishq Kumar, Xiang Yue, Sadhika Malladi, Graham Neubig, and Aditi Raghunathan

URLhttps://arxiv.org/abs/2601.07087. Jacob Mitchell Springer, Sachin Goyal, Kaiyue Wen, Tanishq Kumar, Xiang Yue, Sadhika Malladi, Graham Neubig, and Aditi Raghunathan. Overtrained language models are harder to fine-tune,

work page arXiv
[19]

Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov

URL https://arxiv.org/abs/2503.19206. Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting.J. Mach. Learn. Res., 15(1):1929–1958, January

work page arXiv 1929
[20]

Mitchell Wortsman, Gabriel Ilharco, Samir Yitzhak Gadre, Rebecca Roelofs, Raphael Gontijo-Lopes, Ari S

ISSN 1532-4435. Mitchell Wortsman, Gabriel Ilharco, Samir Yitzhak Gadre, Rebecca Roelofs, Raphael Gontijo-Lopes, Ari S. Morcos, Hongseok Namkoong, Ali Farhadi, Yair Carmon, Simon Kornblith, and Ludwig Schmidt. Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time, 2022a. URLhttps://arxiv.org/abs/2...

work page arXiv
[21]

Qwen3 Technical Report

URLhttps://arxiv.org/abs/2505.09388. Xianjun Yang, Xiao Wang, Qi Zhang, Linda Petzold, William Yang Wang, Xun Zhao, and Dahua Lin. Shadow alignment: The ease of subverting safely-aligned language models,

work page internal anchor Pith review Pith/arXiv arXiv
[22]

14 Preprint

URL https://arxiv.org/abs/ 2310.02949. 14 Preprint. Contributions Lawrence Feng led the project and conducted all the main experiments. Gaurav R. Ghosal, Ziqian Zhong, Jacob Mitchell Springer, and Aditi Raghunathan were involved throughout the project, including project direction, experimental design, analysis, and paper writing. Acknowledgments We gratef...

work page arXiv
[23]

22 Preprint. 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 Retained Post-training Loss 2.2 2.3 2.4 2.5Fine-tuning Loss 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 Retained Post-training Loss 2.1 2.2 2.3 2.4Fine-tuning Loss C4 MusicPile FLAN Unmixed pretraining Mixed pretraining With dropout With replay Figure 12: Dropout and replay applied without pretraining-time mixing (1B).Comp...

2025
[24]

Following (Springer et al., 2025), we have the following initialization: Assumption C.8(Pretrained Initialization Scale).Let (W1(0),W 2(0)) be the parameters at initialization

Next, we will characterize the initialization scale of model before pretraining. Following (Springer et al., 2025), we have the following initialization: Assumption C.8(Pretrained Initialization Scale).Let (W1(0),W 2(0)) be the parameters at initialization. Then we have thatW 1(0) =W 2(0) = exp(−T)Id. Essentially, Assumption C.8 requires that the model pa...

2025
[25]

Next,we will study the dynamics of the non-zero singular values (Lemma A.11 Springer et al

=σi(t)−2ησi(t)(σi(t)2−(σspec,i)2) + 2ηλ(σi(t)2−σi(0)2)(2) As a result, note that whenσ(un)mixed i (0) = 0,σ(un)mixed i (t) = 0for allt. Next,we will study the dynamics of the non-zero singular values (Lemma A.11 Springer et al. (2025)). We will assume that post-training is performed for a sufficient number of steps. Assumption C.13(Sufficient Post-Trainin...

2025