Towards Automated Knowledge Integration From Human-Interpretable Representations

We thank the reviewer for their insightful comments and constructive feedback. Below, we address the questions and concerns raised by the reviewer.

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Weaknesses

W1) Practical applications.

We would like to highlight to the reviewer that our empirical evaluation does, in fact, contain two experiments with real-world data and additional information in natural language.

Section 5.2.1 (setup B) presents an experiment on regression tasks with knowledge representations in a natural language format describing the evolution of the temperature throughout a given day. The natural language descriptions in this experiment resemble those presented in Figure 3, but at the same time are longer, more complex, and the language is specific to the domain of weather forecasting. They include statements like : “temperatures will drop to -5 degrees by early morning”, “ the temperature will gradually increase from 2 to 18”, etc. (see Figure. A.2. for more).

Section 5.2.2 (setup C) presents an experiment on image classification tasks with knowledge represented as textual descriptions of the characteristic features for each bird species that we aim to classify.

Both experiments highlight that providing the additional information significantly improves upon the non-informed predictions. The aim of these experiments is not only to validate the empirical effectiveness of our approach but also to illustrate its potential in practical, knowledge-rich domains where natural language insights may improve low-data learning.

We can draw parallels between these experiments and other domains. To inspire future applied research we have provided additional example applications of informed meta-learning in the below table:

UPDATE: We propose to include the above table in the Appendix of our paper to inspire future applications of informed meta-learning in the real-world.

W2) The purpose of Section 5.1

The purpose of this section is to demonstrate empirically the performance of the informed meta-learning framework as instantiated through the method of INPs in a controlled and fully tractable environment, with no influence of confounding factors that may be present in real-world data. This section also introduces the key metrics used in the subsequent real-world data experiments of section 5.2. While we acknowledge the toy nature of this section, we nevertheless find it useful for building the reader’s intuition, understanding of the method and the training / evaluation pipelines. In this treatment, we follow the convention of the original works on Neural Processes (Garnelo et al., 2018), in which the first introductory experiment is fitted to data sampled from a well-defined stochastic process with subsequent experiments conducted on real-world data generated according the an unknown DGP.

Q4) The paper's motivation and relevance to ICLR.

Thank you for this comment. Knowledge integration indeed remains an important area of research; its success is critical for applications in low-sample and noisy data environments, as noted in our introduction. However, we do not expect many ICLR 2025 paper to focus on this topic, given that conventional knowledge integration methods are often developed in a highly application-specific manner. Such methods are tightly bound to particular knowledge formats and data domains, which limits their relevance for broader ML research audiences.

This is, however, in a sharp contrast to our presented approach of informed meta-learning, which addresses these limitations by offering a flexible approach to knowledge integration. Unlike conventional methods, the informed meta-learning framework is agnostic to the format of knowledge representations and their relationship with the observable data. This makes it applicable across a wide range of domains, where expert knowledge may be present, such as healthcare, finance, or environmental science. We therefore believe that our paper is relevant and of interest to the broader ICLR audience.

Q5) Updating the references

Many thanks for pointing this out. UPDATE: Where applicable, we have now updated all references from preprints to their corresponding conference / journal publications.

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We greatly appreciate the time and effort you’ve dedicated to reviewing our paper. We hope that the resulting additional experiments will strengthen the contributions of this paper and that our responses address your concerns satisfactorily.

Thank you again for your valuable insights and for helping us improve the quality of this work.

The authors of submission #11605

Q2) Figure 4(b): I do not understand how the plot value for {a,b} can be in between {a} and {b}? should {a,b} not be strictly better than {a} and {b}?

