Predicting the Performance of Foundation Models via Agreement-on-the-Line

Aman Mehra^*^1  Rahul Saxena^*^1  Taeyoun Kim^*^1  Christina Baek^1
Zico Kolter^1,2  Aditi Raghunathan^1

Carnegie Mellon University^1, Bosch Center for AI^2

Estimating the out-of-distribution performance in regimes where labels are scarce is critical to safely deploy foundation models. Recently, it was shown that ensembles of neural networks observe the phenomena “agreement-on-the-line”, which can be leveraged to reliably predict OOD performance without labels. However, in contrast to classical neural networks that are trained on in-distribution data from scratch for numerous epochs, foundation models undergo minimal finetuning from heavily pretrained weights, which may reduce the ensemble diversity needed to observe agreement-on-the-line. In our work, we demonstrate that when lightly finetuning multiple runs from a single foundation model, the choice of randomness during training (linear head initialization, data ordering, and data subsetting) can lead to drastically different levels of agreement-on-the-line in the resulting ensemble. Surprisingly, only random head initialization is able to reliably induce agreement-on-the-line in finetuned foundation models across vision and language benchmarks. Second, we demonstrate that ensembles of multiple foundation models pretrained on different datasets but finetuned on the same task can also show agreement-on-the-line. In total, by careful construction of a diverse ensemble, we can utilize agreement-on-the-line-based methods to predict the OOD performance of foundation models with high precision.

Random Head

Data Ordering

Data Subsetting

The ID versus OOD accuracy and agreement trends for CIFAR10 to CIFAR10C “Pixelate” in linear probed CLIP, MNLI to SNLI in full finetuned OPT, and SQuAD to SQuAD-Shifts “Amazon” in full finetuned GPT2 models. Models observe different agreement linear fits depending on the diversity source (columns) used to generate the model ensemble.

§ INTRODUCTION

Foundation models (FM), or large models first pretrained on open world data then finetuned or prompted for a specific downstream task, have proven to be powerful solutions for many common machine learning problems. A notable trait about FMs is that they are far more robust to distribution shift than other deep learning approaches — across image and language benchmarks, they suffer a smaller performance degradation on out-of-distribution (OOD) data, that may vary substantially from the in-distribution (ID) finetuning data [Radford et al., 2021, Radford et al., 2019, Brown et al., 2020, Wortsman et al., 2022, Wang et al., 2023, Devlin et al., 2018]. From clinical decision-making in different hospitals to navigating robots through unseen terrains, FMs are increasingly utilized for tasks prone to distribution shift. However, evaluating these models in OOD settings remains difficult: in many cases, acquiring labels for OOD data is costly and inefficient, while unlabled OOD data is much easier to collect. Although the field has explored other means for estimating OOD accuracy without labeled data, they are not ideal for FMs. A reliable FM performance estimator has the following desirable properties. First, the method must be computationally efficient to account for FMs' large model size. Second, FMs are leveraged for many different tasks (e.g., classification, question-answering, regression), so the method should also be versatile across tasks. Third, as we will see, methods for finetuned FMs may require different model assumptions from neural networks trained from scratch.

Recently, Baek et al., 2022 proposed a promising method for estimating the OOD accuracy of deep networks using the agreement between pairs of these classifiers (i.e., how often two classifiers make the same prediction). For distribution shifts where models observe a strong linear correlation in ID versus OOD accuracy – a common phenomenon in vision and language benchmarks [Miller et al., 2021, Awadalla et al., 2022] – a strong linear correlation also holds for ID versus OOD agreement with extremely similar slopes and intercepts. These effects are referred to as accuracy-on-the-line (ACL) and agreement-on-the-line (AGL) respectively, and together they provide a simple method for estimating OOD accuracy via unlabeled data alone. Namely, without any OOD labels, we can instead measure the linear fit of ID versus OOD agreement as a proxy for the linear fit of ID versus OOD accuracy. We can then verify whether ACL holds using the correlation strength of agreement's linear trend and estimate each model's OOD accuracy by linearly transforming ID accuracy using this approximate slope and bias. This simple approach has shown reliably predict the OOD accuracy of models within a few percentage points across classification and question-answering tasks.

Unfortunately, while the method has several practical advantages, it is unclear whether finetuned FMs also observe the necessary AGL phenomena. Intuitively, a prerequisite to observing AGL is a diverse ensemble of classifiers. Since OOD accuracy falls below ID accuracy, if the linear trend in ID versus OOD agreement is to match that of accuracy, models must also agree much less OOD than ID. For this to happen, errors between any two models must be sufficiently decorrelated.  Baek et al., 2022 observes AGL in ensembles of neural networks trained for hundreds of epochs from scratch where it is conceivable that the stochasticity between training runs leads to large divergences in the weight space, and corresponding models have diverse OOD predictions. However, in the case of finetuned FMs, models are much closer in the weight space. FMs are often either linear probed over the same pretrained weights or full finetuned for a few epochs with a small learning rate and intuitively, such light finetuning may lead to models that “revert back” to their pretrained behavior to make highly correlated predictions OOD.

This raises the question: can we enforce AGL in this paradigm of lightly finetuning heavily pretrained models? In this work, we conduct an extensive study across several modalities, e.g., CLIP-based image classification and LLM-based question-answering, and training regiments, e.g., full finetuning and linear probing, to understand when AGL holds for finetuned FMs. We first investigate whether AGL appears in an ensemble of finetuned models from a single base FM. To collect a deep ensemble, the following sources of diversity can be injected into the finetuning process: 1) random initialization of the linear head; 2) random data ordering; and 3) random data subsetting. We find that not every source of diversity during fine-tuning on ID data manifests in sufficient diversity OOD, breaking the matching linear fits in ID versus OOD accuracy and agreement. Interestingly, finetuning models from different random initializations of the linear head consistently induces AGL in the resulting ensemble across benchmarks. In contrast, neural networks trained from scratch observe AGL irrespective to these diversity sources.

Second, we show that finetuned models from multiple different base FMs can be leveraged for AGL-based performance estimation. As base FMs can be pretrained with entirely different datasets, architectures, and training regiments, the linear trends in ID versus OOD accuracy and agreement may break altogether in such ensembles. Indeed previous works indicate that on vision tasks, FMs pretrained on different image corpora can have different levels of OOD robustness for the same ID performance [Fang et al., 2022, Radford et al., 2021, Taori et al., 2020]. On the contrary, we find that on language tasks, FMs pretrained on different text corpora observe both AGL and ACL across question-answering and text classification tasks.

In total, we develop simple techniques for applying AGL-based performance estimation methods to predict the OOD performance of foundation models. We demonstrate that the AGL phenomenon is not limited to ensembles of neural networks trained from scratch. By simply finetuning FMs from random initializations of the linear head, we can observe the phenomena in FMs across a wide variety of tasks (classification, question-answering) and modalities (vision, language) and training procedures (linear probing, full finetuning). We find that AGL is the only method to accurately estimate the performance of finetuned FMs across all tasks, surpassing other performance estimation baselines by a significant margin as large as $20\%$ mean absolute percentage error.

§ BACKGROUND AND RELATED WORK

§.§ Setup

We are interested in evaluating models that map an input $x \in \sX$ to a discrete output $y \in \sY$. In particular, we finetune foundation models. For a base model $\base$, let $f(\base)$ denote a finetuned version of $\base$. In this work, we consider a variety of foundation models: GPT2 [Radford et al., 2019], OPT [Zhang et al., 2022], Llama2 [Touvron et al., 2023], BERT [Devlin et al., 2018], and CLIP [Radford et al., 2021].

Finetuning strategies. We have access to labeled data from some distribution $\did$ that we use for obtaining $f(\base)$ from $\base$. In this work, we consider the following standard finetuning procedures.

* Linear probing (LP): Given features from the base model $\base_\theta$, we train a linear head $v$ such that the final classifier maps the score $v^\top \base_\theta(x)$ to a predicted class. We randomly initialize $v$ and update $v$ via gradient steps on a suitable loss function. The base model parameters remain frozen. We refer to $v$ as either a linear probe (classification), or span prediction head (question-answering) depending on the task.

* Full finetuning (FFT): We update all parameters of the backbone $\base_\theta$ and the linear head $v$ using a small learning rate. When infeasible to update all parameters, we perform low-rank adaptation (LoRA) [Hu et al., 2021] to reduce the number of trainable parameters while still effectively updating the feature extractor $\base_\theta$. In this work, we do not distinguish between LoRA and FFT as they conceptually achieve the same effect, and seem to show similar empirical trends in our studies.

OOD performance estimation.
Given access to a labeled validation set from $\did$ and unlabeled samples from a related but different distribution $\dood$, our goal is to estimate performance on $\dood$. We consider the standard performance metrics for various tasks: Accuracy $\zerone:\sY \mapsto [0,1]$ for classification, and  Exact Match $\fem:\sY \mapsto [0,1]$ and Macro-averaged F1 score $\fone: \sY \mapsto [0,1]$ for question-answering. We use $\ell$ to denote the appropriate metric in the context.

