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My Third Ph.D. Year

(Thank google translate for a first pass on the translation and thank Katherine Angier for super helpful comments on an earlier draft. See here for the original Mandarin version)

One week after the Liqiu, the heat in New England was finally (temporarily?) relieved. Maybe this year really is hotter than the previous years, or it is the never-ending quarantine makes time feel particularly slow? The sudden pandemic has changed our daily lives as well as the perception of time. Now that Autumn is approaching, suggesting the end of another school year.

Compared with the bewildering exploration in the first two years of my Ph.D., I feel more confused in the third year after settling down with some research directions and getting some results. But what I mean by confusion here may not be as negative as it originally sounds. After all, if you stopped a graduate student at Harvard campus, I believe the majority of them will also say that they are confused about what they are doing. So perhaps the more important thing is then to know what you are confused about? In my case, I mainly doubted my own ability in the first two years of my Ph.D. But now that I’ve crossed the middle line of my Ph.D., it is “what research direction (or even life direction) I want to pursue” that has been constantly mulled over in my mind.

But anyway, before any serious discussion, let’s review what happened this year!

What I’ve done in the past year?

At around this time of last year, I decided to invest more time to study and explore theoretical neuroscience. By some funny coincidences, I started to collaborate with my best friend Brabeeba, a Ph.D. student at MIT. After countless discussion (and debate) in MIT’s student cafeteria, Harvard’s dorm, and even Boston Symphony Orchestra, we successfully solved the research problem we were looking at: giving the first convergence rate analysis for a biologically-plausible learning rule (link). The techniques we developed from that work also brought us to a cute follow-up work (link).

This close collaboration experience with Brabeeba not only brought me more confidence in doing independent research (because this work was completely done by ourselves without any guidance by professors), but also taught me a lot on how to work with other people. Brabeeba and I are very different in research styles to the extent that we are almost completely complementary to each other. On the bright side, such complementation can push the research progress swiftly. Whenever I got stuck, Brabeeba could always come up with some new ideas to circumvent, and whenever he felt that the problem comes to a dead end, it was me to find a way out. However, the difference between us could also bring up undesirable conflicts. Especially when it comes to dividing opinions on research direction or the presentation of writing, we debate all the times. To be honest, sometimes debate is a frustrating process, especially when the other person is your friend and/or your collaborator. But if you changed an angle, as long as the debate is healthy (meaning that no negative words etc.) actually it is very fortunate to have a good and straightforward communication with someone. This helps spotting mistake and ignorance and also provides diversity in the discussion. In the case of working with Brabeeba, in the end we can usually turned the disagreement into a positive force into the research. Also, it is very enjoyable to have a collaborator being one of your best friends!

As for my main research focus in complexity theory, although frustratingly and unexpectedly there is still no progress in most of the directions I’m interested in, luckily one line of works finally has some non-trivial progresses. The starting point is a simple question: how well can you approximate the maximum directed cut of a directed graph (Max-DICUT) with only logarithmic space (when the edges are given in a 1-pass stream)? ( )

Approximating Max-DICUT in the streaming model:

Let $G=(V,E)$ be an input directed graph. A dicut of $G$ is a partition of the vertex set $V=V_{in}\cup V_{out}$ and the value of this dicut is the number of edges going from $V_{out}$ to $V_{in}$ divided by the total number of edges.

The streaming algorithm receives the edge of $G$ in a stream (i.e., edges in $E$ are of the form $i\to j$ for some $i,j\in V$). The streaming algorithm only has logarithmic (in the number of vertices) space. We say the streaming algorithm gives an $\alpha$-approximation for some $\alpha\in[0,1]$ if it outputs a value $v\in[0,1]$ such that the following two criteria holds with good probability: (i) there exists a dicut with value at least $v$ and (ii) $v\geq\alpha\cdot v^*$ where $v^*$ is the maximum dicut value of $G$.

The figure on the left is an input directed graph. But in the streaming model, you only receive edges one by one and do not have the space to store every edges. The goal is to partition the vertex set into two parts that maximizes the number of edges going from one side to the other. In this example, see the figure on the right, every edges go from the black vertex set to the red vertex set. The goal of the streaming algorithm is to give an approximation to the maximum dicut value with limited amount of space and a single pass of input.

The trivial random sampling algorithm gives (1/4)-approximation ( ) while a previous work finds a (2/5)-approximation using a more clever idea. On the hardness side, the best known previous impossibility result only ruled out (1/2)-approximation. Namely, there is a gap between 2/5 and 1/2 in our understanding. In the collaboration with Santhoshini and Sasha, we found that, surprisingly, neither 2/5 nor 1/2 is the true answer: the optimal approximation ratio for Max-DICUT is 4/9. Through related technologies, we thoroughly analyzed the best approximation factor of boolean 2CSP under the streaming model (link)! After Madhu joining us in the beginning of the summer, we have been making significant progress along the further direction (Update: link to this follow-up work).

