How Patterns Dictate Learnability in Sequential Data

We are grateful to the reviewer for the valuable feedback and recommendations. We provide here a detailed answer to the weaknesses and questions.

W1 – Real world dataset

We understand that the empirical part of the paper may appear modest relative to the theoretical contribution, which can make evaluating its practical relevance more challenging. However, the primary focus of our work is indeed theoretical, and as you may have seen, the theoretical contributions—particularly the proofs—are already substantial and technically demanding.

We intentionally chose not to simultaneously tackle the separate and equally challenging problem of experimentally estimating the learning curve and the minimal achievable risk. Indeed, estimating the learning curve is itself a complex theoretical question that warrants dedicated treatment. Addressing it rigorously would require an entirely separate line of work, and we felt that attempting to incorporate this into the current paper would dilute the clarity and focus of our main contributions. For instance, [1] is a full research article dedicated solely to estimating mutual information in sequential settings. Hence, we chose to first address the problem from a theoretical perspective, and plan to tackle the estimation of the learning curve and thus minimal risk in a follow-up article.

That said, we did include experiments on synthetic data, where the estimation of $\mathbf{I}_{\text{pred}}$ is not an issue, since closed-form expressions are available. These experiments serve an illustrative purpose, allowing us to demonstrate the theoretical framework in controlled settings.

Nevertheless, we fully acknowledge the reviewers' concern regarding the lack of real-world evaluations. In response, we applied our framework to the Exchange Rate dataset from GluonTS [2], and show how it can be used on a real world dataset. This dataset contains daily exchange rates (1990–2016) for eight major currencies (AUD, GBP, CAD, CHF, CNY, JPY, NZD, SGD). In this case, we show that this dataset is more limited by data itself rather than by model complexity.

We first computed the MSE for four models: a naive baseline (which predicts the next value as the previous one), a simple feedforward neural network, DeepAR, and the Temporal Fusion Transformer (TFT). The differences in MSE across models are relatively minor. Even the naive model performs comparably to more sophisticated ones, reinforcing the idea that additional architectural complexity does not necessarily translate into meaningful performance gains.

To obtain the minimal achievable risk $\hat{\mathcal{R}}^{\infty}(Q^{*}) = \min_{1 \leq k \leq M} \hat{\mathcal{R}}^{k}(Q_k) - \hat{\Lambda}(k)$, we estimated the learning curve $\Lambda(k)$. While our estimator is admittedly noisy and unstable, the results (shown below) suggest that there is little left to learn in this dataset. Indeed, the actual MSE values lie close to this estimated minimal risk, suggesting that the performance ceiling is due less to model limitations than to the intrinsic unpredictability of the data.

This aligns with the characteristics of the dataset. Unlike many other financial assets, FX rates are driven by the relative economic conditions, monetary policies, and geopolitical factors of two countries, adding significant complexity to their dynamics. This market is also extremely liquid and highly competitive, so any patterns or inefficiencies are quickly detected and eliminated by market participants. These factors contribute to the inherent unpredictability of the dataset.

Table: Comparison between actual MSE and estimated minimal achievable risk for each model

W2 – Estimation Stability

Indeed, as noted above, estimating predictive information in practice is a difficult task—particularly in high-dimensional settings or when the underlying data has weak temporal structure. Our main objective in this work, however, is not to develop a definitive estimator of $\mathbf{I}_{\text{pred}}$, but to provide a theoretical characterization of the limits of learnability from an information-theoretic perspective. We believe that a rigorous and robust estimation procedure for the predictive information is an important and challenging problem in its own right, and we see it as a promising direction for future work.

It is also worth emphasizing that estimating mutual information in sequential settings is a relatively recent and evolving research area—[1] , for instance, was one of the first works to directly address this. While our current estimator is admittedly imperfect, it already yields valuable insights in real-world scenarios, as demonstrated in our experiments on both synthetic and real world datasets. Moreover, our theoretical results suggest new avenues for improving estimation techniques. For example, instead of estimating $\mathbf{I}_{\text{pred}}$ directly, one might focus on estimating the learning curve $\Lambda(k)$, which could offer a more stable and interpretable alternative. We view our contribution as laying the groundwork for such future developments.