§.§ Background on OOD accuracy estimation

There is rich literature on OOD performance estimation for deep networks, with a variety of proposed approaches. Initial works focused on upper bounding the degree of distribution shift through data and/or model dependent metrics, e.g., uniform convergence bounds using $\mathcal{H}$-divergence [Ben-David et al., 2006, Mansour et al., 2009, Cortes et al., 2010, Kuzborskij & Orabona, 2013]. However, these bounds tend to be loose for deep networks  [Miller et al., 2021]. The following works try to estimate the performance exactly.

For classification, a popular approach is to leverage the model's confidence to predict the OOD performance [Hendrycks & Gimpel, 2017, Hendrycks & Dietterich, 2019, Garg et al., 2022, Elsahar & Gallé, 2019, Guillory et al., 2021]. Since deep models are typically overconfident, it is common practice to first calibrate these models in-distribution by temperature scaling. On a related note are uncertainty quantification works that directly try to calibrate models under distribution shift  [Yu et al., 2022, Zou et al., 2023, Podkopaev & Ramdas, 2021]. Confidence based methods are commonly utilized in practice, and favorable for foundation models as they are not computationally intensive and model-agnostic. However, they do have a couple downsides. Namely, they often fail in the presence of large shifts [Garg et al., 2022] and are often only well-defined for accuracy and ill-defined for other common metrics like F1 score. These can be limiting factors for foundation models which are applied to a broad array of tasks. Still, as they are the most common estimation methods, we utilize them as the baselines in our work.

Another approach involves measuring model behavior on known auxiliary tasks to understand how the model will behave under the distribution shift at hand  [Schelter et al., 2020, Deng & Zheng, 2021, Deng et al., 2021]. These approaches however tend to be overfit to specific datasets or modalities. Similar to AGL, there are prediction methods that utilize information from ensembles. Oftentimes a separate “reference” ensemble is trained on some objective to predict the performance of a “target” model [Chuang et al., 2020, Yu et al., 2022, Chen et al., 2021]. These methods generally have a much higher computational cost than AGL. Although AGL also requires at least 3 models to compute agreement between, these models only undergo generic finetuning. Thus, it is better suited approach for evaluating foundation models, especially if off-the-shelf finetuned models are readily available, e.g., from Huggingface (see Section 4).

Overall, there is growing attention towards understanding the safety and reliability of foundation models. To understand the effective robustness of FMs under distribution shift, recent works focus on studying the “accuracy-on-the-line” phenomena [Miller et al., 2021] which we cover in more detail in the next subsection and designing benchmarks that try to expose different failure modes of large models [Malinin et al., 2022, Tran et al., 2022]. On the other hand, unsupervised OOD performance estimation has been underexplored in this modern setting, in terms of new methods and the transferability of old methods to large pretrained models.

§.§ Accuracy and agreement on the line

We are interested in adapting the method “agreement-on-the-line” (AGL) [Baek et al., 2022] for OOD estimation as it obtains state-of-the-art performance estimation across a wide variety of distribution shifts. AGL is based on an earlier observation called “accuracy-on-the-line” (ACL) — across common distribution shift benchmarks, there is a strong linear correlation between the ID and OOD performance of models [Miller et al., 2021, Recht et al., 2018, Recht et al., 2019, Roelofs et al., 2019, Yadav & Bottou, 2019, Taori et al., 2020, Miller et al., 2020]. ACL has been observed in foundation models for image classification, e.g., CIFAR10C [Hendrycks & Dietterich, 2019], ImageNetV2 [Recht et al., 2019], FMoW-WILDS [Koh et al., 2021], and question-answering, e.g., SQuAD-Shifts [Miller et al., 2020]. However, ACL does not always hold, e.g., Camelyon-WILDS [Miller et al., 2021] and SearchQA [Awadalla et al., 2022].

While ACL is a striking phenomenon, it does not immediately provide a practical method to estimate OOD performance—computing the linear fit of ID versus OOD accuracy requires labeled samples from $\dood$. Alternatively,  Baek et al., 2022 suggests that we can estimate this linear trend  exactly using only the agreement between neural networks. Formally, given a pair of models $f_1$ and $f_2$ that map inputs to labels, accuracy and agreement is defined as

\begin{align}
\mathsf{Acc}(f_i) = \E_{x,y \sim \mathcal{D}}[\task(f_i(x), y)], ~~ \mathsf{Agr}(f_1, f_2) = \E_{x,y \sim \mathcal{D}}[\task(f_1(x), f_2(x))],
\end{align}

where $\task$ is the appropriate performance metric of interest. While accuracy requires access to the ground truth labels $y$, note that agreement only requires access to unlabeled data and a pair of models. Baek et al., 2022 observes that when ID versus OOD accuracy is strongly linearly correlated between neural networks, i.e., ACL, then the ID versus OOD agreement of pairs of these models also observe a strong linear correlation with the same linear slope and bias. Furthermore, when accuracies do not show a linear correlation, agreements also do not. This coupled phenomenon is dubbed “agreement-on-the-line” (AGL).

Baek et al., 2022  uses AGL for OOD performance estimation: one may obtain the slope and bias of the agreement line with unlabeled data, and then estimate the OOD performance by linearly transforming the ID validation performance. As in [Baek et al., 2022] and [Miller et al., 2021] we apply probit scaling to the accuracies and agreements, which induces a stronger linear fit. We refer the reader to Baek et al., 2022 for formal AGL-based performance estimation algorithms (ALine-S and ALine-D), which we also provide in Appendix <ref>.

§ PREDICTING OOD PERFORMANCE: SINGLE BASE FOUNDATION MODEL

We first evaluate whether AGL appears in an ensemble of multiple finetuned runs of a single base foundation model. This would enable precise OOD performance estimates for each ensemble member. A practitioner may naively gather a finetuned ensemble by training a couple runs with different seeds or hyperparameters. However, an overriding concern is that even with some randomness in the finetuning process, linear probing or light full finetuning over the same base model may lead to solutions with very correlated predictions. We extensively evaluate the following methods of  introducing diversity into the finetuning process to see what approach (if any) can lead to AGL.

* Random linear heads We initialize the last layer of the network (i.e., the linear head) randomly, instead of via some zero-shot or pre-specified manner.

* Data ordering We present the same training data to each model but shuffle the order of the data, i.e., model observes different minibatches.

* Data subsetting We i.i.d. sample $p\%$ subset of the data to train over.  In the main body, we report models trained on independently sampled $10\%$ of the training data, other proportions of $30\%$ and $50\%$ are reported in Appendix <ref>.

We perturb one source of diversity at a time and study whether AGL occurs in each resulting model ensemble. For each setting, we also vary the number of training epochs to collect models with different ID performances, which is necessary to obtain a meaningful linear correlation in accuracy. The additional randomness induced by varied training epochs does affect any conclusions we make in this section.

§.§ VLM-based Image Classification

We first investigate the effect of diversity source on AGL behavior for vision benchmarks. For image classification, a common pipeline is to finetune over a CLIP [Radford et al., 2021] pretrained foundation model.

CLIP Linear Probing
We finetune over OpenCLIP ViT-B/32 model trained on LAION-2B [Ilharco et al., 2021]. Given its well-established zero-shot capabilities, a popular method of finetuning CLIP is to simply employ linear probing on top of the CLIP representation. We take particular interest in evaluating the OOD performance of an ensemble of linear models trained on top of frozen base model representations.

We evaluate ensembles on synthetic corruptions (CIFAR10C, CIFAR100C, ImageNetC), dataset replication shifts (CIFAR10.1, ImageNetV2), style shifts (OfficeHome), geographical and temporal shifts (FMoW-WILDS, iWildCam-WILDS), and interlaboratory shifts in medicine (Camelyon17-WILDS). iWildCam-WILDS exhibits weak ACL and Camelyon17-WILDS doesn't exhibit any ACL [Miller et al., 2021]. We test on iWildCam-WILDS and Camelyon17-WILDS to verify AGL's negative condition, i.e., when the linear correlation does not exist in ID versus OOD accuracy, it also does not exist in agreement.

ID     OOD

CIFAR10 [Krizhevsky et al., 2009]      CIFAR10C [Hendrycks & Dietterich, 2019], CIFAR10.1 [Recht et al., 2018]

CIFAR100 [Krizhevsky et al., 2009]      CIFAR100C [Hendrycks & Dietterich, 2019]

ImageNet [Russakovsky et al., 2014]      ImageNetC [Hendrycks & Dietterich, 2019], CIFAR10.1 [Recht et al., 2018]

FMoW ID [Koh et al., 2021]     FMoW OOD [Koh et al., 2021]

iWildCam ID [Koh et al., 2021]     iWildCam OOD [Koh et al., 2021]

Camelyon17 ID [Koh et al., 2021]     Camelyon17 OOD [Koh et al., 2021]

OfficeHome [Venkateswara et al., 2017]     All (ID, OOD) pairings of domains Art, ClipArt, Product, Real World

MNLI [Williams et al., 2018]     MNLI-Mismatched [Williams et al., 2018], SNLI [Bowman et al., 2015]

SQuAD [Rajpurkar et al., 2016]     SQuAD-Shifts [Miller et al., 2020]

We evaluate models on the following distribution shift benchmarks.

