Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
Authors: Sunny Sanyal
Hayden Prairie
Rudrajit Das
Ali Kavis
Sujay Sanghavi
Venue: ICML 2025 spotlightposter (2025)
Keywords: Catastrophic Forgetting
Sample Weighting
Fine-tuning
Pre-trained
Primary area: general_machine_learning->transfer_multitask_and_metalearning
TL;DR: We propose a sample-weighting scheme to mitigate catastrophic forgetting during fine-tuning, demonstrating its effectiveness in vision and language tasks while also providing a theoretical analysis for linear models.

Abstract: Fine-tuning a pre-trained model on a downstream task often degrades its original capabilities, a phenomenon known as "catastrophic forgetting". This is especially an issue when one does not have access to the data and recipe used to develop the pre-trained model. Under this constraint, most existing methods for mitigating forgetting are inapplicable. To address this challenge, we propose a *sample weighting scheme for the fine-tuning data* solely based on the pre-trained model's losses. Specifically, we upweight the easy samples on which the pre-trained model's loss is low and vice versa to limit the drift from the pre-trained model. Our approach is orthogonal and yet complementary to existing methods; while such methods mostly operate on parameter or gradient space, we concentrate on the sample space. We theoretically analyze the impact of fine-tuning with our method in a linear setting, showing that it stalls learning in a certain subspace, which inhibits overfitting to the target task. We empirically demonstrate the efficacy of our method on both language and vision tasks. As an example, when fine-tuning Gemma 2 2B on MetaMathQA, our method results in only a $0.8$% drop in accuracy on GSM8K (another math dataset) compared to standard fine-tuning, while preserving $5.4$% more accuracy on the pre-training datasets.

Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
Sunny Sanyal Hayden Prairie"
Rudrajit Das *2
Ali Kavis"
Sujay Sanghavi
Abstract
multiple tasks. Traditionally, this is achieved by training an appropriately large model over datasets of multiple tasks, ensuring that the model jointly learns multiple tasks at once. Unfortunately, it is not viable to repeat this process with every new additional task due to the scale of contemporary models, necessitating effective strategies that can essentially learn without full retraining. A resource-efficiers convention in machine learning is to take a pre-trained model which is trained on some vast and diverse dataset, and fine-tune it on new dataset/task. Such pre-trained models are typically large and expensive to train from scratch but perform well on a variety of tasks while offering a versatile basis for learning a new task.
Fine-tuning a pre-trained model on a downstream task often degrades its original capabilities, a phenomenon known as "catastroohic forgetting". This is especially an issue when one does not have access to the data and recipe used to develop the pre-trained model. Under this constraint, most ex- isting methods for mitigating forgetting are inap- plicable. To address this challenge, we propose a sample weighting scheme for the fine-tuning data solely based on the pre-trained model's losses. Specifically, we upweight the easy samples on which the pre-trained model's s loss is low and vice versa to limit the drift from the pre-trained model. Our approach is orthogonal and yet complemen- tary to existing methods; while such methods mostly operate on parameter or gradient space, we concentrate on the sample space. We theoretically analyze the impact of fine-tuning with our method in a linear setting, showing that it stalls learning in a certain subspace, which inhibits overfitting to the target task. We empirically demonstrate the efficacy of our method on both language and vision tasks. As an example, when fine-tuning method re- sults in only a 0.8% drop in accuracy GSM8K (another math dataset) compared to standard fine- tuning, while preserving 5.4% more accuracy on the pre- pre-training datasets.
Fine-tuning is a delicate process that should ideally serve multiple objectives simultancously; we would like to use the base model and its capabilities to facilitate learning strong model on the downstream task, and in the meantime, preserve the existing abilities of the pre-trained model. On this particular front, the major challenge in standard, unregu- lated teeintunng is the catastrophic forgetting phenomenon. In broad terms, describes the performance decline of the pre-trained model on previously observed data/tasks after fine-tuning on
one. When the learning process for the downstream task interferes with the previously- representations beyond tolerable margins, the pre-trained model loses its prior capabilities and significantly under- performs on previously-earreed tasks.
Gemma 2 2B MetaMathQA,,
Mitigating catastrophic forgetting is an active area of re- search with many fundamental questions awaiting solutions. The key idea is to constrain the fine-tuning process to pre- vent the degeneration of the learned representations while guiding the learning of the new task to augment existing capabilities. The literature on the topic offers various ap- proaches based on the available knowledge pertaining to the pre-traine pre-training process. In fact, mre-trainngg-Se specific data availability and how is treated predominantly dictates the success of mitigating forgetting. In many real-life scenarios, however, the data and the training recipe used for generat- ing the pre-trained model are not available (Radford et al., 2021; Touvron et al., 2023a;b; Grattafiori et al., 2024; Jiang et al., 2023). Naturally, one needs to approach the forgetting phenomenon accordingly to design realistic methods.
1. Introduction
In the modern era of large-scale machine learning, one of the central goals is to design models capable of performing
"qqual contribution. Rudrajit was a PhD student the Univer- sity of Texas at Austin when this work was done. 'University of Texas at Austin 'Google Research. Correspondence to: Sunny Sanyal <sanyalsunny @ @eteeas..euuss Hayden Prairie <hayden- prairie@ utexas..deeee edu>,, Rudrajit Das dasrudrraji @ google.com>>, Ali Kavis <kavis @ www.uin.utexas...om edu>, Sujay Sanghavi <sang- havi www.l.utexas.cduu
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
Therefore, we focus on the case in which we have no ac-