W3 – Model Dependency of the Estimator

We thank the reviewer for this important point. It is true that our estimator of the minimal achievable risk is model-dependent, in contrast to $`EvoRate`$ 's model-free approach in [1]. However, we believe that this dependency is not necessarily a limitation, but rather reflects the different purpose of our framework. While EvoRate focuses on detecting the presence of temporal structure, our aim is to quantify how much of that structure is actually being captured by a given model class.

The core quantity of interest in our framework—the learning curve $\Lambda(k)$—is itself model-independent and provides richer information than EvoRate, as it reflects the data’s intrinsic predictability across context lengths. The connection we draw to model-specific risk should be seen as an added benefit: it enables one to assess the current state of modeling capabilities on a given dataset, and to determine whether improvements are more likely to come from better models or whether the data has already been exhaustively exploited.

Importantly, in our experiments, we observed that the estimated minimal achievable risk is generally stable across different architectures. This suggests that our diagnostic remains robust despite being evaluated within specific model classes.

W4 – Interpretability vs. Accessibility

We appreciate the reviewer’s concern regarding the accessibility of the theoretical content and recognize that the paper is conceptually dense. To address this, we deliberately included intuitive illustrations—such as the lunch meal prediction scenario in the introduction—to ground the theoretical framework in concrete terms. We also designed empirical experiments, including one on a real-world dataset, to directly support and illustrate our theoretical claims. We began with synthetic data precisely because it offers controlled settings where expected behaviors are known, making the theory more digestible for the reader. Additionally, we will revise and clarify some derivations and reinforce the connection between theory and experiments in the final version of the article to further improve interpretability.

We hope we have adequately addressed the reviewer’s questions and concerns regarding this paper.

References

[1] Zeng, Q., Huang, L.-K., Chen, Q., Ling, C. X., & Wang, B. (2025). Towards Understanding Evolving Patterns in Sequential Data. Advances in Neural Information Processing Systems.

[2] Alexandrov, A., et al. (2019). GluonTS: Probabilistic Time Series Modeling in Python. arXiv preprint arXiv:1906.05264.

We thank the reviewer for their thoughtful and constructive feedback. We appreciate the positive assessment of our contributions, as well as the valuable questions and concerns raised. We address them point by point below.

W1 – Illustration on real-world dataset

We acknowledge that our work could benefit from an application on a real world dataset. But the primary focus of our work is theoretical, and as you may have seen, the theoretical contributions—particularly the proofs—are already substantial and technically demanding.

We intentionally chose not to simultaneously tackle the separate and equally challenging problem of experimentally estimating the learning curve and the minimal achievable risk. Indeed, estimating the learning curve is itself a complex theoretical question that warrants dedicated treatment. Addressing it rigorously would require an entirely separate line of work, and we felt that attempting to incorporate this into the current paper would dilute the clarity and focus of our main contributions.

Nevertheless, we agree with the reviewer that a real world application could benefit the overall comprehension of this article. Thus, we applied our estimator of $\mathbf{I}_{\text{pred}}$ to the Exchange Rate dataset from GluonTS [1], in order to assess how much predictive information remains in that data. This dataset contains daily exchange rates (1990–2016) for eight major currencies (AUD, GBP, CAD, CHF, CNY, JPY, NZD, SGD).

We first computed the Mean Squared Error (MSE) for four models: a naive baseline (which predicts the next value as the previous one), a simple feedforward neural network, DeepAR, and the Temporal Fusion Transformer (TFT). The differences in MSE across models are relatively minor. Even the naive model performs comparably to more sophisticated ones, reinforcing the idea that additional architectural complexity does not necessarily translate into meaningful performance gains.

To better understand this plateau, we estimated the learning curve $\Lambda(k)$. This curve is approximately flat and close to zero (see table). While our estimator is admittedly noisy and unstable, the results (shown below) suggest that there is little left to learn in this dataset.

We then computed the minimal achievable risk using: $\hat{\mathcal{R}}^{\infty}(Q^{\*}) = \min_{1 \leq k \leq M} \hat{\mathcal{R}}^{k}(Q_k) - \hat{\Lambda}(k)$.

Interestingly, the actual MSE values lie close to this estimated bound, suggesting that the performance ceiling is due less to model limitations than to the intrinsic unpredictability of the data.This aligns with the characteristics of the dataset. Unlike many other financial assets, FX rates are driven by the relative economic conditions, monetary policies, and geopolitical factors of two countries, adding significant complexity to their dynamics.

