Tuesday, April 20, 2010

The intersection of two halfspaces has high threshold degree

Consider a Boolean function $f: \{-1,1\}^n \mapsto \{-1,+1\}$. It is well known and also easy to show that $f$ can be represented exactly by a real polynomial of degree atmost $n$. In other words, there exists a real polynomial $p(x)$, such that $p(x) = f(x), \,\, \forall x \in \{-1,1\}^n$. One could also ask the following question: how well can a function be approximated by a polynomial? There are multiple notions of approximation and the one we will look at in this talk is called sign representation. We say that a polynomial $p(x)$ sign represents a Boolean function $f$, if $sgn(p(x)) = f(x), \,\, \forall x \in \{-1,1\}^n$. The sign degree of a function $f$ is defined as the least degree of a real polynomial $p$, such that $p$ sign represents $f$.

Such questions concerning approximation of functions by polynomials have been studied by Math and CS researchers since the $1960$'s. Some of the earliest papers on these topics are by Newman and Minsky and Papert. Although interesting in its own right, this line of work has found numerous applications in various areas of CS theory such as communication complexity, quantum computing, complexity theory etc. For more details, see this excellent tutorial by Scott Aaronson. Sign representation by polynomials also has a very nice implication for machine learning. Roughly speaking, if a function has sign degree $d$, then it can be learned in time $n^{O(d)}$ under a standard model of learning. This connection has been exploited heavily in recent years to come up with learning algorithms for many non-trivial problems. Such algorithms are sometimes also called ``perceptron based algorithms''.

However, one notorious class of functions for which no efficient learning algorithm is known till date is the class of intersection of two halfspaces. A halfspace over $\{-1,1\}^n$ is a Boolean function defined as $f(x) = sgn(a_0 + \sum_i a_ix_i)$. The intersection of two halfspaces is a function $h(x,y) = f(x) \land g(y)$, where both $f$ and $g$ are halfspaces over $\{-1,1\}^n$. Clearly, by definition the sign degree of a halfspace is $1$. Minsky and Papert in 1969, showed that the sign degree of the intersection of two halfspace is $\omega(1)$. In their paper they also introduced a very important technique, called ``symmetrization'', which can be used to lower bound the sign degree of certain functions. The next non-trivial improvement came in 2003, when O'Donnell and Servedio showed that the sign degree of the intersection of two halfspaces is $\Omega(\frac {\log n} {\log \log n})$. Although a massive improvement, this still does not rule out sub-exponential time ``perceptron based'' algorithms for learning the intersection of two halfspaces.

Last year in an award winning paper presented at FOCS, Alexander Sherstov showed that this bound is $\Omega(\sqrt{n})$. Also, in a very recent follow-up work he has managed to show that the sign degree of the intersection of two halfspaces is in fact $\Theta(n)$. Hence, perceptron based approaches cannot be used to get any non-trivial algorithm for this class of functions. This in turn means that some new ideas will be required to make progress on the problem of learning the intersection of two halfspaces. On Thursday, we will prove Sherstov's result from FOCS'09 which shows that the sign degree of the intersection of two halfspaces is $\Omega(\sqrt{n})$. The talk will be self-contained.

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