Paper: The (co)limit of metabeliefs (ICLR-2024)

Reviewer comment (Theory):
Could the authors provide more concrete examples or real-world applications of their theory?

Author response:
We thank the reviewer for their comments. 

To respond to the questions: 
(1) Real-world implications of our theory, would be, for example, to choose the appropriate level of metabeliefs for specific problems, offer guarantees on approximation given a particular choice of level, and reason about compositions of problems.  
(2) We are unaware of any comparable effort to formulate the problem of metabeliefs. If the reviewer has specific suggestions, those would be appreciated.  
(3) Yes, we do plan to write subsequent papers on specific applications! As suggested in the introduction, we believe there are quite significant implications across learning, reinforcement learning, and social inference. In each case, engaging with the literate will require additional notion and technical machinery, and we therefore believe it best to tackle these applications in subsequent papers. If the reviewer raises their rating a bit, that would help us move faster toward that goal!

Thanks again!

Reviewer comment (Novelty):
How does this work compare and contrast with other mathematical approaches to metabeliefs or similar constructs in machine learning?

Author response:
We thank the reviewer for their comments. 

To respond to the questions: 
(1) Real-world implications of our theory, would be, for example, to choose the appropriate level of metabeliefs for specific problems, offer guarantees on approximation given a particular choice of level, and reason about compositions of problems.  
(2) We are unaware of any comparable effort to formulate the problem of metabeliefs. If the reviewer has specific suggestions, those would be appreciated.  
(3) Yes, we do plan to write subsequent papers on specific applications! As suggested in the introduction, we believe there are quite significant implications across learning, reinforcement learning, and social inference. In each case, engaging with the literate will require additional notion and technical machinery, and we therefore believe it best to tackle these applications in subsequent papers. If the reviewer raises their rating a bit, that would help us move faster toward that goal!

Thanks again!

Reviewer comment (Experiments):
Are there plans to test the presented theories empirically, or to develop algorithms/tools based on this framework?

Author response:
We thank the reviewer for their comments. 

To respond to the questions: 
(1) Real-world implications of our theory, would be, for example, to choose the appropriate level of metabeliefs for specific problems, offer guarantees on approximation given a particular choice of level, and reason about compositions of problems.  
(2) We are unaware of any comparable effort to formulate the problem of metabeliefs. If the reviewer has specific suggestions, those would be appreciated.  
(3) Yes, we do plan to write subsequent papers on specific applications! As suggested in the introduction, we believe there are quite significant implications across learning, reinforcement learning, and social inference. In each case, engaging with the literate will require additional notion and technical machinery, and we therefore believe it best to tackle these applications in subsequent papers. If the reviewer raises their rating a bit, that would help us move faster toward that goal!

Thanks again!

Reviewer comment (Novelty):
The categorical and geometrical results, while natural, do not have foreseeable applications to machine learning (that I am aware of or can imagine). This is not unexpected for an exploratory paper to propose a new modeling perspective (colimit of topological spaces), but stronger connections to machine learning definitely help.

Author response:
We thank the reviewer for their comments. To respond to the reviewer's question, we are not aware of any comparable methods of modeling metabeliefs to which we may compare. If the reviewer has specific examples in mind, those would be appreciated. 

We agree that understanding how the colimit might be useful in machine learning is important. We believe that applications are beyond the scope of the current paper, given the already significant technical content. To improve the clarity of the connection, we have elaborated the example in Section 4: Computing distance in the colimit: rock-paper-scissors. In particular, we give the solution to the $3\times 3$ optimization problem, and simplify the integration to remove the infima. We hope this helps make potential connections more clear.

Reviewer comment (Experiments):
While the authors argue that colimit of probability spaces (metrized under the Wasserstein $p$-distance) can model the geometry (and topology) of metabeliefs in co-operative and competitive games, are there examples showing that this modeling is better than other modeling for metabeliefs in terms _applications to machine learning?_

Author response:
We thank the reviewer for their comments. To respond to the reviewer's question, we are not aware of any comparable methods of modeling metabeliefs to which we may compare. If the reviewer has specific examples in mind, those would be appreciated. 

We agree that understanding how the colimit might be useful in machine learning is important. We believe that applications are beyond the scope of the current paper, given the already significant technical content. To improve the clarity of the connection, we have elaborated the example in Section 4: Computing distance in the colimit: rock-paper-scissors. In particular, we give the solution to the $3\times 3$ optimization problem, and simplify the integration to remove the infima. We hope this helps make potential connections more clear.

Reviewer comment (Theory):
Page 5, Lemma 6 — The authors do not prove that the alleged metric is positive definite (that \( W(\mu,\nu) > 0 \) if \( \mu \neq \nu \)), which is a requirement of the usual definition of a metric. Clarification is needed whether they require this and, if so, whether their map \( W \) satisfies this property.