Table: Comparison between actual MSE and estimated minimal achievable risk for each model

W2 – Summary of Contributions

This work addresses two fundamental questions in sequential modeling:

1. What is the minimal achievable risk when modeling sequential data?
2. Is poor predictive performance due to model limitations or to the intrinsic unpredictability of the data?

To address these questions, we introduce a practical framework grounded in information theory. A central element of this framework is the learning curve $\Lambda(k)$, which captures the improvement in predictive accuracy when the context length increases from $k$ to $k+1$. This measure is derived from the predictive information $I_{\text{pred}}(k, k')$ (defined as the mutual information between a block of $k$ past observations and a block of $k'$ future observations ) through this formula :

$\Lambda(k) = \lim_{k' \to \infty} \left[ I_{\text{pred}}(k+1, k') - I_{\text{pred}}(k, k') \right]$

Using this quantity, we introduce an estimator of the minimal achievable risk — the lowest possible prediction error attainable by any model — through a bound involving the $k$-order forecasting risk. This estimator is defined as follows:

$\hat{\mathcal{R}}^{\infty}(Q^{*}) = \min_{1 \leq k \leq M} \hat{\mathcal{R}}^{k}(Q_k) - \hat{\Lambda}(k)$,

where $\hat{\mathcal{R}}^k(Q)$ is the empirical risk of a model $Q$ using a context of length $k$.

This leads to a simple yet effective diagnostic tool to address the second question: is poor performance due to model limitations or to the data itself? If the actual model risk $\hat{\mathcal{R}}^k(Q)$ is close to the estimated minimal risk $\hat{\mathcal{R}}^\infty(Q^*)$, it indicates that either the data contains little predictive structure or that this structure has already been captured—suggesting that increasing model complexity is unlikely to yield better results. Conversely, a substantial gap between these two quantities signals that the model could be missing exploitable patterns in the data, pointing to potential model improvements.

Q1 - Corollary 4.4 Illustration

Due to recent NeurIPS policies, we are unfortunately unable to include new plots during the rebuttal period. However, we performed additional experiments to validate Corollary 4.4 using a concrete example that illustrates the asymptotic consistency of our dimension estimator.

For this additional illustration, we follow the same setup as the experiment in Section 5.3, which is based on the Ising spin sequence example from [2]. We generated a sequence of $10^9$ samples with block size $M = 400{,}000$, and computed the learning curve up to $k = 20$. We restricted the analysis to $k \leq 20$ since $2^{20} \approx 10^6$ corresponds to the number of distinct words of length 20; beyond this point, entropy estimation $S(20)$ - via empirical frequencies - becomes unreliable due to insufficient sampling.

The table below reports the values of our dimension estimator defined as $\hat{p}(k) = 2k \cdot \hat{\Lambda}(k)$ :

This clearly shows that our estimator converges to the real dimension p = 1.

[1] provides both theoretical and empirical support for this behavior. In particular, they show that $\hat{p}(n) = 2n \cdot \Lambda(n) = p - \frac{1}{2n} + o\left(\frac{1}{n}\right)$, confirming that our estimator is consistent and can be used to estimate dimension of parameters, with convergence rate $(1/n)$.

Q2 - Estimating the true window size (model order) from small windows

Let us first restate the question to ensure we have understood it correctly.

Given an autoregressive (AR) process of order $p$, Proposition 4.2 in our paper shows that the learning curve $\Lambda(k)$ becomes exactly zero for all $k \geq p$. The reviewer asks whether it is possible to estimate this value of $p$ without having to compute $\Lambda(k)$ for all values up to $p$—in particular, using only small window sizes $k \ll p$.

We propose a simple extrapolation-based method to approximate $p$ from a small number of evaluations. The learning curve $\Lambda(k)$ is decreasing and converges to zero, and under mild conditions—such as for Gaussian AR processes—it is also convex. We exploit this structure by computing $\Lambda(k)$ for $k = 1, \dots, M$, where $M$ is a user-defined limit, and fitting a simple parametric function such as a rational function $f(k) = a / (k + b)$ or an exponential decay. These function classes are chosen to reflect the smooth and typically convex decay of $\Lambda(k)$ as stated in [2]. Once the function is fitted, we define an estimator $\hat{p}$ as the first $k$for which $f(k) < \Lambda(1)/N$, with $N$ another tunable parameter controlling tolerance.