In ensembles with diverse random initializations, ACL and AGL holds across benchmarks in linear probed CLIP models. Similar to [Baek et al., 2022], neither ACL nor AGL holds for the Camelyon17-WILDS

Across vision benchmarks, linear probed CLIP models observe ACL meaning there is a strong linear correlation in the ID versus OOD performance. Similarly, on the same datasets, we can observe a corresponding strong linear correlation in agreement across ensembles injected with diversity in linear head initialization, data ordering, and data subsetting (Figures <ref>). However, we find that only ensembles with diverse initialization leads to  AGL where the linear trend of agreement and accuracy matching slope and bias (Figure <ref>). See Appendix <ref> for results on other datasets. In model ensembles obtained by data ordering and data subsetting, we observe a consistent trend where the agreement trend observe a much higher slope close to the diagonal $y = x$ line. These results are not specific to linear probing alone. In full finetuned CLIP models, we also observe that random linear heads induce the most reliable AGL behavior (See Appendix <ref>). Note that this setting is still notably different from [Baek et al., 2022] where models are heavily trained for tens to hundreds of epochs often with a large learning rate, which causes AGL behavior to be more robust to the source of diversity used to induce the ensemble (See Appendix <ref>).

§.§ LLM-based Question-Answering and Text Classification

We conduct a similar systematic investigation of AGL in finetuned runs of a single base language model. Similar to CLIP linear probing, we find that AGL cannot be observed without random head initialization in language models evaluated on text classification and question-answering tasks.

Full Finetuned Language Models
We evaluate over a collection of $450$ full finetuned runs of several base FMs: GPT2-Medium [Radford et al., 2019] and OPT-125M [Zhang et al., 2022]. Models are full finetuned for up to $20$ epochs with a small learning rate ($\leq 1e^{-6}$). Hyperparameters specifics can be found in Appendix <ref>. We do not conduct a linear probing study for question-answering as it leads to poorly performing models. For text classification, we also conduct a linear probing study in Appendix <ref>.

Datasets We test models on a text classification shift from MNLI [Williams et al., 2018] in the GLUE benchmark [Wang et al., 2018] to MNLI-Mismatched [Williams et al., 2018] and SNLI [Bowman et al., 2015]. We also evaluate extractive question-answering models on the shift from SQuAD v1.1 [Rajpurkar et al., 2016] to  SQuAD-Shifts [Miller et al., 2020].

We evaluate models on accuracy for text classification and F1 score for question-answering. Similar to our findings in CLIP, in both text classification and question-answering benchmarks, ensembles of full finetuned LLMs observe AGL when models are trained from different randomly initialized linear or span heads while data ordering and data subsetting observe an agreement trend closer to the diagonal $y = x$ line (Figure <ref> and Appendix <ref>). We note that with full finetuning, the differences in AGL behavior between diversity sources are not as stark as with linearly probed models. In some sense, how model diversity is achieved is becomes increasingly less important for observing AGL as the base model parameters also diverge, with ensembles of models heavily trained from scratch at the extreme [Baek et al., 2022].

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Source of Diversity     CIFAR10C     SQuAD-Shifts Amazon     SQuAD-Shifts Reddit     SNLI

Random Linear Heads     14.64     6.34     3.48     11.7

Data Ordering     37.01     10.30     9.59     15.4

Data Subsetting     35.85     16.21     13.94     15.5

ALine-D MAPE ($\%$) of different sources of diversity for CLIP linear probing and GPT2-Medium/OPT-125M full finetuning. We average the score over all corruptions for CIFAR10C.

§.§ Summary and Implications

Across image and language modalities, we demonstrate that ensembles of finetuned FMs can also observe  agreement-on-the-line like to heavily trained CNN's [Baek et al., 2022]. A natural hypothesis about finetuning a single base FM may be that heavy pretraining and light finetuning leads to downstream models with highly correlated behavior under distribution shift. However, simply randomizing the initialization of the linear head alone induces sufficiently decorrelated models for observing AGL. The diversity in the ensemble becomes important when predicting the OOD performance of models using downstream AGL-based methods. In Table <ref>, we show that AGL-based methods can only accurately predict the OOD performance of models in ensembles with diverse initialization, while methods fail for ensembles with diverse data subsetting or ordering.

Furthermore, our findings contrast to previous works that suggest AGL is a neural-network specific phenomenon [Baek et al., 2022, Lee et al., 2023], unlike ACL which is model agnostic [Miller et al., 2021]. Specifically, Baek et al., 2022 reports that linear models trained on top of the flattened CIFAR10 images do not observe AGL. However, we find that, on top of CLIP features, linear models can exhibit AGL with random initialization. Previous works on the Generalization Disagreement
Equality [Jiang et al., 2022] also contend that data subsetting leads to the most diversity in model predictions. Specifically, in-distribution, the agreement rate between pairs of models was shown to equal their expected accuracy in ensembles with diverse data subsetting, while diverse random initialization leads to slightly higher agreement [Jiang et al., 2022, Nakkiran & Bansal, 2020]. On the other hand, out-of-distribution, we find that randomizing the linear head is important for diverse predictions and AGL behavior.

§ PREDICTING OOD PERFORMANCE: MULTIPLE FOUNDATION MODELS

Alternatively, we consider a scenario where the ensemble consists of multiple base foundation models. First, it is unclear whether ACL itself holds amongst such models. The base models are heavily pretrained on different data corpuses which may cause respective downstream models to have different ID versus OOD accuracy trends or “effective robustness” [Fang et al., 2022]. On vision tasks, for example, linear probing over CLIP, EfficientNet [Tan & Le, 2019], ViT [Dosovitskiy et al., 2020], and BYOL [Grill et al., 2020] observe varying robustness trends [Radford et al., 2021].  Second, even when ACL does indeed hold, it is unclear whether these model ensembles will also observe AGL. Here the problem is seemingly different from the single base model setting: any pair of foundation models finetuned from different base models may agree too little, or OOD agreement rate may vary across model pairs depending on the similarity of the pretraining corpus, breaking the linear correlation of agreement entirely. Yet, we observe that for language models and tasks, ensembles of finetuned FMs from a wide range of base models observe both ACL and AGL.

Models We finetune models from OPT-125M, OPT-350M, OPT-1.3B [Zhang et al., 2022], GPT2, GPT2-Medium, GPT2-Large, GPT2-XL [Radford et al., 2019], GPT-Neo-135M [Black et al., 2021], Llama2-7B [Touvron et al., 2023], Alpaca-7B [Taori et al., 2023], and Vicuna-7B [Chiang et al., 2023]. We full finetune OPT and GPT models and LoRA finetune Llama, Alpaca, and Vicuna. These models are pretrained on different mixtures of  BookCorpus [Zhu et al., 2015], Stories [Trinh & Le, 2018], PILE [Gao et al., 2020], CCNews v2 corpus, and PushShift.io Reddit [Baumgartner et al., 2020]. Alpaca and Vicuna are additionally instruction-finetuned over Llama2.

AGL can be observed between models finetuned from different base models (Llama, GPT, OPT) for the F1 score for question-answering shift (SQuAD to SQuAD-Shifts) and accuracy for text classification (MNLI-Matched to MNLI-Mismatched and SNLI).

§.§ Results

We investigate the AGL behavior of an ensemble of foundation models finetuned from diverse base models in Figure <ref> for question-answering. First note that base LLM models pretrained on different text corpora lead to finetuned models that lie on the same linear trend in accuracy. Unlike the different accuracy trends observed by different vision foundation models [Radford et al., 2019], we suspect that the pretraining datasets for the language models in our study observe much more homogeneity. Second, the ID versus OOD agreement between pairs of models in this ensemble, including those between different base foundation models, is also strongly correlated and the slope and intercept closely matches that of accuracy. In other words, ensembles of different base models also observe AGL without any special regularization for ensemble diversity.

§ ESTIMATING OOD ACCURACY USING AGL IN DIVERSE ENSEMBLES

By constructing a diverse ensemble of foundation models, we can leverage AGL to extract precise estimates of model performances under distribution shift. We construct diverse ensembles by collecting models trained from randomly-initialized heads (Section 3) and different base models (Section 4). For image classification, our model collection consists of just the former, i.e., linear models over CLIP representations. For text classification and question-answering, we include GPT, OPT, Llama models individually full finetuned from differently initialized heads. Table <ref>, we compare the Mean Absolute Percentage Error (MAPE) of AGL-based prediction algorithms, ALine-S and ALine-D  [Baek et al., 2022], to other baselines.

We compare against confidence based methods ATC [Garg et al., 2022], AC [Hendrycks & Gimpel, 2017] and DOC-Feat [Guillory et al., 2021] and  Naive Agreement which directly uses agreement between model pairs as a proxy for their average accuracy [Jiang et al., 2022, Madani et al., 2004]. For confidence based methods, we first temperature scale the models using ID validation data, and pick the lower error rate from the estimations obtained with and without temperature scaling. Since confidence baselines are designed to estimate the accuracy metric on classification tasks, there are several limitations when attempting to naively apply the baselines to estimate performance on question-answering. First, there is no easy analogous formulation of confidence baselines for the F1 score, so we estimate the exact-match score instead for fair comparison. On the other hand, note that AGL can predict the performance of models across metrics accuracy, F1, and exact-match. Second, extractive question-answering is a joint classification task where models predict both the start and end token index of the answer span in the context. More details for how we calibrate baselines for this setting is provided in Appendix <ref>.