Author response:
$W_p^{\infty}$ is indeed positive definite. We have added this proof to the proof of Lemma 6 given in the appendix: ``Let $\mu\in\mathcal{P}_p^m(\mathcal{X})$ and $\nu\in\mathcal{P}_p^n(\mathcal{X})$ with $m\geq n$ such that $\mu$ and $\nu$ are not single-atom Dirac distributions (so they could be either different distributions or points in $\mathcal{X}$). By definition, $W_p^{\infty}([\mu],[\nu])=W_p^m(\mu,\nu_m)$, where $\nu_m$ is the unique element in the intersection $[\nu]\cap\mathcal{P}_p^m$. Then because $W_p^m$ is a metric, $W_p^m(\mu,\nu_m)\geq 0$ with $W_p^m(\mu,\nu_m)=0$ if and only if $\mu=\nu_m$. However, if $\mu=\nu_m$, then because we have chosen $m$ so that $\mu$ is not a single-atom Dirac distribution, $\nu_m$ is not a single-atom Dirac distribution. Therefore since $\nu_m\in [\nu]$ and $[\nu]$ is the unique non-single-atom Dirac distribution (or point in $\mathcal{X}$) in $[\nu]$, we have $\mu=\nu_m=\nu$."

Reviewer comment (Writing):
Page 1, first paragraph following the abstract — Typo: "... to sequences of beliefs quantify uncertainty ..." should read something like "... to sequences of beliefs that quantify uncertainty ..." (add "that" or an alternative).

Author response:
We also fix all of the typos in the list provided, and wish to once again thank the reviewer for providing such a detailed list.

Reviewer comment (Writing):
Page 1, fourth paragraph — Typo: "... one must play of the three options." should read "... one must play one of the three options.".

Author response:
We also fix all of the typos in the list provided, and wish to once again thank the reviewer for providing such a detailed list.

Reviewer comment (Writing):
Page 1, fifth paragraph — Typo: "geoemtric" should read "geometric". In the following sentence, the word "gives" is superfluous.

Author response:
We also fix all of the typos in the list provided, and wish to once again thank the reviewer for providing such a detailed list.

Reviewer comment (Writing):
Page 2, paragraph following Theorem 1 — Potential numbering error: "Parts (2)--(6)" should be "Parts (2)--(5)" since there are only 5 parts in Theorem 1.

Author response:
We also fix all of the typos in the list provided, and wish to once again thank the reviewer for providing such a detailed list.

Reviewer comment (Theory):
Page 3, Lemma 3 — In the proof, the authors use a result that guarantees each \( \mathcal{P}^m(\mathcal{X}) \) is \( T_1 \) if \( \mathcal{X} \) is \( T_1 \). While plausible, a proof or reference is needed to confirm this standard result.

Author response:
The T1 claim may be found in Parthasarathy (2005) "Introduction to Probability and Measure", though the statement can also be amended to change T1 to, say, locally compact and metrizable, and the statement is easy to prove directly.

Reviewer comment (Theory):
Page 4, proof of Theorem 4 — Theorem 1 part (4) addresses *compact* spaces, not the weaker notion of *paracompact* spaces. Clarification is needed on how this inference is made, or if it suffices for later applications that spaces are compact in addition to being paracompact.

Author response:
Many of the applications (particularly to machine learning) implicitly use compact spaces; however in the statement of part 4 of Theorem 1, "compact" can be changed to "paracompact," and the statement is still true (albeit weaker). We have amended this.

Reviewer comment (Writing):
Page 5, right-hand side of Equation (10) — The part after "|" is not a condition, which reads oddly. Suggest adding that \( \tilde{\nu} \) is in the sets shown, if correct.

Author response:
We have fixed the right-hand side of Equation 10 by removing the unnecessary and confusing portions of the notation: the beginning and ending curly braces and the $\nu\in\mathcal{P}_p^i(\mathcal{X})$.

Reviewer comment (Theory):
Page 5, Equation (11) — Clarification is needed on why \( \Delta^i \) commutes with \( \bigcap \) and \( \bigcup \). This may be a standard result, but a reference is requested.

Author response:
$\Delta^i$ commutes with $\bigcup$ since for any function $f$ and sets $A,B$, $f(A\cup B)=f(A)\cup f(B)$. To make the passage from equation 11 to equation 12 more clear, we have added a few lines to the proof (including why $\Delta^i$ commutes with $\bigcap$). This addition begins immediately following Equation 11, "Applying $\Delta^i$..." and continues until just before Equation 14, terminating with, "Now, Equation 13 becomes:".

Reviewer comment (Writing):
Page 8, subsection "Competitive/Cooperative games" — There is a missing dot between the first two sentences. Also, "In this sense, both agents beliefs ..." should read "In this sense, both agents' beliefs ...".

Author response:
We also fix all of the typos in the list provided, and wish to once again thank the reviewer for providing such a detailed list.