Giannis Iakovidis

Ilias Diakonikolas, Giannis Iakovidis, Mingchen Ma

We study the task of agnostic learning of multiclass linear classifiers under the Gaussian distribution. Given labeled examples $(x, y)$ from a distribution over $\mathbb{R}^d \times [k]$, with Gaussian $x$-marginal, the goal is to output a hypothesis whose error is comparable to that of the best $k$-class linear classifier. While the binary case $k=2$ has a well-developed algorithmic theory, much less is known for $k \ge 3$. Even for $k=3$, prior robust algorithms incur exponential dependence on the inverse of the desired accuracy in both complexity and representation size. In this work, we develop new structural results for multiclass linear classifiers and use them to design fully polynomial-time robust learners with dimension-independent error guarantees. Our first result shows that the standard multiclass perceptron algorithm requires super-polynomially many samples and updates, even with clean labels and Gaussian marginals, revealing a basic obstruction absent in the binary case. Our main positive result is a pairwise improper-learning framework which yields an efficient learner with error $\widetilde O(k^{3/2}\sqrt{\mathrm{opt}})+ε$ for general $k$. Additionally, we develop a sharper localization-based framework which leads to error $O(\mathrm{opt})+ε$ for $k=3$, and error $\mathrm{poly}(k)\mathrm{opt}+ε$ for geometrically regular $k$-class linear classifiers.

Ilias Diakonikolas, Giannis Iakovidis, Daniel M. Kane et al.

We study the algorithmic task of testably learning general Massart halfspaces under the Gaussian distribution. In the testable learning setting, the aim is the design of a tester-learner pair satisfying the following properties: (1) if the tester accepts, the learner outputs a hypothesis and a certificate that it achieves near-optimal error, and (2) it is highly unlikely that the tester rejects if the data satisfies the underlying assumptions. Our main result is the first testable learning algorithm for general halfspaces with Massart noise and Gaussian marginals. The complexity of our algorithm is $d^{\mathrm{polylog}(\min\{1/γ, 1/ε\})}$, where $ε$ is the excess error and $γ$ is the bias of the target halfspace, which qualitatively matches the known quasi-polynomial Statistical Query lower bound for the non-testable setting. The analysis of our algorithm hinges on a novel sandwiching polynomial approximation to the sign function with multiplicative error that may be of broader interest.

Ilias Diakonikolas, Giannis Iakovidis, Daniel M. Kane et al.

We study the basic task of mean estimation in the presence of mean-shift contamination. In the mean-shift contamination model, an adversary is allowed to replace a small constant fraction of the clean samples by samples drawn from arbitrarily shifted versions of the base distribution. Prior work characterized the sample complexity of this task for the special cases of the Gaussian and Laplace distributions. Specifically, it was shown that consistent estimation is possible in these cases, a property that is provably impossible in Huber's contamination model. An open question posed in earlier work was to determine the sample complexity of mean estimation in the mean-shift contamination model for general base distributions. In this work, we study and essentially resolve this open question. Specifically, we show that, under mild spectral conditions on the characteristic function of the (potentially multivariate) base distribution, there exists a sample-efficient algorithm that estimates the target mean to any desired accuracy. We complement our upper bound with a qualitatively matching sample complexity lower bound. Our techniques make critical use of Fourier analysis, and in particular introduce the notion of a Fourier witness as an essential ingredient of our upper and lower bounds.