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
cess to the eestraining-speccgie information during the fine- tuning process; we call it the data-oblivious setting. The only piece of information available during fine-tuning is in- deed the pre-trained model. Therefore. one needs to devise a strategy to regulate and guide the fine-tuning process to preserve the pre-trained model capabilities while learning the new task the absence of prior knowledge, Under this challenging setting, we present an answer to the question:
90-
61.17
37.73
Accura
50-
40
669 Fine Buning Act. (Agg. over (osks
30
me me-tuming Acc. [rop-1 miganat-Ak) Avarage
Can we design principled method that mitigates forgetting during fine-tuning in the data-oblivious setting?
Standard
I-Reg.
WISE-FT
FLOW (Ours)
Figure I. FLOW versus standard fine-tuning (FT) and relevant baselines for a ResNet-50 model pre-trained on ImageNet- IK (from Table 1). FLOW achieves the best average accuracy (hetween pre-training and target fine-tuning accuracies).
In this paper, we propose Fine-tuning with Pre-trained Loss- Oriented Weighting (FLOW) to mitigate catastrophic for- getting in the data-oblivious setting. Our key insight is upweighting the "easy" samples which the pre-trained model's 's loss is low and vice versa. We believe that boosting the samples on which the pre-trained model performs well (i.e., has low loss) will introduce supervised bias to the gra- dient updates in favor of the pre-trained model. Intuitively, this will prevent the parameters from deviating too much from the initial pre-trained state, thus mitigating forgetting.
FLOW for linear models. In particular, the covariance matrix of the fine-tuning data weighted by FLOW has a small eigenvalue and training is stalled along the corresponding eigenvector, impeding overfitting to the fine-tuning task (see Remark 7.4).
Some prior papers assign more importance to samples with larger losses to accelerate the training process (Loshchilov & Hutter, 2015; Shrivastava et al., 2016; Katharopoulos & Fleuret, 2017; Kawaguchi & Lu, 2020; Das et al., 2024). We follow the reciprocal reasoning; we tweak the fine-tuning process in favor of the pre-trained model by assigning larger weights to samples with smaller pre-trained loss values. We elaborate on this while stating our contributions next.
We end this section with preview of the comparison of our method FLOW with some relevant baselines (in the data-oblivious setting) in Figure 1.
2. Related Work
2.1. Mitigating Catastrophic Forgetting
We begin by summarizing the vast literature on catastrophic forgetting with a focus on prior works most relevant to our proposed setting. For streamlined presentation, we survey prior work in two settings data-aware and data-oblivious. Due to space limitations, we refer the readers to Appendix A for a more detailed and explanatory review of the literature.
1. To mitigate forgetting, we propose FLOW, which fine-tunes the pre-trained model using a sample-wise weighted loss. Inspired by robust optimization ideas, we derive the 7th sample's weight be exp(-E,/T), where & is the 7th sample's 's pre-trained loss and t is parameter which we as (e in practice. Thus, our method is essentially parameter-free.
1. DATA-AWARE APPROACHES
The majority of the approaches for mitigating forgetting assume task task-specific knowledge access to different extents; either (a subset of) the pre-training dataset itself or some in- formation/statistic computed from pre-training data. Below, we describe the data-aware approaches based on how they make use of task-specific knowledge.
2. We demonstrate the superiority of FLOW over relevant baselines (model averaging, & regularization, LoRA, etc.) in both vision and language model experiments. For instance, ResNet-50 fine-tuned with FLOW on six image classification datasets achieves ~ 17% higher average accuracy (over pre-training and fine-tuning data) than standard fine-tuning, while also surpass- ing other relevant baselines (see Table 1). When fine- tuning Gemma 2 2B on math datasets, the correspond- ing improvement of FLOW over standard fine-tuning is ~ 4% (see Table 2).
Regularization- based methods. This line of work aims to preserve existing capabilities by keeping the parameters close to the pre-trained model. The key idea is to introduce task-specific regularization penalize modifications along the "important" directions for the old tasks (Ahn et al., 2019). Kirkpatrick et al. (2016) introduces the elastic weight con- solidation (EWC) algorithm, which estimates the important directions by approximating the Fisher information matrix. Several variants of EWC have been proposed (Schwarz et al., 2018: Ritter al., 2018; Lee al., 2020; Liu et al., 2018).
3. We also empirically show that combining FLOW with existing methods for mitigating forgetting improves the performance of the base methods (see Tables 3 and 4).
4. We theoretically analyze the effect of fine-tuning with
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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
Zenke et al. (2017); Aljundi et al. (2018) infer the impor- tance of each parameter by their variational effect the outputs. In a similar spirit, Lee et al. (2017) aims to match the posteriors of the pre-trained and fine-tuned models.
model and that of the fully fine-tuned model for the new task. MA and more sophisticated model merging variants have
been studied in relevant context to forgetting (Lubana et al.,
2021; Wortsman et al., 2021; Ilharco et al., 2023; Lin et al.,
2023; Kleiman et al., 2025). Some recent works Chen et al.
Optinization-driven methods. Another perspective to mitigating forgetting is guiding the optimization process by constraining the algorithms directly as opposed to manip- ulating the loss function. The core idea is to keep track of "important directions" for the old tasks, and train on the new task "orthogonally." This could be done by storing prior data samples or gradients in a buffer (Lopez-Paz & Ranzato, 2017: Farajtabar et al., 2020; Chaudhry et al., 2019a) or by incrementally expanding the subspace of important direc- tions without storing task-specific information (Zeng et al., 2019; Wang et al., 2021; 2023b).
(2024b); Panda et al. (2024) introduce different strategies to selectively update a subset of parameters
pre-training data-agnostic manner. Finally, Biderman al. (2024) has shown that LoRA (Hu al., 2022) could be effective for mitigating catastrophic forgetting in transformers. Unlike the methods discussed above which focus on the parameter or gradient space, ours focuses on the sample space.
2.2. Sample Selection and Weighting
Sample-wiee wise importance selection/weighting has been stud- ied in optimization papers (Needell et al., 2014; Zhao & Zhang, 2015; Alain et al., 2015; Stich et al., 2017) and ML papers (Loshchilov & Hutter, 2015; Shrivastava et al., 2016; Katharopoulos & Fleuret, 2017: 2018; Kawaguchi & Lu, 2020; Das al., 2024) speed up the optimizatiion/rrannng process by reducing the variance of the gradient updates. Such papers advocate focusing on "hard" samples with high- gradient norms or losses. In contrast, we focus on "casy" samples to mitigate forgetting. Another line of work focuses on robust learning under uncertain data distributions. Distri- butionally robust optimization (DRO) proposes to minimize the worst-case weighted loss, where the sample weights are constrained or regularized (Ben-Ta Tal et al., 2013; Levy et al., 2020; Duchi & Namkoong. 2021; Qi et al., 2021). Some recent works (Xie et al., 2024; Chen et al., 2024a; Anony- mous, 2025) propose dynamic sample-weighting strategies for LLM training based on the previously discussed idcas.
Replay-based methods. A more direct approach is to store old task samples in buffers and introduce them into the training process for the new task to refresh task-specific representations periodically. There are several components to such methods. Some prior work focus on data selection based on the nature of old data access (Rebuffi et al., 2017: Aljundi et al., 2019; Bang al., 2021; Chaudhry et al., 2019b; Isele& & Cosgun, 2018; De Lange& & Tuytelaars, 2021; Borsos et al., 2020; Tiwari et al., 2021) (e.g., streaming versus on-demand). Another important perspective is the re-introduction strategy of the stored information into the fine-tuning process (Silver& & Mercer, 2002; Li & Hoiem, 2016; Triki et al., 2017; Lee et al., 2019b; Dhar et al., 2019; Rebuffi et al., 2017; Riemer et al.. 2019; Chaudhry et al., 2019b; De Lange& & Tuytelaars, 2021; Tiwari al., 2021).
Architecture-driven methods. Another technique to limit interference between tasks is to allocate separate trainable set of parameters per task. This could be done by initializing a sub-networks per new task (Rusu et al., 2016; Aljundi et al., 2017; Collier et al., 2020: Rajasegaran et al., 2019: Ramesh & Chaudhari, 2021; Wang et al., 2023a; 2022a), gradually expanding the parameters of base network (Yoon
3. Notation and Definitions
1(.) denotes the indicator variable. For any n E N, the set m. is denoted by [n]] Vectors and matrices are in lowercase and uppercase bold font, respectively. The fp norm of a vector v is denoted by ||v||p- The inner product between two vectors v and v' is denoted as (v, v'). A set of n linearly independent n-dimensional vectors {ui..... un is said to be orthonormal basis for RD if (u., Uy) [l j). A vector [V1...., vz)" is said to belong to the n-dimensional probability simplex A, if Σ = 1 and v, 2 0Vi E [n]. For any E N. I, denotes the identity matrix of dimension n.
et al., 201 8; Ostapenko et al., 2019; Hung et al., 2019), or segregating fixed model into task -asc-specific subsets (Mallya et al., 2018; Kang et al., 2022; Serra et al., 2018; Wortsman et al., 2020; Mallya & Lazebnik, 2017; Mustafa B Gurbuz, 2022; Jung et al., 2020). The main downside with this line of work that task identities must be known for inference to (de)activate relevant sub- netwoksss (Aljundi et al., 2017).
2.1.2. DATA-OBLIVIOUS APPROACHE
In the less-explored data-oblivion setting, it is particularly challenging to devise a principled approach, as there is no access to any data-specific information, except for the pre-trained model. One line of work explores the simple idea of "model averaging" (MA) which essentially does convex combination of the parameters of the pre-trained
4. Proposed Algorithm
Our proposed algorithm consists of two main steps: (i) com- puting weights for the samples based their respective pre-trained loss values; and (ii) fine-tuning with weighted loss wherein the per-sample losses are scaled by their respec-
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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
tive weights. The sample-wise weights are computed once and used throughout the entire fine-tuning process. We for- mally state our proposed fine-tuning protocol in Algorithm and delve into its design details in the sequel.
model's losses {[)())) (and not the current model's losses (f:(0))[_1) We would like design л so that:
1. for all * j such that f:(0)) S f,(0"), T; Tj
Algorithm 1 Fine-tuning with Pre-trained Loss-Oriented Weighting (FLOW)
2. does not concentrate around one few samples but rather spreads uniformly over the samples.
Input: Pre-trained model 0", dataset ((x, Y,) for the new task, and temperature parameter т.
These two requirements can be enforced by minimizing the following function (w.r.t. л)) involving negative entropic regularization:
(0) + Th sample's loss at 0, with a non-negative loss function (e.g.. cross-entropy loss).
1. Compute sample weights: w; = exp ( 1:(0")
g(T)) = Tf +
T log T..
(I)
2. Weighted loss: [(0) - Ei=1 wifi
3. Fine-tune with weighted loss: 0- == arg min (()).
Here T >> 0 is a parameter controlling the extent of the sec- ond requirement which facilitated by the entropy term. We now state the minimizer of g(T)) (proof is in Appendix C).
0
Output: Fine-tuned model 0-
Proposition 4.3. Let nt" [ T.."
arg min ((T).
Remark 4.1. Depending on the setting, our model might have task-specific components, such as per-task prediction heads (e.g., in vision). Algorithm I can be slightly modi- fied in the presence of task-specific components to enhance performance. Refer to Appendix B for these modifications.
A.
Then we have पे = 1/2 exp (:10%) where Z is the nor- malizing factor.
Modulo the normalizing factor Z (it does not matter when optimizing w.r.t. 0), note that wi; and L()) in Algorithm 1 are equivalent to गद and Lx (0), respectively.
Remark 4.2. As a heuristic prescription, we set median (f))))) in all our experiments (unless otherwise stated), which leads to consistently good performance. Thus, our algorithm is essentially parameter-free in practice.
Distributionally robust optimization (DRO) perspective. Our formulation above is motivated by prior work on DRO (Qi et al., 2021), but it is exactly the opposite of DRO in spirit. Specifically, in our setting, Qi et al. (2021) consider the following min-mac max problem:
Algorithm design. Our main intuition is that can con- trol forgetting by not drifting away too much from the pre- trained model (i.e., 0*) during fine-tuning. In the pres- ence of pre-training data, this is done by introducing data- dependent constraints on the parameter space or gradient space. Since we have no access to pre-training data, we redirect our focus towards strategies on the sample space depending only on the pre-trained model.
min max 0 TEA
Σ T.((0)) 1=1
m3 log T..
(2)
The first term in Eq. (2) is the worst-case weighted loss at i, while the second term (i.e., entropic regularization) pro- motes uniform weights. The optimal solution to the inner max function w.r.t. π turns out to be मं x exp [.(0) Note that this is essentially the inverse of our weighting function (modulo the normalizing factor) because it as- signs higher weight to samples with larger losses (i.c.. the "hard" samples). The weighting function of DRO would be very conducive to forgetting because it focuses more on the "hard" samples. Further, our weighting function is static (or one-shot) it depends only on the losses at 0* On the other hand, the weighting function of DRO is dynamic (i.c., it depends on the current point 0). In fact, after plugging in the optimal value of æ into Eq. (2) and simplifying, the
To that end, we propose to infer the casiness of each sample of the fine-tuning dataset with respect to the pre-trained model, based on the per-sample losses f0(0''ss (see Alg. 1). We say that the 7th sample is "casy" if f((O') is "small". In- tuitively, prioritizing the "easy" samples during fine-tuning would limit the drift from --. On the other hand, over- focusing on the "easy" samples would probably lead to poor performance on the fine-tuning task. Thus, it is important to strike a balance.
Let us formalize these ideas mathematically. For fine-tuning on the new task, let us consider the objective function L., = 1 (0), where π = . Tn. _T is a static design-choice E A, (i.c., Σn= n. = 1 and क. 2 0 V [n)) which we allow only depend on the pre-trained
.
DRO objective reduces to ming Σn exp [.(0) this is noticeably different from our objective L(0) in Algorithm 1.
'This is not formal definition and so "small" is not quantified.
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can be Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
5. Experimental Setup
5.2. Language Model Experiments
We empirically evaluate the performance of FLOW (Algo- rithm 1) on vision and language tasks, showcasing its ef- fectiveness across different model architectures and modali- ties. Here, explain details of our experiments: baselines, model architectures, datasets, and evaluation metrics.
We follow a similar setup to Biderman et al. (2024); Chen et al. (2024b), where a language model's s general capabilittiss are evaluated before and after fine-tuning on a mathematical reasoning dataset. All training for language experiments is done with HuggingFace peft (Mangrulkar et al., 2022), t z frnsfrrmers (Wolf et al., 2020), datasets (Lhoest et al., 2021), and accelerate (Gugger et al., 2022).
Baselines. In our language and vision experiments, we compare FLOW against relevant baselines in the data- oblivious setting, namely, standard fine-tuning (fine-tuning with vanilla unweighted loss), fz-regularization [following Kirkpatrick al. (2016)], and WiSE-FFT (Wortsman al., 2021) (model averaging of pre-trained and standard fine- tuned models). Additionally, we compare against linear probing (fine-tuning only the classification head, keeping the body frozen) and low-rank adaptation (LoRA) (Hu et al., 2022) in language experiments. More details on the base- lines can be found in Appendix G.1.
Models. We use Gemma 2B (Team al., 2024) and Llama 3.2 3B (Grattafiori al., 2024) our base language models. Further details on training found in Appendix G.2.
Datasets. Following previous work (Biderman et al., 2024; Chen et al., 2024b), we fine-tune on MetaMathQA (Yu et al., 2023), mathematical reasoning dataset that bootstrapped from the training set of GSM8K (Cobbe et al., 2021) and MATH (Hendrycks et al., 2021b) using LLM. We train with all 395K samples in MetaMathQA.
5.1. Vision Experiments
Evaluation metrics. To evaluate the validity of FLOW, we break down our metrics into general capability and targer fine-tuning evaluations. To evaluate general capabilities, we again follow a similar setup to Chen al. (2024b), where we use commonsense reasoning, 5-shot MMLU (Hendrycks et al., 2021a), and 3-shot MBPP (Austin et al., 2021) met- rics. To evaluate the target domain, we use 5-shot GSM8K (Cobbe et al., 2021). All evaluations are performed with Am-evaluat seattiinnhhaeee (Gao et al., 2024). More details on evaluation and the commonsense metric can be found in Appendix G.3. Similar to vision, we also report the average of general capabilities and the target fine-tuning accuracies unified metric.
We compare the performance of FLOW and associated base- lines a transfer learning setup.
Models. We experimented with ResNet-18 and ResNet-50 (Wightman et al., workshop) pre-trained on Imagenet- IK (IN-1K).
Datasets. We used seven widely-used image classifica- tion datasets: CIFAR-10 (Krizhevsky, 2009), CIFAR-100 (Krizhevsky, 2009), Flowers (Nilsback & Zisserman, 2008), Caltech101 (Li et al., 2022), Cars (Krause et al., 2013), and Dogs (Parkhi et al., 2012).
Evaluation metries. Vision models are trained with task- specific parts. such as classification head (head) and batch- norm (BN); see Appendix B for how FLOW works with with task-specific parts. Forgetting is measured by how much the model's 's top-1 validation accuracy on ImageNet- IK (subsequently referred to as IN-1K accuracy) reduces after fine-tuning. We report the fine-tuning performance in terms of average fine-tuning accuracy over all the re- lated datasets following (Goyal et al.. 2023; Ilharco et al., 2023). For IN-1K evaluation after fine-tuning, we replace the task-specific components of the fine-tuned model with their pre-trained counterparts. An extended discussion on experimental details, evaluation, and hyper-parameters are in Appendix G.4. We also report the average of IN-1K accu- racy and averaged fine-tuning accuracy for each method; this is reasonable unified metric to evaluate the performance of a method jointly on the pre-training and fine-tuning data.
6. Experimental Results
6.1. Comparing FLOW and Related Baselines
For our vision experiments on ResNets, Table lists the accuracies of all the baselines and FLOW. We observe similar trends for both ResNet models, so we discuss the results for the larger ResNet-50 model here. The pre-trained ResNet-50 model achieves top-1 accuracy of 79.02% on ImageNet- 1K's validation set. Standard fine-tuning experi- ences a significant 42. 11% drop in IN-IK accuracy, while achieving an average fine-tuning accuracy of 91.78% across the target datasets. In contrast, FLOW suffers only a 2.93% drop in IN-1K accuracy and exhibits a reasonable 86.25% average accuracy on target fine-tuning datasets, demonstrat- ing a significant improvement over standard fine-tuning. Overall, FLOW's average on IN-IK and target domain ac- curacy is 16.83% % higher than standard fine-tuning.
*Our code is publicly available here.
Going beyond standard fine-tuning, our results in Table I show that FLOW comprehensively outperforms other base-
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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
general capabilities and achieving good target fine-tuning performance. Additional details commonsense reasoning results are in Appendix 1.1 and an ablation for our choice of sample weighting in LLMs is in Appendix I.2.
Table 1. Performance of FLOW with ResNet vision models. Bolded and underlined values indicate the best and second-best accuracies within each column (and for each model). Deltas (in color) for IN-1K and target performance are computed w.r.t. the pre-trained and standard fine-tuned models. FLOW attains the best average accuracy and is better than the sccond-bess method (lincar probing) by 2. 2.94% and 3.44% for ResNet- 18 and ResNet- 50, respectively.
In summary, FLOW strikes a good balance between learn- ing a new task and retaining knowledge from pre-training.
6.2. Combining FLOW with Baselines
Method
IN-IK Acc.
Target Ace.
Average
To complement our results in Tables and 2, we investigate the performance of baselines when combined with FLOW.
Pre-trained
69.76
(+0.00)
ResNet--18
Standard FT
19.58 (50.18)
89.07
(+0.0)
54.60
Lincar Probe
69.76 (48.0)
73.57
(-15550)
71.63
In the vision setting, we consider uniform model averaging with WiSE-FT (with a 0.5) and report its performance with and without FLOW in Table 3. Interestingly, averaging the pre-trained IN-1K KK model and the fine-tuned model ob- tained with FLOW improves over standard WiSE-FT (i.c., averaging the pre-trained IN-1K model and the standard fine-tuned model) by 4.18% and 4.52% for ResNet-18 and ResNet-50, respectively, in average performance.
fy-Reg.
34.78
(34.08)
88. 12
(4995
61.45
WiSE-FT
54.15
(11561)
80.23
(4.84)
67.19
FLOW (Ours)
65.21
(44.55
83.93
(-5.14)
74.57
Pre-trained
79.02 (40.00)
ResNet-50
Standard FT
36,91 (-22.11)
91.78
(40.00)
64.34
Linear Probe
79.02 (+0.00)
76,45
(15:33)
77.73
f2-Reg.
44.78 (334244)
91.58
(.020)
68.18
WISE-FT
61.65 (-37.37)
81.38
(-10.40)
71.52
FLOW (Ours)
76.09
(-2.98)
86.25
(-5.53)
81.17
Further, as seen in Table 4, FLOW boosts the performance of other baselines in language modeling. When combined with éz-regularization, we observe improvements in gen- eral capability between 0.5% and 1.80%, with only a 0.83% reduction in GSM8K performance. Furthermore, the integra- tion of FLOW with LoRA yields even stronger results, en- hancing general capability performance by 1.07% to 3.40%, while simultancously improving GSMSK performance by 1.06%. Further details and discussion combining FLOW with fe-reuularization and LoRA are in Appendix I.1.
lines. Interestingly, despite its simplicity, linear probing is the second-best method. Overall, FLOW outperforms other baselines, when averaging IN-1K and target fine-tuning accuracies, by a 3.44% % advantage over the closest com- petitor, linear probing. Although linear probing completely prevents forgetting, it learns significantly less during fine- tuning compared to FLOW. The accuracies on individual fine-tuning datasets and corresponding accuracies for IN-IK can be found in Appendix H.
7. Theoretical Analysis
Additionally, Appendix H.1, we compare FLOW with a distillation-basee approach for mitigating forgetting called "learning without forgetting" (LwF) (Li & Hoiem, 2016) VIT-B/16, which is a larger model than ResNets and on the Food101 (Bossard et al., 2014) dataset, which is relatively larger than each of the six datasets here. In summary, FLOW outperforms LwF in terms of average accuracy; while LwF has the edge on tue-ttning performance, FLOW does better on the forgetting front.
Here we consider linear pre-training and fine-tuning tasks and theoretically analyze the effect of fine-tuning with our proposed method FLOW (Alg- 1). Specifically, we compare the non-asymptotic trajectories of FLOW and vanilla fine- tuning. A key insight of our analysis is that FLOW stalls training in a certain direction, impeding overfitting to the fine-tuning task (see Remark 7.4). We also demonstrate that FLOW goes beyond the simple idea of model averaging (see Remark 7.5).
Our language model results are in Table 2. Results for Gemma 2 2B show that FLOW helps preserve (and even somewhat enhance) the general capabilities of the pre- trained model. Specifically, compared to standard fine- tuning, FLOW improves general capability accuracy by 2.52% in commonsense reasoning. 3.73% in MMLU, and 10.00% in MBPP, with a minor degradation of 0.83% in GSM8K. We see similar trend in our Llama 3.2 3B ex- periments. Furthermore, while alternative baselines show specific strengths (such as WiSE-FT's general capability per- formance and fz-regularization's 's target fine-tuning perfor- mance), FLOW outperforms all baselines, on average, for both models, striking the best balance between preserving
We begin by describing the problem setting.
Pre-training task: The label y E RP for a d-dimensional data point x ~ P is given by = (0.,x), x). where O, E Rd is the ground-truth model. Let 'D denote the joint distribution of (x,y), y), where x ~ P. Let Σ = Ex-p xx be the data covariance matrix. Without loss of generality, let Σ 7 I,-
Fine-tuning task: The label ý E R for a d-dimensional data point x P is given by ý = (0. x). where 0, e Rd is the
'Our insights carry over to neural networks following the dy- namics of lincar models under gradient descent (Lee et al., 2019a).
6