Note that ALine-S and ALine-D only provide estimation guarantees in circumstances where the linear correlation in agreement is strong. Consistent with [Baek et al., 2022], we filter out datasets with low correlation strength $R^2 \leq 0.95$. These shifts include iWildCam-WILDS, Camelyon-WILDS, and a few corruptions in CIFAR10C, CIFAR100C, and ImageNetC. We evaluate ALine-S/D for these specific failure cases in Appendix <ref>. Across datasets that observe a strong linear correlation in agreement, ALine-S and ALine-D provide precise estimates of OOD performance in finetuned foundation models, surpassing other baselines by a large margin. In particular, they perform notably better on question-answering task SQuAD, with the next best confidence method achieving as large as $20\%$ higher error.

The MAPE (%) of predicting OOD performance using AGL-based ALine and other baseline methods. We collect a diverse ensemble by randomizing the linear initialization and including multiple base models. $^*$We filter out shifts with low correlation in agreement.

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OOD Dataset     ALine-D     ALine-S     Naive Agr     ATC     AC     DOC-Feat

CIFAR10C$^*$     5.44     4.73     17.39     6.90     11.49      11.91

CIFAR10.1 v6     1.95     1.99     16.95     2.60     4.93     5.36

CIFAR100C$^*$     7.17     6.79     17.66     8.18     17.58     14.96

ImageNetC$^*$     15.03     14.17     32.27     15.9     22.83     13.42

ImageNetV2 MatchFreq     8.44     8.43     22.21     3.02     15.53     8.43

fMoW-WILDS     14.26     7.29     141.31     12.17     19.87     8.76

OfficeHome-Art      17.78     13.94     40.60     38.52     19.68     44.40

OfficeHome-ClipArt     14.04     11.70     33.44     33.81     14.97     28.77

OfficeHome-Product     14.64     11.78     39.88     84.18     63.05     75.44

OfficeHome-Real     12.65     10.18     36.20     27.72     21.85     28.85

SQuAD-Shifts Reddit     3.61     3.48     26.56     19.06     30.94     9.18

SQuAD-Shifts Amazon     3.61     4.93     26.46     24.35     34.93     7.31

SQuAD-Shifts NYT     1.64     1.75     23.46     4.01     25.96     2.80

SQuAD-Shifts New Wiki     6.33     6.58     25.24     5.18     25.96     7.50

MNLI Mismatched     0.55     0.41     0.55     12.00     0.63     0.51

SNLI     2.9     2.1     2.5     6.3     3.8     8.4

§ CONCLUSION

We develop methods for extending AGL to foundation models to enable OOD performance prediction in this emerging paradigm. We find that utilizing AGL for performance estimation requires a careful tuning of ensemble diversity. Unlike the original paradigm of AGL, where models observed tens or hundreds of epochs of training on the ID dataset, we find that randomness in specific optimization choices, especially linear head initialization, is crucial for foundation models. In fact, in contrast to Baek et al., 2022, we find that linear models can also observe AGL, specifically in the CLIP representation space, suggesting that AGL may not be a neural network specific phenomena.

Our conclusion on AGL also sheds light on the robustness of foundation models. First, our experiments show that light finetuning alone can corrupt models to have diverse behaviors.  Next, in contrast to vision models, where previous works show different forms of pretraining lead to different slopes in the linear correlations [Radford et al., 2021], a term that is often called “effective robustness”, we find that all the language models we evaluate, e.g., OPT, GPT2, GPT2-Neo, Alpaca, Llama, lie on the same accuracy line. This is particularly intriguing because it goes against the common wisdom that the pretraining data influences the models' effective robustness. We leave these questions for future analysis.

[Awadalla et al., 2022]
Anas Awadalla, Mitchell Wortsman, Gabriel Ilharco, Sewon Min, Ian Magnusson, Hannaneh Hajishirzi, and Ludwig Schmidt.
Exploring the landscape of distributional robustness for question answering models.
arXiv preprint arXiv:2210.12517, 2022.

[Baek et al., 2022]
Christina Baek, Yiding Jiang, Aditi Raghunathan, and J Zico Kolter.
Agreement-on-the-line: Predicting the performance of neural networks under distribution shift.
Advances in Neural Information Processing Systems, 35:0 19274–19289, 2022.

[Baumgartner et al., 2020]
Jason Baumgartner, Savvas Zannettou, Brian Keegan, Megan Squire, and Jeremy Blackburn.
The pushshift reddit dataset.
In Proceedings of the international AAAI conference on web and social media, volume 14, pp.  830–839, 2020.

[Ben-David et al., 2006]
Shai Ben-David, John Blitzer, Koby Crammer, and Fernando Pereira.
Analysis of representations for domain adaptation.
Advances in neural information processing systems, 19, 2006.

[Black et al., 2021]
Sid Black, Leo Gao, Phil Wang, Connor Leahy, and Stella Biderman.
GPT-Neo: Large Scale Autoregressive Language Modeling with Mesh-Tensorflow, March 2021.
URL <https://doi.org/10.5281/zenodo.5297715>.
If you use this software, please cite it using these metadata.

[Bowman et al., 2015]
Samuel R Bowman, Gabor Angeli, Christopher Potts, and Christopher D Manning.
A large annotated corpus for learning natural language inference.
arXiv preprint arXiv:1508.05326, 2015.

[Brown et al., 2020]
Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al.
Language models are few-shot learners.
Advances in neural information processing systems, 33:0 1877–1901, 2020.

[Chen et al., 2021]
Jiefeng Chen, Frederick Liu, Besim Avci, Xi Wu, Yingyu Liang, and Somesh Jha.
Detecting errors and estimating accuracy on unlabeled data with self-training ensembles.
arXiv preprint arXiv:2106.15728, 2021.

[Chiang et al., 2023]
Wei-Lin Chiang, Zhuohan Li, Zi Lin, Ying Sheng, Zhanghao Wu, Hao Zhang, Lianmin Zheng, Siyuan Zhuang, Yonghao Zhuang, Joseph E. Gonzalez, Ion Stoica, and Eric P. Xing.
Vicuna: An open-source chatbot impressing gpt-4 with 90%* chatgpt quality, March 2023.
URL <https://lmsys.org/blog/2023-03-30-vicuna/>.

[Chuang et al., 2020]
Ching-Yao Chuang, Antonio Torralba, and Stefanie Jegelka.
Estimating generalization under distribution shifts via domain-invariant representations.
arXiv preprint arXiv:2007.03511, 2020.

[Cortes et al., 2010]
Corinna Cortes, Yishay Mansour, and Mehryar Mohri.
Learning bounds for importance weighting.
Advances in neural information processing systems, 23, 2010.

[Deng & Zheng, 2021]
Weijian Deng and Liang Zheng.
Are labels always necessary for classifier accuracy evaluation?
In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp.  15064–15073. IEEE Computer Society, 2021.

[Deng et al., 2021]
Weijian Deng, Stephen Gould, and Liang Zheng.
What does rotation prediction tell us about classifier accuracy under varying testing environments?
arXiv preprint arXiv:2106.05961, 2021.

[Devlin et al., 2018]
Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.
Bert: Pre-training of deep bidirectional transformers for language understanding.
arXiv preprint arXiv:1810.04805, 2018.

[Dosovitskiy et al., 2020]
Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al.
An image is worth 16x16 words: Transformers for image recognition at scale.
arXiv preprint arXiv:2010.11929, 2020.

[Elsahar & Gallé, 2019]
Hady Elsahar and Matthias Gallé.
To annotate or not? predicting performance drop under domain shift.
In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pp.  2163–2173, 2019.

[Fang et al., 2022]
Alex Fang, Gabriel Ilharco, Mitchell Wortsman, Yuhao Wan, Vaishaal Shankar, Achal Dave, and Ludwig Schmidt.
Data determines distributional robustness in contrastive language image pre-training (clip).
In International Conference on Machine Learning, pp.  6216–6234. PMLR, 2022.

[Gao et al., 2020]
Leo Gao, Stella Biderman, Sid Black, Laurence Golding, Travis Hoppe, Charles Foster, Jason Phang, Horace He, Anish Thite, Noa Nabeshima, et al.
The pile: An 800gb dataset of diverse text for language modeling.
arXiv preprint arXiv:2101.00027, 2020.

[Garg et al., 2022]
Saurabh Garg, Sivaraman Balakrishnan, Zachary C Lipton, Behnam Neyshabur, and Hanie Sedghi.
Leveraging unlabeled data to predict out-of-distribution performance.
International Conference on Learning Representations, 2022.

[Grill et al., 2020]
Jean-Bastien Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre Richemond, Elena Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Guo, Mohammad Gheshlaghi Azar, et al.
Bootstrap your own latent-a new approach to self-supervised learning.
Advances in neural information processing systems, 33:0 21271–21284, 2020.

[Guillory et al., 2021]
Devin Guillory, Vaishaal Shankar, Sayna Ebrahimi, Trevor Darrell, and Ludwig Schmidt.
Predicting with confidence on unseen distributions.
In Proceedings of the IEEE/CVF international conference on computer vision, pp.  1134–1144, 2021.

