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In-Context Learning through the Bayesian Prism

Madhur Panwar, Kabir Ahuja, Navin Goyal

arXiv:2306.04891v2cs.LGcs.CL

TL;DR

The paper investigates whether Bayesian prediction can explain transformers’ in-context learning, where prior work had not extensively verified this account across function families. It extends meta-ICL to mixtures of task families and tests transformers against Bayesian predictions. The experiments find frequent agreement with Bayesian prediction, while also identifying generalization and deviations from it.

  • Problem

    How far Bayesian prediction explains in-context learning across function families remains empirically under-verified, despite its relevance to understanding ICL’s inductive bias.

    Source: §1 · p. 2

  • Method

    The paper extends meta-ICL to hierarchical mixtures of function classes and tests transformers on diverse linear and nonlinear problems against Bayesian predictors where tractable.

    Source: §1 · p. 2, §2.1 · p. 4

  • Results

    Across linear and nonlinear inverse problems, transformers in-context learn and generalize to unseen functions, with errors and implied weights agreeing with Bayesian predictors where comparisons are tractable.

    Source: §7 · pp. 9–10

  • Takeaways & Limitations

    The Bayesian perspective helps explain how pretraining distribution shapes ICL inductive bias and how transformers handle task mixtures, while generalization to new tasks can involve apparent deviations from Bayesian inference.

    Source: §1 · p. 3, §8 · p. 10

    The study focuses on continuous-function settings; how strongly its Bayesian account applies to real-world LLMs remains open, especially given demonstration-order effects.

    Source: §8 · p. 10

Abstract

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In-context learning (ICL) is one of the surprising and useful features of large language models and subject of intense research. Recently, stylized meta-learning-like ICL setups have been devised that train transformers on sequences of input-output pairs $(x, f(x))$. The function $f$ comes from a function class and generalization is checked by evaluating on sequences generated from unseen functions from the same class. One of the main discoveries in this line of research has been that for several function classes, such as linear regression, transformers successfully generalize to new functions in the class. However, the inductive biases of these models resulting in this behavior are not clearly understood. A model with unlimited training data and compute is a Bayesian predictor: it learns the pretraining distribution. In this paper we empirically examine how far this Bayesian perspective can help us understand ICL. To this end, we generalize the previous meta-ICL setup to hierarchical meta-ICL setup which involve unions of multiple task families. We instantiate this setup on a diverse range of linear and nonlinear function families and find that transformers can do ICL in this setting as well. Where Bayesian inference is tractable, we find evidence that high-capacity transformers mimic the Bayesian predictor. The Bayesian perspective provides insights into the inductive bias of ICL and how transformers perform a particular task when they are trained on multiple tasks. We also find that transformers can learn to generalize to new function classes that were not seen during pretraining. This involves deviation from the Bayesian predictor. We examine these deviations in more depth offering new insights and hypotheses.

1 INTRODUCTION

The paper asks whether Bayesian prediction can explain transformers’ in-context learning, extending study from individual function families to mixtures of task families. It reports evidence that Bayesian prediction explains task-mixture behavior and pretraining-dependent inductive bias, alongside generalization and deviations from that account.

1 INTRODUCTION

  • Transformers can learn in context from mixtures of function families in HMICL, an extension of the MICL setup.The paper studies high-capacity models on mixtures such as linear functions and decision trees.Source: §1 · p. 2
  • Across diverse linear and nonlinear families, the paper finds direct and indirect evidence that high-capacity transformers often mimic the Bayesian predictor.The task families and priors were chosen to enable explicit Bayesian inference or provide strong baselines.Source: §1 · p. 2
  • The authors argue task-mixture learning follows naturally from Bayesian prediction, without separate algorithm selection, and report agreement at all prompt lengths across more mixtures than prior work.For Gaussian mixture models, they also compare with the exact Bayesian predictor.Source: §1 · p. 3
  • When lower frequencies dominate pretraining, transformers favor lower-frequency functions in context; with equally represented frequencies, they show no frequency preference.The paper relates this finding to simplicity bias and says the ICL inductive bias is determined by the pretraining distribution.Source: §1 · p. 3
  • With sufficient task diversity during pretraining, transformers generalize to new HMICL tasks, while some deviations from Bayesian prediction arise from multitask generalization or insufficient model capacity.The paper reports early generalization to many tasks followed by forgetting, and hypothesizes that lower-capacity transformers may be attempting Bayesian inference.Source: §1 · p. 3

2 BACKGROUND

The paper defines MICL as learning a sampled function from in-context examples and evaluates predictions on unseen prompts. HMICL extends this setup by sampling among multiple function classes before generating each prompt.