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
Table Performance of FLOW with LLMs. After fine-tuning Gemma 2 2B and Llama 3.2 3B on MetaMathQQ, we compare the target fine-tume tuning performance (GSM8K) with general capability performance. Bolded and underlined values indicate the best and second besttd-best results within each column (and for each model). Deltas (in color) for general capability metrics and fine-tuning metrics are computed w.r.t the pre-trained and standard fine-tuned model's accuracy, respectively. We see that FLOW, on average, has the best performance on general capabilities and target domain, achieving within ~ 0.8% (Gemma 2 2B) and 1.4% (Llama 3.2 3B) of standard mee-uunin'ss s target performance, while significantly mitigating the degradation of general pre-training capabilities in comparison to other basclines.
General Capability Acc.
Target Ace.
Method
Commonsense
MMLU
MBPP
GSM8K
Average
Gemma 2B
Pre-trained
57.23 (10.00)
49.59 (10.00)
28.40 (40.00)
24.49 (:3889)
40.79
Standard Fine-tuning
55.07 (-2.16)
45.59 (-400)
16.80 (-11.80)
63.38 (40.0)
46.31
WiSE-FT (a = 0.5)
57.28 (+0.05)
50.13 (+0.54)
25.60 (-280)
53.30 (:1008)
47.60
LoRA (r = 64)
55.67 (-1.6)
44.28 (5.31)
25.80 (-260)
60.43 (2295)
47.05
f2- t--Reuulaizattonn
57.01 (-0.22)
48.43 (-1.16)
24,80 (3360)
62.85 (0.53)
49.19
FLOW (Ours)
57.59 (40.36)
49.31 (-0.28)
26.80 (-1.60)
62.55 (4083)
49.98
Llama 3.2 3B
Pre-trained
54,48 (+0.00)
54.34
(+0.00)
38,00 (+0.00)
26.01 (-4094)
44.28
Standard Fine-tuning
50.68 (3380)
45.29 (9905)
17.80 (-20.20)
66.95 (+0.00)
46.10
WiSE-FT (a = 0.5)
54.54 (40.04)
53.33 (-1.01)
34,60 (:3.40)
57.01 (9.94)
50.75
LoRA (r = 64)
53.10 (-1,88)
50.95 (-3.39)
34.00 (44000)
63.84 (3.15)
51.66
f Re-Rggularizaton
53.60 (-0888
51.28 (33.06)
33.60 (-4,40)
66.87 (40.8)
52.30
FLOW (Ours)
54.30 (-0.18)
51.86
(:2.48)
36.00 (-2.00)
65.58 (-1.37)
52.87
Table WISE-FT with FLOW vs. WiSE-FT vision. "WiSE- FT+" denotes WiSE-FT with FLOW in the table. Comparison here is the same setting as Table 1. Note that WiSE-FT+ is significantly better than WiSE-FT.
Table 4. Ez-Reg//LoRA FLOW vs. E2-Reg./LoRA language. "L2+" and "LoRA+" denote fz-Reg. with FLOW and LoRA with FLOW, respectively. The results helow are for Gemma 2B in the same setup Table 2. We let A1, A2, A3, and B1 represent Commonsense, MMLU, MBPP, and GSM8K, respectively. We see that t and LoRA+ are better than l and LoRA, respectively.
Method
IN-1K
Target
Average
Method
A1
A2
A3
B1
Avg.
WiSE-FT
54.15
80.23
67.19
l
57.01
48.43
24.80
62.85
49.19
ResNet-18
FT+
68.71
74.03
71.37
l2z
57.53
49.38
26.60
62.02
49.79
WiSE-FT 61.65
81.38
71.52
LoRA
55.67
44.28
25.80
60.43
47.05
ResNet-50
WiSE-FT+
78.29
73.80
76.04
LoRA+
56.74
47.68
28.80
61.49
49.31
ground-truth model. Let D denote the joint distribution of (x, where x ~ P. Also, let
relationship between E and e (i.e. the difference between the optima of the pre-training and fine-tuning tasks).*
e = 0. 0 e/|le=
For a model parameterized by 0 E Rdd let
and e be a unit vector orthogonal to e. We consider the case of p = (O,,, E), where
e (0) = EEp (y x)) " = -0--.) (0(00-)-(0- ((0-))..
S == Ia plco +CLCT).
(3)
erry (0) == ED (5 (0,
x) )²| = -0-00) T 1( (0000,)) 0.) (4)
where E 1) is a constant. Note that E is the data covariance matrix here.
be the population errors the pre- pre-training and fine fine-tuning tasks, respectively. Also, the total error with @ on the two tasks is denoted by err (0) = ert, (0) + err, (0).
Remark 7.1 (Regarding E). We study the case of E as given in Eq. (3) because is the minimal analytically tractable case where can show that FLOW goes be- yond model averaging (MA) (see Remark 7.5). Specifically, if 0 and Σ- = La, then FLOW reduces to MA. Moreover, for an arbitrary Σ, characterizing the digen-spectrun of the matrix dictating the trajectory of FLOW becomes in- tractable. For the analysis to be tractable, we need some
We assume that initially, we learn 0. with the pre-training data; so 0, is our pre-trained model. Note that
er
(0.) (f.)
err (0.) e Ee |el2
(5)
where the last step follows by using Σ from Equation (3).
*See Appendix E. for more details.
7

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
We start fine-tuning starting from O.. Specifically, we as- sume access to the population(X, y) D of the fine-tuning task, but we lose access to the pre-training data.
Note that (we use Σ > Id below):
erTi,x (ottoo(0.) (0.) - err, e (0.) = =e -e 2 ||||
(13)
Vanilla fine-tuning (FT); We minimize (0) (Eq. (4)) with gradient descent (GD) starting from 9. using a constant learning rate 7j. Our iterate k at the Tth iteration is given by (using the value of E from Eq. (3) and O. - 0. e):
On the other hand, the non-zero eigenvalues and corre- sponding cigenvectors of Q are not straightforward to compute. We do this computation in Lemma F.3 with the re- re-parameterization of = V 8(1-p²) (1-8)(1-9p²) for some 8 E 10. Using this in Eq. (11) and simplifying, we get for FLOW:
0x- = 0, + [1 2T Ia + pee
ce
)) e. (6)
FLOW: For some temperature T, the weight of (x D ( (0),2))))
K A 32 p2
e-3p 77 T
Ōk 0.+
I|e|@r,
is wxx y) exp
We minimize
1 +
B22p² p²
1 B2p²
(14)
2 0 == BB
w (xx (5 (0, x.)
(7)
where π1 =
1+8pp T+
and 222 = 0² (1-1))) (1
Remark 7.4 (FLOW's trajectory). Note that we can con- tral \1 by varying 3. Specifically, we can make X1 arbitrar- ily close to 1 by choosing small enough B. On the other 1-3 hand, I-3p" < 1+00" xn and so, Do < pZS.. Hence, beyond certain number of iterations K, Eq. (14) becomes:
with GD starting from O. using constant learning rate i7. Suppose our iterate at the T iteration is 0K-
1/2
Theorem 7.2 (FLOW). Let p= (+2|ee|B/
Then:
1+
0x= 0. + [Is- - -2nEE)) e,
(8)
0x = 0x := =0.+ + (KK, (c ( -ppllel22.) (15)
where E' = p[a - Q) with
with 7(K = [1
Because we can control X by varying B3, we can control (K, B). Thus, we can stall convergence along (e
X
Q = (1 - 1²² )ee²
jee + +p²(1 - -1
lla - pp- (co + +ele)) (9)
251), impeding the con-
vergence of 0. to 0..
Remark 7.5 (FLOW goes beyond model averaging). If we perform model averaging between 0_ and Q, with pa- rumeter w E [0, 11, then our averaged model is:
We prove Thm. 7.2 in Appendix D. The main technical challenge is the evaluation of E', viz., the covariance matrix of the weighted fine-tuning data; see Lemma F.1 for this.
Now, we are going to compare vanilla FT (6) with ग = 1 and FLOW (8) with 7 = 1.
Omz((w) = w + (1 - -)O = 0. + we.
(16)
Zp We believe these are com- parable learning rates for vanilla FT and FLOW because the resultant matrices (Eqs. (10) and (11) dictating the convergence of both methods have exactly two non-zero eigenvalues and the corresponding eigenvectors lie in the span of and T Plugging in 7 1/2 into Eq. (6), we get:
Comparing the above with Eq. (15), we see that FLOW goes beyond model averaging because of the component along
1 But we can make 0x (Eq. (15)) + Owe (w) by choosing 3 + 0 and K such that <(, B) + w. So, we expect FLOW to be at least as powerful as model averaging.
As per Lemma F.4, the minimum total error on both tasks with optimally tuned model averaging is given by:
k = 0, + pke, e, with P -p((e pе|| + teee")
(10)
for vanilla FT. Plugging in fj 2n into Eq. (8), we get:
Σe Σe + 1
min err (Oavg(w))) = we(0.1)
||el| < |el|
0x= = 0_ + Q*e, e, with Q given by Eq. (9)
(11)
c
(17)
for FLOW. The non-zero eigenvalues of P are Fp and the corresponding eigenvectors are - (e + @T). Using this in v2
where recall that Σ is the covariance matrix of the pre- training data. On the other hand, using Eqs. (5) and (13)
(10) and simplifying, we get for vanilla FT:
min err (0.). eIT ( (0.) 1|e112.
(18)
0x = 0.+ (I(K is evenee - 1(K is odd) [ellze)). (12)
'The corresponding temperature is 28(1-~2)
(1- 3
Remark 7.3 (Vanilla FT). Since p < 1, 0g converges to 0. rapidly, and we cannot impede this convergence.
Thii direction is the top cigenvector of Q. Since Σ = v(a=
this is also the eigenvector of E' with the smallest eigenvalue.
8