[Hendrycks & Dietterich, 2019]
Dan Hendrycks and Thomas G. Dietterich.
Benchmarking neural network robustness to common corruptions and perturbations.
In 7th International Conference on Learning Representations, ICLR, 2019.

[Hendrycks & Gimpel, 2017]
Dan Hendrycks and Kevin Gimpel.
A baseline for detecting misclassified and out-of-distribution examples in neural networks.
In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings, 2017a.

[Hendrycks & Gimpel, 2017]
Dan Hendrycks and Kevin Gimpel.
A baseline for detecting misclassified and out-of-distribution examples in neural networks.
International Conference on Learning Representations, 2017b.

[Hu et al., 2021]
Edward J Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen.
Lora: Low-rank adaptation of large language models.
arXiv preprint arXiv:2106.09685, 2021.

[Ilharco et al., 2021]
Gabriel Ilharco, Mitchell Wortsman, Ross Wightman, Cade Gordon, Nicholas Carlini, Rohan Taori, Achal Dave, Vaishaal Shankar, Hongseok Namkoong, John Miller, Hannaneh Hajishirzi, Ali Farhadi, and Ludwig Schmidt.
Openclip, July 2021.
URL <https://doi.org/10.5281/zenodo.5143773>.
If you use this software, please cite it as below.

[Jiang et al., 2022]
Yiding Jiang, Vaishnavh Nagarajan, Christina Baek, and J Zico Kolter.
Assessing generalization of sgd via disagreement.
International Conference on Learning Representations, 2022.

[Koh et al., 2021]
Pang Wei Koh, Shiori Sagawa, Henrik Marklund, Sang Michael Xie, Marvin Zhang, Akshay Balsubramani, Weihua Hu, Michihiro Yasunaga, Richard Lanas Phillips, Irena Gao, et al.
Wilds: A benchmark of in-the-wild distribution shifts.
In International Conference on Machine Learning, pp.  5637–5664. PMLR, 2021.

[Krizhevsky et al., 2009]
Alex Krizhevsky, Geoffrey Hinton, et al.
Learning multiple layers of features from tiny images.

[Kuzborskij & Orabona, 2013]
Ilja Kuzborskij and Francesco Orabona.
Stability and hypothesis transfer learning.
In International Conference on Machine Learning, pp.  942–950. PMLR, 2013.

[Lee et al., 2023]
Donghwan Lee, Behrad Moniri, Xinmeng Huang, Edgar Dobriban, and Hamed Hassani.
Demystifying disagreement-on-the-line in high dimensions, 2023.

[Loshchilov & Hutter, 2017]
Ilya Loshchilov and Frank Hutter.
Decoupled weight decay regularization.
arXiv preprint arXiv:1711.05101, 2017.

[Madani et al., 2004]
Omid Madani, David Pennock, and Gary Flake.
Co-validation: Using model disagreement on unlabeled data to validate classification algorithms.
Advances in neural information processing systems, 17, 2004.

[Malinin et al., 2022]
Andrey Malinin, Neil Band, Ganshin, Alexander, German Chesnokov, Yarin Gal, Mark J. F. Gales, Alexey Noskov, Andrey Ploskonosov, Liudmila Prokhorenkova, Ivan Provilkov, Vatsal Raina, Vyas Raina, Roginskiy, Denis, Mariya Shmatova, Panos Tigas, and Boris Yangel.
Shifts: A dataset of real distributional shift across multiple large-scale tasks, 2022.

[Mansour et al., 2009]
Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh.
Domain adaptation: Learning bounds and algorithms.
arXiv preprint arXiv:0902.3430, 2009.

[Miller et al., 2020]
John Miller, Karl Krauth, Benjamin Recht, and Ludwig Schmidt.
The effect of natural distribution shift on question answering models.
In International conference on machine learning, pp.  6905–6916. PMLR, 2020.

[Miller et al., 2021]
John P Miller, Rohan Taori, Aditi Raghunathan, Shiori Sagawa, Pang Wei Koh, Vaishaal Shankar, Percy Liang, Yair Carmon, and Ludwig Schmidt.
Accuracy on the line: on the strong correlation between out-of-distribution and in-distribution generalization.
In International Conference on Machine Learning, pp.  7721–7735. PMLR, 2021.

[Nakkiran & Bansal, 2020]
Preetum Nakkiran and Yamini Bansal.
Distributional generalization: A new kind of generalization.
arXiv preprint arXiv:2009.08092, 2020.

[Podkopaev & Ramdas, 2021]
Aleksandr Podkopaev and Aaditya Ramdas.
Distribution-free uncertainty quantification for classification under label shift.
In Uncertainty in artificial intelligence, pp.  844–853. PMLR, 2021.

[Radford et al., 2019]
Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al.
Language models are unsupervised multitask learners.
OpenAI blog, 10 (8):0 9, 2019.

[Radford et al., 2021]
Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al.
Learning transferable visual models from natural language supervision.
In International conference on machine learning, pp.  8748–8763. PMLR, 2021.

[Rajpurkar et al., 2016]
Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang.
Squad: 100,000+ questions for machine comprehension of text.
arXiv preprint arXiv:1606.05250, 2016.

[Recht et al., 2018]
Benjamin Recht, Rebecca Roelofs, Ludwig Schmidt, and Vaishaal Shankar.
Do cifar-10 classifiers generalize to cifar-10?
arXiv preprint arXiv:1806.00451, 2018.

[Recht et al., 2019]
Benjamin Recht, Rebecca Roelofs, Ludwig Schmidt, and Vaishaal Shankar.
Do imagenet classifiers generalize to imagenet?
In International conference on machine learning, pp.  5389–5400. PMLR, 2019.

[Roelofs et al., 2019]
Rebecca Roelofs, Vaishaal Shankar, Benjamin Recht, Sara Fridovich-Keil, Moritz Hardt, John Miller, and Ludwig Schmidt.
A meta-analysis of overfitting in machine learning.
Advances in Neural Information Processing Systems, 32, 2019.

[Russakovsky et al., 2014]
Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael S. Bernstein, Alexander C. Berg, and Li Fei-Fei.
Imagenet large scale visual recognition challenge.
CoRR, abs/1409.0575, 2014.
URL <http://arxiv.org/abs/1409.0575>.

[Schelter et al., 2020]
Sebastian Schelter, Tammo Rukat, and Felix Biessmann.
Learning to validate the predictions of black box classifiers on unseen data.
In Proceedings of the 2020 ACM SIGMOD International Conference on Management of Data, pp.  1289–1299, New York, NY, USA, 2020. Association for Computing Machinery.
ISBN 9781450367356.

[Tan & Le, 2019]
Mingxing Tan and Quoc Le.
Efficientnet: Rethinking model scaling for convolutional neural networks.
In International conference on machine learning, pp.  6105–6114. PMLR, 2019.

[Taori et al., 2020]
Rohan Taori, Achal Dave, Vaishaal Shankar, Nicholas Carlini, Benjamin Recht, and Ludwig Schmidt.
Measuring robustness to natural distribution shifts in image classification.
Advances in Neural Information Processing Systems, 33:0 18583–18599, 2020.

[Taori et al., 2023]
Rohan Taori, Ishaan Gulrajani, Tianyi Zhang, Yann Dubois, Xuechen Li, Carlos Guestrin, Percy Liang, and Tatsunori B. Hashimoto.
Stanford alpaca: An instruction-following llama model.
<https://github.com/tatsu-lab/stanford_alpaca>, 2023.

[Touvron et al., 2023]
Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, Dan Bikel, Lukas Blecher, Cristian Canton Ferrer, Moya Chen, Guillem Cucurull, David Esiobu, Jude Fernandes, Jeremy Fu, Wenyin Fu, Brian Fuller, Cynthia Gao, Vedanuj Goswami, Naman Goyal, Anthony Hartshorn, Saghar Hosseini, Rui Hou, Hakan Inan, Marcin Kardas, Viktor Kerkez, Madian Khabsa, Isabel Kloumann, Artem Korenev, Punit Singh Koura, Marie-Anne Lachaux, Thibaut Lavril, Jenya Lee, Diana Liskovich, Yinghai Lu, Yuning Mao, Xavier Martinet, Todor Mihaylov, Pushkar Mishra, Igor Molybog, Yixin Nie, Andrew Poulton, Jeremy Reizenstein, Rashi Rungta, Kalyan Saladi, Alan Schelten, Ruan Silva, Eric Michael Smith, Ranjan Subramanian, Xiaoqing Ellen Tan, Binh Tang, Ross Taylor, Adina Williams, Jian Xiang Kuan, Puxin Xu, Zheng Yan, Iliyan Zarov, Yuchen Zhang, Angela Fan, Melanie Kambadur, Sharan Narang, Aurelien Rodriguez, Robert Stojnic, Sergey Edunov, and Thomas
Llama 2: Open foundation and fine-tuned chat models, 2023.

[Tran et al., 2022]
Dustin Tran, Jeremiah Liu, Michael W. Dusenberry, Du Phan, Mark Collier, Jie Ren, Kehang Han, Zi Wang, Zelda Mariet, Huiyi Hu, Neil Band, Tim G. J. Rudner, Karan Singhal, Zachary Nado, Joost van Amersfoort, Andreas Kirsch, Rodolphe Jenatton, Nithum Thain, Honglin Yuan, Kelly Buchanan, Kevin Murphy, D. Sculley, Yarin Gal, Zoubin Ghahramani, Jasper Snoek, and Balaji Lakshminarayanan.
Plex: Towards reliability using pretrained large model extensions, 2022.