2 BACKGROUND

  • MICL samples a function and input-output examples for a prompt, then trains a transformer to predict the function value on the next input using squared-error loss.At test time, loss@k measures error on unseen prompts with k in-context examples.Source: §2 · p. 3

2.2 MODEL AND TRAINING DETAILS

  • Models use squared-error training on sampled function prompts, with a 12-layer, 8-head, 256-hidden-unit decoder-only transformer and curriculum learning by increasing task complexity.Experiments generally use 500k training steps and batch size 64.Source: §2 · p. 3, §2.2 · p. 4

2 BACKGROUND

  • With squared-error training, the ideal model’s prediction is the posterior mean estimator (PME), the Bayes-optimal prediction for a prompt.The paper computes this predictor for each prompt length.Source: §2 · pp. 3–4

2.1 HIERARCHICAL META-ICL

  • HMICL samples a function class according to mixture probabilities before sampling a function and generating the prompt; MICL is its special case with one class.The hierarchical PME combines class-specific PMEs using prompt-dependent weights.Source: §2.1 · p. 4

3 TRANSFORMERS CAN IN-CONTEXT LEARN TASK MIXTURES

On task mixtures, transformers can match the mixture Bayesian predictor in a tractable Gaussian-mixture regression setting and generalize across more complex mixtures. The results support an account in which Bayesian prediction jointly handles task recognition and solution.

3.1 GAUSSIAN MIXTURE MODELS (GMMS)

  • Transformer prediction errors and implied weights almost exactly match the Gaussian-mixture PME, converging to the prompt component’s zero error after d examples.The component-specific PME for the other task’s component worsens as more prompt examples arrive.Source: Figure 1 · p. 5, §3.1 · p. 5
  • Across mixtures of linear inverse problems and mixtures including neural networks or decision trees, transformers generalize to new functions and match single-class transformer performance depending on the prompt distribution.The experiments include mixtures of two or three function classes.Source: §3.1 · p. 5
  • In the Gaussian-mixture experiments, Bayesian prediction intertwines recognizing the task with solving it rather than requiring separate task recognition.The authors connect this implication to the mixture PME in Equation 1.Source: §3.1 · p. 5

4 SIMPLICITY BIAS IN ICL?

The Fourier-series experiments show that ICL’s frequency bias depends on the pretraining distribution: mixtures favor frequencies that occur more often, while single-class training shows no such preference. The authors interpret this pattern through Bayesian prediction, in which pretraining-distribution biases are reflected in the posterior.

Measuring inductive biases

  • The authors recover a transformer’s implied frequencies by applying a Discrete Fourier Transform to its predictions on test inputs given the in-context examples.The resulting coefficients indicate the dominant frequencies in the function represented by the model’s predictions.Source: §4 · p. 6

Results

  • In single-class Fourier training, transformers show no frequency preference; in mixtures, they favor lower frequencies at short prompts but recover the gold frequencies with enough examples.The mixture’s lower frequencies occur in more function classes, while higher frequencies are rarer.Source: §4 · p. 6
  • The mixture’s frequency bias tracks training prevalence, and high-frequency-biased pretraining also produces complexity bias during ICL.The paper reports additional experiments biasing the pretraining distribution toward high frequencies.Source: §4 · p. 6

Implications

  • The authors attribute ICL’s simplicity bias, or its absence, to the pretraining distribution: the biases in the prior are reflected in the posterior.They argue that transformers add no extra inductive bias when they emulate the Bayesian predictor.Source: §4 · pp. 6–7

5 MULTI-TASK GENERALIZATION

In the HMICL monomial setup, increasing task diversity improves generalization to unseen feature sets, though at a cost to in-distribution performance. The section also reports close agreement with the Bayesian predictor for the single-class monomial problem.

