It has been known for a long time that the classical spherical perceptrons can be used as storage memories. Seminal work of Gardner, \cite{Gar88}, started an analytical study of perceptrons storage abilities. Many of the Gardner's predictions obtained through statistical mechanics tools have been rigorously justified. Among the most important ones are of course the storage capacities. The first rigorous confirmations were obtained in \cite{SchTir02,SchTir03} for the storage capacity of the so-called positive spherical perceptron. These were later reestablished in \cite{TalBook} and a bit more recently in \cite{StojnicGardGen13}. In this paper we consider a variant of the spherical perceptron that operates as a storage memory but allows for a certain fraction of errors. In Gardner's original work the statistical mechanics predictions in this directions were presented sa well. Here, through a mathematically rigorous analysis, we confirm that the Gardner's predictions in this direction are in fact provable upper bounds on the true values of the storage capacity. Moreover, we then present a mechanism that can be used to lower these bounds. Numerical results that we present indicate that the Garnder's storage capacity predictions may, in a fairly wide range of parameters, be not that far away from the true values.

Perceptrons have been known for a long time as a promising tool within the neural networks theory. The analytical treatment for a special class of perceptrons started in seminal work of Gardner \cite{Gar88}. Techniques initially employed to characterize perceptrons relied on a statistical mechanics approach. Many of such predictions obtained in \cite{Gar88} (and in a follow-up \cite{GarDer88}) were later on established rigorously as mathematical facts (see, e.g. \cite{SchTir02,SchTir03,TalBook,StojnicGardGen13,StojnicGardSphNeg13,StojnicGardSphErr13}). These typically related to spherical perceptrons. A lot of work has been done related to various other types of perceptrons. Among the most challenging ones are what we will refer to as the discrete perceptrons. An introductory statistical mechanics treatment of such perceptrons was given in \cite{GutSte90}. Relying on results of \cite{Gar88}, \cite{GutSte90} characterized many of the features of several types of discrete perceptrons. We in this paper, consider a similar subclass of discrete perceptrons and provide a mathematically rigorous set of results related to their performance. As it will turn out, many of the statistical mechanics predictions obtained for discrete predictions will in fact appear as mathematically provable bounds. This will in a way emulate a similar type of behavior we observed in \cite{StojnicGardGen13,StojnicGardSphNeg13,StojnicGardSphErr13} when studying spherical perceptrons.

What does it mean to claim that a physical or natural system computes? One answer, endorsed here, is that computing is about programming a system to behave in different ways. This paper offers an account of what it means for a physical system to compute based on this notion. It proposes a behavioural characterisation of computing in terms of a measure of programmability, which reflects a system's ability to react to external stimuli. The proposed measure of programmability is useful for classifying computers in terms of the apparent algorithmic complexity of their evolution in time. I make some specific proposals in this connection and discuss this approach in the context of other behavioural approaches, notably Turing's test of machine intelligence. I also anticipate possible objections and consider the applicability of these proposals to the task of relating abstract computation to nature-like computation.

Discriminative latent-variable models are typically learned using EM or gradient-based optimization, which suffer from local optima. In this paper, we develop a new computationally efficient and provably consistent estimator for a mixture of linear regressions, a simple instance of a discriminative latent-variable model. Our approach relies on a low-rank linear regression to recover a symmetric tensor, which can be factorized into the parameters using a tensor power method. We prove rates of convergence for our estimator and provide an empirical evaluation illustrating its strengths relative to local optimization (EM).

The strategy of early stopping is a regularization technique based on choosing a stopping time for an iterative algorithm. Focusing on non-parametric regression in a reproducing kernel Hilbert space, we analyze the early stopping strategy for a form of gradient-descent applied to the least-squares loss function. We propose a data-dependent stopping rule that does not involve hold-out or cross-validation data, and we prove upper bounds on the squared error of the resulting function estimate, measured in either the $L^2(P)$ and $L^2(P_n)$ norm. These upper bounds lead to minimax-optimal rates for various kernel classes, including Sobolev smoothness classes and other forms of reproducing kernel Hilbert spaces. We show through simulation that our stopping rule compares favorably to two other stopping rules, one based on hold-out data and the other based on Stein's unbiased risk estimate. We also establish a tight connection between our early stopping strategy and the solution path of a kernel ridge regression estimator.

