Week 1 · May 26–30
This week marked the beginning of the DIMACS REU program. I moved into Rutgers housing
and met my roommates, Ryan, Arnav, and Herbert. I also met my research teammates, Eli and
Roger, as well as the rest of the REU cohort during orientation.
Following orientation, we met with our Dr. Gao and Daniel to discuss prior work completed
through the program and potential directions for our summer research. Much of the week was
spent conducting a broad literature review. Our professor shared us resources related to nine
potential directions we could take the project in, ranging from Social behaviors of LLMs to
Repeated Learning with Bayesian models. We explored several research avenues and ultimately
narrowed our focus to the three primary questions stated in the project description.
Week 2 · May 31–June 6
On Tuesday, all of the REU groups gave short presentations introducing their projects. It was
really fun presenting and learning about everyone else's projects. Our presentation is linked
under resources.
I began the week by reviewing the network constructions studied by the 2023 REU and thinking
about whether the butterfly network could be modified to become more robust to adversarial
agents. While reading William's 2025 REU research log, I came across an insightful observation:
Bayesian agents that are unaware of adversaries may become confused by the resulting state of
the world. This motivated me to explore alternative aggregation rules where agents do not need
to know about the existence of adversaries.
I considered using a weighted-majority-vote aggregation method, where the weightings were
determined based on vertex learning rates. However, it is not known if there exists an efficient
algorithm for computing the learning rate of a vertex in a network with a fixed decision ordering
under majority-vote (or bayesian inference). The 2024 REU showed that finding the optimal
learning rate over all possible orderings is NP-hard, and they were unable to identify a
certificate demonstrating that the corresponding decision problem lies in NP. This suggests that
computing learning rates for a single vertex may itself be computationally difficult.
While exploring this question, I proved that a related variant of the problem is #P-hard
through a reduction from #Monotone-2SAT. The variant I considered allows agents to have
asymmetric and unbounded private signals. I do not believe that my proof strategy can be
extended to the original problem, since my construction relies on AND and OR gadgets that are
implemented using unbounded signals.
Meanwhile, Eli showed that determining the action taken by a Bayesian agent in a sequential
learning network, given a fixed ordering and the actions of all previous agents, is NP-hard.
Throughout the week, Eli, Roger, and I met up a lot in the office.
During our meeting with Professor Gao and Daniel, we were encouraged to write up our reductions
and continue exploring our three main research directions by establishing simple theoretical
guarantees. They also introduced several related topics worth investigating, including repeated
sequential learning systems, continuous variants of majority vote, and potential adversarial environments.
Week 3 · June 7–June 13
This week, I focused on turning last week's hardness ideas into a more formal writeup. I wrote
up my reduction showing that computing the learning rate is #P-hard, and then spent time
reading more about #P-hardness reductions to see whether the argument could be adapted to a
more realistic setting where q∈(0,1) and the signal accuracy q is homogeneous across all
vertices.
I also wrote up a separate proof showing that, under weighted majority vote, the learning rate of
the network with a fixed ordering is monotone with respect to the learning rate of any individual
vertex. As a group, we found the following observation interesting: Learning rate with Bayesian
inference is monotone with respect to future agents but not the private signal strength (example
in 2025 REU of Bayesian inference failing when private signal strength was too high), while
Majority vote has the opposite properties.
Eli wrote up his proof showing that Bayesian inference is NP-hard, while Roger continued
experimenting with multipass sequential learning. In our meeting with Professor Gao and
Daniel, we mainly discussed our writeups.
One interesting issue we discussed was a potential "exploit" in weighted majority vote protocols.
If weights are allowed to vary continuously, the protocol can effectively change the topology of
the network by assigning weight 0 to certain neighbors. For example, even in a complete graph,
a weighted majority protocol could force a superconstant number of early agents to reveal only
their private signals, after which the rest of the network could aggregate and copy their
information. This makes the protocol much more powerful than ordinary majority vote in a way
that may not reflect the intended network structure. To address this, Professor Gao suggested
studying weighted majority vote with discrete labels, such as "high" and "low," rather than
allowing arbitrary continuous weights. She also encouraged us to look further into multipass
sequential learning, since that direction appears to be less explored in prior work.
Week 4 · June 14–June 20
This week, I spent most of the week writing up a proof showing that the learning rate is #P-hard
even with fixed, homogenous, bounded signals. A part of the proof relies on a matrix being
invertible. Eli and I believe the matrix is invertible but don't have a proof for it. As a group,
we are shifting away from building complexity results related to past works and want to focus on
exploring more into newer ideas. Eli has been working on learning with incentives. Roger and I
have been working on multipass learning.
Week 5 · June 21–June 27
On Monday we had culture day at DIMACS. I learned about tattoos, Ohio, salsa dancing, and
the history of Czechoslovakia. Allison and Lucy made Buckeye Candy while Martin and Sofia
made bramboráky, which was really delicious.
Later this week, Roger and I flew to UC Santa Barbara to attend the 2026 ACTION Annual
Review and Knowledge Expo. I was able to meet Edward, a participant from last year's REU
who worked on our project, as well as Tim Robinson. It was interesting to observe professors
coming together to showcase the work of the institute, answer questions from NSF evaluators
on the spot, and use feedback to discuss future directions. The keynote talks about
cybersecurity for robots and about enhancing cybersecurity defense through offensive were very
informative.
For multipass learning, I showed that actions will eventually stabilize. I also showed an example
of a network with a fixed ordering such that the learning rate is higher with single pass
compared to multipass.
Week 6 · June 28–July 4
This week, Eli, Roger, and I gave a mid project presentation to our friends at UCSB who were
just starting their program. This week I mainly spent time formalizing and typing up my results
about multipass related to stability, canonical networks, and examples where multipass performs
worse. Roger and I both conjecture that random order multipass always perform better than
random order single pass. Eli was able to fix up the earlier majority vote learning rate is #P hard
proof with an elegant method that is both tiebreaker independent and circumvents having to
prove invertibility for a very complicated family of matrices.
