Paper: An Invex Relaxation Approach for Minimizing Polarization from Fully and Partially Observed Initial Opinions (ICLR-2024)

Reviewer comment (Novelty):
What is the contribution of the proposed framework to solve this problem? It seems to be the use of projected gradient descent. But I think the projected gradient descent is also very standard in optimization. So what is the novelty of this method?

Author response:
Firstly, we thank the reviewer for taking their valuable time to provide feedback.

Running first-order optimization algorithms on non-convex problems only guarantees us a local minimum. Studying the properties of these local minima and proving an even stronger result that there exists a global minima improves our theoretical understanding of these problems. This, in turn, motivates the design of efficient algorithms for such problems.   There is a wide range of non-convex practical problems which have been studied in this context. These include, but are not limited to, principal components analysis, canonical correlation analysis, orthogonal tensor decomposition, phase retrieval, dictionary learning, matrix sensing, matrix completion, and other nonconvex low-rank problems. A few references are provided below:

Lee, J.D., Simchowitz, M., Jordan, M.I. and Recht, B., 2016, June. Gradient descent only converges to minimizers. In Conference on learning theory (pp. 1246-1257). PMLR.

Ge, R., Huang, F., Jin, C. and Yuan, Y., 2015, June. Escaping from saddle points—online stochastic gradient for tensor decomposition. In Conference on learning theory (pp. 797-842). PMLR.

Our contribution here is similar in this aspect, where we theoretically demonstrate the characteristics of polarization function under the popular  FJ dynamics, which is believed to be non-convex. By showing that polarization (and the general class of $M \rightarrow$ $s^TM^{-k}s$ functions) are invex, a special class of non-convex functions, we ensure there are no saddle points, and every local minimum is a global minimum for this class of functions. This information is valuable in algorithm design as it ensures that first-order methods are not stuck at local minima or saddle points. The objective function for $k=2$, $M \rightarrow$ $s^TM^{-2}s$, is conjectured to have only one local minimum [Citation: Xi Chen et.al., Page 10]. Ours is the first theoretical result that validates this conjecture. 

Not only do we theoretically demonstrate the global optimality scenarios for various practical cases of polarization, including stubborn actors and multi-period polarization, but we also deal with scenarios where only partial information about initial opinions is available. Our relaxations provided in equations (5), (6), and (24) aim to identify the edges that provably minimize polarization. 

Also, observe that most of the literature is focussed on minimizing convex objectives such as Polarization and Disagreement index or understanding Disagreement among users (references are provided in Section 3: Prior Work). 

[Citation: Miklos Z. Rácz et al., page 12] studied this problem in a discrete setting and provided heuristics to minimize polarization. 
Ours is the first continuous optimization study that provides theoretical guarantees in this aspect and outperforms the existing approaches. 

Thus, our contributions are multi-fold: 

1) Rather than modelling the problem in a discrete setting, we provide a continuous relaxation backed up with theoretical guarantees for polarization under known initial opinions, including the realistic cases of stubborn actors and multi-period.
2) We also study a realistic problem setting that has not been previously studied, wherein the observer has access to only a subset of initial opinions. We show that in these cases, too, PGD can attain the global minimum. Note that this scenario can also be extended to stubborn actors and multi-period. 


We hope to have addressed the reviewer's questions. We would be happy to answer any further questions.