The outlying property detection problem is the problem of discovering the properties distinguishing a given object, known in advance to be an outlier in a database, from the other database objects. In this paper, we analyze the problem within a context where numerical attributes are taken into account, which represents a relevant case left open in the literature. We introduce a measure to quantify the degree the outlierness of an object, which is associated with the relative likelihood of the value, compared to the to the relative likelihood of other objects in the database. As a major contribution, we present an efficient algorithm to compute the outlierness relative to significant subsets of the data. The latter subsets are characterized in a "rule-based" fashion, and hence the basis for the underlying explanation of the outlierness.

We consider how selfish agents are likely to share revenues derived from maintaining connectivity between important network servers. We model a network where a failure of one node may disrupt communication between other nodes as a cooperative game called the vertex Connectivity Game (CG). In this game, each agent owns a vertex, and controls all the edges going to and from that vertex. A coalition of agents wins if it fully connects a certain subset of vertices in the graph, called the primary vertices. Power indices measure an agent's ability to affect the outcome of the game. We show that in our domain, such indices can be used to both determine the fair share of the revenues an agent is entitled to, and identify significant possible points of failure affecting the reliability of communication in the network. We show that in general graphs, calculating the Shapley and Banzhaf power indices is #P-complete, but suggest a polynomial algorithm for calculating them in trees. We also investigate finding stable payoff divisions of the revenues in CGs, captured by the game theoretic solution of the core, and its relaxations, the epsilon-core and least core. We show a polynomial algorithm for computing the core of a CG, but show that testing whether an imputation is in the epsilon-core is coNP-complete. Finally, we show that for trees, it is possible to test for epsilon-core imputations in polynomial time.

In this article we introduce the Arcade Learning Environment (ALE): both a challenge problem and a platform and methodology for evaluating the development of general, domain-independent AI technology. ALE provides an interface to hundreds of Atari 2600 game environments, each one different, interesting, and designed to be a challenge for human players. ALE presents significant research challenges for reinforcement learning, model learning, model-based planning, imitation learning, transfer learning, and intrinsic motivation. Most importantly, it provides a rigorous testbed for evaluating and comparing approaches to these problems. We illustrate the promise of ALE by developing and benchmarking domain-independent agents designed using well-established AI techniques for both reinforcement learning and planning. In doing so, we also propose an evaluation methodology made possible by ALE, reporting empirical results on over 55 different games. All of the software, including the benchmark agents, is publicly available.

The (constrained) minimization of a ratio of set functions is a problem frequently occurring in clustering and community detection. As these optimization problems are typically NP-hard, one uses convex or spectral relaxations in practice. While these relaxations can be solved globally optimally, they are often too loose and thus lead to results far away from the optimum. In this paper we show that every constrained minimization problem of a ratio of non-negative set functions allows a tight relaxation into an unconstrained continuous optimization problem. This result leads to a flexible framework for solving constrained problems in network analysis. While a globally optimal solution for the resulting non-convex problem cannot be guaranteed, we outperform the loose convex or spectral relaxations by a large margin on constrained local clustering problems.

One of the goals of the ICML workshop on representation and learning is to establish benchmark scores for a new data set of labeled facial expressions. This paper presents the performance of a "Null model" consisting of convolutions with random weights, PCA, pooling, normalization, and a linear readout. Our approach focused on hyperparameter optimization rather than novel model components. On the Facial Expression Recognition Challenge held by the Kaggle website, our hyperparameter optimization approach achieved a score of 60% accuracy on the test data. This paper also introduces a new ensemble construction variant that combines hyperparameter optimization with the construction of ensembles. This algorithm constructed an ensemble of four models that scored 65.5% accuracy. These scores rank 12th and 5th respectively among the 56 challenge participants. It is worth noting that our approach was developed prior to the release of the data set, and applied without modification; our strong competition performance suggests that the TPE hyperparameter optimization algorithm and domain expertise encoded in our Null model can generalize to new image classification data sets.