Using More Data to Speed-up Training Time

In many recent applications, data is plentiful. By now, we have a rather clear understanding of how more data can be used to improve the accuracy of learning algorithms. Recently, there has been a growing interest in understanding how more data can be leveraged to reduce the required training runtime. In this paper, we study the runtime of learning as a function of the number of available training examples, and underscore the main high-level techniques. We provide some initial positive results showing that the runtime can decrease exponentially while only requiring a polynomial growth of the number of examples, and spell-out several interesting open problems.

Large-Scale Convex Minimization with a Low-Rank Constraint

We address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation guarantees. Each iteration of the algorithm involves (approximately) finding the left and right singular vectors corresponding to the largest singular value of a certain matrix, which can be calculated in linear time. This leads to an algorithm which can scale to large matrices arising in several applications such as matrix completion for collaborative filtering and robust low rank matrix approximation.

Regularization Techniques for Learning with Matrices

There is growing body of learning problems for which it is natural to organize the parameters into matrix, so as to appropriately regularize the parameters under some matrix norm (in order to impose some more sophisticated prior knowledge). This work describes and analyzes a systematic method for constructing such matrix-based, regularization methods. In particular, we focus on how the underlying statistical properties of a given problem can help us decide which regularization function is appropriate. Our methodology is based on the known duality fact: that a function is strongly convex with respect to some norm if and only if its conjugate function is strongly smooth with respect to the dual norm. This result has already been found to be a key component in deriving and analyzing several learning algorithms. We demonstrate the potential of this framework by deriving novel generalization and regret bounds for multi-task learning, multi-class learning, and kernel learning.

Learning Kernel-Based Halfspaces with the Zero-One Loss

We describe and analyze a new algorithm for agnostically learning kernel-based halfspaces with respect to the \emph{zero-one} loss function. Unlike most previous formulations which rely on surrogate convex loss functions (e.g. hinge-loss in SVM and log-loss in logistic regression), we provide finite time/sample guarantees with respect to the more natural zero-one loss function. The proposed algorithm can learn kernel-based halfspaces in worst-case time $\poly(\exp(L\log(L/\epsilon)))$, for $\emph{any}$ distribution, where $L$ is a Lipschitz constant (which can be thought of as the reciprocal of the margin), and the learned classifier is worse than the optimal halfspace by at most $\epsilon$. We also prove a hardness result, showing that under a certain cryptographic assumption, no algorithm can learn kernel-based halfspaces in time polynomial in $L$.

Online Learning of Noisy Data with Kernels

We study online learning when individual instances are corrupted by adversarially chosen random noise. We assume the noise distribution is unknown, and may change over time with no restriction other than having zero mean and bounded variance. Our technique relies on a family of unbiased estimators for non-linear functions, which may be of independent interest. We show that a variant of online gradient descent can learn functions in any dot-product (e.g., polynomial) or Gaussian kernel space with any analytic convex loss function. Our variant uses randomized estimates that need to query a random number of noisy copies of each instance, where with high probability this number is upper bounded by a constant. Allowing such multiple queries cannot be avoided: Indeed, we show that online learning is in general impossible when only one noisy copy of each instance can be accessed.

Efficient Learning with Partially Observed Attributes

We describe and analyze efficient algorithms for learning a linear predictor from examples when the learner can only view a few attributes of each training example. This is the case, for instance, in medical research, where each patient participating in the experiment is only willing to go through a small number of tests. Our analysis bounds the number of additional examples sufficient to compensate for the lack of full information on each training example. We demonstrate the efficiency of our algorithms by showing that when running on digit recognition data, they obtain a high prediction accuracy even when the learner gets to see only four pixels of each image.