Paper: A representation-learning game for classes of prediction tasks (ICLR-2024)

Reviewer comment (Theory):
An algorithm is proposed to learn the representation for the general case. However, the evidence, from the theoretical or the practical perspective, is insufficient to determine the applicability of the proposed method.

Author response:
Our main goal here is to propose a principled framework for learning representations based on partial knowledge of downstream prediction tasks. We aimed for analytical exploration of its fundamental properties, and thus focused on analysis of the linear-MSE setting, specifically Theorem 3. We believe that this provides a strong evidence for the validity of the framework. The proof of this result is rather convoluted (see our answer to d) below), and thus is also a main contribution of the paper. Algorithm 1 was developed to show that the framework is viable beyond the linear case, even in general cases (loss function, representation rules). Following this approach, we have focused the paper on the core of the framework and basic settings, rather than on practical baselines. Going beyond basic settings inherently requires some domain-knowledge, e.g., determining the class of functions $\{\cal F\}$, and we believe that this deserves a separate inquiry. We hope that this would be achieved in future research, both by us, and by experts in various domains. We note, however, that the incremental algorithmic approach we take follows a widely acceptable solution in various ML problems, as we discuss in depth on page 2 (“Algorithmic contribution”), and in Appendix B, pages 18-19 (“Game-theoretic formulations in statistics and machine-learning” and “Incremental learning of mixture models”).

Reviewer comment (Novelty):
The analysis for the linear case seems to be adapted from the game theory. Could you please clarify the technical contribution, if any, in the analysis?

Author response:
The main part of the analysis in the linear case is the proof of Theorem 3 (which is rather long, about 7 pages). Its goal is to find the optimal solution for the representation in mixed strategies. The main technical difficulty is that solving the mixed minimax problem directly, that is, solving

$\min\_{\boldsymbol{R}}\max_{f\in{{\mathcal F}_{S}}} \mathbb{E}\left[\mathsf{regret}(\boldsymbol{R},f \mid \Sigma\_{\boldsymbol{x}})\right]$

is intractable. However, the minimax theorem from game theory implies that its value equals the maximin problem, that is, to 
$$\max_{\boldsymbol{f}}\min_{\boldsymbol{R}}\mathbb{E}\left[\mathsf{regret}(\boldsymbol{R},f\mid\Sigma_{\boldsymbol{x}})\right]$$ 
where now the function can be random too. This problem is still not trivial, but we managed to solve it. This resulted $\boldsymbol{f}^{\*}$, the function part of the saddle-point $(\boldsymbol{R}^{\*},\boldsymbol{f}^{\*})$, which solves both the minimax and maximim problems. We then exploited the maximin solution as a certificate to find the optimal representation $\boldsymbol{R}^{\*}$, because the minimax value is never smaller than the maxinim value. Thus, if one finds a random representation for which

$\max_{f\in{\cal {\cal F}_{S}}}\mathbb{E}\left[\mathsf{regret}(\boldsymbol{R},f\mid\Sigma\_{\boldsymbol{x}})\right]$

equals the maxinim value, this guarantees that this random representation is the sought $\boldsymbol{R}^{\*}$. We find this by first making an educated guess regarding a simple structure that $\boldsymbol{R}^{\*}$ should posses. We then optimized over this structure, and shown it is reduced to solving an equation $Ap=b$. We proved that any solution to this equation leads to a representation whose regret is no larger than the maximin value, and is thus $\boldsymbol{R}^{\*}$, the saddle-point representation. To complete the proof, we show that a solution to $Ap=b$ must exist, and that the support of this solution is at most $\ell^{*}+1$. This leads to an optimal random representation $\boldsymbol{R}^{\*}$ that is drastically simpler than a general probability distribution over $\mathbb{R}^{d\times r}$, and reinforces the effectiveness of the approach.

Reviewer comment (Theory):
Is it possible to show the game will reach some situation, such as an equilibrium? Representations from the equilibrium may represent a negotiated outcome from both players.

Author response:
As follows from our answer to d), the obtained solution $(\boldsymbol{f}^{\*},\boldsymbol{R}^{\*})$ is exactly the equilibrium of the game. If both sides agree to use this solution, no side can improve its payoff by a one-sided change of his/her strategy.