---

Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
Remark 7.6 (Error comparison). By comparing Eqs. (17) and (18), we see that optimally tuned model averaging at- tains smaller total error than both O. fi.e., the pre-trained model) and 0, to which vanilla FT converges rapidly (Re- mark 7.3). More importantly, following our discussion in Remark 7.5, we conclude that optimally tuned FLOW's total error is at least as good as the one in Eq. (17).
funded in part by the Swiss National Science Foundation (SNSF) under grant number P500PT 217942. The authors are grateful to anonymous reviewers for their feedback on improving this paper.
Impact Statement
This paper presents work whose goal is to advance the field of machine learning. There potential societal conse- quences of our work, none of which we feel must be specifi- cally highlighted here.
8. Conclusion
In this paper, we studied the problem of catastrophic for- getting in pre-trained models during fine-tuning when we do not have access to the pre-training data. To mitigate this issue, we proposed FLOW, method which upweights easy samples based on the pre-trained loss values. Empirically, we showed that FLOW, on average, outperforms relevant baselines and is also complementary to these baselines in both vision and language settings. We also theoretically analyzed FLOW for linear models.
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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
Appendix
Table of Contents
Appendix A: Extended Related Work
Appendix B: Our Algorithm in the Presence of Task-Specific Model Components
Appendix C: Proof of Proposition 4.3
Appendix D: Proof of Theorem 7.2
Appendix E: Difficulty in the Analysis with General Covariance Matrix Σ
Appendix F: Lemmas Used and Their Proofs
Appendix G:
Experimental Details
Appendix G.I: Baseline Details
Appendix G.2: Language Model Hyper-Parameters
Appendix G.3: Language Model Evaluation Details
Appendix G.4: Vision Model Implementation Details
Appendix H: Detailed Vision Results and Ablations
Appendix H.I: Comparison with Distillation-Banl Method for Mitigating Forgetting
Appendix Additional Language Model Results and Ablations
Appendix L.:: Extended Commonsense Reasoning Results
Appendix L.2: Tosen-wise Sample Weighting Ablations
- Appendix L3: Extended Weight Averaging Results
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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
A. Extended Related Work
The majority of the approaches for mitigating forgetting assume task-specific knowledge access to different extents; either (a subset of) the pre-traimm pre-training dataset itself or some information/statistie computed from pre-training data. Below, we describe the data-aware approaches based on how they make use of task-specific knowledge.
regularization-basee methods. This line of work aims to preserve performance on previously learned tasks by keeping the (fine-tuned) model parameters close to the pre-trained model. The key idea is to introduce task- regularization in the fine-tuning phase which will penalize updates along the "important" directions for the old tasks (Ahn et al., 2019). Kirkpatrick et al. (2016) introduces the elastic weight consolidation (EWC) algorithm, which estimates the important direction per-task by calculating a diagonal approximation to the Fisher information matrix (FIM), which acts as the weight matrix for the regularization term. Several variants of EWC have been subsequently proposed (Schwarz et al., 2018 8; Ritter et al., 2018; Lee et al., 2020; Liu al., 2018). Zenke al. (2017); Aljundi al. (2018) adopt online strategies infer the importance of each parameter by their variational effect on the model outputs. In spirit similar to EWC, Lee et al. (2017) incrementally matches the posterior of the pre-trained model and the new task by assuming Gaussian posteriors.
Optimization-driven methods. Another perspective to mitigating forgetting is guiding the optimization process by constraining the algorithms directly as opposed to manipulating the loss function. The core idea is to keep track of "important directions" for the old tasks, and train on the new task "orthogonally." This could be done by storing prior data samples or gradients in buffer (Lopez-Pat Paze & Ranzato, 2017; Farajtabar et al., 2020; Chaudhry et al., 2019a) or by incrementally expanding the subspace of important directions without storing task-specific information (Zeng et al., 2019; Wang et al., 2021; 2023b).
Replay-based methods. Drawing inspiration from the complementary learning systems theory (Mcclelland et al., 1995). a more direct approach is to introduce samples from old tasks into the training process for the new task. Samples are selected in a streaming fashion or by manually crafting a subset on demand, stored in dedicated buffers and replayed during the fine-tuning. The intuition is that the task specifie representations are refreshed periodically through historical data.
Replay-based methods consist of two fundamental components: data selection and data reiteration mechanisms. When the data is received in streaming fashion, information has to be buffered online (Riemer et al., 2019; Chaudhry et al., 2019b; Isele& & Cosgun, 2018; De Lange & Tuytelaars, 2021). In the case when datasets are available on demand, Rebuffi et al. (2017) selects samples which are "representative" of their respective class, while others focus on inducing diversity (Aljundi et al., 2019; Bang et al., 2021) and balance (Borsos et al., 2020; Tiwari et al., 2021) across buffered data. For the scenarios in which storage is limited, Caccia et al. (2020); Wang et al. (2022b) develop compression methods for buffered data. As a complimentary component to the data selection process, how the buffered data is replayed plays a significant role in the success of such methods. A fundamental idea, which has several interpretations across the board, is knowledge distillation (Hinton et al., 2015). Prior work argues that augmenting on the forgetting front (Lopez-Paz & Ranzato, 2017; Chaudhry et al., 201 9a; Rebuffi et al., 2017; Jung et al., 2017; Triki et al., 2017; Li & Hoiem, 2016; Lee et al., 2019b; Dhar et al., 2019).
with knowledge distillation shows great performance
An orthogonal research direction focuses on maintaining a generative model that could reliably output pseudo-samples that are representative of the dataset of the old tasks (Kemker & Kanan, 2018; Wu et al., 2018). Note that generative approaches are prone to scalability issues and distribution shifts.
Archivecture-driven methods. Another technique to limit the interference between tasks is allocating separate trainable set of parameters per task. This could be done by initializing sub-networks per new task (Rusu al., 2016; Aljundi et al., 2017; Collier et al., 2020; Rajasegaran et al., 2019; Ramesh & Chaudhari, 2021; Wang et al., 2023a; 2022a), gradually expanding the parameters of a base network (Yoon et al., 2018; Ostapenko et al., 2019; Hung et al., 2019), or segregating a fixed model into task-specific subset of parameters (Mallya et al., 2018; Kang et al., 2022; Serra et al., 2018; Wortsman et al., 2020; Mallya& & Lazebnik, 2017; Mustafa B Gurbuz, 2022; Jung et al., 2020). While some parameters are task-specific, parts of the overall model could be shared to enable knowledge transfer. The main downside associated is that the task identity must be available during inference (de)activate relevant sub-networks, hindering versatility. Aljundi al. (2017) develop dedicated strategies to overcome the need for task identification by automatizing task-specific parameter activation.
FLOW and its connection to other ML applications. Our approach has connections to factuality in LLM training (Ghosal et al., 2024; Gekhman et al., 2024). For a fine-tuning task on a factual knowledge-based dataset, samples that are not properly represented in the pre-training distribution could be considered hard. In the data-oblivious setting, one principled way would be to rank hardness based on pre-training loss values; therefore, FLOW would identify such samples and guide
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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
the fine-tuning process such that those samples will not unexpectedly deteriorate the performance.
Although fundamentally different in its objective, machine unlearning is another avenue of application for our sample weighting approach which could be interpreted as means of soft unlearning. While unlearning and forgetting sound similar, they are not the same in spirit; in unlearning we deliberately induce "forgetting" on some samples, whereas in our context, forgetting is an undesirable side effect that we want to avoid. Hence, it might not be straightforward extend techniques of one into another, one could extend our sample-wise weighting scheme with appropriate modifications to help selectively unlearn specific samples.
B. Our Algorithm in the Presence of Task-Specific Model Components
Suppose our model is parameterized by 0 = UUV, where U is the common/shared part of the model for all tasks (i.e., this part remains the same for all tasks), and V is the task-specific part of the model. In particular, in our vision experiments, the models have task-specific prediction heads (i.e., softmax layers) and batch-norm (BN). The modified version of Algorithm I in the presence of task dask-specific components is stated Algorithm 2. The main differences from Algorithm 1 are steps (i) worth mentioning that if
and (iv) - these steps optimize the task-specific part for the new task with uniform weighting. model consists of task-specific prediction heads which is the case in our vision experiments then steps (i) and (iv) just vanilla linear probing with the pre-trained body and the body learned after fine-tuning, respectively.
Algorithm 2 Fine-tuning with Pre-trained Loss- Cass-Oriented Weighting (FLOW)
Input: Pre-trainee model ( = u(") U v(1), dataset {(x,. H= for the new task, and temperature parameter T.
fi, V) 8th sample's loss at 0 U U V, with non-negative loss function (e.g., cross-entropy loss).
Step (i) Fine-tune task-specific part for new task with vanilla unweighted loss: (2) := argmin v >Σ f((U!) V).
Step (ii) Compute sample weights: w; exp (-f( v/T)
Step (iii) Fine-tune full model with weighted loss: U v!! == argmin u.v E wrJi(U,
Step (iv) Fine-tune task-specific part for new task using the learned common part with vanilla unweighted loss: v! := argmin Eis f( V).
Output: New model for
U v!!) New/fine-tunin task is 0(2)_ U 2) (2))
Original/pre-training task is 0 (2)
Remark B.1. In all our vision experiments (with task-specific parts), we set 7 (e( v(²))) (similar to Remark 4.2).
C. Proof of Proposition 4.3
Proof. We wish to minimize g(T) T. = n. log subject to Σ T. The proof is straightforward application of Lagrangian multipliers. is enough to enforce >i 1 only (T; 2 0 for all i E [n] will also follow). For that, the Lagrangian function is:
and n, for all i [m].
n -1).
((T = Σ Tlf((0)) +
Σ
कy log n. + A
Σ m 1=1
(19)
where A is the Lagrangian multiplier. Now, at the optimal point n:"
[TT [ ......... x
we must have:
aJ
=f(00) + + + log m3) + + 0,
(20)
0T; m
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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
for all E [n]. Simplifying, get:
1 exp
[i(0")
(21)
Z
where = exp ((1+ ?)) is the normalizing constant. To have E T5 = 1, we get = =E exp (-1:W))) Also,
note that good with the non-negativity constraints.
D. Proof of Theorem 7.2
Proof. Note that:
eñr 0 (0 0.)
E (xx (.)
(22)
Also, after plugging in y = (0., x),, we get:
((0, - -0., xx) x
w(X,3) ÿ) exp
Recall c: = 0. - 0, and c: Tellz
Suppose T = al|e|12. for some a > O. Then wxx,)) cxp (- [(,,))
and we can
focus on
Σ' : RxN exp (- ( x
((e,
xX
(23)
a
Let (142)) 1/2 = = (7+2|eell
1/2
As per Lemma F.1, we have:
'' = ( QQ,
(24)
where
Q
(1 n² eee
+ + 12 je ह pn² (εe +ē
(25)
So if minimize ( with GD starting from O = 0. and using constant learning rate î7. iterate 0x at the K iteration satisfies:
0 (Is-2 ---2"" (0. 0.) =(1+ 21j2"" e,
(26)
where the last step follows by recalling that 0, 0, e, and E' is given by Equation (24).
E. Difficulty in the Analysis with a General Covariance Matrix Σ
We will first derive the weighted (fine-tuning) data covariance matrix Σ' in the context of Theorem 7.2 for a general (fine-tuning) data covariance matrix Σ. Specifically, following the proof of Theorem 7.2, have:
((e, ((e,x))2
E E ( exp
XX
(27)
T
Note that x 5/1/22, where z Nū Iz). Using this above, we get:
((e, ((e,))))))) )1/2)))
(S52/22,2) 2))²
= $1/2 exp
zz 11/2 51/27
exp
2z
51/2
(28)
where the last step follows by using the symmetry of Σ. Let ai In that case, we have:
m||/1/20|13 for some > 0. Also, let =
((r, 2)) 2
- 13 where M : E exp
zz
(29)
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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
Suppose (rig 1= is orthonormal basis for the subspace of IRd orthogonal r; (Fi.). ( [d and (T_.j. T1,k) = [ = k) j, E [d - 1]. Note that (r,r..1...., ri,d-1) forms an orthonormal basis for R Then, as per Lemma F.5, we have that r is an eigenvector of M with eigenvalue (xt2) 3/2 and each FL, is an eigenvector of M with
eigenvalue (a+2) 1/2 For brevity, let μ -aat2) 1/2 Then, we can write:
M M prrr' +
= =
+ (Ta-rr").
(30)
where the last step follows because (r,r (id- if forms an orthonormal basis for Rd, due to which rr T +
Li.rr La. Simplifying Equation (30) a bit, we get:
M / (I. -1-1111p (1-12)rrT)
(31)
Plugging this into Equation (29) and recalling that = (/1/20//1111 //////////// we get:
Eec E
Σ' =B= where B
Σ 5 (1-1144 112
(32)
Ee
Equation (32) is the weighted covariance matrix for general Σ.
Remark E.1 (Difficulty with general E). is hard proceed with the analysis after this point because it is difficult characterize the eigen-spectrum of B in general, without assuming any relation between E and This is what we meant in Remark 7.1.
F. Lemmas Used and Their Proofs
Lemma F.1. the proof of Theorem 7.2, recall that T al|el12. Then, we have:
(),
'' R xNp
exp
xx
= 11 Ia (1 1e T - p² - + (eo] + [ tele)))
Cx
where μ = (122) 1/2
1/2
Proof. Recall that e and t are orthogonal to each other and both are unit-norm. Suppose for,3, CLA:.... CLLd:...,e_.ll {Pيية is an orthonormal basis for the (d - ddddiemnsaa subspace Rd orthogonal to e and e Thus, (e, E1,3 1. B1,1 {p,يع an orthonormal basis for d Then using Lemma F.2, we can write:
d
x = z (pz) + V1- p22 El + Σ 2erss
(33)
/=3
where (2,1-11 AN 1).
Using independence and zero-mean nature of {2,]d [ we get:
27 a
z
S'
exp
2
cc
+
exp
Zi
pzz
/1 p²zz)
[oc] + T
T=T:
T=
2 pz2
(- z
+
E
exp
p7z + v - p22
l + Σ exp
E
2
el. (34)
a
a
3=3
T=
T=44
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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
Note that (we the independence of 21 and 22):
T2 pTi V
2216
exp
[| pTı,
(35)
and
zi
T3 = p²T, 2pv p²
ExP
21
E Z2 +(1
p² TT
1 ppT + (1 (1- T²)
(36)
0
In the above two equations, we have again used the independence 2ι and 7:. Now we will compute Ti and T4. We have:
( 2))
3/2
T1
zxexp
(37)
2m
a + 2
and
1
) )
((a a
1/2
T4
22 -00
exp
2
dzi
(38)
Recall that / 100=((0))) (a42) ) 1/2 Plugging this into Equations (35) to (38) gives us:
Ti = n To - ph² Ta = p²pr² + (1 p²) and T4 = μ.
(39)
Plugging this into Equation (34) gives us:
d '' = preee T + (Ee ( +ē + (p²p² + +1 (1- (1) 3 e + E
(40)
j=3
Recall that (č, εT, G1,3 Ci...... tpTd} is an orthonormal basis for Rd Thus, Eソ EloeT = - cc this above, we get:
- Using
Σ' = 11 Ia Ia- (1 p²) - (2)e + (eel +e el))
(ee
(41)
This finishes the proof.
Lemma F22 Suppose {,,01, 01,3, 11,4..... EL,a} 3 is orthonormal basis for Rdd If ~ NiOo, E), then we can write:
d
210 (pz) v1 -1---12)00 Σ ZoLss
(42)
j=3
where (7,)4=1 NO, 1).
Proof. I is as per Equation (42), then clearly x is a zero-mean Gaussian. All that remains to show is that
E
XX
S
L p ee +
Using independence and zero-mean mean nature {2 we get:
XX =
ce
+
E Zi
pz)
+ vI - p1=p222)
)
(ce) +clo+)+ + +B (pr +
pz=p2z) 2
113
(A)
el.e T.
(43)
1=
23

---

Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
Note that (we the independence of 2 and 22):
(A) - p p
Elz
E
[22]
P
(44)
and
B = p" 1
+2p
p²
E
21 E
22 0
+(1
p²)
22
(45)
=0
Plugging this into Equation (43), we get:
E
ee T + p(eee] +ele)) Ialal Σ Geizels
(46)
Recall that (e, 61,3 ...4..... (i,a is an orthonormal basis for Rd Thus, Σj=3 Bi = I1 e6 this above, we get:
- e Using
E XX
Ia + p +ale = Σ.
(47)
This finishes the proof.
Lemma F.3. Recall that
Q
p²2 ec (()_))))__ pp² c3 t
Let
3(1- - p²)
11 (1+ B)(1 - /p22)
for some 3 E (0,1), 1]. In that case, the eigenvalues of Q are:
R 1+ 3p3
1 3 and X2 =p² / 1-B (1-3
11
and the corresponding eigenvectors are:
3p
3p
11+ 6
σ and 2 =
c1
V B2p²
V 82p²
52p
Proof. Q is a rank-2 2 matrix and its two eigenvectors will be in the span of é and e In particular, an eigenvector of Q is of the form @e @ee_bb bb where E 2 is an cigenvector of the 2 x 2 matrix:
n²)
pp2 p²1 (11)+2)
A
(48)
Also, the corresponding eigenvalues of Q are the corresponding eigenvalues of A. It can be verified that the eigenvalues of A are:
R (1 p11
(1 p²) - 22)
+
0²
(49)
and
2 (1 (1+p²)(11 122)
(1 2) 22)2
(50)
The corresponding eigenvectors of A are:
b
6
(51)
62 + b7.2
b1.2
24

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
where b1,1 (1-p")(1 (1-p")(1_=") + (1 (--2))(1-- p² and 21,2 pp-, and
12,
22
(52)
(b2 + 1/
62
where 12,1 (1--²)(1-1) (1-)))
+ pd and 12, =pp?. Thus, the eigenvalues of Q are X₁ and X2; the corresponding eigenvectors Vi fe, e. [b and V== ee, bb2. Note that:
(1 (1-p2)a(1 2
(1 - p²) -2)
p²))(1 n2²² + p < (1 (1-p²) - 122) + pp²
2
2
Let us set / [|= 1 + p²px+
(1-
(1-
+ Bpp²x2, u,,,K for some 3 E (0, 1]. That gives us:
2
3(1 - p2)
p V (1 + 3)(1 - -Bp2)
(53)
In that case, we have:
and π22 =p² / 1-
2, 1+ 3p² +
(54)
1- Bp2
Also,
1-p² b1,1 (1+3)(
3p(1 - p²) and b
(² - p2)
b1.2 62,2
(55)
3p2)
(1+ B1 Bp²)²
(1 3) BP²)
Therefore,
6,
-8p] and b2 VI+ -1
1
Bp
(56)
VI+ 82p²
Recall that the eigenvalues of Q are XS and Ê2, and the corresponding eigenvectors are
3p ē and v22 |,,
Bp
1
v b
ē
E1.
VI+ 8
VI 1 8
VI+ 52p
VI
+
32 pz
Finally, recall that
8(1-2))
[1+0)[1-3p-]
Lemma F4. Recall that the averaged model with parameter ws as defined in Equation (16) was
Oavg (w) = wO, + (1 - -w)0 0, we.
We have:
ē Ee é Σe +
min erf (@ag(w)) wE(0,1]
/1el|2.
(57)
where recall that E is the covariance matrix of the pre-training data.
Proof. We have:
erT (Omgi (w)) eIt, (0mg(w)) + eerzz (Oang (w)) (0ang (w) - 0.) E(Omn(() -m.)
Swn (w)
-0.).
(58)
Plugging in the value of (avg(w) and using the value of Σ from Equation (3) above, we get:
em (w)( (w)) = (1 we T Ee + w
(59)
It can be verified (with elementary calculus) that the optimal value of w that minimizes the RHS in Equation (59) is w e De Plugging this into Equation (59) and simplifying bit yields the desired result.
2e+llel?
25

---

Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
Lemma F.5. Suppose a 0 and r Rd is a unit-norm vector, ie., ||r||2 1. Let
((r, 2
: :
2Z
where u ~ N( Ia). is an eigenvector of M with eigenvalue of TRd orthogonal to all have eigenvalues (++2 )
)/12. a+?
Further, the eigenvectors of M in the subspace
Proof. We have:
((r, 23) 2
Mr
E
exp
(r. z)/
(60)
Suppose (ri,) 1=1 is an orthonormal basis for the subspace orthogonal to r; so (rig,r) = 0 V E [d 1] and (r jr... r1,A) = 1(j= = =k) [d 1]. Then, note that:
z = (r, 2) +
z)r
(61)
1=1
Since Nūss Ia), (r, 2) and ((ragj 2) 2 are i.idd. (N, 1). Using all of this in Equation (60), we get:
exp (- ((r,2))²
d
exp (- ((r,z))
E
Mr
E
(tr, 2))
r
E
(r. 2) (r 2)
rij
(62)
α
a
=0 ((r, 2) and (ra. 2) are independent)
- 72 72 r
Ez~A((011
exp
(because (r, ~ 1)),
(63)
exp (-2² (- + )
dz
(64)
2m
2
-00
3/2
r.
(65)
0 +
(=42) 3/2
So r is an eigenvector of M with eigenvalue
Next, note that:
(r, 2
exp (
2(x)
((I)) zi)) FL.1
Mr =
exp
( 2) (r, z))
r+ E
a
=0
d
((r, 2)) a
exp
(r (r_,1 z) (rig 2)
ri.. (66)
In the above equation, the first and last terms are because (r, z) and ((ri,,, 2) are i.i.d. N/00 1): using this fact again,
26

