Embedding Hard Learning Problems Into Gaussian Space

We give the first representation-independent hardness result for agnostically learning halfspaces with respect to the Gaussian distribution. We reduce from the problem of learning sparse parities with noise with respect to the uniform distribution on the hypercube (sparse LPN), a notoriously hard problem in theoretical computer science and show that any algorithm for agnostically learning halfspaces requires n^Omega(log(1/epsilon)) time under the assumption that k-sparse LPN requires n^Omega(k) time, ruling out a polynomial time algorithm for the problem. As far as we are aware, this is the first representation-independent hardness result for supervised learning when the underlying distribution is restricted to be a Gaussian. We also show that the problem of agnostically learning sparse polynomials with respect to the Gaussian distribution in polynomial time is as hard as PAC learning DNFs on the uniform distribution in polynomial time. This complements the surprising result of Andoni et. al. 2013 who show that sparse polynomials are learnable under random Gaussian noise in polynomial time. Taken together, these results show the inherent difficulty of designing supervised learning algorithms in Euclidean space even in the presence of strong distributional assumptions. Our results use a novel embedding of random labeled examples from the uniform distribution on the Boolean hypercube into random labeled examples from the Gaussian distribution that allows us to relate the hardness of learning problems on two different domains and distributions

Efficient Learning of Linear Separators under Bounded Noise

We study the learnability of linear separators in $\Re^d$ in the presence of bounded (a.k.a Massart) noise. This is a realistic generalization of the random classification noise model, where the adversary can flip each example $x$ with probability $\eta(x) \leq \eta$. We provide the first polynomial time algorithm that can learn linear separators to arbitrarily small excess error in this noise model under the uniform distribution over the unit ball in $\Re^d$, for some constant value of $\eta$. While widely studied in the statistical learning theory community in the context of getting faster convergence rates, computationally efficient algorithms in this model had remained elusive. Our work provides the first evidence that one can indeed design algorithms achieving arbitrarily small excess error in polynomial time under this realistic noise model and thus opens up a new and exciting line of research. We additionally provide lower bounds showing that popular algorithms such as hinge loss minimization and averaging cannot lead to arbitrarily small excess error under Massart noise, even under the uniform distribution. Our work instead, makes use of a margin based technique developed in the context of active learning. As a result, our algorithm is also an active learning algorithm with label complexity that is only a logarithmic the desired excess error $\epsilon$

A PTAS for Agnostically Learning Halfspaces

We present a PTAS for agnostically learning halfspaces w.r.t. the uniform distribution on the $d$ dimensional sphere. Namely, we show that for every $\mu>0$ there is an algorithm that runs in time $\mathrm{poly}(d,\frac{1}{\epsilon})$, and is guaranteed to return a classifier with error at most $(1+\mu)\mathrm{opt}+\epsilon$, where $\mathrm{opt}$ is the error of the best halfspace classifier. This improves on Awasthi, Balcan and Long [ABL14] who showed an algorithm with an (unspecified) constant approximation ratio. Our algorithm combines the classical technique of polynomial regression (e.g. [LMN89, KKMS05]), together with the new localization technique of [ABL14]