Eli Yablon

Week 1

I moved in to Rutgers and met my fellow REU participants. John, Roger, and I met with Professor Gao to discuss potential research directions, and she suggested some papers for us to read.

Week 2

John, Roger, and I spent the beginning of the week making a slideshow about our topic for our fellow REU students, which we presented on Tuesday. For the rest of the week, I worked on trying to understand a 2019 paper in which they proved that even being able to compute the outputs of Bayesian agents in learning networks is PSPACE-hard. They assumed a more generalized setting in which agents' signals could vary in strength and could be inhomogeneus (i.e. a private signal of $1$ could be a much stronger indicator that the ground truth is $1$ than a private signal of $0$ for the ground truth of $0$). I spent some time modifying the proof so that it also held for homogeneus, equal-strength signals across all agents. John came up with an interesting construction to prove that computing learning rates under majority vote dynamics (assuming homogeneus but varying strength signals) was #P-hard via a reduction to the monotone 2SAT problem.

We hope to use these results to be able to better understand the computational complexity of Bayesian network learning and hopefully resolve the hardness conjecture on random orderings from last year.

Week 3

I spent the beginning of this week writing up my reduction from last week, which took a while as there were a lot of details to iron out and a lot of messy algebra to complete. I then spent some time thinking of how to extend John's result to Bayesian networks, or at least to majority-vote networks in which every agent's signal had probability $p>1/2$ of being correct, but had no success in either case.

I then began thinking about information exchange networks with incentives. This comes from the observation that if we wanted to achieve asymptotic truth learning in a network we should just have the first $N$ agents announce their private signals, and then have every agent that comes after use these observations to predict the ground truth. For sufficiently well-connected graphs this protocol should always work by the law of large numbers. However, this doesn't really represent reality, as every individual usually just gets a reward if they predict the correct ground truth and nothing if they don't. Thus, a network in which everyone uses Bayesian reasoning to announce their signals is a nash equilibrium in a sense, and that's why it's a more accurate reflection of reality.

Week 4

I started off the week by finding an incentive system in which asymptotic truth learning occurs almost surely for a complete graph. The gist is that if we tell all the agents that the $n$th person who announces the correct answer will receive $1/n$ dollars, and anyone who announces the wrong answer will receive nothing, then an information cascade can't occur indefinitely. Thus, an infinite amount of agents will end up revealing their private signal, which will then lead to learning for everyone.

John was able to figure out how to prove that computing the learning rate in networks with fixed, homogeneus signal strength is #P-hard, though his proof relies on proving that a certain matrix is invertible, which we haven't exactly been able to do yet. However, Professor Gao advised us to investigate the learning with incentives idea further, as well as the idea of multipass learning, in which agents have multiple rounds in which they announce what they think the most likely ground truth is.

Week 5

This week was a pretty laid-back week since John, Roger, and Professor Gao all went to Santa Barbara for the ACTION conference. John, Roger, and I made a presentation on our results so far to give to all the other ACTION students, as we were the farthest along in our program compared to everyone else at ACTION.

I then worked a little bit more on the learning with incentives idea. The result from last week suggested that any incentive system in which we tell all agents that the $n$th person who announces the correct answer will receive $f(n)$ dollars, where $f(n)$ monotonically decreases to zero, will lead to asymptotic truth learning. However, I was able to prove this isn't actually true, as there are some functions which decay too fast, preventing the agents on converging on some truth since there is too much incentive to be a "devil's advocate."

Week 6

After talking to Professor Gao at the beginning of the week, I decided to focus the rest of my efforts on proving computational complexity results. After all, computational complexity results are the area in which we've made the most progress, and it's also the area I enjoy the most since it is the most theory heavy. I started the week by reviewing at John's results from a few weeks ago proving #P-hardness, thinking about how to make the proof complete (it hinged on proving some matrix was invertible, which we were unable to do). Also, I noted that the proof was also heavily tie-breaker dependent, meaning that the proof only worked assuming a very specific tie breaking rule, so I thought a little about how to make the proof tie-breaker independent. Eventually, I was able to resolve both of these difficulties by abandoning the matrix altogether to complete the proof of #P-hardness and for it to be tiebreaker independent.

Week 7