Results

  • For a single degree-2 monomial feature set, transformer error curves closely follow the OLS_S Bayesian-predictor baseline at all prompt lengths, in both ID and OOD evaluation.The feature set S defines the basis used to form the function class.Source: §5 · p. 7

Extending to HMICL setting

  • HMICL samples from a uniform mixture of K fixed monomial feature-set classes and evaluates transfer on feature sets absent from that mixture.Each feature set has size D, and K is the task diversity of the pretraining distribution.Source: §5 · p. 7

Results

  • As task diversity K increases, transformers approach OLSΦM and LassoΦM performance on unseen feature sets, while ID and OOD performance eventually become identical.At small K, OOD generalization is poor; the authors report similar results for Fourier-series function classes.Source: §5 · p. 8
  • The OOD improvement with increasing K comes at a cost to ID performance, and the resulting transfer to unseen classes deviates from Bayesian behavior on the pretraining distribution.The authors report analogous multi-task generalization results for Fourier-series classes.Source: §5 · p. 8

Comparison with prior work

  • The authors describe their HMICL setting as more general than prior multi-task MICL work, with new-class generalization occurring similarly to new-task generalization in that work.They note that the prior work emphasized deviation from the Bayesian predictor.Source: §5 · p. 8

6 DEVIATIONS FROM BAYESIAN INFERENCE?

The section examines deviations from Bayesian prediction through pretraining dynamics and capacity limits. It reports that some models first generalize across task distributions and later forget that generalization, and proposes gradient descent as a capacity-constrained approximation to Bayesian inference.

6.1 ICL TRANSFORMER FIRST GENERALIZES THEN MEMORIZES DURING PRETRAINING

  • The forgetting study uses noisy linear regression with d = 8 and prompt length p = 15, sampling training tasks from a fixed pretraining set and defining the full task distribution as standard normal.Functions are f(x) = w^T x + ϵ, with ϵ sampled from a zero-mean Gaussian of variance 0.25.Source: §6.1 · pp. 8–9
  • For task diversities 2^7 to 2^11, OOD loss first reaches the ID loss and matches Ridge, then worsens during pretraining and ends between dMMSE and Ridge.The authors call this the Gaussian forgetting region; task diversities 2^1 to 2^3 show no generalization or forgetting, while 2^12 to 2^20 show full generalization without forgetting.Source: §6.1 · p. 9
  • At the final checkpoint, task diversities up to 2^6 agree with dMMSE, diversities from 2^12 onward agree with Ridge, and the intermediate range agrees with neither.The paper identifies the intermediate diversities 2^7 to 2^11 as both the Gaussian forgetting region and the transition region.Source: §6.1 · p. 9

6.2 GRADIENT DESCENT AS A TRACTABLE APPROXIMATION OF BAYESIAN INFERENCE

  • The authors hypothesize that gradient descent appears in ICL because it may be the best approximation to Bayesian inference within a transformer’s capacity constraints.They connect this proposal to evidence that increasing transformer capacity moves linear-regression behavior toward Bayesian inference.Source: §6.2 · p. 9

7 SUMMARY OF FURTHER RESULTS

Across linear and nonlinear inverse problems in MICL and HMICL, the authors report in-context learning and generalization to unseen functions. Where Bayesian inference is tractable, they compare transformer behavior with the exact predictor; otherwise, they use numerical solutions or strong baselines.

Summary of further results

  • Across linear and nonlinear inverse problems in MICL and HMICL, transformers in-context learn and generalize to unseen functions, with agreement to Bayesian predictors or strong baselines where comparisons are available.The tested linear problems include dense, sparse, sign-vector, low-rank, and skewed-covariance regression; nonlinear problems include Fourier series, degree-2 monomials, random Fourier features, and Haar wavelets.Source: §7 · pp. 9–10
  • When exact Bayesian inference is tractable, the authors compare transformers with the exact predictor; otherwise, they use MCMC-based numerical solutions or strong baselines when sampling does not converge.For linear problems, they also compare the weights of the function implied by the transformer with the predictor or baseline.Source: §7 · pp. 9–10
  • For Dense Regression, transformer performance is independent of in-context example permutation across prompt lengths, while HMICL generalization may depend on normalizing outputs across function classes.The paper says Bayesian inference is order-independent for the problem classes in its setup and experimentally verifies this result for Dense Regression.Source: §7 · p. 10

8 CONCLUSION

The paper argues that Bayesian prediction helps explain ICL’s inductive biases and task-mixture behavior, while generalization to unseen tasks can involve deviations from Bayesian inference. It also reports that removing positional encodings improves length generalization in dense and sparse regression without materially affecting training-regime performance.

8 CONCLUSION

  • The Bayesian perspective unifies evidence about ICL’s inductive bias and task mixtures, but the paper says its extent and relevance to real-world LLMs remain open questions.The study focuses on continuous functions and treats transformers as black boxes.Source: §8 · p. 10

A.1 PME THEORETICAL DETAILS

  • Under squared-error loss, the posterior mean estimator is the optimal prediction for a prompt.For a mixture of function classes, its prediction combines the class-specific PMEs using prompt-dependent posterior weights.Source: §A.1 · p. 16

A.2 THE CURIOUS CASE OF POSITIONAL ENCODINGS.