[Trinh & Le, 2018]
Trieu H Trinh and Quoc V Le.
A simple method for commonsense reasoning.
arXiv preprint arXiv:1806.02847, 2018.

[Venkateswara et al., 2017]
Hemanth Venkateswara, Jose Eusebio, Shayok Chakraborty, and Sethuraman Panchanathan.
Deep hashing network for unsupervised domain adaptation.
In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp.  5018–5027, 2017.

[Wang et al., 2018]
Alex Wang, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel R. Bowman.
GLUE: A multi-task benchmark and analysis platform for natural language understanding.
In the Proceedings of ICLR.

[Wang et al., 2023]
Dequan Wang, Xiaosong Wang, Lilong Wang, Mengzhang Li, Qian Da, Xiaoqiang Liu, Xiangyu Gao, Jun Shen, Junjun He, Tian Shen, et al.
Medfmc: A real-world dataset and benchmark for foundation model adaptation in medical image classification.
arXiv preprint arXiv:2306.09579, 2023.

[Williams et al., 2018]
Adina Williams, Nikita Nangia, and Samuel R. Bowman.
A broad-coverage challenge corpus for sentence understanding through inference, 2018.

[Wortsman et al., 2022]
Mitchell Wortsman, Gabriel Ilharco, Jong Wook Kim, Mike Li, Simon Kornblith, Rebecca Roelofs, Raphael Gontijo Lopes, Hannaneh Hajishirzi, Ali Farhadi, Hongseok Namkoong, et al.
Robust fine-tuning of zero-shot models.
In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.  7959–7971, 2022.

[Yadav & Bottou, 2019]
Chhavi Yadav and Léon Bottou.
Cold case: The lost mnist digits.
Advances in neural information processing systems, 32, 2019.

[Yu et al., 2022]
Yaodong Yu, Stephen Bates, Yi Ma, and Michael Jordan.
Robust calibration with multi-domain temperature scaling.
Advances in Neural Information Processing Systems, 35:0 27510–27523, 2022a.

[Yu et al., 2022]
Yaodong Yu, Zitong Yang, Alexander Wei, Yi Ma, and Jacob Steinhardt.
Predicting out-of-distribution error with the projection norm, 2022b.

[Zhang et al., 2022]
Susan Zhang, Stephen Roller, Naman Goyal, Mikel Artetxe, Moya Chen, Shuohui Chen, Christopher Dewan, Mona Diab, Xian Li, Xi Victoria Lin, Todor Mihaylov, Myle Ott, Sam Shleifer, Kurt Shuster, Daniel Simig, Punit Singh Koura, Anjali Sridhar, Tianlu Wang, and Luke Zettlemoyer.
Opt: Open pre-trained transformer language models, 2022.

[Zhu et al., 2015]
Yukun Zhu, Ryan Kiros, Rich Zemel, Ruslan Salakhutdinov, Raquel Urtasun, Antonio Torralba, and Sanja Fidler.
Aligning books and movies: Towards story-like visual explanations by watching movies and reading books.
In Proceedings of the IEEE international conference on computer vision, pp.  19–27, 2015.

[Zou et al., 2023]
Yuli Zou, Weijian Deng, and Liang Zheng.
Adaptive calibrator ensemble for model calibration under distribution shift.
arXiv preprint arXiv:2303.05331, 2023.

§ APPENDIX

§.§ Details Regarding OOD Performance Estimation Baselines

§.§.§ AGL-Based Estimation Methods: ALine-S/D

ALine algorithms are the AGL-based performance estimation methods proposed in [Baek et al., 2022]. When the AGL phenomenon occurs, i.e., models observe a strong linear correlation in both ID versus OOD agreement and accuracy with matching slopes and biases, algorithms ALine-S and ALine-D effectively apply the linear transformation calculated using agreements to map the ID performances to OOD performance estimates. We describe the algorithms in more detail below.

AGL Provided a collection of models $\mathcal{F} = \{f_1, f_2, ..., f_n\}$, AGL suggests that ID versus OOD accuracy observe a strong linear correlation if and only if ID versus OOD agreement observes a strong linear correlation and when they do, the slopes and biases match: $\forall f_i, f_j \in \mathcal{F}$ where $i \neq j$

\begin{equation}
\begin{gathered}
\Phi^{-1}(\text{Acc}_{\text{OOD}}(f_i)) = a \cdot \Phi^{-1}(\text{Acc}_{\text{ID}}(f_i)) + b  \\
\Updownarrow \quad \\
\Phi^{-1}(\text{Agr}_{\text{OOD}}(f_i, f_j)) = a \cdot \Phi^{-1}(\text{Agr}_{\text{ID}}(f_i, f_j)) + b
\end{gathered}
\label{eq:AGL}
\end{equation}

$\Phi^{-1}$ is the probit transform used to induce a better linear fit as used in [Baek et al., 2022] and [Miller et al., 2021]. Provided access to $\text{Acc}_{\text{ID}}(f_i), \text{Agr}_{\text{ID}}(f_i, f_j), \text{Agr}_{\text{OOD}}(f_i, f_j) \ \forall i, j$, we'd like to estimate $\text{Acc}_{\text{OOD}}(f_i)$ for all $f_i \in \mathcal{F}$.

ALine-S The algorithm ALine-S simply estimates the the slope $a$ and bias $b$ of accuracy by computing the linear fit of agreement.

\begin{equation}\label{eq:your_label}
\hat{a}, \hat{b} = \arg \min_{a, b \in \mathbb{R}} \sum_{i \neq j} \left( \Phi^{-1}(\hat{\text{Agr}}_{\text{OOD}}(f_i, f_j)) - a \cdot \Phi^{-1}(\hat{\text{Agr}}_{\text{ID}}(f_i, f_j)) - b \right)^2
\end{equation}

With $\hat{a}$ and $\hat{b}$, we estimate ${\text{Acc}_{\text{OOD}}}(f_i) \approx \hat{a} \cdot  {\text{Acc}_{\text{ID}}}(f_i) + \hat{b}$. This method is called Aline-S.

ALine-D This method instead constructs the following system of linear equations. Provided the relation in Equation <ref>, one can derive that for any $f_i, f_j \in \mathcal{F}$,

\begin{equation}
\begin{aligned}
&\frac{1}{2}\left(\Phi^{-1}(\text{Acc}_{\text{OOD}}(f_i)) + \Phi^{-1}(\text{Acc}_{\text{OOD}}(f_j))\right) \\
&\approx \Phi^{-1}(\text{Agr}_{\text{OOD}}(f_i, f_j)) + \hat{a} \cdot \left(\frac{\Phi^{-1}(\text{Acc}_{\text{ID}}(f_i)) + \Phi^{-1}(\text{Acc}_{\text{ID}}(f_j))}{2} - \Phi^{-1}(\text{Agr}_{\text{ID}}(f_i, f_j))\right)
\end{aligned}
\end{equation}

Treating $\text{Acc}_{\text{OOD}}(f_i) \ \forall i$ as unknown variables, note that the right hand side is known and we can construct a linear system of equations using all $n \choose 2$ pairs of models. The algorithm employs least squares to solve this approximate system of linear equations.

§.§.§ Temperature Scaling for Confidence-Based Estimation Methods

We compare ALine against confidence-based methods ATC [Garg et al., 2022], AC [Hendrycks & Gimpel, 2017], and Doc-Feat [Guillory et al., 2021]. These methods notably perform better after calibrating the models ID by temperature scaling.

Classification For classification tasks, we optimize a temperature $T$ for each model $f$ on the cross-entropy loss over the in-distribution validation data.

\begin{align}
\min_T \sum_{x, y} \mathsf{CE}(\sigma(f(x) \exp(T)), y)
\end{align}

where $\sigma(\cdot)$ is the softmax.

Question-Answering For extractive question-answering tasks, the model has to predict two labels – the start and end token index $y = [y_s, y_e]$ of the context span that answers the question. For each model, we attach a span prediction head $v = [v_s, v_e] \in \R^{d \times 2}$ on top of the  base  $\mathsf{B}_\theta(x) \in \R^{d \times N}$ where $N$ is the token length of $x$. $s(x) = v_s^\top \mathsf{B}_\theta(x)$ and $e(x) = v_e^\top \mathsf{B}_\theta(x)$ predict the start and end token index, respectively.

We're interested in evaluating question-answering models on the exact match (EM) objective,

\begin{align}
\mathsf{EM}(\hat{y}, y) = \indicator{\hat{y}_s = y_s} \cdot \indicator{\hat{y}_e = y_e}
\end{align}

EM treats question-answering as a classification problem over $N \times N$ choices of start and end index pairs. This allows us to utilize confidence-based methods that are designed for classification tasks. We can calculate the model confidence for index pair $[i, j]$ as $\sigma(s(x))_i \cdot \sigma(e(x))_j$.