---

Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
get:
(-2-)
2 --
I
Mr
111
exp (0..1)
R (22)
(67)
=1
00
1
1
exp
2
+
dz
F1,l
(68)
2
2
a
1/2
r1,1-
(69)
+ +
Similarly, we can show that for j = (2....................
1}, we have:
E Mr
a
1/2
Fi.j-
(70)
+ 2
So for all j E [d - 1J. ri.j is an eigenvector of M with eigenvalue orthogonal to all have eigenvalues a++2
(a+2)
1/2
Thus, the eigenvectors of M in the subspace
1/2
G. Experimental Details
In this section, we further discuss the experimental setup of FLOW's usage in both language and vision settings, specifically covering the following:
Appendix G.1: Baseline Details.
Appendix G.2: Language Model Hapar-Pasameters.
Appendix G.3: Language Model Evaluation Details.
Appendix G.4: Vision Model Implementation Details.
G.1. Baseline Details
In this section, we further discuss the baselines mentioned in Section 5.
Linear Probing: In our vision experiments, we define linear probing as freezing the body of the pre-trained model, initializing new (task-specific) head and batch normalization layers, and training only the new head and batch normalization layers.
le regularization: Based on Kirkpatrick et al. (2016), we perform la regularization as bascline in the data-oblivious setting. Specifically, the fe-reuularized loss is:
(()) Σ f((0) 1 0-
(71)
where fs is the 1hh sample's loss, 0" is the pre-trained model, and À is the regularization parameter. Intuitively, as À increases, our model stays closer to the pre-trained model, mitigating forgetting at the expense of target domain performance.
LoRA (Hu et al., 2022): Recently, Biderman et al. (2024) showed that fine-tuning language models with LoRA (Hu et al., 2022) effectively mitigates forgetting. Following similar setup as us, Biderman et al. (2024) fine-tuned language models on MetaMathQA (Yu et al., 2023) and then evaluated the fine-tuned model on several general capability tasks, viz., HellaSwag (Zellers et al., 2019), ARC-c (Clark et al., 2018), and WinoGrande (Sakaguchi et al., 2019), and one target domain task, viz., GSM8K (Cobbe al., 2021). Further details about experimental hyper-parameters can be found Appendix G.2.
27

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
WiSE-FT (Wortsman et al., 2021): We also consider model averaging as a baseline, specifically focusing on WiSE-FT (Wortsman al., 2021). WiSE-FT'ss is simply the convex combination of the model parameters shared between the tasks, while the task-specific parts are not averaged. Specifically, we perform model averaging between the pre-trained model and the fine-tuned model. The convex combination parameter a of WiSE-FT is set to 0.5 in our experiments, as we cannot optimize a in the data-oblivious setting.
Learning without Forgetting (Li & Hoiem, 2016): This work considers using distillation- based loss to mitigate forgetting when the data from the training of previous tasks is not available. Initially, they record the responses yo on the new task images with the old tasks' model parameters, then train the model using a combination of fine-tuning loss, distillation loss, and model regularization. For new tasks, they use standard cross-entropy loss, while for old tasks they employ distillation loss that encourages the updated model's responses to match the recorded responses Yo of the original model. Their proposed method has a distillation loss scaling factor X and a temperature parameter T in the distillation loss, introducing extra tunable parameters. Their loss induces joint optimization of shared parameters 0,, old task parameters O.. and new task parameters 0, using only new task data. Note that unlike FLOW, they also update the old task parameters 0,.
G.2. Language Model Hyper- ParaaeePasss
For both Gemma 2 2B (Team al., 2024) and Llama 3.2 3B (Grattafiori al., 2024), we run hyper-parameter sweeps on learning rates for each baseline. For standard fine-tuning, lg regularization, and FLOW, do learning rate sweep in [lc-4, 2e-5, 1e-5, 5e-6], and for LoRA (r = 64) we do a sweep in [2e-4, 2e-1], following the learning rates used in (Biderman et al., 2024). We then select the learning rate that results in the best GSM8K (Cobbe et al., 2021) accuracy, oblivious to general capability metrics. We report the hyper-parameters used for our Gemma 2B (Team et al., 2024) experiments in Table 6 and for Llama 3.2 3B (Grattafiori et al., 2024) in Table 7.
The hyper--ppaaaeee parameerretterss used to train Gemma 2 2B in our experiments. Note that the learning rate selected is based on the best results on GSM8K after mee-tuning the method MetaMathQA.
Table 6.
Hyper-parameter
Standard Fine-tuning
LoRA (r = 64)
R2-Reg.
FLOW (Ours)
Learning Rate
le-5
2e-4
5e-6
5e-6
Learning Rate Scheduler
Cosine
Batch Size
128
Optimizer
AdamW
Weight Decay
0.00
Warmup Ratio
0.03
Epochs
2
Max Sequence Length
1024
Seed
42
Table The hyper- parameters used to train Llama 3.2 in our experiments. Note that the learning rate selected based on the best results on GSM8K after fine-tunin fine-tuning the method on MetaMathQA.
Hyper-parameter
Standard Fine-tuning
LoRA (r 64)
fg-Reg.
FLOW (Ours)
Learning Rate
2e-5
2e-4
le-5
le-5
Learning Rate Scheduler
Cosine
Batch Size
128
Optimizer
AdamW
Weight Decay
0.00
Warmup Ratio
0,03
Epochs
Max Sequence Length
1024
Seed
42
28

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
For our WiSE-FT (Wortsman al., 2021) model averaging experiments, we use a 0.5. For our LoRA (Hu al., 2022) experiments,
64. For & regularization use À le which is taken from (Chen al., 2024b). Most training hyper-parameters for our language experiments are taken from Chen et al. (2024b), with the introduction of learning rate sweeps.
G.3. Language Model Evaluation Details
As described in Section 5.2, we create commonsense reasoning metric composed of the following six metrics: ARC-e (Clark et al., 2018), ARC-c (Clark et al., 2018), HellaSwag (Zellers et al., 2019), PIQA (Bisk et al., 2020), SIQA (Sap et al., 2019), and OBQA (Mihaylov et al., 2018). On top of the commonsense metric, we evaluate MMLU (Hendrycks et al., 2021a) and MBPP (Austin et al., 2021) to estimate the general capabilities of a language model and to measure the effects of catastrophic forgetting when fine-tuning a model on MetaMathQA (Yu et al., 2023). We additionally use GSM8K (Cobbe al., 2021) to evaluate the target fine-tuning performance of a given fine-tuning method. We provide a brief describe each of these evaluation metrics:
1. HellaSwag (Zellers et al., 2019): A benchmark designed test commonsense reasoning. HellaSwag presents context followed by several plausible endings, and the model must choose the most appropriate continuation.
2. ARC Easy (Clark et al., 2018): : A benchmark part of the A12 reasoning challenge designed to test basic scientific reasoning and knowledge. ARC Easy presents 5,197 multiple-choice science questions drawn from grade 3-9 standardized tests, where each question typically includes a brief scientific scenario or statement followed by four possible answer choices.
3. ARC Challenge (Clark al., 2018): A benchmark part of the A12 reasoning challenge designed test advanced scientific reasoning and knowledge application. ARC Challenge presents 2,590 multiple-choice science questions drawn from grade 3-9 standardized tests, where each question typically includes a scientific scenario or phenomenon followed by four possible answer choices. The questions in ARC Challenge are significantly more challenging than ARC Easy.
4. PIQA (Bisk et al., 2020): benchmark designed to evaluate physical commonsense understanding in natural language. PIQA presents goal and two possible solutions, requiring models choose the most appropriate solution that demonstrates understanding of everyday physical interactions.
5. SIQA (Sap et al., 2019): A benchmark designed to evaluate social commonsense intelligence and emotional reasoning. SIQA presents a social situation context followed by question and three possible answers, requiring models to demonstrate an understanding of social interactions, emotional responses, and behavioral implications.
6. Open Book QA (Mihaylov et al., 2018): A benchmark designed to assess understanding of elementary science concepts in an open-book exam format. OBQA presents 5,957 multiple-choice questions paired with a small "book" of 1,326 science facts, requiring models to combine these facts with common knowledge to arrive correct answers.
7. MMLU (Hendrycks et al., 2021a): A benchmark designed to evaluate massive multitask language understanding. MMLU presents approximately 16,000 multiple-choice questions spanning 57 subjects including mathematics, philoso- phy, law, and medicine, requiring models to demonstrate broad knowledge and reasoning capabilities.
8. MBPP (Austin et al., 2021): benchmark designed evaluate basic Python programming capabilities. The entire MBPP dataset presents 974 Python programming problems, where each problem includes a natural language task description and three test cases written as assert statements, requiring models generate functionally correct Python code solutions.
9. GSMSK (Cobbe et al.. 2021): A A benchmark designed to evaluate multi-step mathematical reasoning capabilities. The GSMSK test set contains 1,000 grade school math word problems, where each problem requires 2-8 steps to solve using basic arithmetic operations (addition, subtraction, multiplication, division).
We follow the standard evaluation process for each of these datasets and specifically use Im-evaluat - (Gao et al., 2024) to evaluate our experiments.
29

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
G.4. Vision Model Implementation Details
We performed an extensive hyper-parameter search over six learning rates (Irs = [0.05, 0.01, 0.005, 0.001, 0.0005, 0.0001].. two models, and six datasets (i.e., 72 total rums per method) for standard fine-tuning, linear probing, and FLOW. We chose over all the target (fine-tuning) datasets. Since our method
the best learning rates associated with the highest average is data oblivious, we do not use the validation set of ImageNet-1K IK other than for evaluation. For training models with fz-regularization, we adapted the same learning rates and other related hyperparameters used for standard fine-tuning. We searched for À using one dataset and ResNet50 model (A = [0.002, 0.00001, 0.00002) and chose 0.002 based on average accuracy over target data. We chose (a 0.05) for WiSE-FT following Wortsman et al. (2021). We present all the important training details in Table for the ResNet models.
Table 8. Hyperparameter configurations for finetuning ResNet18 and ResNet50 on the image classification datasets.
Model
Hyperparameters
CIFARIO
CIFARI00
Flowers102
Caltech101
Dogs
Cars
# GPUs
1 A6000
Optimizer
SGD
LR Schedule
Cosine (except for Linear probing)
Weight Decay
0.0005
Seed
42
À for lg-Reg.
0.002
for WiSE-FT.
0.05
7 (temperature) for FLOW
median loss
Epochs
20
25
25
30
30
30
ResNer50 18
LR-Standard fine-tuning
5E-3
IB-2
5E-2
5E-3
1E-3
5E-2
LR-Linear probing
5E-3
5E-3
5E-2
1B-2
5B-3
5E-2
LR-FLOW
1E-3
5E-3
5E-2
1E-2
5B-3
1E-2
LR-Standard fine-tuning
5E-3
1E-3
IE-2
5E-3
5E-4
5E-2
LR-Linear probing
5E-2
5E-2
5E-2
5E-2
IE-2
5E-2
LR-FLOW
5E-4
1B-3
1E-2
1E-2
5E-3
1F-2
Hyperparameter Details for VIT-B/16. We fine-tuned for epochs Food-101. We used learning rate of 5e-5 for standard fine-tuning, fg-regularization (with À = 0.002), and our method FLOW. For linear probing, we used higher learning rate of le-3. For LwF, we used the same learning rate as standard fine-tuning (5e-5), 7 = 2, and Åo = 5e-4, Here we used a higher value of T for FLOW and set it equal to 80th percentile of the pre-trained loss values.
Datasets
1. ImageNet-1K (Russakovsky et al., 2015) serves as the pre-training dataset for all our vision base models. It is widely used large-scale image classification dataset, consisting of over million images spanning 1000 classes.
2. CIFAR-10 (Krizhevsky, 2009) is widely used dataset for image classification tasks. It consists of 60,000 32x32 color images divided into ten classes, with 6,000 images per class.
3. CIFAR-100 (Krizhevsky, 2009) extends CIFAR-10 by providing 100 classes containing 600 images each. This dataset is used for ine-grained image classification tasks.
4. Caltech101 (Li et al., 2022) comprises images of diverse range of objects across 101 categories with diverse set of image classes.
5. Flowers100 (Nilsback & Zisserman, 2008) comprises 102 categories of flowers, with each category containing between 40 to 258 images. This dataset is commonly used for fine-grained image classification and flower recognition tasks.
30

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
6. Cars (Krause al., 2013) refers the Stanford Cars dataset, which includes 16,185 images of 196 classes of cars. provides rich resource for fine-grained car classification task.
7. Dogs (Parkhi et al., 2012) pertains to the Stanford Dogs dataset, containing 20,580 images of 120 breeds of dogs. This dataset is widely used for fine-grained dog breed classification and recognition tasks.
8.
Food101 (Bossard et al., 2014) is large-scale dataset for food classification containing 101 categories with 1,000 images per class, commonly used to evaluate models on fine-grained object recognition tasks.
31