To demonstrate this approach, we extended the experiment from Section 5.2 of our paper. We generate a 5-dimensional Gaussian AR process with autoregressive coefficient $\rho = 0.9$, and vary the true order $p$. We evaluate $\Lambda(k)$ for $k = 1$ to $10$, fit the rational function $f(k)$, and estimate $\hat{p}$ as above with $M=10$ and $N = 10$. Results are summarized below:

This method achieves surprisingly accurate estimates, even for large values of $p$, despite using only the first 10 points of the learning curve. Of course, extrapolation becomes harder as $p$ grows, but the approach remains effective and tunable: one can adjust the number of observed points $M$, the threshold $N$, or the fitting function class to balance cost and precision.

As an alternative, one could also apply a root-finding method (e.g., Newton’s method) to estimate where $\Lambda(k) \approx 0$. While such techniques may reduce the number of required evaluations, they still require accessing large $k$-values close to $p$, limiting their practical gain in high-order regimes.

We thank the reviewer for this insightful question. These techniques may also prove useful in real-world applications where the learning curve is currently estimated via predictive information. Leveraging extrapolation strategies like the one discussed here could help refine learning curve estimation directly, beyond mutual information bounds.

References

[1] Alexandrov, A., et al. (2019). GluonTS: Probabilistic Time Series Modeling in Python. arXiv preprint arXiv:1906.05264.

[2] Bialek, W., Nemenman, I., & Tishby, N. (2001). Predictability, complexity, and learning.

We thank the reviewer for their overall positive evaluation and for highlighting both the strengths and limitations of our work. We appreciate the comments regarding empirical scope, experimental coverage, and reproducibility, and we address each of these points below.

W1 – Limited empirical scope

It is true that our paper does not include any experiments on real-world datasets and we understand that it might be challenging to fully assess the impact of our work without those. The primary focus of our work is indeed theoretical, and as you may have seen, the theoretical contributions—particularly the proofs—are already substantial and technically demanding.

We intentionally chose not to simultaneously tackle the separate and equally challenging problem of experimentally estimating the learning curve and the minimal achievable risk. Indeed, estimating the learning curve is itself a complex theoretical question that warrants dedicated treatment. Addressing it rigorously would require an entirely separate line of work, and we felt that attempting to incorporate this into the current paper would dilute the clarity and focus of our main contributions. For instance, [1] is a full research article dedicated solely to estimating mutual information in sequential settings. Hence, we chose to first address the problem from a theoretical perspective, and plan to tackle the estimation of the learning curve and thus minimal risk in a follow-up article.

That said, we did include experiments on synthetic data, where the estimation of $\mathbf{I}_{\text{pred}}$ is not an issue, since closed-form expressions are available. These experiments serve an illustrative purpose, allowing us to demonstrate the theoretical framework in controlled settings.

Nevertheless, we fully acknowledge the reviewers' concern regarding the lack of real-world evaluations. In response, we applied our framework to the Exchange Rate dataset from GluonTS [2], and show how it can be used on a real world dataset. This dataset contains daily exchange rates (1990–2016) for eight major currencies (AUD, GBP, CAD, CHF, CNY, JPY, NZD, SGD). In this case, we show that this dataset is more limited by data itself rather than by model complexity.

We first computed the MSE for four models: a naive baseline (which predicts the next value as the previous one), a simple feedforward neural network, DeepAR, and the Temporal Fusion Transformer (TFT). The differences in MSE across models are relatively minor. Even the naive model performs comparably to more sophisticated ones, reinforcing the idea that additional architectural complexity does not necessarily translate into meaningful performance gains.

To obtain the minimal achievable risk $\hat{\mathcal{R}}^{\infty}(Q^{*}) = \min_{1 \leq k \leq M} \hat{\mathcal{R}}^{k}(Q_k) - \hat{\Lambda}(k)$, we estimated the learning curve $\Lambda(k)$. While our estimator is admittedly noisy and unstable, the results (shown below) suggest that there is little left to learn in this dataset. Indeed, the actual MSE values lie close to this estimated minimal risk, suggesting that the performance ceiling is due less to model limitations than to the intrinsic unpredictability of the data.

This aligns with the characteristics of the dataset. Unlike many other financial assets, FX rates are driven by the relative economic conditions, monetary policies, and geopolitical factors of two countries, adding significant complexity to their dynamics. This market is also extremely liquid and highly competitive, so any patterns or inefficiencies are quickly detected and eliminated by market participants. These factors contribute to the inherent unpredictability of the dataset.