Reviewer comment (Experiments):
Although Examples 6 -- 9 help us to understand the nature of Algorithm 1, one still cannot know Algorithm 1's applicability. Would it be better to compare Algorithm 1 with baselines to show the representations output from Algorithm 1 are better than the existing methods?

Author response:
As we have answered to point c), we focused this paper on the general approach, theoretical guarantees on fundamental settings, and a general algorithm. However, even though it is just a basic setting, in Example 9 we compare our method to PCA and show a significant improvement. PCA is also clearly not the state-of-the-art, but it is a classic and principled method that enables initial comparisons, and evidence of the potential of the method. 

In summary, our main contributions are a general framework, an analytic solution in basic setting (linear-MSE) and its proof, and a “meta-algorithm” which shows the viability of the method in the general cases. Hopefully, applications will be explored in depth in the future. In light of the above clarification, we would appreciate if you could re-evaluate our contributions, and finally thank you for your effort in reviewing the paper.

Reviewer comment (Presentation):
The meat of the paper is mostly in what is effectively the supplementary material (after the references). The authors have albeit put substantial effort in trying to condense the ideas involved in the proofs as well as illustrate using good examples in the main section. However, it seems to be a losing battle, and any useful perusal of this paper must involve substantial parts of the supplementary.

Author response:
Our approach in writing the paper was to first present the general framework and its importance; second, explore it theoretically in basic linear settings; and third, propose an efficient algorithmic for the general case, which cannot be covered by theoretical analysis. We believed that all these components are vital, and thus deferred the technical aspects to the appendix, leaving them to in-depth readers. We understand that this comes at a cost, and we will re-inspect the paper from this aspect. We would be happy to take any suggestions for improvement (e.g., what important points are missing in the body of the paper), and will incorporate them into an updated version.

Reviewer comment (Experiments):
So far the current results only focus on simple cases where the classifiers are linear. Ideally, some attempts to cope with nonlinear classifiers or nonlinear feature mapping would be appreciated.

Author response:
* **Response to the weakness**: Our proposed Algorithm 1 is applicable to any differential representation, not just linear. This algorithm alternates between optimizing an adversarial function (phase 1) and a new representation rule (phase 2). We have not focused on this in the body of the paper, but the optimization of the new representation rule in phase 2 is discussed in Appendix G.2 in detail. Algorithm 3 proposed there for the phase 2 optimization of the representation is based on computing gradients of the loss with respect to the representation rule (line 20). Therefore, the algorithm is applicable as long as these gradients can be computed (sub-gradients may also suffice). In Appendix H.4 we conducted a preliminary experiment with a one hidden-layer NN architecture, and showed the improvement of the regret. In future research we plan to explore our algorithm for non-linear representations in depth.

Reviewer comment (Theory):
Is it possible to extend the first result (linear regression) to kernelized classifiers? If not, can you explain why?

Author response:
* **Response to the question**: Yes, extension to representations that are based on infinite-dimensional features is possible. In Appendix F, we provide one possible such extension, to a Hilbert space setting (this is mentioned on page 5, in the remark before Example 4). In this setting, the original feature vector $x\in\mathbb{R}^{d}$ is mapped to a vector $R(x)=(\psi_{1}(x),\ldots,\psi_{r}(x))^{\top}\in\mathbb{R}^{r}$ where $\\{\psi_{i}(x)\\}$ are non-linear feature-maps, taken from a Hilbert space of such maps. Our theoretical results on pure and mixed representations are generalized to this setting in Theorems 19 and 20. One can also consider the opposite setting – a linear representation followed by a non-linear (kernelized) predictor. Generalizing the theoretical results to this case is interesting, but appears to be challenging. This is because it requires obtaining a closed-form expression for $\mathsf{regret}(R,f\mid P_{\boldsymbol{x}})$, as was obtained for the linear case in Lemma 16, Appendix E.2. In turn, this requires understanding how the non-linear feature map affects a linear representation $z=Rx$, which is difficult in general. In future research, it would be interesting to furnish conditions that allow to analyze such cases too. By the way, referring to the weakness above, this setting is another case in which the representation is non-linear.