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My Third Ph.D. Year
17 Aug 2020
(中文版原文連結)
One week after Liqiu, the heat in New England was finally relieved, if only temporarily. Perhaps this year is truly hotter than previous years, or perhaps the never-ending quarantine has made time feel particularly slow. The sudden pandemic has changed our daily lives as well as our perception of time. Now that autumn is approaching, it suggests the end of another school year.
Compared to the bewildering exploration in the first two years of my Ph.D., I feel more confused in the third year, despite having settled on some research directions and achieved some results. But what I mean by “confusion” here may not be as negative as it sounds. After all, if you stopped a graduate student on the Harvard campus, I believe the majority of them would also say they are confused about what they are doing. So perhaps the more important thing is 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. Now that I’ve crossed the midpoint of my Ph.D., it is “what research direction (or even life direction) I want to pursue” that has been constantly on 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 last year, I decided to invest more time studying and exploring theoretical neuroscience. By some funny coincidence, I started to collaborate with my best friend, Brabeeba, a Ph.D. student at MIT. After countless discussions (and debates) in MIT’s student cafeteria, Harvard’s dorm, and even the 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. The techniques we developed from that work also led us to a cute follow-up work.
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 from professors), but also taught me a lot about how to work with other people. Brabeeba and I have very different research styles to the extent that we are almost completely complementary to each other. On the bright side, such complementation can push 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 came to a dead end, it was me who found 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 time. To be honest, sometimes debate is a frustrating process, especially when the other person is your friend and/or your collaborator. But if you change your perspective, as long as the debate is healthy (meaning that no negative words are used), it can be very fortunate to have good and straightforward communication with someone. This helps to spot mistakes and ignorance and also provides diversity in the discussion. In the case of working with Brabeeba, we can usually turn disagreements into positive forces in the research. Additionally, it is very enjoyable to have a collaborator who is also one of your best friends!
As for my main research focus in complexity theory, frustratingly and unexpectedly, there is still no progress in most of the directions I’m interested in, but luckily one line of work finally has some non-trivial progress. 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 edges 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 hold 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 edge. The goal is to partition the vertex set into two parts that maximize the number of edges going from one side to the other. In this example, as shown in the figure on the right, every edge goes 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 a limited amount of space and a single pass of input.
The trivial random sampling algorithm gives a (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 a (1/2)-approximation. Namely, there is a gap between 2/5 and 1/2 in our understanding. In 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. After Madhu joined us at the beginning of the summer, we have made significant progress in this direction.
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, the algorithm can simply output 1/4 knowing that (i) there exists a dicut with such a 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 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 postdoc in the theoretical physics department at Harvard, heard that we were 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 of the algorithm.
In the ongoing follow-up project, we use theory and numerics to demonstrate the performance of our algorithm, due to some fundamental obstacles in the theoretical analysis. This kind of research methodology is very rare in theoretical computer science but is typical in theoretical physics. In the past, I always felt that if the result could not 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 physics-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 the research problem. If it is a fundamental theoretical and mathematical problem, the requirement for mathematical rigor must be 100%. However, for more practical or interdisciplinary problems, finding a beautiful balance between theory and experiment may provide better insights.
New confusion？
This year, I have conducted research 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 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, worrying and being stuck in this paradox is not going to help. I guess what I can do is reflect on this issue regularly. What is complexity theory for me? Is it an arena to prove my intellectual power? Or does it carry 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 on the current path, I would 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 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 that 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 lifestyle 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 to 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 the 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 in the next five or ten years. Boaz’s advice was simple, but it was an enlightening moment for 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 path. Maybe some people can see the development in the next few years, but in most cases, the future steps highly depend 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 being 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 not to 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, what makes oneself happy is sometimes not that obvious. Maybe the answer (if there’s any) is hidden in the confusion by wild thoughts!?