  • Removing positional encodings substantially improves length generalization in dense and sparse regression while maintaining virtually the same performance within the training regime.The models were trained with a maximum of 40 in-context examples.Source: §A.2 · p. 17, Figure 5 · p. 17

A.3 EXPERIMENTAL SETUP

The experiments examine transformer ICL across linear inverse problems and compare model predictions with established baselines and Bayesian predictors where available. The setup includes several priors and structural constraints on the weights, as well as a task that does not satisfy convex geometry conditions.

A.3 EXPERIMENTAL SETUP

  • Models use Adam and curriculum training; results average errors over 1,280 prompts, with 90% confidence intervals from 1,000 bootstrap trials.The project used 32 NVIDIA V100 GPUs and about 30,000 GPU hours.Source: §A.3 · p. 17

B LINEAR AND NON-LINEAR INVERSE PROBLEMS

  • Transformers track dense-regression baselines, reveal sparse and sign-vector weight structure, and closely match numerical PME solutions for low-rank and sign-vector regression.For sparse and sign-vector tasks, the recovered structure appears at prompt lengths above 10.Source: B LINEAR AND NON-LINEAR INVERSE PROBLEMS · p. 18, Figure 10 · p. 21
  • The study evaluates single-task ICL on linear and nonlinear regression problems, comparing transformers with Bayesian predictors where computation is possible.The task-mixture setting is discussed separately.Source: B LINEAR AND NON-LINEAR INVERSE PROBLEMS · p. 18

B.1 LINEAR INVERSE PROBLEMS

  • Linear inverse problems estimate parameters from few, sometimes fewer-than-parameter-count, measurements; structural constraints such as sparsity or low rank can make recovery possible.The paper encodes problem-specific structure in the prior on the weights.Source: §B.1 · p. 20

B.1.1 FUNCTION CLASSES AND BASELINES

  • The linear task families vary the weight prior or structure: dense and skewed Gaussian, sparse, sign-vector, and low-rank weights, with OLS, Lasso, norm-minimization, or nuclear-norm baselines.For dense and skewed Gaussian regression, PME and MAP coincide with constrained minimum-norm solutions.Source: §B.1.1 · p. 20
  • FZR tests whether transformers can handle weights that violate convex geometry conditions used by some exact-recovery techniques.Its weight vectors are formed by duplicating vectors whose entries come from {−2, −1, 1, 2}.Source: §B.1.1 · pp. 20–21

B.1.2 RESULTS

Across linear tasks, transformers track Bayesian or strong-baseline predictions, and Fourier-series results show similar agreement with the ideal predictor. Random Fourier Feature performance is mixed: it weakens as input dimension grows, despite handling many features at lower dimensions.

B.1.2 RESULTS

  • Transformers closely follow the skewed-covariance PME and align with strong task-specific baselines, while low-rank regression sometimes performs better than its baseline.They outperform OLS on the reported tasks and can solve them with fewer than d samples.Source: §B.1.2 · p. 22
  • FZR-trained transformers outperform Sign-Vector-trained transformers on their respective prompts, but the reverse holds on out-of-distribution dense-regression prompts.The authors conclude that transformers do not require convexity conditions on weight vectors for this task.Source: §B.1.2 · p. 23

B.2 NON-LINEAR FUNCTIONS

  • For nonlinear tasks, the study maps inputs to features and performs linear regression over those features, with a Gaussian-prior PME available for this setup.The feature map may represent more complex input-output relationships than linear functions of the original inputs.Source: §B.2 · p. 23

B.2.1 FOURIER SERIES

  • Fourier-series predictions align with OLS and recover the gold maximum frequency as examples increase, without a preference for lower frequencies; larger models show better agreement.Figures compare loss, predicted functions, and frequency distributions.Source: §B.2.1 · pp. 23–24, Figure 12 · p. 24, Figure 13 · p. 25

B.2.2 RANDOM FOURIER FEATURES

  • Random Fourier Feature results are mixed: performance is good at input dimensions 1 and 4 but substantially worse than RFF-OLS at dimension 10, with little loss improvement after about 15 examples.The reported difficulty is governed primarily by input dimension, while models can handle large numbers of random features at lower dimensions.Source: §B.2.2 · p. 27

B.2.3 DEGREE-2 MONOMIAL BASIS REGRESSION

The degree-2 monomial and Haar-basis experiments find that transformers closely mimic the corresponding OLS/PME predictors, including on monomial-basis shifts. The paper distinguishes HMICL from single-family MICL to study how models perform different function-family tasks.