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
H. Detailed Vision Results and Ablations
First, we present the detailed version of the results Table in Tables 9 and 10.
Table 9, ResNets: Target accuracies on each of the six datasets for the results in Table 1.
Method
CIFAR-10
CIFAR-100
Flowers-100 102
Caltech-101
Dogs
Cars
Average
ResNet18
Linear probing
81.32
60.06
87.20
91.15
78.50
43.23
73.57
Standard FT
96.15
83.42
92.45
94.02
80.47
87.91
89.07
Pg-Regularization
95.53
81.82
92.11
94.23
80.27
84.78
88.12
WiSE-FT
91.47
65.90
87.28
91.40
82.48
62.88
80.23
FLOW (Ours)
88.25
78.95
90.01
93.05
86.20
67.17
83,93
ResNet50
Linear prohing
86.62
67.80
83.64
93.45
85.76
41.97
76.45
Standard FT
97.61
86.11
91.74
96.02
89.26
89.94
91.78
2--Reuularizatonn
97.50
85.77
91.67
95.85
89.29
89.42
91.58
WiSE-FT
94.65
72.55
71.95
93.73
92.52
62.89
81.38
FLOW (Ours)
91.11
79.42
86.78
94.45
91.16
74.59
86.25
Table 10. ResNets: Top-1 ImageNet-11K accuracy after fine-tuning each target dataset for the results in Table 1.
Method
CIFAR-10
CIFAR-100
Flowers-
Caltech-101
Dogs
Cars
Average
ResNet18
Linear probing Standard FT
69.76
69.76
69.76
69.76
69.76
69.76
69.76
19.93
0.39
6.48
34.17
56.38
0.17
19.58
22 12-Regularization
37.86
29.86
19.34
46.67
58.34
16.64
34.78
WiSE-FT FLOW (Ours)
62.24
47.65
49.98
64.70
67.34
33.03
54.15
69.02
52.64
67.80
68.32
67.78
65.74
65.21
Linear probing
79.02
79.02
79.02
79.02
79.02
79.02
79.02
ResNer50
Standard FT
16.89
35.95
61.01
40.51
66.93
0.21
36.91
Eg S--Reullrizationn
33.98
47.16
62.85
43.42
67.03
14.27
44.78
WiseFT (a = 0.5)
61.40
73.04
76.33
73.25
77.36
8.55
61.65
FLOW (Ours)
78.26
75.13
78.60
73.38
78.55
72.64
76.09
In Table 11, we present small ablation to compare the pre-training and fine-tuning performances with different values of in FLOW; recall that we prescribed selecting 7 to be the median pre-training loss value. As we see, if we could tune т, then FLOW's results would be even better.
Table 11. FLOW ablation with different values of Pre-training and fine-tuning accuracies as function of set to different percentiles the pre-training losses. The model is ResNet-50, pre- pre-training datasct is ImageNet- (IN-IK) IK), and the fine-tuning dataset is Caltech101 (Target). So if we could tune r, then FLOW's results would further improve.
IN-1K Accuracy
Target Accuracy
Average
Percentile (%
68.51
91.15
79,83
10
64.13
91.01
77.57
30
54.72
91.80
73.26
50
45.59
92.91
69.25
70
20.51
94.02
57.27
90
Further, in Figure 2, we present another ablation study where we compare FLOW with different values of r, WiSE-FT with different values of the convex combination parameter, and random selection where we train on a random subset the fine-tuning data to limit the drift from the pre-trained model.
H.1. Comparison with Distillation-Base Method for Mitigating Forgetting
Here, we compare our method FLOW against a distillation based method for mitigating forgetting called "learning without forgetting" (LwF) (Li & Hoiem, 2016) in vision. More details about this method can be found in Appendix G. but
32

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
mmgennee 1k Top Accuracy (96)
70
60
Pretrained model Jeses IN-1K accuracy when finetuned
Pretrained
Full Finetuned
Random Selection
30
FLOW (Ours)
20
10
Finetune Accuracy (%)
Figure 2. Comparison of FLOW with different values of r and some other baselines with different hyper-parameter values. This plot is for ResNet-50 the Stanford cars dataset. FLOW's plot (in red) is with {100 20, 30, 40, 50) percentile of the per-sample losses. As the name "random selection" implies, we just pick random subset the fine-tuning data and train on this subset to limit the drift from the pre-trained model. To have correspondence with our choice of for FLOW, we pick random (10, 20, 30, 40, 50) of the data in "random selection". As we see, FLOW significantly outperforms other methods.
the important thing to note is that it updates the head corresponding to the pre-training data (which FLOW does not do) and also comes with more tunable parameters, additional memory, and evaluation overhead compared to FLOW. We consider the ViT-B/16 (Dosovitskiy et al., 2021) model pre-trained on Imagenet- (IN-1K) and single large fine-tuning dataset, Food101 (Bossard al., 2014). The evaluation metric is the same as Section 5.1. Hlyper-paaamete details are in Appendix G.4.
In Table 12, we list the accuracies of the pre-trained model, standard FT, linear probing, fz-regularization, LwF, and FLOW. Note that FLOW outperforms LwF despite its simplicity. Specifically, FLOW achieves better performance on the forgetting front, while LwF does better on the fine-tuning task.
Table Comparison with distillation- method (LwF)) Li Hoiem (2016). Bolded underlined values indicate the best and second-hest accuracies within each column (and for each model). Deltas (in color) for IN-1K and target performance are computed w.r.t. the pre-trained and standard fine-tuned models. Note that FLOW outperforms despite not updating head the pre-training data, unlike LwF, and being more efficient than LwF, which comes with more tunable parameters, additional memory, and evaluation overhead.
Method
IN-1K Accuracy
Target Accuracy
Average
Pre-trained
81.10
(+0.00)
VII-B/16
Standard FT
56.11
(22999
91.60
(+0.00)
73.86
Linear Probe
81.10
(40.0)
83.86
(-7.74)
82.48
l f2-Reg.
59.18
(21.92)
91.66
(40.06)
75.42
LwF
76.39
(4471)
91.23
83.81
FLOW (Ours)
77.94
(3.16)
90.57
(-1.3)
84.26
I. Additional Language Model Results and Ablations
In this section, we discuss expanded results and further ablations of FLOW within our language experiments, specifically covering the following:
Appendix I.1: An expanded table of results on commonsense reasoning tasks along with other baselines.
Appendix 1.2: An additional ablation on loken-wise weighting scheme for fine-tuning with language data.
Appendix L.3: An expanded set of plots and results for the combination of FLOW with weight averaging techniques such as Wise-FT (Wortsman al., 2021).
33

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
I.1. Extended Commonsense Reasoning Results
As discussed in Section 5.2 and Appendix G.3, we evaluate FLOW and the other baselines various commonsense reasoning tasks within fine-tuning with the procedure described in Section 5.2. We include the exact results of these evaluation metries for various baselines and FLOW in Table 13. We also include the results of commonsense reasoning metrics for the ablation combining FLOW with LoRA and e ez-regularization in Table 14.
Table 13, Extended commonsense reasoning metrics for FLOW and other baselines within language modeling. The performance on commonsense reasoning evaluations when tun-ttning Gemma 2B (Team al., 2024) and Llama 3.2 3B (Grattafiori et al., 2024) MetaMathQA (Yu et al., 2023). We include the target domain evaluation GSM8K (Cobbe et al., 2021) for convenience. The results show that FLOW effectively mitigate catastrophic forgetting while still getting strong performance
target fine-tuning task.
Method
ARC-e
ARC-c
HellaSwag
PIQA
SIQA
OBQA
Average
GSM8K
Pre-trained
80.18
46.84
54.95
78.67
51.33
31.40
57.23
24.49
Gemma 2 2B
Standard Fine-tuning
76.09
42.07
54.41
76.99
48.06
32.00
55.07
63.38
WiSE-FT
79.55
46.42
56.43
78.24
51.08
32.00
57.28
53.30
LoRA = 64)
77.78
44.37
54.59
76.99
50.51
29.80
55.67
60.43
2 2--Reuularization
79.08
45.99
56.21
77.20
50.97
32.60
57.01
62.85
FLOW (Ours)
79.76
47.18
56.23
77.69
51.48
33.20
57.59
62.55
Pre-trained
74.54
42.15
55.31
76.66
47.03
31.20
54.48
26.01
Llama 3.2 3B
Standard Fine-tuning
70.03
34.22
52.02
74.16
45.24
28.40
50.68
66.95
WiSE-FT
75.63
40.79
55.18
76.93
47.34
31.40
54.54
57.01
LoRA (r 64)
71.38
37.88
55.01
76.55
47.39
30.40
53.10
63.84
f2-Regularization
73.57
38.91
54.939
76.12
47.24
30.80
53.60
66.87
FLOW (Ours)
74.96
39.68
55.39
76.01
47.80
32.00
54.30
65.58
Table 14. Extended commonsense reasoning metrics for combining FLOW with other baselines. The performance on commonsense reasoning ovaluations when fine-tuning Gemma 2B (Team ct al., 2024) baselines in conjunction with FLOW on MetaMathQA (Yu al., 2023). We include the larget domain evaluation GSM8K (Cobbe et al., 2021) convenience. The results show FLOW can
effectively be used in conjunction with other methods that mitigate catastrophic forgetting.
Method
ARC-e
ARC-e
HellaSwag
PIQA
SIQA
OBQA
Average
GSM8K
LoRA (r 64)
77.78
44.37
54.59
76.99
50.51
29.80
55.67
60.43
LoRA (r 64) FLOW
79.50
45.39
55.27
77.31
51.18
31.80
56.74
61.49
bg- Jy-Regularization
79.08
45.99
56.21
77.20
50.97
32.60
57.01
62.85
2- Uz-Regularization + FLOW
79.67
47.10
56.38
77.48
51.13
33.40
57.53
62.02
Table 13 shows clear trend that FLOW, can strongly mitigate catastrophic forgetting in comparison to standard fine-tuning, For Gemma 2B (Team et al., 2024), we can see that FLOW only has 0.8% reduction the performance of the target fine-tuning while average maintaining the commonsense reasoning abilities of the pre-trained model, a 2.52% increase see that FLOW again maintain the
standard fine-tuning. For Llama 3.2 3B (Grattafiori al., 2024), commonsense reasoning abilities of the base pre-trained model while only having 11.4% drop on target fine-tuning performance. Overall, FLOW strikes a strong balance between general capabilities and target fine-tuning performance compared to other baselines.
For experiments with Gemma 2B (Team et al., 2024), FLOW can on average maintain the best scores on commonsense reasoning tasks. Performing only 0.8% and 0.3% worse on GSM8K (Cobbe al., 2021) in comparison standard fine-tuning and f2 regularization, FLOW can improve on commonsense reasoning metries by 2.42% and 0.58% respectively. Interestingly, in our Llama 3.2 3B (Grattafiori et al., 2024) experiments, we found that WiSE-FT (Wortsman et al., 2021) performed the strongest in preventing catastrophic forgetting of commonsense capabilities (+0..04 over the pre-trained model); however, this came at the cost of significant decrease in GSM8K (Cobbe et al., 2021) accuracy (-9.94 under standard fine-tu tuninnn In comparison, FLOW effectively mitigated forgetting in commonsense reasoning metrics (-0.18 under the pre-trained model), while achieving significantly higher accuracy in GSM8K (Cobbe et al., 2021) (-1.37 under standard fine-tuning).
34

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
1.2. Token-wise Sample Weighting Ablations
In the language experiments, "sample" for FLOW can be defined as an entire sequence or an individual token. The experiments in the main paper treat sequence sample; in that case, the per-sample loss is the average loss over the tokens in the sequence. We call this sequence-wise re- re-weighting. Instead, one could treat token case the per-sample loss is just the token's 's loss. We call this token-wise re-weighting. We run small ablation on both sequence-wise and loken-wise re-weighting by following a similar experimental setup as Section 5.2. We train Gemma 2 2B (Team et al., 2024) on MetaMathQA (Yu et al., 2023) and evaluate it on several general capability and target domain evaluations. The results of this experiment are in Table 15.
sample in which
Table 15. The performance of Gemma 2B on general capabilities metrics compared to target domain performance (GSM8K) when training MetaMathQA, Pre-trained is the base model performance of Gemma 2 2B, Standard is the performance after full end-to-end fine-tuning, Sequence is our sequence sample weighting schema with FLOW, and Token is our token sample weighting schema with FLOW. Bold and underlined values indicate the best and second best results respectively within each evaluation metrie.
Method
ARC-e
ARC-e
HellaSwag
PIQA
SIQA
OBQA
MMLU
MBPP
GSM8K
Base
80.18
46.84
54.95
78.67
51.33
31.40
49.59
28.40
24.49
Standard
76.09
42.07
54.41
76.99
48.06
32.80
45.59
16.80
63.38
Sequence
79.76
47.18
56.23
77.69
51.48
33.20
49.31
26.80
62.55
Token
79.38
45.90
53.95
78.29
51.28
31.80
48.75
22.00
23.73
Histogrem Sequercee Weights
Hislagram Toben Weights
14000
17020
2.0
1.5
0000
0000
1.4
2000
0.0
0.5
Figure 3. Histograms comparing the sample-wise distribution of weights in sequence-wise re-weighting schema for FLOW and token-wise distribution of weights token-wise ro-weighting schema for FLOW. The sequence-wise weight distribution is given on the left, while the token-wise weight distribution is given the right.
While token-wise sample re-weighting performs comparably or slightly worse than sequence-wise sample re-weighting in terms of the catastrophic forgetting of general capabilities Gemma 2B, struggles to effectively learn the fine-tuning target domain of GSMSK. To further understand this problem, we compare the weight distributions between sequence-wise and moken-wise re- wewihihtting schema in Figure We can see that the sequence weights appear Gaussian, while most the 1. We speculate that token-wiss re-weighting will force any token not commonly appearing in
token weights are either the pre-training data have a high loss or perplexity, which combined with our algorithm, will heavily down-weight them to almost zero. We further speculate that these tokens are essential to improving the performance of our target fine-tuning task and that using FLOW with a token-wiss wise scheme over-regularizes, preventing any meaningful learning of the target task. As sequence-wic wise re-weighting significantly outperforms loken-wise re- re-weighting, we recommend using sequence-wii wise re- weighting in FLOW for language models.
35

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Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting
I.3. Extended Weight Averaging Results
As discussed in Section 6, we further combine FLOW with WiSE-FT (Wortsman al., 2021) mitigate the effects of catastrophic forgetting when fine-tume fine-tuning. In this section, report the full results of combining FLOW and WiSE-FT to prevent catastrophic forgetting with Gemma 2 2B.
COMBC Teact Match Commormmnun
CEVEX Scect Match VNLJ Accuracy
GIMBI tsuct Matcch HETP Paas
20-
Out Wos
Kear Maan
13-
15.3
Du.t
20
45
50
60
teact Netsh Euse Accunacy
GIMBE
Accurass
truck sawas Accunwcy
Oxs Woo
Full Trestniigg
Oys
Iiee-1
Wiee
Outy Wos
Pull naturng
nd fretunnng
CONOX Crac) Matcc
GSVOK
GSMBC Match SIG4
GSMBK Match OBOX Accuracy
22.
32.5
13.25
10.0
35.6
12.00
00.0
17.39
17.35
Wos-TT
Bane More:
Bewe Hoad
Bave Meet
77.00
Dnn
Figure 4. FLOW is complementary with model averaging (WiSE-FT) in language modeling. We compare WiSE-FT (Wortsman et al., 2021) with standard model fine-tuning and with FLOW after fine-tuning Gemma 2B on MetaMathQA. We use varying a [0, 1] for WiSE-FT. The results indicate that combining Wise-FT with FLOW outperforms vanilla WiSE-FT with standard fine-tuning.
36