Thank you for pointing this out. Indeed, it is expected that upon successful knowledge integration, providing more knowledge (here knowledge about parameters a an b as opposed to just a or just b), should result in better predictions. In Figure 4b), the improvement for $\mathcal{K} \sim \\{b\\}$ and $\mathcal{K} \sim \\{a\\}$ is within the standard error of the improvement when $\mathcal{K} \sim \\{a, b\\}$, meaning the difference is not statistically significant. The reported mean improvement is sensitive to outliers as the variance of relative values is generally higher than that of absolute values. In addition to providing average relative $\Delta \text{AUC}$ across testing tasks, we include the average values of the negative log-likelihood across diferent knowledge types in Table A.2 in the Appendix. Comparison of the average loss across tasks agrees with our expectations: providing knowledge about 2 parameters results in better predictions than just a single parameter. Although for the case of $\{b\}$ and $\{a, b\}$ these differences are not significant—this is may be caused by the fact that the value of $b$ combined with a 1 or 2 datapoints almost fully determines the shape of the underlying curve, making the learning of the relationships between knowledge of type $\\{a, b\\}$ challenging, as for tasks with $\mathcal{D}_C \neq \varnothing$, the value of $a$ may appear redundant for the model.

We also note that, in practice, the learned prior $p_\theta(f \vert \mathcal{K})$, and the posterior predictive $p_\theta(y \vert x, \mathcal{D}_C, \mathcal{K})$ , are only a neural approximation of the true, underlying distribution, as learned based on the finite meta-training set ot tasks. The quality of this approximation will depend on the the number of meta-training tasks, the train-test distribution shift, and the chosen model hyperparameters (see the remarks section of the paper). Thus, in practice, we may observe empirical results deviating from the theoretically optimal behaviours. In this instance, while providing knowledge about both parameters $a$ and $b$ should be better on average across all tasks, we may find individual tasks in which improvement will is not strictly better than just providing the value of a single parameter. This is one of the disadvantages of using meta-learned priors instead of conventional model-based approaches, in which accurate knowledge integration can be guaranteed by design. We make this point in the Limitations section of our paper and in the analysis of the results from section 5.2.1 (lines 449-454).

Q3) Research contributions of Section 3.

The research contributions of this section are as follows:

• Section 3.2: Formally defining and casting knowledge-conditioned meta learning as a way of performing automated knowledge integration. This view point on meta-learning is novel.
• Examples 1 and 2 present possible ideas for concrete implementation of informed meta-learners, laying our foundations for future work and the later introduced method of INPs.
• Section 3.2.1. motivates theoretically the proposed approach.
• Section 3.2.1. highlights the challenges associated with practical implementation of informed meta-learners pointing at aspects worth paying attention to during empirical evaluation of informed meta-learners (this section is tightly linked with the questions investigated in the empirical study of section 5 and the additional experiments presented in section 5). It also points at the directions for future work.

W3) Lack of comparison to model-based approaches

Thank you for your comment. We would like to highlight that the primary advantage of informed meta-learning lies in its universality and applicability to scenarios where model-based approaches to knowledge integration are infeasible or require significant human effort. Our experiments are designed to demonstrate the feasibility of knowledge integration via informed meta-learning, and we compare INPs against the non-informed baseline of NPs to validate the effectiveness of meta-learned, knowledge-dependent priors.

Unlike informed meta-learning, model-based approaches require knowledge representations to be easily translatable into explicit inductive biases, such as functional representations, parameter priors, or regularisers for the loss function. In the specific case of Section 5.1, such a model-based approach is indeed feasible: it is easier to fit an explicit regression model with fixed parameter values derived from $\mathcal{K}$’s  than to learn the relationship between $\mathcal{K}$’s and the neural network’s parameters from data. Consequently, Bayesian linear regression would naturally outperform INPs in this well-defined environment of tasks. This scenario, however, is not representative of the expected use case of informed meta-learning, in which the underlying data-generating processes are unknown and translation of knowledge into explicit inductive biases is challenging.

In Section 5.2, we demonstrate the usage of informed meta-learning with real-world time series and image data and where knowledge is represented in natural language. In these cases, model-based approaches to knowledge integration are inapplicable. To the best of our knowledge, informed meta-learning is the only feasible method for integrating knowledge irrespective of its format or its relationship to the ground truth DGP.

In summary, the major advantage of informed meta-learning it its versatility and universality across diverse scenarios, knowledge, and data formats. That said, we acknowledge that this universality comes with trade-offs in performance: when a model-based approach is available, it should be preferred, as it guarantees the correctness of knowledge integration by design instead of relying on a neural approximation learned based on a finite number of meta-training tasks. We highlight this trade off in the final section of the paper (lines 510-517).

