## Lower bounds in communication complexity and learning theory via analytic methods

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A central goal of theoretical computer science is to characterize the limits
of efficient computation in a variety of models. We pursue this research objective
in the contexts of communication complexity and computational learning theory.
In the former case, one seeks to understand which distributed computations require
a significant amount of communication among the parties involved. In the latter
case, one aims to rigorously explain why computers cannot master some prediction
tasks or learn from past experience. While communication and learning may seem
to have little in common, they turn out to be closely related, and much insight into
both can be gained by studying them jointly. Such is the approach pursued in this
thesis. We answer several fundamental questions in communication complexity and
learning theory and in so doing discover new relations between the two topics. A
consistent theme in our work is the use of analytic methods to solve the problems at
hand, such as approximation theory, Fourier analysis, matrix analysis, and duality.
We contribute a novel technique, the pattern matrix method, for proving lower
bounds on communication. Using our method, we solve an open problem due to Krause and Pudlák (1997) on the comparative power of two well-studied
circuit classes: majority circuits and constant-depth AND/OR/NOT circuits.
Next, we prove that the pattern matrix method applies not only to classical
communication but also to the more powerful quantum model. In particular,
we contribute lower bounds for a new class of quantum communication
problems, broadly subsuming the celebrated work by Razborov (2002) who
used different techniques. In addition, our method has enabled considerable
progress by a number of researchers in the area of multiparty communication.
Second, we study unbounded-error communication, a natural model with applications
to matrix analysis, circuit complexity, and learning. We obtain
essentially optimal lower bounds for all symmetric functions, giving the first
strong results for unbounded-error communication in years. Next, we resolve
a longstanding open problem due to Babai, Frankl, and Simon (1986) on
the comparative power of unbounded-error communication and alternation,
showing that [mathematical equation]. The latter result also yields an unconditional,
exponential lower bound for learning DNF formulas by a large class of algorithms,
which explains why this central problem in computational learning
theory remains open after more than 20 years of research.
We establish the computational intractability of learning intersections of
halfspaces, a major unresolved challenge in computational learning theory.
Specifically, we obtain the first exponential, near-optimal lower bounds for
the learning complexity of this problem in Kearns’ statistical query model,
Valiant’s PAC model (under standard cryptographic assumptions), and various
analytic models. We also prove that the intersection of even two halfspaces
on {0,1}n cannot be sign-represented by a polynomial of degree less than [Theta](square root of n), which is an exponential improvement on previous lower bounds
and solves an open problem due to Klivans (2002).
We fully determine the relations and gaps among three key complexity measures
of a communication problem: product discrepancy, sign-rank, and discrepancy.
As an application, we solve an open problem due to Kushilevitz and
Nisan (1997) on distributional complexity under product versus nonproduct
distributions, as well as separate the communication classes PPcc and UPPcc
due to Babai, Frankl, and Simon (1986). We give interpretations of our results
in purely learning-theoretic terms.

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