Reviews:
[{"review_id": "ldYeltsA7e", "reviewer": "Reviewer_txXi", "summary": "The paper contributes the following:\n - a weighting strategy (called FLOW) for finetuning examples in the data-oblivious setting where there is no access to pre-training examples, that upweights examples with small loss, and aimed at mitigating forgetting of the pretraining task.\n - a benchmark against other strategies on vision and language tasks, which demonstrates a higher mean accuracy between the pre-training task and fine-tuning task, or put differently, less forgetting.\n - a theoretical analysis in a linear setting trained by minimizing the population l2 loss (meaning access to infinite data) using gradient descent with constant learning rate.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the positive assessment of our work! We address your questions below. \n\n**Number of fine-tuning epochs:** For the language experiments, this was chosen based on existing literature recommendations for fine-tuning epochs. Following the observation of [1], we trained for 2 epochs due to compute constraints. For vision, we selected the number of epochs to achieve a fine-tuning accuracy comparable to that of standard fine-tuning on specific models and tasks, aligning with the performance reported in [2].\n\n**Theoretical Claims:** \nOur theory setting is a typical linear setting, commonly used for analysis purposes. However, as mentioned in footnote 2, our insights carry over to wide neural networks following the dynamics of linear models under gradient descent (Lee et al., 2019). For a general choice of $\\widetilde{\\Sigma}$, we derived the weighted covariance matrix $\\widetilde{\\Sigma}'$ impacting the dynamics of FLOW in Appendix E (see eq. 32). Note that $\\widetilde{\\Sigma}'$ depends on $\\widetilde{\\Sigma}$ and $e = \\theta_{\\ast} - \\widetilde{\\theta}_{\\ast}$; the latter is the difference between the optimal solutions of the pre-training and fine-tuning tasks. \n\nUnfortunately, as we explained in Remark E.1, it is hard in general to characterize the eigen-spectrum of $\\widetilde{\\Sigma}'$. But informally, *more is the alignment between $e$ ($= \\theta_{\\ast} - \\widetilde{\\theta}_{\\ast}$) and the eigenvectors of $\\widetilde{\\Sigma}$, slower the convergence along a direction that is aligned to $e$, inhibiting overfitting to the fine-tuning data*. Formalizing this intuition is left for future work.\n\n**Questions For Authors:**\n\n**1.** Please see Fig. 3 (in Appendix I) for the distribution of sample weights of Gemma 2 2B, when we train with sequence-wise weights (results in the main paper) as well as token-wise weights (which we did as an ablation in the Appendix). Note that the distribution is Gaussian-like in the sequence-wise case, while it is concentrated in the token-wise case; as a result, the latter didn’t work very well. We’ll also produce similar plots for our vision experiments in the future. \n\n**2.** Indeed, the samples with high pre-training losses have lower accuracy when using FLOW compared to standard fine-tuning (FT). \n\n| Datasets       | Samples with top 10% highest losses | Standard FT | FLOW (Ours) |\n|---------------|-----------------|---------|-------|\n| CIFAR10       | 1000            | 86.60   | 30.70 |\n| CIFAR100      | 1000            | 56.40   | 21.30 |\n| Stanford cars | 805             | 71.30   | 13.175 |\n\nIn fact, this behavior is in line with the premise of our approach, namely, sacrificing performance on the hard examples from the fine-tuning data to maintain performance on the pre-training data.\n\n\n**3.** If the fine-tuning examples have high losses with the pre-trained model ($\\theta_{\\ast}$), it probably indicates that there is no solution in the vicinity of $\\theta_{\\ast}$ for which the performance on the fine-tuning data is good. Without any access to the pre-training data, all we can hope to do is to remain \"close\" to $\\theta_{\\ast}$ to have decent performance on the pre-training data. In that case, all the methods that we are aware of should struggle to achieve good performance jointly on pre-training and fine-tuning data. Regarding the distribution of the weights, the value of the pre-training losses doesn’t really matter; what matters is their relative spread because the temperature $\\tau$ is chosen as the median of the pre-training losses.\n\n---\n\n[1]: Biderman et al., 2024. “LoRA Learns Less and Forgets Less.” \n\n[2]: Wightman et al., 2021. “Resnet strikes back: An improved training procedure in timm.”", "claims_and_evidence": "The theoretical claims are rigorously prooved as far as I can tell. They apply to a particular toyish setup, so the empirical evaluation on actual data is welcome.", "methods_and_evaluation_criteria": "The chosen benchmark are particularly relevant. A question comes to mind however: how is the trade-off chosen between carrying on fine-tuning, vs stopping at some point. The number of fine-tuning iterations looks as a very important hyperparameter, whose choice should be discussed in more details.", "theoretical_claims": "I skimmed through the proof which looked correct to me, and the exposition is clear and easy to follow through.\n\nOverall, this part however lacks a more thorough discussion of the setup. What is implied by this particular choice of distribution on x, and on y ? Is it somehow representative of actual tasks ? I understand that this specific choice of covariance matrix \\hat{\\Sigma} what chosen because it makes it possible to derive analytical training dynamics, but how does it translate to the relationship between the pre-training and fine-tunins tasks in actual settings ?", "experimental_designs_or_analyses": "As state above, I would appreciate a discussion regarding the stopping criterion for the fine-tuning task, which is in effect a tradeoff between pretraining accuracy and finetuning accuracy.", "relation_to_broader_scientific_literature": "The previous literature was appropriately cited.", "essential_references_not_discussed": "Not applicable.", "other_strengths_and_weaknesses": "The paper is clearly written and comprehensive, yet compact. I appreciate the fact that the proposed method and experiments are supported by some analytical results, even in a simplified setting.\n\nUnless any other reviewer finds a major flaw in the methodology or proofs, I recommend acceptance.", "other_comments_or_suggestions": "Not applicable.", "questions_for_authors": "1. What does the distribution of example weights actually look like in your experiments? Is it concentrated on only a few examples?\n\nSome questions that came to mind while reading your paper,\n 2. Is the accuracy on minority examples (i.e. rare instances or \"difficult\" sub-patterns in the distribution, think i.e. fairness related issues) lower when finetuning using your training strategy, since corresponding examples would probably be downweighted.\n 3. What happens when the pretraining and finetuning tasks are very different, and all finetuning examples get a high loss ?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "I9KaHQcsh5", "reviewer": "Reviewer_Ej68", "summary": "The work relates to the issue of catastrophic forgetting in finetuned models wherein the pre-trained knowledge of the model is wiped out substantially after finetuning, and proposes a method that up-weighs the samples in the finetuning dataset that incur a low loss value (in contrast with existing approaches), in order to retain the pre-training knowledge. The up-weighing scale is theoretically determined and fixed at the beginning of pre-training and the method shows promising improvements over the baselines on known vision and language benchmarks. Theoretical analyses are provided to compare against vanilla finetuning under linear task regimes to evaluate convergence behavior.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the positive assessment of our work! We address your questions below. \n\n**Ablation for $\\tau$:** In the table below, we show the pre-training and fine-tuning accuracies as a function of $\\tau$ when the model is ResNet-50, the pre-training dataset is ImageNet-1K, and the fine-tuning dataset is Caltech 101. The last column represents the percentile of pre-training losses, which we use as the value of $\\tau$.\n\n| **Pre-trained IN-1K (Top-1 Acc)** | **Fine-tuning Caltech101 (Acc)** | **Percentile (%)** |\n|--------------|------------|----------|\n| 68.51       | 91.15      | 10         |\n| 64.13       | 91.01      | 30         |\n| 54.72       | 91.80      | 50         |\n| 45.59       | 92.91      | 70         |\n| 20.51       | 94.02      | 90         |\n\nIn our experiments, we used $\\tau = 50^\\text{th}$ percentile. While smaller $\\tau$ works better here, in general, it seems very difficult to “guess” the optimal value of $\\tau$ without any access to the pre-training data. But this is an interesting point which we will investigate in the future. Thanks for bringing this up!\n\n**Regarding Machine Unlearning (MU):** Thanks for pointing out papers on MU! [1, 2] fall within the more prominent category of weighted updates in the parameter space. As we discussed in our paper, there are several ideas in the parameter space for mitigating forgetting. *Our approach is orthogonal to such ideas as we perform weighting in the sample space*. One could extend the ideas of [1, 2] for forgetting and apply them with our sample-weighting approach (as we did with existing baselines in Section 6.2). Similarly, as you pointed out, one could extend our sample-weighting scheme for MU. These are great future directions! However, it is worth mentioning that while MU and forgetting sound similar, they are not the same in spirit; so it may not be straightforward to extend techniques of one into another. In MU, we deliberately (and ideally, provably) want to induce “forgetting” on some samples, whereas in our context, forgetting is an undesirable side effect that we want to avoid as much as possible. However, we will discuss the connections between MU and [1,2] in the next version of our paper.\n\n**Comparison between adaptive and fixed weighing scenarios and also from multi-task finetuning experiments:** These are great points! We opted for the fixed weighting scheme due to two reasons: (a) a principled adaptive scheme in the absence of pre-training data was not clear to us (specifically, in the way we derived Alg. 1 in Section 4), and (b) to keep the fine-tuning procedure light-weight. The multi-task finetuning setting may require a bit more thought, especially when the tasks appear sequentially – specifically, how the weights are chosen for each new task. This is left for future work.\n\n**Questions For Authors:**\n\n**1.** Potentially, yes! As we discussed above, a combination of our approach and the approach of [1] may lead to better performance.\n\n**2.** This is a great direction! Potentially, our sample weighting scheme could be extended to the machine unlearning problem.\n\n**3.** If there are *out-of-distribution* outliers with large pre-training losses, then our weighting scheme would assign them low weights and so the influence of such outliers would be curtailed. Hence, our algorithm handles such bad samples. On the other hand, if by outliers, you mean in-distribution samples with large pre-training losses, then it seems difficult to do well on such samples in general if we wish to also do reasonably well on the pre-training data, especially without any access to the pre-training data. The premise of our work is that such samples should be considered detrimental to the pre-training performance as they’d make the model drift away from the pre-training weights.\n\n**4.** Answered at the top.\n\n**5.** We believe our language experiments are on reasonably large-scale models and datasets. As for vision, please see our response to Reviewer 1DND, where we show additional results with (i) a ViT-B/16 model on Food101 which is much a harder dataset with 101k images, and (ii) CLIP ViT-B/32 (OpenAI) which is pre-trained with the contrastive loss but we fine-tune the image encoder with a cross-entropy loss.\n\n**6.** The premise of this work is that there is some kind of alignment between the pre-training and fine-tuning tasks; otherwise, it is very hard to find a model that does well on both tasks. Moreover, we assume that the pre-training losses are partially reflective of the “hardness” of the fine-tuning examples; otherwise, in the absence of any pre-training data, this seems like a very hard problem. In the case of a pre-trained generative model, the extension should be straightforward. Consider BERT, which is trained using masked language modeling. We would do linear probing of the [CLS] token’s output embedding, followed by applying Algorithm 2 (Appendix B).", "claims_and_evidence": "The claims made in the submission are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are appropriate for the problem at hand.", "theoretical_claims": "The proofs were checked for Appendix C, D, E.", "experimental_designs_or_analyses": "Yes, the experimental designs are sound. The experiments augmenting existing methods with FLOW are convincing of the method's merits. The work could benefit from an ablation for the choice of $\\tau$ as the median value of the per-sample losses.", "relation_to_broader_scientific_literature": "The work is pertinent to the finetuning and catastrophic forgetting literature as mentioned in the manuscript. Additionally the method may be relevant to machine unlearning literature [1, 2] wherein weighing samples from the retain/forget sets using the method in addition to masking based strategies [1] could enhance performance.\n\n\n[1] Fan, Chongyu, et al. \"Salun: Empowering machine unlearning via gradient-based weight saliency in both image classification and generation.\" arXiv preprint arXiv:2310.12508 (2023).\n\n[2] Jia, Jinghan, et al. \"Model sparsity can simplify machine unlearning.\" Advances in Neural Information Processing Systems 36 (2023): 51584-51605.", "essential_references_not_discussed": "Could incorporate machine unlearning literature where finetuning is actively investigated.", "other_strengths_and_weaknesses": "The work is original, thorough and simple. The manuscript is well written and structured. The work could benefit from a comparison between adaptive and fixed weighing scenarios and also from multi-task finetuning experiments.", "other_comments_or_suggestions": "Minor typo in line 247. LP and Linear Probing were both mentioned in Table 8.", "questions_for_authors": "1. Could the work benefit further in terms of performance and efficiency if it incorporated a saliency mask in the pre-trained network inspired by [1].\n\n2. Could the work be re-purposed for machine unlearning whereby there exists a \"forget set\" that the network is tasked for forgetting in addition to retaining it performance over the \"retain set\"? It could be a significant contribution to the unlearning literature if demonstrated.\n\n3. How would the method handle outliers if the weights assigned to them were low based upon the loss values?\n\n4. Is it possible to justify the choice of $\\tau$ through a small scale ablation?\n\n5. When applied to larger scale dataset/model scenarios, how may the method perform?\n\n6. How would one approach a finetuning task using this method if the task is fairly/completely detached from the pre-training task? For instance, if finetuning a generative model over a classification task. Would the values for the sample losses be meaningful?\n\n\n\n\n[1] Fan, Chongyu, et al. \"Salun: Empowering machine unlearning via gradient-based weight saliency in both image classification and generation.\" arXiv preprint arXiv:2310.12508 (2023).", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "cgxm6nomU3", "reviewer": "Reviewer_QV3i", "summary": "This paper examines how we can mitigate catastrophic forgetting during fine-tuning. They propose that prioritizing \"easy\" examples (i.e. those that the base model already has low loss on). Concretely, this allows mitigating forgetting without significant access to the pretraining dataset -- as would be required by a replay-based setup. The authors test their method on an Imagenet based vision task, as well as fine-tuning large language models for mathematical reasoning. Additionally, the authors demonstrate that their method has compositional benefits when combined with other robust fine-tuning approaches (such as WISE-FT). Finally, this paper examines a theoretical setup which formalizes the gains from prioritizing easier examples, by preventing overfitting along noise directions of the finetuning dataset.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for recognizing the novelties of the simple design of our sample-wise weighting scheme. We appreciate the constructive and encouraging feedback. Please find our responses to your major comments below.\n\n**Wider range of fine-tuning tasks, especially for language modeling:** While we agree that our language modeling section could benefit from additional fine-tuning domains (e.g., instruction tuning, question answering, safety alignment, etc.), we are unfortunately unable to present any further experiments due to compute and time constraints. More specifically, for example, we don’t have LLM-as-a-judge set up to evaluate instruction tuning. \n\nWe appreciate you pointing out several of these fine-tuning domains, but we would like to clarify that *we did not intend to focus on this method for specific language tasks*. Rather, we have proposed this method oblivious to any particular architectural or data modality, with theoretical intuitions for linear models. However, these are great suggestions for our algorithm’s applications – with potential adaptations – to specific important language tasks. Thanks!\n\nWe additionally include results on fine-tuning the image encoder of a pre-trained (OpenAI) multi-modal CLIP ViT-B/32 model. We use zero-shot ImageNet-1K accuracy as our pre-trained metric and then train a classification head on top of the image encoder of the CLIP model. We present their results in the table below. As you see, FLOW performs the best even here by a decent margin.\n\n\n| Method                        | IN-1K (Top-1 Acc) | Target Acc | Average |\n|-------------------------------|-------|------------|---------|\n| Pre-trained                   | 61.66 | 64.76      | 63.21   |\n| Standard FT                 | 1.98   | 88.80      | 45.39   |\n| Linear probe                 | 61.66 | 84.84      | 73.25   |\n| **FLOW (Ours)**          | 60.76 | 90.53      | **75.64**   |\n\nWe follow a similar experimental setup to section 5.1 in the paper, fine-tuning on 7 different downstream tasks. Due to a lack of space, we are omitting the rest of the details here.\n\n**Relationship to prior works on factuality (Ghosal et al., 2024; Gekhman et al., 2024) and the effect of training with unfamiliar/unpopular facts:** This is an interesting perspective regarding our weighting scheme and on what basis the samples could be “easy” or “hard”. We came up with FLOW with a more general perspective and problem scope in mind, without a specific consideration for language modeling and factuality. However, we do see the connection. Thank you for bringing it to our attention.\n\nWithin the context of factuality and LLMs, for a fine-tuning task on some factual knowledge-based dataset, which has samples that are not properly represented in the pre-training distribution, the respective samples could be considered hard, and vice versa. In a data-oblivious setting, as we do not have access to pre-training data, one way (probably the only principled way) would be to rank hardness based on pre-trained losses. Intuitively, such “hard” samples would have high losses, and our algorithm would give them smaller weights. Given the apparent connections, FLOW should intuitively behave in parallel with the findings of (Ghosal et al., 2024; Gekhman et al., 2024). We will include these discussions in the next version of our paper.", "claims_and_evidence": "The author's principal claim is that prioritizing easy examples via their loss weighting can enable a better tradeoff between fine-tuning in distribution performance and mitigating catastrophic forgetting from fine-tuning. They support their claim using an Imagenet finetuning task and a mathematical reasoning task. In general, the experiments that were conducted were well done and the authors compare to a variety of existing robust finetuning methods including full-ft, weight interpolation, and regularization. The results in vision setting wee especially remarkable with massive reductions in the forgetting while maintaining a lower, but comparable in-distribution performance. The results in LLM setting experienced smaller level of forgetting, but seemed to have similar relative trends. While the experimental evaluations are a promising start, the experiments are relatively limited and it would be nice to see a wider range of fine-tuning tasks represented. Particularly in the language modeling setting, for example, the claim would be better supported if also established on common fine-tuning tasks such as instruction following, safety training, etc. In particular, the authors make the particularly strong claim that \"FLOW strikes a good balance between learning a new task and retaining knowledge from pre-training\". However, I find that (particularly for the language modeling setting), this cannot be established simply from one mathematical reasoning task. Rather, the claim would be better supported by a suite of tasks that better leverage a wider section of the LLMs capabilities (such as tasks requiring factual/world knowledge, tasks requiring simply linguistic capabilities, etc).", "methods_and_evaluation_criteria": "Generally, the authors did a good job of selecting appropriate baseline methods which does make their result interesting and compelling. As I mentioned in \"Claims and Evidence\", the number of finetuning tasks/benchmarks sampled could be made a bit more diverse to comprehensively support their claim. Of course, the current math finetuning setting is an important component of this.", "theoretical_claims": "I briefly examined the theoretical claims and they appear reasonable to me...haven't carefully checked correctness.", "experimental_designs_or_analyses": "In general, I did not identify any major weaknesses of the experimental designs.", "relation_to_broader_scientific_literature": "Robust fine-tuning and catastrophic forgetting are crucial issues in the foundation model era. As authors mention, there have been multiple classes of approaches to including (regularization/capacity constrained finetuning and replay based techniques). In this work, the authors make the very valid observation that replay-based techniques are infeasible given the closed nature of pretraining corpora and also the scale. This paper makes an exciting observation that rather than carefully craft novel fine-tuning algorithms and regularizers we can simply reweight the finetuning data to mitigate forgetting. This result dos appear novel in my view. However, it does reflect certain results found in the knowledge-based question answering setting about the role of different data points in finetuning [1,2]", "essential_references_not_discussed": "As I mentioned above, the results are highly related to findings in language model factuality and it would be good for the authors to discuss these relationships [1,2].\n\n\n[1] Ghosal, Gaurav, Tatsunori Hashimoto, and Aditi Raghunathan. \"Understanding finetuning for factual knowledge extraction.\" arXiv preprint arXiv:2406.14785 (2024).\n\n[2] Gekhman, Zorik, et al. \"Does fine-tuning LLMs on new knowledge encourage hallucinations?.\" arXiv preprint arXiv:2405.05904 (2024).", "other_strengths_and_weaknesses": "Overall, I do think this is a strong paper that taps into a somewhat unexplored topic of how finetuning data point choice affects learning and forgetting. While prior works have studied this more specifically in factuality and knowledge setting, a more general characterization is not present to my knowledge. While limited, the experimental results are promising. Additionally, the theoretical results are thought provoking and well done in my opinion.", "other_comments_or_suggestions": "Not applicable", "questions_for_authors": "1) Can authors also run their method on finetuning tasks such as instruction following, safety refusal, or context-based question answering for the LLM setup. Having some additional finetuning tasks for LLMs would greatly strengthen the paper. \n\n2) Can authors contextualize/discuss the relationship between their results and prior works on factuality and the effect of training with unfamiliar/unpopular facts?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ycnL1YdPTJ", "reviewer": "Reviewer_1DND", "summary": "This paper proposes a sample weighting scheme for fine-tuning data based solely on the loss of the pre-trained model.\nThe proposed method weights samples with low loss of the pre-trained model (i.e., easy samples) to suppress the divergence from the pre-trained model.\nExisting methods operate in the parameter or gradient space, but the proposed method is characterised by its emphasis on the sample space.\nFurthermore, this paper theoretically analyses the effect of fine-tuning in the linear setting of the proposed method, and shows that learning stops in a specific subspace and that overfitting to the target task is suppressed.\nIn the experiments, the effectiveness of the proposed method is discussed in both language and visual tasks.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their detailed evaluation and encouraging feedback. We respond to your questions and feedback below:\n\n**Performance for larger/different models and larger datasets:** We believe that our language experiments are indeed large-scale, and the size of our vision datasets is comparable to prior works such as [A, B]. However, we appreciate your feedback, and in light of that, we performed extra experiments on larger models (ViT-B/16 and CLIP ViT-B/32) and a larger dataset (Food 101). We provide the details and results below:\n\n* **ViT-B/16 model on Food101:** As requested, we performed experiments with a ViT pre-trained on ImageNet-1k (IN-1K). Also, Food 101 has twice the number of samples (101k samples) as CIFAR (50k samples), the image sizes are very different, and are corrupted with random noise. We follow a similar experimental setup to section 5.1 in the paper. Due to a lack of space, we are omitting the rest of the details here. Here are the results:\n\n| Method                 | IN-1K (top-1) | Food101 | Average |\n|------------------------|---------------|---------|---------|\n| Pre-trained           | 81.10         | ---     | ---     |\n| Standard FT          | 56.11         | 91.60   | 73.86   |\n| Linear probe         | 81.10         | 83.86   | 82.48   |\n| L2 reg              | 59.18         | 91.66   | 75.42   |\n| **FLOW (Ours)**                | 77.94         | 90.57   | **84.26** |\n\nFLOW exhibits the same behavior as in our previous experiments in the paper – *meaningfully outperforming the baselines*.\n\n* **CLIP ViT-B/32:** Here, we fine-tune the image encoder of a pre-trained (OpenAI) multi-modal CLIP ViT-B/32 model. We use zero-shot ImageNet-1K accuracy as our pre-trained metric and then train a classification head on top of the image encoder of the CLIP model. We follow a similar experimental setup to section 5.1 in the paper, fine-tuning on 7 different downstream tasks. Due to a lack of space, we are omitting the rest of the details here. Please see the results below. \n\n| Method                        | IN-1K (Top-1 Acc) | Target Acc | Average |\n|-------------------------------|-------|------------|---------|\n| Pre-trained                   | 61.66 | 64.76      | 63.21   |\n| Standard FT                 | 1.98   | 88.80      | 45.39   |\n| Linear probe                 | 61.66 | 84.84      | 73.25   |\n| **FLOW (Ours)**          | 60.76 | 90.53      | **75.64**   |\n\n*FLOW performs the best by a decent margin*.\n\n**Distillation paper [1]:** Based on the feedback, we compared the distillation-based idea (abbreviated as LwF in [1]) against ours for the new ViT-B/16 experiment on Food101. We used the hyper-parameters for LwF suggested by [1]. We observe that our method FLOW outperforms LwF despite the simplicity; FLOW achieves better performance on the forgetting front, while LwF has higher accuracy on the fine-tuning task. \n\n| Method                 | IN-1K (top-1) | Food101 | Average |\n|------------------------|---------------|---------|---------|\n| Pre-trained           | 81.10         | ---     | ---     |\n| Standard FT          | 56.11         | 91.60   | 73.86   |\n| LwF [1]         | 76.39         | 91.23   | 83.81   |\n| **FLOW (Ours)**                | 77.94         | 90.57   | **84.26** |\n\nWe believe our method has clear advantages on the efficiency front as well. Note that **the LwF method comes with more tunable parameters, additional memory, and evaluation overhead**. They have a distillation loss scaling factor $\\lambda_o$ and “softening parameter” $T$ that rescales the outputs as $y’_i = (y’_i)^{1/T} / \\sum_k^l (y’_k)^{1/T}$ for an $l$-dimensional output vector. In contrast, our method only has $\\tau$, for which we use a prescribed value. Second, the logits obtained per sample for the pretrained model body and head must be stored to compute the distillation loss, which is infeasible in the language modeling setting. *Our method shows comparable performance with easier tuning and better storage efficiency*. Thank you for pointing out this line of work. Combining the distillation idea with our sample selection idea is also an interesting future work. \n\nWe hope to have resolved your concerns, and we're happy to discuss further. *If you’re satisfied, we sincerely hope you will raise your score*!\n\n----\n\n[A]: Wang et al., 2025. “LiNeS: Post-training Layer Scaling Prevents Forgetting and Enhances Model Merging” \n\n[B]: Wortman et al., 2022. “Robust fine-tuning of zero-shot models”", "claims_and_evidence": "This paper proposes a method for weighting samples with low loss from a pre-trained model to prevent deviation from the pre-trained model.\nThe effectiveness of this method is discussed from an experimental perspective, but there are some shortcomings in the experiment.\nFor example, in the experiment on the vision task, only the experiment using ResNet is discussed. In addition, the scale of the dataset is small.\nTherefore, it is difficult to judge the effectiveness of the proposed method compared to conventional methods.\nOn the other hand, the effectiveness of the proposed method is also discussed theoretically. I think that the theoretical analysis is technically sound.", "methods_and_evaluation_criteria": "I think the theoretical analysis of linear regression is technically sound.\nHowever, in order to demonstrate its effectiveness in more practical application scenarios, it is necessary to conduct experiments on large datasets and demonstrate its effectiveness with various models, including ViT.", "theoretical_claims": "I think the theoretical analysis in sec.7 and appendix is technically sound.\n(However, please note that I have not completely proved all of the mathematical formulas in this paper myself.)", "experimental_designs_or_analyses": "One of the strengths of this paper is that it discusses the effectiveness of the proposed method in both vision tasks and language tasks.\nHowever, as mentioned above, in the experiments for vision tasks, only ResNet was used. In addition, the scale of the dataset is small.\nIn the experiments, many of the experiments were conducted using small-scale datasets. For example, if a larger dataset than CIFAR is used, will the proposed method work effectively?\nIn the class identification task, there are many proposals for larger datasets than CIFAR, etc.\nThe vision model is a small model such as ResNet18 and ResNet50. Can you also argue for the effectiveness of the proposed method in larger transformers such as ViT?\nIn addition, the method of simple distillation, e.g., [1], is also a standard approach that does not use past data (i.e., data before fine-tuning). Is the proposed method more effective than the method of simple distillation?\n\n[1] Li,Z. andHoiem,D. Learningwithout forgetting. In Leibe,B.,Matas, J.,Sebe,N.,andWelling,M. (eds.) ECCV(4),volume9908ofLectureNotesinComputer Science,pp.614–629.Springer,2016.", "relation_to_broader_scientific_literature": "Yes.\nThe proposed method and evaluation criteria are appropriate as a problem setting for lifelong learning.\nThe main contribution of this paper (the upweighting framework) is closely related to several existing methods cited in this paper, as well as general methods such as LoRA.\nExperimental comparisons are made with these conventional methods, and the effectiveness of the proposed method compared to conventional methods is discussed.", "essential_references_not_discussed": "The reference meets the minimum required standards.", "other_strengths_and_weaknesses": "This paper proposes a method for preventing deviation from pre-trained models by weighting simple samples with low loss.\nThe proposed method is simple but effective, and can be applied to various tasks such as image classification tasks and language tasks.\nThe theoretical analysis is technically sound.\nHowever, there are not enough experiments on image tasks.\nAlso, the introduction is too long (especially after line 82).\nInstead of shortening the introduction, I think that experimental supplements and additions on image tasks are necessary in the main text.", "other_comments_or_suggestions": "n/a", "questions_for_authors": "In the experiments, many of the experiments were conducted using small-scale datasets. For example, if a larger dataset than CIFAR is used, will the proposed method work effectively?\nThe vision model is a small model such as ResNet18 and ResNet50. Can you also argue for the effectiveness of the proposed method in larger transformers such as ViT?\nIn addition, the method of simple distillation, e.g., [1], is also a standard approach that does not use past data (i.e., data before fine-tuning). Is the proposed method more effective than the method of simple distillation?\n\n[1] Li,Z. andHoiem,D. Learningwithout forgetting. In Leibe,B.,Matas, J.,Sebe,N.,andWelling,M. (eds.) ECCV(4),volume9908ofLectureNotesinComputer Science,pp.614–629.Springer,2016.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "n/a"}}]