Questions

Q1) There seems to be no empirical application with K = natural language for a regression problem in the paper.

We would like to note that the temperature prediction experiment presented in section 5.2.1., setup B, is precisely a regression problem where knowledge is presented in natural language. The presented sentence describe the evolution of the temperature curve throughout the day, including expressions like “temperatures will drop to -5 degrees by early morning”, "the temperature will gradually increase from 2 to 18”, etc. (see Figure. A.2. for sample knowledge representations). These kinds of descriptions are in a close resemblance to the ones presented in Figure 3. In fact, they are even more complex and detailed.

Dear reviewer gpLZ,

Thank you for getting back to us regarding our response to your review. We really appreciate your engagement in the rebuttal process.

UPDATE: Following up on your comment regarding Bayesian regression, we have now included an additional experiment comparing (I)NPs with exact, model-based inference. These results are available in the Appendix A.8.4. of the updated manuscript. We hope that this addition will be of interest to the readers!

In terms of the weather forecast experiment, naturally, we will make appropriate adjustments to the presentation of this experiment based on the final comments received.

Once again, thank you for dedicating your time to reviewing our paper.

Kind regards,

The Authors

W2) Construction and properties of knowledge representations

The construction of knowledge representations used for meta-training is an important aspect to be considered for a successful deployment of informed meta-learning in practice. This construction, is however, inherently application-specific and dependent on the representations of knowledge expected to be observed at test time—as provided by a domain expert or derived from a knowledge repository (wikipedia, textbooks). Further, the underlying meaning of knowledge representations that is learnable in our proposed data-driven manner is strictly dependent on the available data for meta-training. For instance, if we were to learn what “increasing” means in the context of regression tasks, our meta-training dataset must contain negative examples of non-increasing functions. To generalise this argument, we can assume that at test time we are interested in providing information that has up to $d$ intrinsic factors of variability, i.e. that the information contained in $\mathcal{K}$’s can be essentially compressed to a $d$-dimensional representation. In the synthetic regression example we have $d=3$ : slope, oscillation frequency, bias. In the example of bird classification, $d$ corresponds to the maximum number of attributes that a given knowledge representation can describe (eye color, beak length, etc.). To ensure that all properties of sample $f$’s associated with each of the $d$ underlying dimensions of $\mathcal{K}$ are learned adequatelly, the meta-training dataset should contain tasks that span this space sufficiently, as succesful extrapolation to previously unobserved regions of the knowledge space, let alone dimensions, cannot be guaranteed.

The point about knowledge complexity and the resulting requirement of more training tasks for successful meta-learning has been mentioned in section 3.2.2 Finite sample approximation and demonstrated empirically in Appendix A.8.2. We also hint at the usage of generative models for synthetic data generation and augmentation as a plausible strategy for mitigating the challenge of meta-training set collection in section 3.2.2 Existence of the meta-training set and Appendix 2.2.2. This strategy has also been employed in our experiments on real-world datasets in sections 5.1 and 5.2, where knowledge representations are generated with GPT-4 aiming to simulate natural language descriptions that a real human expert may provide. We hope these experiments can inspire future applications across a wider range of domains, where natural language descriptions can guide numerical predictions such as healthcare, finance, or retail.

W3) Typos and axes in Figure 6

UPDATE: Thank you for pointing this out. We have now updated this in the manuscript.

We thank the reviewer for their insightful comments and constructive feedback. We have carefully considered each point of feedback and provide our point-by-point responses below.

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Weaknesses

W1) Additional theory: strict inequality and more knowledge ⇒ stronger priors

Thank you for your questions regarding the theory which prompted us to include a deeper discussion on the conclusions from the presented Theorem 1. UPDATE: We have provided an extended theoretical discussion regarding the aspect of necessary conditions for the inequality of Theorem 1 to be strict, and a formalisation of the statement “more information leads to stronger priors”. This is now available in the Appendix A.1.3. of the updated manuscript. To summarise this discussion, we show that:

