What is the minimum amount of resources—time, space, energy, layers in a neural net—required to solve a given computational problem? This fundamental question is the essence of complexity theory and the main driving force behind the research of Omri Weinstein, a theoretical computer scientist who joins Columbia’s Computer Science Department this semester as assistant professor.
While practitioners might be content to use empirical trial and error in the search for increasingly improved heuristics, Weinstein and other theorists aim to answer the above question in a rigorous mathematical manner. Inherent to this research endeavor is proving lower bounds, that is, showing that no algorithm—out of the infinitely many that exist for a given problem—can solve it with fewer than X resources. In this way, theorists begin to unravel the limitations of what can and cannot be done with respect to a given problem. Reasoning simultaneously about all possible algorithms for a problem requires developing “universal” tools and techniques that are independent of any particular algorithm; otherwise researchers are left with the hopeless task of ruling out each individual algorithm of the uncountable number that exists.
The new toolbox Weinstein is developing comes from the field of communication
and information theory. “Communication over a channel is one of the few models of computation that we actually understand and for which we have simple and powerful analytical tools,” says Weinstein. “Computation on the other hand, is a beast which we are still very far from understanding. It is conceivable that there is some unimaginably clever algorithm for solving the Traveling Salesman problem that we are not yet aware of, since we have few tools to reason about processing of information. However, we can capture some of the hardness of the problem by splitting the input between two (or more) parties and asking them to solve the resulting distributed problem over a communication channel. At this point, we can use tools from information theory, combinatorics, and algebra to reason about the problem, bringing it to our home court.”
The machinery Weinstein cites is mostly drawn from information theory, which was established by Claude Shannon some sixty years ago in the context of the data compression problem. In the simplest setup, Alice wishes to transmit to Bob a random message, say a huge database of high-resolution pictures, using minimum communication bits. However, most real-life (as well as theoretical) distributed scenarios are inherently interactive, with processors exchanging small chunks of data over multiple rounds (one natural example is adaptive trading in economic markets such as stock exchanges). Classic information theory does not readily lend itself to the interactive setup; in particular, compressing conversations turns out to be notoriously challenging. Much of Weinstein’s work can be viewed as an interactive extension of Shannon’s classic information theory (aka “information complexity”), where the goal is to understand how information behaves in interactive scenarios, and to develop new information-theoretic methods in communication complexity, tailored for computer science applications: “Many computational models, such as data streaming, circuit design, and economic markets, have no apparent relation to communication problems,” says Weinstein. “Using clever reductions, though, it turns out that many of these models have some intrinsic communication problem implicitly ‘embedded’ within them.”
A prototypical example of this implicit connection are data structures, which enable efficient storage and retrieval of queries in a database. Data structures may not seem to have anything to do with communication, but can in fact be viewed, in some sense, as a communication protocol between the memory (which stores the database) and the processor (which holds a query). Indeed, consider the distributed communication problem in which Alice holds the database and Bob holds a query, and the goal of the players is to answer Bob’s query with respect to Alice’s database. Clearly, neither party can solve the problem on its own; hence communication must take place. Weinstein illustrates a surprisingly simple and beautiful connection to communication complexity: “If there was a too-good-to-be-true data structure that uses too little memory and has fast access time when retrieving an answer to a query, Alice and Bob could use this hypothetical data structure to construct a very efficient communication protocol for the aforementioned distributed problem, by jointly simulating the data structure: in each communication round, Bob asks Alice for the next memory-address his query prescribes, and Alice responds with the content of this memory cell in the data structure. If the data structure has ‘fast’ retrieval time, this would imply a low-communication protocol; hence time-space lower bounds for data-structures can be obtained via lower bounds for the underlying communication problem. Again, this is typically much easier, since the communication model has much more structure.”
Information complexity has gained popularity in recent years and led to some breakthrough results on several long-standing computational problems, some by Weinstein and his colleagues, some by other scientists. His paper Direct Products in Communication Complexity, which investigated the fundamental problem of “whether solving k copies of a single problem is k times harder than solving a single copy,” led to a breakthrough in understanding the limits of parallel computing, showing that solving multiple copies of any two-party distributed problem cannot be remedied much by parallelism (i.e., parallelizing cannot reduce communication significantly). Says Weinstein, “It is hard for me to imagine a proof that does not use information theory here, as this language is much more expressive than previous analytical tools and captures the exact properties needed to solve the problem; to me this paper provides further evidence that information theory is the ‘right’ language to study communication problems.”
A tangent line of research in Weinstein’s work is studying computational aspects of economics. A 2015 paper looked to find the minimum running time required to find an approximate Nash equilibrium in two-player games, one of the most important problems in algorithmic game theory. His papers on this subject played a role in the recent breakthrough results, showing that finding Nash equilibria is some games requires a huge amount of time and communication, thereby undermining the consensus over the basic definition of equilibrium as a practical economic solution concept.
While much of his work is theoretical and abstract, Weinstein is also looking into applied problems, especially in an era where rapid evolution in technology is constantly posing new challenges that require new techniques. With Columbia colleagues, Weinstein is studying an asymmetric compression problem that naturally arises from the Internet of Things (IoT) model. Here, millions of weak tags—low resource, low memory, low power devices—are required to convey messages to a central (powerful) gateway in an extremely efficient manner, both in terms of bandwidth and processing power. Compression obviously plays a key role in this model. Alas, standard compression schemes rely on the assumption that both the encoder and the decoder know the underlying prior distribution of the message to be sent, an assumption that is not realistic on the tag side (as they do not have the space nor the energy to store expensive statistics). Says Weinstein, “Can we nevertheless exploit the fact that the gateway knows the prior distribution, even if the tags don’t?” Perhaps surprisingly, the answer is “yes.” Weinstein and his co-authors developed a low-communication encoding scheme, which further admits efficient decoding time of the message on the gateway side. Once again, information theory and communication complexity are lurking beneath this fascinating problem.
“Omri’s research strengths and interests, especially his expertise in the nascent field of information complexity, complement and enhance the scope of our existing efforts in theoretical computer science. We are thrilled he has decided to join the Computer Science Department at Columbia, and we welcome him and look forward to working together!” said Rocco Servedio, Professor of Computer Science and a member of the department’s theory group.
Weinstein received his PhD from Princeton University in 2015 under the supervision of Mark Braverman, and holds a BSc in mathematics and computer science from Tel-Aviv University. He was named a 2016 Simons Society Junior Fellow and a Siebel Scholar in 2015; in 2013, he was awarded a Simons Graduate Fellowship.