B.2.3 DEGREE-2 MONOMIAL BASIS REGRESSION

  • The monomial family represents functions as linear combinations of degree-2 monomials, with coefficients sampled from a standard normal distribution.The tested subfamily selects a subset S of the full monomial basis, equivalently setting other coefficients to zero.Source: §B.2.3 · p. 27
  • On degree-2 monomial prompts, transformers closely follow OLSS in-distribution and out-of-distribution, even surpassing it when OLSS exhibits double descent.For larger basis shifts, transformer errors also rise near k = |S|, while the transformer still follows OLSS when the test basis has no terms in common with the training basis.Source: Figure 17 · p. 28, Figure 18 · p. 28, §B.2.3 · p. 29

B.2.4 HAAR WAVELET BASIS REGRESSION

  • On Haar-wavelet regression, the transformer closely mimics OLSH errors and converges to OLSH at k = 32, suggesting it simulates the PME.With Gaussian weight priors, OLSH is the PME.Source: Figure 19 · p. 29, §B.2.4 · p. 30

C.1 WHY HMICL?

  • HMICL keeps the function-family hierarchy explicit, enabling analysis of whether and how models perform distinct meta-tasks.The authors argue that this finer-grained view is closer to ICL settings where models may perform many meta-tasks.Source: §C.1 · p. 30

C.2 GAUSSIAN MIXTURE MODELS (GMMS)

Across Gaussian-mixture, more complex task-mixture, and Fourier-mixture experiments, transformers adapt their in-context predictions to the prompt’s task distribution. Where the paper compares them with tractable Bayesian predictors, their behavior supports the PME account, while Fourier results link frequency bias to the training mixture.

C.2 GAUSSIAN MIXTURE MODELS (GMMS)

  • As prompt evidence accumulates, the mixture PME’s component weights shift toward the prompted task, and the transformer’s probed first weight dimension closely mimics that evolution.For equal mixture weights, the first dimension moves from approximately 0 toward +3 or −3 according to the prompted component; specially constructed prompts that reveal no component information leave it near 0.Source: §C.2 · pp. 32, 36, Figure 22 · p. 33, Figure 23 · p. 34
  • In Gaussian-mixture regression, transformer errors agree with the mixture PME and are smaller than OLS errors in the over-determined region, supporting that the transformer simulates the mixture PME rather than OLS.In the over-determined case, the solution is unique and all predictors have errors approximately zero.Source: Figure 21 · p. 32, §C.2 · p. 36
  • With unequal component weights, the mixture PME starts closer to the more heavily weighted component, and transformer probed weights follow that PME.The transformer also converges to the mixture PME and prompted-component PME at k = 10 in the p = 20 model.Source: Figure 24 · p. 35, Figure 25 · p. 35, §C.2 · p. 36

C.3 MORE COMPLEX MIXTURES

  • On dense/sparse regression mixtures, the transformer follows OLS and its dense-regression single-task model on dense prompts, and Lasso and its sparse-regression model on sparse prompts.The mixture model’s implied weights also agree with single-task models and the OLS/Lasso baselines.Source: §C.3 · p. 37, Figure 26 · p. 37, Figure 27 · p. 37
  • Across sign-vector, skewed-covariance, three-task, and heterogeneous mixtures, transformers adapt to the prompted task; the three-task model approaches single-task errors once sufficient examples are provided, still under 20.The heterogeneous dense/decision-tree and decision-tree/neural-network mixtures follow their corresponding single-task models on prompts from each task.Source: §C.3 · pp. 37, 40, 41, Figure 28 · p. 38, Figure 29 · p. 39

C.4 FOURIER SERIES MIXTURE DETAILED RESULTS

  • On Fourier mixtures, transformer performance lies between the gold-frequency and maximum-frequency OLS baselines; it does much better than the latter for k < 20, while the former baseline performs better.With few examples, predictions favor lower frequencies, but sufficient examples recover gold frequencies; the paper traces this simplicity bias to the training mixture, where lower frequencies are more common.Source: §C.4 · p. 41

C.4.1 COMPLEXITY BIASED PRE-TRAINING

The authors test whether frequency bias follows the frequency makeup of the training mixture. A high-frequency-biased mixture produces complexity bias at short prompts, but is harder to learn than the simplicity-biased mixture.