Table: Comparison between actual MSE and estimated minimal achievable risk for each model

W2 – Expressiveness of the experiments

For the $(k, k')$ combinations, please refer to our response to Question 1.
Regarding the second experiment, the results hold consistently across all values of $p$. We chose to report only $p = 5$ and $p = 10$ in the article to maintain clarity and avoid cluttering the figure. For the third experiment, we already included several cross-validation block sizes ( $M = 10^4, 10^5, 10^6, 10^7$ ) to illustrate the robustness of the results across different magnitudes. That said, our conclusions remain valid for other values of $M$ as well. We do observe, however, that the estimation of the learning curve $\Lambda(k)$ becomes less stable when $M \ll 10^4$, due to higher variance in the empirical estimator of $\mathbf{I}_{\text{pred}}$

W3 – Reproducibility (for the last experiment)

Here are the exact model and training details:

Model architectures

• MLP: 2 hidden layers (64, then 32 neurons), ReLU activations, output layer with 2 units (softmax).
• LSTM: Single-layer LSTM (32 hidden units), fully connected layer mapping to 2 output classes.

Training setup

• Adam optimizer (lr = $10^{-3}$)
• Batch size = 128
• Max epochs = 1000
• Early stopping with patience = 10
• 50 batches per epoch
• 80/20 train-validation split

We will explicitly add these details in the final manuscript.

Q1 – On the fixed pair $(k, k')$

We selected the fixed pair $(k, k')$ = $(5, 10)$ based on two main considerations.

First, we needed to satisfy the constraint $k'$ > $k$, and since the rest of the paper focuses on estimating the learning curve $\Lambda(k)$ we aimed for $k'$ to be sufficiently large to approximate the limit $k' \to \infty$. Second, for autoregressive Gaussian processes with order $1$ < d < $20$, we wanted $k$ and $k'$ to be of the same order of magnitude as $d$, which motivated our choice of $k$ = $5$ and $k'$ = $10$. While this choice is somewhat arbitrary, other settings such as $(k, k')$ = $(10, 20)$ would also be valid.

Importantly, our conclusions hold for other combinations. We repeated the first experiment with other $(k, k')$ pairs and observed consistent results. Due to character limits, we report only the results for $(k, k') = (10, 20)$ here, but we can provide additional tables upon request. We had to remove some columns from the original experiment due to space and formatting constraints.

Table: Estimated predictive information $\hat{I}_{\text{pred}}$ across models and GP kernels for $(k, k') = (10,20)$

Q2 – Meaning of the "critical zone" in Figure 2

The "critical zone" represents values of the universal learning curve $\Lambda(k)$ below a small threshold (here set at $0.02$).

In practice, estimating $\Lambda(k)$ doesn't yield exact zeros due to inherent variance. Therefore, choosing precisely the context length $k$ at which the learning curve becomes negligible is challenging.

To address this, we introduce a practical cutoff—this "critical zone"—below which we consider the learning curve effectively zero. In our example, the threshold ($0.02$) was chosen to be small relative to the typical magnitude of $\Lambda(k)$, yet large enough to account for estimation noise.

While somewhat arbitrary, this choice helps clearly identify the region where additional context no longer meaningfully improves predictions.

Q3 – Reference correction (line 569)

Thank you for catching this error. The missing reference is [3]. We will correct this citation in the final version.

We thank the reviewer again for their constructive feedback. We believe the clarifications above, as well as the new real-world experiment on the Exchange Rate dataset, help to address the concerns raised regarding empirical scope and reproducibility.

References

[1] Zeng, Q., Huang, L.-K., Chen, Q., Ling, C. X., & Wang, B. (2025). Towards Understanding Evolving Patterns in Sequential Data. Advances in Neural Information Processing Systems.

[2] Alexandrov, A., et al. (2019). GluonTS: Probabilistic Time Series Modeling in Python. arXiv preprint arXiv:1906.05264.

[3] Kallenberg, O. (2002). Foundations of Modern Probability. Springer.

We would like to thank the reviewer for the careful reading and constructive feedback. Below we address each question in turn.