We jointly optimize a separate temperature for the start and end logits $T_s$ and $T_e$ and we minimize the cross-entropy loss over the in-distribution validation data.

\begin{align}
\min_T \sum_{x, y} \mathsf{CE}(\sigma(s(x) \exp(T_s)) \sigma(e(x) \exp(T_e))^\top , y)
\end{align}

§.§.§ Comparison to ProjNorm

In this section, we present a comparison with ProjNorm [Yu et al., 2022], a method that yields a score which is shown to be correlated with the OOD performance of the model. We study the same setting as presented in Section <ref> where we estimate OOD performance of foundation models pretrained on different text corpora. Unlike AGL, ProjNorm doesn't provide an estimate of the OOD performance hence we compare the linear correlation between the predicted value and OOD performance. From Table <ref> it can be seen that estimates from ALine-D are more strongly correlated with OOD performance than ProjNorm.

Correlation coefficient between OOD Accuracy and Prediction for ALine-D and ProjNorm
max width=

Method     SQuAD-Shifts Amazon     SQuAD-Shifts Reddit

ALine-D     0.98     0.98

ProjNorm     0.64     0.79

§.§.§ Failure Datasets with Low Linear Correlation

In our comparison with baselines in Section <ref>, we filter out datasets with a low correlation coefficient $\leq 0.95$ in ID vs OOD agreement. When the linear correlation of agreement is weak, AGL tells us that the correlation is also low for accuracy, and AGL-based methods are not guaranteed to be reliable in such circumstances. We provide ID vs OOD accuracy and agreement scatter plots for all datasets in Appendix <ref>.

Below, we separately provide the comparison with baselines for the excluded datasets. We generally find that the baseline ATC [Garg et al., 2022] is significantly better in circumstances where AGL-based methods are unreliable.

The MAPE (%) of predicting OOD performance using AGL-based ALine and other baseline methods. We collect a diverse ensemble by randomizing the linear initialization and including multiple base models.

max width=

OOD Dataset     Agreement $R^2$     ALine-D     ALine-S     Naive Agr     ATC     AC     DOC-Feat

CIFAR10C Gaussian Noise     0.85     65.59     56.67     41.85     42.77     74.60     75.40

CIFAR10C Glass Blur     0.89     44.24     44.97     31.24     33.58     79.45     80.36

CIFAR10C Shot Noise     0.89     45.02     37.60     27.70     28.25     49.23     49.90

CIFAR10C Speckle Noise     0.90     36.67     30.13     23.45     22.72     43.46     44.12

CIFAR100C Gaussian Noise     0.93     34.98     32.21     28.88     21.18     69.04     63.81

CIFAR100C Glass Blur     0.93     66.03     63.45     42.41     25.47     109.73     103.70

ImageNetC Gaussian Noise     0.89     50.10     47.26     73.47     13.21     54.86     42.66

ImageNetC Glass Blur     0.87     74.20     71.84     100.32     15.97     74.12     59.60

ImageNetC Impulse Noise     0.88     62.86     59.68     87.83     16.54     63.23     49.99

ImageNetC Shot Noise     0.88     53.89     51.12     78.37     14.66     57.47     44.58

iWildCam-WILDS     0.85     22.05     25.29     46.42     37.25     57.31     69.58

Camelyon17-WILDS     0.59     10.14     6.44     13.26     6.46     8.76     8.90

§.§ Finetuning Hyperparameters

We state here the hyperparameters used to finetune the models for diversity experiments reported in Section <ref>.

§.§.§ Linear Probing over CLIP for Vision Tasks

We train all linear probes using SGD. Models are trained for different timesteps to achieve an even distribution of ID accuracies.

CLIP Linear Probing
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Dataset     Hyperparameters

CIFAR10     Learning Rate: $5 \times 10^{-4}$
Batch Size: 1028

22cmCIFAR100     Learning Rate: $1 \times 10^{-3}$
Batch Size: 1028

22cmImageNet     Learning Rate: $1 \times 10^{-1}$
Batch Size: 1028

22cmOfficeHome     Learning Rate: $1 \times 10^{-3}$
Batch Size: 200

22cmFMoW-WILDS     Learning Rate: $1 \times 10^{-3}$
Batch Size: 200

22cmiWildCam-WILDS     Learning Rate: $1 \times 10^{-3}$
Batch Size: 200

22cmCamelyon17-WILDS     Learning Rate: $1 \times 10^{-3}$
Batch Size: 200

§.§.§ Full Finetuning GPT, OPT, BERT on Language Tasks

We use AdamW [Loshchilov & Hutter, 2017] to full finetune language models. We keep the learning rate small. Models are trained for different timesteps to achieve an even distribution of ID accuracies.

Full Finetuning GPT2-Medium on SQuAD
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Source of Diversity     Hyperparameters

45cmInitialization +  Ordering
Learning rate: $2 \times 10^{-7}$

Weight Decay: $1\times 10^{-5}$

Batch Size: 32

Max Epochs:  $20$

45cmSubsetting (10, 30 $\%$ of data)
Learning rate: $6, 4 \times 10^{-7}$

Weight Decay: $1\times 10^{-5}$

Batch Size:

Max Epochs:  $20$

Full Finetuning OPT-125M on SQuAD
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Source of Diversity     Hyperparameters

45cmInitialization +  Ordering
Learning rate: $4 \times 10^{-7}$

Weight Decay: $1\times 10^{-5}$

Batch Size: 32

Max Epochs:  $20$

45cmSubsetting (10, 30, 50 $\%$ of data)
Learning rate: $40, 12, 8 \times 10^{-7}$

Weight Decay: $1\times 10^{-5}$

Batch Size: 32

Max Epochs:  $20$

Full Finetuning BERT-Uncased on SQuAD
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Source of Diversity     Hyperparameters

45cmInitialization +  Ordering
Learning rate: $2 \times 10^{-7}$

Weight Decay: $1\times 10^{-5}$

Batch Size: 32

Max Epochs:  $20$

45cmSubsetting (10, 30, 50 $\%$ of data)
Learning rate: $20, 6, 4 \times 10^{-7}$

Weight Decay: $1\times 10^{-5}$

Batch Size: 32

Max Epochs:  $20$

Full Finetuning GPT2-Medium on MNLI
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Source of Diversity     Hyperparameters

45cmInitialization +  Ordering
Learning rate: $5 \times 10^{-4}$

Weight Decay: $1\times 10^{-5}$

Batch Size: 128

Max Epochs:  $10$

45cmSubsetting (10$\%$ of data)
Learning rate: $5 \times 10^{-3}$

Weight Decay: $1\times 10^{-5}$

Batch Size: 128

Max Epochs:  $10$

Full Finetuning OPT-125M on MNLI
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Source of Diversity     2cOPT-125M

Varied     Fixed

45cmInitialization +  Ordering
Learning rate: $1 \times 10^{-3}$

Weight Decay: $1\times 10^{-5}$

Batch Size: 128

Max Epochs:  $10$

45cmSubsetting (10$\%$ of data)
Learning rate: $1 \times 10^{-2}$

Weight Decay: $1\times 10^{-5}$

Batch Size: 128

Max Epochs:  $10$

§.§ More Experiments on the Effect of Diversity Source with CLIP

§.§.§ OfficeHome Linear Probing Diversity Experiments

In Section <ref>, we examine how diversity source impacts whether AGL is observed in linear probed CLIP models from CIFAR10 to CIFAR10C. Here, we perform the same experiment on Office-Home [Venkateswara et al., 2017], which consists of 4 domains or image styles (“Art”, “Clip Art”, “Product”, and “Real World”) for 65 common objects. We train models on one domain and treat the remaining three domains as OOD. Similarly, only Random Initialization yields AGL or matching slopes in accuracy and agreement, and as a result, the corresponding MAPE of estimating the OOD performance of this diverse ensemble is the smallest.

Random Head

Data Ordering

Data Subsetting

ID vs OOD accuracy and agreement of linear probed CLIP models on OfficeHome Art (top row), Product (middle row), and Real World (bottom row). The figure title is the OOD domain.

The ALine-S MAPE(%) for ensembles trained on each domain of OfficeHome.
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OfficeHome ID domain     Source of Diversity     OOD Estimation MAPE(%)

Art     Random Linear Heads     14.09

Data Ordering     20.67

Data Subsetting     120.85

ClipArt     Random Linear Heads     12.23

Data Ordering     28.45

Data Subsetting     78.49

Product     Random Linear Heads     13.63

Data Ordering     110.45

Data Subsetting     92.97

Real     Random Linear Heads     9.95

Data Ordering     33.36

Data Subsetting    3078

§.§.§ AGL appears in Random Head CLIP Ensembles across Datasets

We report the strength of AGL in linear probed CLIP models with randomly initialized linear heads across all datasets discussed in Section <ref>.

AGL and ACL for all C10C shifts with random head initialization finetuning.

AGL and ACL for the C10.1 shifts with random head initialization finetuning.

AGL and ACL for the C100C shifts with random head initialization finetuning.

AGL and ACL for the ImageNetC shifts with random head initialization finetuning.

AGL and ACL for the ImageNet V2 shifts with random head initialization finetuning.

AGL and ACL for 3 benchmarks from the WILDS dataset with random head initialization finetuning.

AGL and ACL for the OfficeHome ClipArt, Product, Real shifts with random head initialization finetuning over OfficeHome Art.