C.4.1 COMPLEXITY BIASED PRE-TRAINING

  • The alternate Fourier basis spans frequencies from minimum frequency n0 through N, and the mixture varies n0 so that higher frequencies are more consistently represented.In this mixture, frequency N appears in every class, while frequency 1 appears only in the class beginning at 1.Source: §C.4.1 · p. 41
  • At short prompt lengths, the N = 5 high-frequency-biased model exhibits a strong bias toward higher frequencies near 5.This is the reported complexity-bias pattern for the trained model.Source: §C.4.1 · p. 42, Figure 32 · p. 42
  • The high-frequency-biased mixture is harder to learn: the N = 10 model fails, and even N = 5 converges more slowly than the simplicity-biased mixture.The authors suggest that simplicity bias in pretraining data may lead to more efficient training.Source: §C.4.1 · p. 42

C.5 CONDITIONS NECESSARY FOR MULTI-TASK ICL

In the dense/sparse regression mixture, transformers learned both tasks when training signals were balanced by task weighting or output normalization; without curriculum, longer training also eventually enabled both. The appendix further examines prompt normalization and Fourier multi-task generalization.

C.5 CONDITIONS NECESSARY FOR MULTI-TASK ICL

  • Increasing the sparse-task weight to αSR = 0.87 or normalizing task outputs enabled multi-task ICL, unlike the initial mixture where transformers followed OLS for both tasks.Dense regression had higher output variance, which the authors hypothesized gave it a disproportionately strong training signal.Source: §C.5 · p. 42, Figure 34 · p. 44
  • Without curriculum, the model eventually learned both tasks after longer training, as indicated by a sharp dip in sparse-regression evaluation loss.The appendix compares the 500k and 800k checkpoints and plots sparse-regression loss during training.Source: §C.5 · pp. 42, 44, Figure 35 · p. 45
  • For Fourier-series mixtures, the transformer exhibits simplicity bias at low frequencies as maximum frequency and in-context example count vary.Figure 33 measures the frequencies inferred by the transformer across values of M and k.Source: Figure 33 · p. 43
  • A model trained on an unnormalized dense/SVR mixture failed on normalized prompts, following OLS for both tasks; the authors leave this OOD setting for future work.The unnormalized tasks had the same output variance, and the model showed multi-task ICL on unnormalized prompts.Source: §C.5 · p. 46, Figure 36 · p. 47

D.2 FOURIER SERIES MULTI-TASK

  • For Fourier-series multi-task evaluation, ID performance degrades and OOD performance improves as task diversity K increases; from K = 100 onward, they become identical and similar to LassoΦN and BPproxy.Table 4 reports ID and OOD evaluations.Source: Table 4 · p. 47, §D.2 · p. 48

D DETAILS REGARDING MULTI-TASK GENERALIZATION EXPERIMENTS

D.3 DETAILS ON THE PHENOMENON OF FORGETTING

In noisy linear regression, OOD generalization and forgetting depend on task diversity: some models improve before forgetting, while the highest-diversity group maintains ID-level OOD loss. Forgetting is robust across several settings, and its extent increases with input dimension.

D.3 DETAILS ON THE PHENOMENON OF FORGETTING

  • For task diversities 21–23, OOD loss never improves and ends at or worse than its initial value, agreeing with dMMSE at the end of training.Source: §D.3 · p. 49
  • For task diversities 24–26, OOD loss improves then worsens and remains above ID loss; for 27–211, it reaches ID loss before worsening, while 212–220 maintains matching ID/OOD losses.At the minimum, the 27–211 group agrees with Ridge; by the end it lies between dMMSE and Ridge, while 212–220 agrees with Ridge throughout.Source: §D.3 · p. 49, Figure 37 · p. 50, Figure 38 · p. 51
  • The authors hypothesize that forgetting reflects a simplicity bias favoring a simpler continuous Gaussian distribution over a discrete pretraining distribution with unnecessary details.They propose that pretraining-distribution simplicity varies with task diversity, with the 21–23 group as an exception.Source: §D.3 · p. 49
  • Forgetting persists across model sizes, learning rates, schedules, and position encodings, while its extent is directly proportional to input dimension when signal-to-noise ratio is held constant.For larger task diversities, bigger models exhibit higher OOD loss over training; for smaller diversities, model size shows no clear trend.Source: §D.3 · p. 49, Figure 39 · p. 53, Figure 40 · p. 54
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