1. The inequality of Theorem 1 can be made strict under the assumption of positive mutual information between the target $y$ and knowledge $\mathcal{K}$ at the location $x$, formalised as $\mathbb{I}(y; \mathcal{K} | x) := \mathbb{H}[p(y \vert x)] - \mathbb{E}_{p(\mathcal{K})}\mathbb{H}[p(y \vert x, \mathcal{K})] > 0$
2. We can re-exprees the above in terms of the mutual information of $\mathcal{K}$ and $f$ to arrive at an equivalent condition of $\mathbb{I}(f; \mathcal{K}) - \mathbb{E}\_{p(y \vert x)}\mathbb{I}[(f; \mathcal{K} \vert x, y)] > 0$, where $\mathbb{I}(f; \mathcal{K}) = \mathbb{H}[p(f)] - \mathbb{E}\_{p(\mathcal{K})}\mathcal{H}[p(f \vert \mathcal{K})]$ and $\mathbb{I}(f; \mathcal{K} \vert x, y) = \mathbb{H}[p(f \vert x, y)] - \mathbb{E}\_{p(\mathcal{K})}\mathbb{H}[p(f \vert \mathcal{K}, x, y)]$. Then, the condition $\mathbb{I}(y; \mathcal{K} | x) > 0 \Leftrightarrow \mathbb{I}(f; \mathcal{K}) - \mathbb{E}\_{p(y \vert x)}\mathbb{I}[(f; \mathcal{K} \vert x, y)] > 0$ requires that:
   • The mutual information of $f$ and $\mathcal{K}$ is strictly positive, i.e. $\mathbb{I}(f ; \mathcal{K}) > 0$, which should trivially be satisfied for any valid knowledge representation by the defintion of knowledge as information about the ground-truth data generating process.
   • Observing $y$ at a specified location $x$, brings sufficient information about the underlying $f$, e.g., the observational noise of $y$’s cannot be too high. If it is not the case, $\mathbb{E}_{p(y \vert x)}\left[\mathbb{I}(f; \mathcal{K} \vert x, y)\right]$ is high, bringing it closer to $\mathbb{I}(f; \mathcal{K})$ and thus making $\mathbb{I}(y; \mathcal{K})$ closer to 0.
3. We also formalise the meaning of more informative knowledge representations and how such more informative representation reduce the divergence between the posterior predictive $p(y \vert x, \mathcal{D}_C, \mathcal{K})$ and the ground-truth marginal $p(y \vert x, f)$.

Questions

Q1) I would add "particularly" to the first sentence of the abs

UPDATE Thank you for your suggestion, this has now been added in the abstract.

Q2) Fig. 1 caption (indicies on Ks)

UPDATE Thank you for your suggestion, we have now added a brief comment explaining the relationships between the function and knowledge indices.

Q3) Assumptions of the stochastic process and conditional independence.

The stochastic process convention. The assumption that an underlying function is sampled from a stochastic process is powerful and very common in probabilistic models, especially in nonparametric Bayesian approaches like Gaussian Processes (GPs), allowing models to naturally handle uncertainty in their predictions. While modelling with GPs can be restrictive, due to the strong assumption of the stochastic process being a Gaussian, methods like Deep Kernel Learning or Neural Process, enable far greater flexibility. In particular, NPs discussed in this paper, can approximate arbitrarily complex stochastic processes that are learned based on real-world data, imposing little to no assumptions on the properties of sample functions.

Model uncertainty. To address your example with the sinusoid vs. linear function, the fact that many different sample functions can fit the observed data equally well is exactly the problem of inductive bias specification in low sample regimes. With only a few observations we are unable to decide with certainty whether a linear or a sinusoid function is a better fit, unless we posses additional expert knowledge about the underlying data generating process, or we can collect more data. This, however, does not imply that there does not exist an underlying function $f$ that gave rise to these observations. The assumption we make here and which is the underlying principle of most of machine learning approaches, is that $f$ can be recovered with certainty as the number of observed data points grows.

Conditional independence. In terms of the conditional independence assumption, we argue that this is a rather a natural assumption to make. Namely, if we know the ground-truth form of $f$ that gave rise to the noisly-observed dataset $\mathcal{D}$, in order to describe the properties of $f$, we need not observe individual datapoints $(x_i, y_i) \in \mathcal{D}$. The information contained in $\mathcal{K}$, can be derived purely from $f$.