Author rebuttal:

Meta-review: This paper tackles fine-tuning, and argues it is important to both perform well on the pre-train dataset and the new finetune dataset, but without access to the dataset used during pre-training. They propose to upweight 'easy samples' in order to achieve better performance. 

Reviewers all agree to accept the paper, and the authors rebutted many of the points reviewers raised in a convincing manner. Therefore this publication should be accepted. 

Reviewers agree that this paper is well-written, the experiments are good, and the setup is important. There were concerns about not enough image experiments, but additional experiments were added during the rebuttal. 

Two reviewers (txXi and Ej68) brought up the point about outliers / fairness issues of forgetting 'minority populations' when using this technique. I expect the authors to comment on this in a limitations and in the broader impacts. I trust the authors will do this given they added results already in the rebuttal. 

Reviewer 1DND brings up Learning without Forgetting. In general, this fine-tuning setup (not forgetting the pre-train data) is very similar to a 2-task continual learning setting. The authors have added LwF results (although it is not clear what hyperparameter selection procedure was used), and in general refer to continual learning literature in their related works. However, their other baselines are very simple and do not consider continual learning. Typically, continual learning methods (like LwF) perform very well in a 2-task setting because it is easy to tune the hyperparameter to trade-off forgetting vs learning. That said, a benefit of this work may be that there is little-to-no tuning of hyperparameters required.
Decision: Accept (spotlight poster)