We study the problem of completing a binary matrix in an online learning setting. On each trial we predict a matrix entry and then receive the true entry. We propose a Matrix Exponentiated Gradient algorithm [1] to solve this problem. We provide a mistake bound for the algorithm, which scales with the margin complexity [2, 3] of the underlying matrix. The bound suggests an interpretation where each row of the matrix is a prediction task over a finite set of objects, the columns. Using this we show that the algorithm makes a number of mistakes which is comparable up to a logarithmic factor to the number of mistakes made by the Kernel Perceptron with an optimal kernel in hindsight. We discuss applications of the algorithm to predicting as well as the best biclustering and to the problem of predicting the labeling of a graph without knowing the graph in advance.

Neural Information Processing Systems http://nips.cc/

We consider learning under the constraint of local differential privacy (LDP). For many learning problems known efficient algorithms in this model require many rounds of communication between the server and the clients holding the data points. Yet multi-round protocols are prohibitively slow in practice due to network latency and, as a result, currently deployed large-scale systems are limited to a single round. Despite significant research interest, very little is known about which learning problems can be solved by such non-interactive systems. The only lower bound we are aware of is for PAC learning an artificial class of functions with respect to a uniform distribution (Kasiviswanathan et al., 2008).

Neural Information Processing Systems http://nips.cc/

Summary of Contributions: UPDATE: I have read the author rebuttal and my review is unchanged. This work considers the role of interaction in (distribution-free) PAC learning with local differential privacy (LDP). In this model the learning algorithm only has access to examples that have been locally randomized in a differentially private way; the learner may specify the index of the example and the randomizer. If the learning algorithm makes all of its queries in advance, it is non-interactive, and if it makes its queries independent of the labels returned by the oracle, it is label-non-adaptive. The main results of this paper show that PAC learning with a label-non-adaptive LDP algorithm is characterized (up to polynomial factors) by the margin complexity MC(C) of the class, which is the inverse largest margin of separation achievable by linearly separable (into positive and negative examples) embedding of the domain.

We consider learning under the constraint of local differential privacy (LDP). For many learning problems known efficient algorithms in this model require many rounds of communication between the server and the clients holding the data points. Yet multi-round protocols are prohibitively slow in practice due to network latency and, as a result, currently deployed large-scale systems are limited to a single round. Despite significant research interest, very little is known about which learning problems can be solved by such non-interactive systems. The only lower bound we are aware of is for PAC learning an artificial class of functions with respect to a uniform distribution (Kasiviswanathan et al., 2008).

We study the problem of completing a binary matrix in an online learning setting. On each trial we predict a matrix entry and then receive the true entry. We propose a Matrix Exponentiated Gradient algorithm [1] to solve this problem. We provide a mistake bound for the algorithm, which scales with the margin complexity [2, 3] of the underlying matrix. The bound suggests an interpretation where each row of the matrix is a prediction task over a finite set of objects, the columns. Using this we show that the algorithm makes a number of mistakes which is comparable up to a logarithmic factor to the number of mistakes made by the Kernel Perceptron with an optimal kernel in hindsight. We discuss applications of the algorithm to predicting as well as the best biclustering and to the problem of predicting the labeling of a graph without knowing the graph in advance.

We consider learning under the constraint of local differential privacy (LDP). For many learning problems known efficient algorithms in this model require many rounds of communication between the server and the clients holding the data points. Yet multi-round protocols are prohibitively slow in practice due to network latency and, as a result, currently deployed large-scale systems are limited to a single round. Despite significant research interest, very little is known about which learning problems can be solved by such non-interactive systems. The only lower bound we are aware of is for PAC learning an artificial class of functions with respect to a uniform distribution (Kasiviswanathan et al., 2008).

We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to approximate, rather then exactly represent, a given hypothesis class. We show that such notions are not only sufficient for learning using linear predictors or a kernel, but unlike the exact variants, are also necessary. Thus they are better suited for discussing limitations of linear or kernel methods.

We give an online algorithm and prove novel mistake and regret bounds for online binary matrix completion with side information. The bounds we prove are of the form $\tilde{\mathcal{O}}({\mathcal{D}}/{\gamma^2})$. The term ${1}/{\gamma^2}$ is analogous to the usual margin term in SVM (perceptron) bounds. More specifically, if we assume that there is some factorization of the underlying $m\times n$ matrix into $\mathbf{P} \mathbf{Q}^{\top}$ where the rows of $\mathbf{P}$ are interpreted as ``classifiers'' in $\Re^d$ and the rows of $\mathbf{Q}$ as ``instances'' in $\Re^d$, then $\gamma$ is is the maximum (normalized) margin over all factorizations $\mathbf{P} \mathbf{Q}^{\top}$ consistent with the observed matrix. The quasi-dimension term $\mathcal{D}$ measures the quality of side information. In the presence of no side information, $\mathcal{D} = m+n$. However, if the side information is predictive of the underlying factorization of the matrix, then in the best case, $\mathcal{D} \in \mathcal{O}(k + \ell)$ where $k$ is the number of distinct row factors and $\ell$ is the number of distinct column factors. We additionally provide a generalization of our algorithm to the inductive setting. In this setting, the side information is not specified in advance. The results are similar to the transductive setting but in the best case, the quasi-dimension $\mathcal{D}$ is now bounded by $\mathcal{O}(k^2 + \ell^2)$.