Q1 – GluonTS

The example mentioned in line 32 refers to the Exchange Rate dataset from GluonTS [1], which contains daily exchange rates (1990–2016) for eight major currencies (AUD, GBP, CAD, CHF, CNY, JPY, NZD, SGD).

In this dataset, we observe that even state-of-the-art models — including Transformer-based and probabilistic architectures — offer only marginal improvements over simpler baselines such as DeepAR. This suggests a plateau in forecasting performance despite increasing model complexity.

To support this observation, we computed Exchange dataset MSE for four models: a naive baseline (which predicts the next value as the previous one), a simple feedforward neural network, DeepAR, and the Temporal Fusion Transformer (TFT) and applied our framework to this dataset. As shown in the table below, the differences in MSE across models are relatively minor. Even the naive model performs comparably to more sophisticated ones, reinforcing the idea that additional architectural complexity does not necessarily translate into meaningful performance gains.

To obtain the minimal achievable risk $\hat{\mathcal{R}}^{\infty}(Q^{*}) = \min_{1 \leq k \leq M} \hat{\mathcal{R}}^{k}(Q_k) - \hat{\Lambda}(k)$, we estimated the learning curve $\Lambda(k)$. While our estimator is admittedly noisy and unstable, the results (shown below) suggest that there is little left to learn in this dataset. Indeed, the actual MSE values lie close to this estimated minimal risk, suggesting that the performance ceiling is due less to model limitations than to the intrinsic unpredictability of the data.

This aligns with the characteristics of the dataset. Unlike many other financial assets, FX rates are driven by the relative economic conditions, monetary policies, and geopolitical factors of two countries, adding significant complexity to their dynamics. This market is also extremely liquid and highly competitive, so any patterns or inefficiencies are quickly detected and eliminated by market participants. These factors contribute to the inherent unpredictability of the dataset.

Table: Comparison between actual MSE and estimated minimal achievable risk for each model

Q2 – Notations

Thank you for pointing this out. We will revise the final version to avoid overloading notation (e.g., $\mathcal{F}_k$ vs. $F$) and clarify the use of logarithms throughout.

Q3 – Figure 1 legend

In the final version, we will clarify that blue regions indicate negative bias (underestimation), and that color intensity reflects the magnitude of this bias.

Q4 – Related work section

We are grateful to the reviewer for bringing up the notion of prospective learning, which was not initially part of our analysis, and for pointing us to relevant references. We will incorporate the following discussion into the "Related Work" section:

Recent developments in statistical learning theory increasingly emphasize learning in contexts where classical assumptions, such as independent and identically distributed (i.i.d.) samples, no longer hold. In particular, extending traditional frameworks to accommodate stochastic processes with temporal dependencies or evolving distributions has become a critical direction for research. [2] directly addresses this by proposing learners that output sequences of hypotheses aimed at minimizing future risk—an approach well-suited for non-stationary settings where the optimal predictor may itself change over time. Complementary to this, [3, 4] offer general characterizations of learnability under arbitrary stochastic processes, framing it in terms of regret minimization or universal consistency, and showing its deep connections with online learning. These frameworks share a common goal: to define when learning is possible in complex, non-i.i.d. environments. However, they primarily rely on external performance criteria—such as risk or regret—without explicitly analyzing the structural properties of the data itself. Our work complements these approaches by introducing an intrinsic, information-theoretic perspective: we characterize learnability in terms of the presence of stable and identifiable patterns within the data-generating process. This allows us to bridge high-level learnability guarantees with the underlying structure of the data, providing new insights into how and why generalization is achievable beyond traditional settings.

We hope the clarifications above resolve the reviewer’s concerns and further illustrate how our framework complements existing work on sequential learning. We appreciate the insightful feedback and look forward to refining the final manuscript accordingly.

References

[1] Alexandrov, A., et al. (2019). GluonTS: Probabilistic Time Series Modeling in Python. arXiv preprint arXiv:1906.05264.

[2] De Silva, A., Ramesh, R., Yang, R., Yu, S., Vogelstein, J. T., & Chaudhari, P. (2024). Prospective Learning: Learning for a Dynamic Future. arXiv preprint arXiv:2411.00109.

[3] Dawid, A. P., & Tewari, A. (2020). On learnability under general stochastic processes. arXiv preprint arXiv:2005.07605.

[4] Hanneke, S. (2021). Learning whenever learning is possible: Universal learning under general stochastic processes. Journal of Machine Learning Research, 22(130), 1-116.