UPDATE Thank you for raising these questions. We appreciate that the assumptions made in our paper may seem restrictive or only useful for the theory. We have now added additional references and intuitions behind these assumptions to motivate their validity.

Q4) Nice concrete examples in section 3. but formatting is hard to read

UPDATE Thank you for your comment. We have used the default theorem latex environment to define the example environment. We agree, however, that the italics make it difficult to read and we have changed it to a standard font in the update version of the manuscript.

Q5) INP and NP acronyms.

UPDATE Thank you for this suggestion. We have now made sure that the NP and INP acronyms are properly introduced in section 4 so that they are better linked across sections.

Q6) Unclear how the aggregator operator works.

Addressed in W3).

Q7) For the dist shift experiments, what was the K you provided?

As specified around the lines 356-357, the knowledge presented in the distribution shift experiment specifies the ground-truth value of the parameter $b$

Thank you for taking the time to review our work. We appreciate the reviewer’s detailed feedback on our submission and we address their concerns point by point in our response below.

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Weaknesses

W1) Missing references to conditional meta-learning

References to conditional meta-learning, while not explicitly named as such, have been made in the Appendix A.4. Extended Related Work under the Meta-learning with meta-data subsection (e.g.: Kaddour et al., Tseng et al., Iwata & Kumagai). We agree, however, that the main body of the paper should include an explicit reference as well.

UPDATE: Following the reviewer’s comment we have now included additional references to conditional meta-learning, including them in the main body of the paper when conditional meta-learning is mentioned for the first time and in the related work section.

W2) Lack of comparisons to ‘baselines’ discussed earlier on

Automated vs. model-based approaches. The alternative methods for knowledge integration discussed in the paper are model based and necessitate the involvement of a human engineer to interpret the knowledge in a given representation $\mathcal{K}$ and manually encode it within the learning algorithm (lines 112-113). This dependence on manual algorithm design is particularly limiting in cases where the underlying functional form of the data generating processes is unknown, making conventional knowledge integration challenging and n many cases simply infeasible. In contrast, the focus of this paper is on automating the knowledge integration process. This is especially relevant for applications where such functional forms are complex or undefined, and the task-specific knowledge is difficult to represent in a form of an exact mathematical equation or a constraint—e.g. qualitative proporties described in human natural language. To the best of our knowledge, the framework of informed meta-learning presented here is the only viable approach for achieving fully automated and controllable integration of such forms knowledge.

INPs vs. alternative implementations of informed meta-learning. Within this paradigm, we present the method of Informed Neural Processes method primarily for illustrative purposes, aiming to inspire further research in this area. Whilea systematic evaluation of alternative knowledge-conditioned meta-learning methods, including conditional MAML (Example 1) and approaches like in-context learning with LLMs (lines 527-528), may be valuable for practitioners, such a benchmarking effort would constitute a substantial undertaking that could warrant its own dedicated study. The current paper thus aims to serve as a foundational reference point for future work.

W3) The way K is given to the model is unclear

We made a conscious decision of keeping this paper conceptual in nature—with its main contributions being the proposed new way of thinking about meta-learning as an enabler for automated and controllable inductive bias specification and the accompanying theory. The implementation details have been purposefully postponed to the Appendix, where for every experiment, we provide details on how knowledge is encoded (see Appendix A.5 and A.7).

UPDATE: To strike a better balance between theory and practice we have included more detail about the implementation details in the main body of the paper. Following the reviewer’s suggestion we have included a short explanation on the aggregating operator when it is introduced in the main body of the paper—in most experiments the aggregation is a simple summation of the data embedding and the knowledge embedding vectors.

W4) Conditioning physics-informed neural nets should be cited

Thank you for your comment regarding conditioning physics-informed neural networks. In fact, the paper already includes references to conditional meta-learning for PINNS in the Appendix A.4. Extended Related Work under the Meta-learning with meta-data subsection (Huang et al. (2022); Belbute-Peres et al. (2021)). UPDATE: We have included an additional reference to conditional PINNs in our related work section.