The (1/4)-approximation algorithm for Max-DICUT: Note that if we assign each vertex to be either black or red with equal probability, then each edge is a cut edge with probability 1/4. Namely, for a random dicut, the expected cut value is 1/4. Hence, then algorithm can simply output 1/4 knowing that (i) there exists a dicut with such dicut value and (ii) the output value is at least 1/4 of the maximum dicut value (because the dicut value is at most 1).

Finally, in the collaboration with Boaz and Xun, we tried to challenge Google’s quantum supremacy experiment. At first, Boaz and I had a simple classical algorithm, but the analysis was not very satisfying. Xun, a post doc in the theoretical physics department at Harvard, heard that we are working on this problem and introduced the tensor network technique (which is a very common tool in theoretical physics) to us. It turns out that tensor networks can directly give a rigorous analysis for the algorithm (link).

In the ongoing follow-up project, due to some fundamental obstacles in the theoretical analysis, we partly use theory and partly use numerics to demonstrate the performance of our algorithm. This kind of research methodology is very rare in theoretical computer science but is very typical in theoretical physics. In the past, I always felt that if the result cannot be rigorously demonstrated in mathematics from the beginning to the end then it does not count as a theoretical research. However, after intensively working with Xun this time, I gradually started to appreciate this kind of physic-style research methodology. Sometimes, a theoretical research with a bunch of assumptions might not provide more insights than a research with part of rigorous mathematics and part of experiments/simulations. But of course this depends on what the research problem is. If it is a fundamental theoretical and mathematical problem, the requirement for mathematical rigor must of course be 100%. However, for more practical or interdisciplinary problems, finding a beautiful balance between theory and experiment may be able to provide better insights.

New confusion?

This year, I have done researches in three very different directions and fortunately have achieved some partial results. These experiences have built up confidence in myself, but what immediately followed was thinking about my future direction. First, how to allocate my time and how to prioritize these different directions. Second, what kind of research do I appreciate and want to pursue. Although the problems in complexity theory are indeed very attractive to me, they sometimes also give me a sense of emptiness like playing intellectual games. This may be an unavoidable contradiction/paradox of pure theoretical research. On one hand, if you only look at the big problem without taking small steps, you may be stepping around at the same place forever; on the other hand, if you spend all your time on small incremental things, you might never touch the real fundamental problem. However, keep worrying and being stuck in this paradox is not going to help. I guess what I can do is to reflect on this issue regularly. What is the complexity theory for me? Is it an arena to prove my intellectual power? Or is it carrying a sense of mission to solve fundamental problems?

Therefore, at the beginning of the year, I started to have the idea of ​pursuing another Ph.D. (in Algebraic Geometry or related fields) in pure mathematics, mainly because after learning more mathematics, I deeply felt that I’m just touching the surface of real mathematics. Furthermore, much of the deep mathematics has not been (seriously) considered and applied to complexity theory. Also, learning these mathematics requires the right “language”. If I stayed in the current path, I will not have the opportunity to learn these languages ​​well. It’s like it is (almost) impossible to learn English well if you are always in a non-English native environment. But having said that, pursuing a Ph.D. in mathematics is just one of the possible ways to achieve the goal, maybe there is another better way? Anyway, it’s too early to decide. I can wait until this time next year to worry about it…

Facing the future

People are always changing. This is probably my biggest awakening in the past few years. The changes can be in the ability, vision or even in personality, and values. This tells us that what we think we want in the future at this moment may not necessarily be what we really want in the future. In my case, it’s quite funny in retrospect that the reason why I went to academia was not because I like doing research (after all, compared with most people, I started to do research relatively late), but because I love learning new things. At first glance, the academic road seems to provide a life style of life-long learning that can make me happy. Fortunately, after many years, I also found myself enjoying the feeling of exploring the unknown in research. In particular, the final results of several projects were very different from what I expected at the beginning, which made me deeply attracted by the mysterious nature of doing research. This is even more true in the research direction. A year ago, I couldn’t expect that I would be working on these research problems that I mentioned above. Similarly, I believe it is impossible to predict what I will be doing a year later.

A few weeks ago, during a walking meeting with Boaz, I told him that I had recently encountered a career crisis: I don’t know which direction to go, and I don’t know where I will be going to in the next five or ten years. Boaz’s advice was simple, but it was an enlightening moment to me. He told me to just think about what I’m going to do in the next year! Maybe it’s because there are too many people’s examples to look at before the Ph.D., so I can always plan my future plans in advance. However, when it comes to the stage of independent research, everyone must create their own path. Maybe some people can see the development in the next few years, but in most cases the future steps are highly depending on the immediate next step. In the end, everyone is changing, and the world is also changing (e.g., the sudden pandemic), so in addition to be bothered by the plan for long-term future, a very important ability is to find a good direction in each moment and work as hard as possible. This is probably also true for other life topics other than doing research.

My friends always told me don’t overthink. Indeed, thinking less may make myself less annoyed and have a happier life. But for me at least, thinking more and deeper often makes me more aware of what I really want. While we all want to live happily, but what makes oneself happy is sometimes not that obvious. Maybe the answer (if there’s any) is hidden in the confusion by wild thoughts!?