ID vs OOD accuracy and agreement with OH ClipArt ID
AGL and ACL for the OfficeHome Art, Product, Real shifts with random head initialization finetuning over OfficeHome ClipArt.

AGL and ACL for the OfficeHome ClipArt, Art, Real shifts with random head initialization finetuning over OfficeHome Product.

AGL and ACL for the OfficeHome Art, ClipArt, Product shifts with random head initialization finetuning over OfficeHome Real.

§.§.§ Diversity Matters in Full Finetuned CLIP Ensembles

We show that light full finetuning over CLIP also observes similar effects of diversity source on the strength of AGL. We verify this on shifts from CIFAR10 to CIFAR10C.

Random Head

Data Ordering

Data Subsetting

ID vs OOD accuracy and agreement of full finetuned CLIP models on shift from CIFAR10 to CIFAR10C “JPEG Compression” (top row) and “Pixelate” (bottom row) shifts. Only Random Initialization (Column a) yields AGL or matching slopes in accuracy and agreement.

The average MAPE(%) of ALine estimates of OOD performance of full finetuned CLIP models across all 19 CIFAR10C shifts. As can be seen from Figure <ref>, only ensembles with diverse random initialization consistently results in smallest MAPE values.
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Source of Diversity     CIFAR10C MAPE(%)

Random Linear Heads     29.37

Data Ordering     69.98

Data Subsetting     33.57

§.§.§ Any diverse ensemble display AGL in models heavily trained from scratch

On the other hand, we demonstrate that in models heavily trained from scratch, AGL can be observed irrespective of the diversity source. Consistent with the models in [Baek et al., 2022], we train ResNet18 on CIFAR10 from scratch, varying the different sources of randomness. These models are trained heavily with SGD with learning rate of $1\times10^{-2}$, batch size 128, and weight decay of $1 \times 10^{-5}$ for up to 200 epochs. We do not use any data augmentation.

Effect of Diversity Source on ResNet18 from CIFAR10 to CIFAR10C-Snow

§.§.§ Effect of Training Dataset Size

To rid of any confounding factors from data subsetting observing a smaller amount of data, we evaluate all randomness sources on different training dataset sizes. We track the effect of diversity in random initialization, data ordering, and data subsetting for different portions of the training data (100$\%$, 50$\%$, 30$\%$, $10\%$). For each percentage $x\%$, Random Initialization and Data Ordering ensembles are trained on the same randomly sampled $x\%$ proportion of the data while each model in the Data Subsetting ensemble observe different randomly sampled $x\%$ portions

Random Head

Data ordering

ID vs OOD accuracy and agreement of linear probe CLIP models finetuned on 100% of CIFAR10 training data evaluated on the CIFAR10C “JPEG Compression” (top row) and “Pixelate” (bottom row) shifts

Random Head

Data ordering

Data Subsetting

ID vs OOD accuracy and agreement of linear probe CLIP models finetuned on 50% of CIFAR10 training data evaluated on the CIFAR10C “JPEG Compression” (top) and “Pixelate” (bottom) shifts

Random Head

Data ordering

Data Subsetting

ID vs OOD accuracy and agreement of linear probe CLIP models finetuned on 30% of the CIFAR10 training data evaluated on the CIFAR10C “JPEG Compression” (top) and “Pixelate” (bottom) shifts

Random Head

Data Ordering

Data Subsetting

ID vs OOD accuracy and agreement of linear probe CLIP models finetuned on 10% of the CIFAR10 training data and evaluated on the CIFAR10C “JPEG Compression” (top) and “Pixelate” (bottom) shifts

§.§ More Experiments on the Effect of Diversity Source for Extractive Question Answering

§.§.§ Single-Base Diversity Experiments Using OPT and BERT

In Section <ref>, we report the effect of diversity source in full finetuned GPT2-Medium models for the extractive QA task SQuAD to SQuAD-Shifts. In this section, we also provide the same experiments on OPT-125M and BERT. As observed in GPT, using Random Heads yields the strongest AGL behavior and achieves the smallest ALine-D MAPE.

In Table <ref> and <ref>, we report the MAPE of OOD performance estimation of models trained with Random Initialization and Data Ordering using $100\%$ of training data and Data Subsetting with $10\%$ of training data.

The average MAPE ($\%$) of ALine-D OOD performance estimates of OPT-125M models full finetuned on SQuAD.
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Source of Diversity     SQuAD-Shifts Amazon     SQuAD-Shifts Reddit

Random Head     6.54     5.43

Data Ordering     11.37     8.70

Data Subsetting     11.15     9.65

The average MAPE ($\%$) of ALine-D OOD performance estimates of BERT models full finetuned on SQuAD.
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Source of Diversity     SQuAD-Shifts Amazon    SQuAD-Shifts Reddit

Random Head     14.70     8.62

Data Ordering     16.55     9.16

Data Subsetting     22.13     18.64

§.§.§ Diversity in Full Finetuned GPT2 for Different Data Portions

We track the effect of diversity in random initialization, data ordering, and data subsetting for different portions of the training data (100$\%$, 50$\%$, 30$\%$, $10\%$). For each percentage $x\%$, Random Initialization and Data Ordering ensembles are trained on the same randomly sampled $x\%$ proportion of the data while each model in the Data Subsetting ensemble observe different randomly sampled $x\%$ portions.

Random Head

Data Ordering

ID vs OOD accuracy and agreement of models finetuned on SQuAD from a single pretrained GPT2 model with 100% of the training data

Random Head

Data Ordering

Data Subsetting

ID vs OOD accuracy and agreement of models finetuned on SQuAD from a single pretrained GPT2 model with 50% of the training data

Random Head

Data Ordering

Data Subsetting

ID vs OOD accuracy and agreement of models finetuned on SQuAD from a single pretrained GPT2 model with 10% of the training data

§.§.§ Diversity in Full Finetuned OPT for Different Data Portions

We track the effect of diversity in random initialization, data ordering, and data subsetting for different portions of the training data (100$\%$, 50$\%$, 30$\%$, $10\%$). For each percentage $x\%$, Random Initialization and Data Ordering ensembles are trained on the same randomly sampled $x\%$ proportion of the data while each model in the Data Subsetting ensemble observe different randomly sampled $x\%$ portions.

Random Head

Data Ordering

ID vs OOD accuracy and agreement of models finetuned on SQuAD from a single pretrained OPT-125m with 100% of the training data

Random Head

Data Ordering

Data Subsetting

ID vs OOD accuracy and agreement of models finetuned on SQuAD from a single pretrained OPT-125m with 50% of the training data

Random Head

Data Ordering

Data Subsetting

ID vs OOD accuracy and agreement of models finetuned on SQuAD from a single pretrained OPT-125m with 30% of the training data

Random Head

Data Ordering

Data Subsetting

ID vs OOD accuracy and agreement of models finetuned on SQuAD from a single pretrained OPT-125m with 10% of the training data

§.§.§ Diversity in Full Finetuned BERT for Different Data Portions

We track the effect of diversity in random initialization, data ordering, and data subsetting for different portions of the training data (100$\%$, 50$\%$, 30$\%$, $10\%$). For each percentage $x\%$, Random Initialization and Data Ordering ensembles are trained on the same randomly sampled $x\%$ proportion of the data while each model in the Data Subsetting ensemble observe different randomly sampled $x\%$ portions.

Random Head

Data Ordering

ID vs OOD accuracy and agreement of models finetuned on SQuAD from a single pretrained BERT with 100% of the training data

Random Head

Data Ordering

Data Subsetting

ID vs OOD accuracy and agreement of models finetuned on SQuAD from a single pretrained BERT with 50% of the training data

Random Head

Data Ordering

Random Head

ID vs OOD accuracy and agreement of models finetuned on SQuAD from a single pretrained BERT with 30% of the training data

Random Head

Data Ordering

Random Head

ID vs OOD accuracy and agreement of models finetuned on SQuAD from a single pretrained BERT with 10% of the training data

§.§ More Experiments on the Effect of Diversity Source for Text Classification

§.§.§ Diversity Source for Linear Probing

In Section <ref>, we reported the effect of diversity source for full finetuned OPT. Here, we demonstrate similar results on text classification shift from MNLI-matched to SNLI with linear probed GPT2-Medium (Figure <ref>) and OPT-125M (Figure <ref>). Similarly, random initialization observes the strongest AGL behavior. In Table <ref>, we report the average MAPE of OOD performance estimation using ALine across models.

Random Head

Data Ordering

Data Subsetting

ID vs OOD accuracy and agreement of models finetuned on MNLI from GPT2-Medium. Random Head and Data Ordering ensembles are trained on $100\%$ of the training data while Data Subsetting ensemble is on $10\%$.

Random Head

Data Ordering

Data Subsetting

ID vs OOD accuracy and agreement of models finetuned on MNLI from OPT-125M. Random Head and Data Ordering ensembles are trained on $100\%$ of the training data while Data Subsetting ensemble is on $10\%$.

The average MAPE (%) of ALine-D performance estimates of accuracy on SNLI.
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Model     GPT2-Medium     OPT-125M

Random Linear Heads     5.9     5.4

Data Ordering     6.7     12.4

Data Subsetting     5.3     8.6