arXiv subjects

Amnon Shashua

Publications and source records attributed to Amnon Shashua.

47 records · Page 3Linked to original sources

arXiv2016

Convolutional Rectifier Networks as Generalized Tensor Decompositions

Convolutional rectifier networks, i.e. convolutional neural networks with rectified linear activation and max or average pooling, are the cornerstone of modern deep learning. However, despite their wide use and success, our theoretical understanding of the expressive properties that drive these networks is partial at best. On the other hand, we have a much firmer grasp of these issues in the world of arithmetic circuits. Specifically, it is known that convolutional arithmetic circuits possess the property of "complete depth efficiency", meaning that besides a negligible set, all functions that can be implemented by a deep network of polynomial size, require exponential size in order to be implemented (or even approximated) by a shallow network. In this paper we describe a construction based on generalized tensor decompositions, that transforms convolutional arithmetic circuits into convolutional rectifier networks. We then use mathematical tools available from the world of arithmetic circuits to prove new results. First, we show that convolutional rectifier networks are universal with max pooling but not with average pooling. Second, and more importantly, we show that depth efficiency is weaker with convolutional rectifier networks than it is with convolutional arithmetic circuits. This leads us to believe that developing effective methods for training convolutional arithmetic circuits, thereby fulfilling their expressive potential, may give rise to a deep learning architecture that is provably superior to convolutional rectifier networks but has so far been overlooked by practitioners.

cs.NE↗

arXiv2016

On the Sample Complexity of End-to-end Training vs. Semantic Abstraction Training

We compare the end-to-end training approach to a modular approach in which a system is decomposed into semantically meaningful components. We focus on the sample complexity aspect, in the regime where an extremely high accuracy is necessary, as is the case in autonomous driving applications. We demonstrate cases in which the number of training examples required by the end-to-end approach is exponentially larger than the number of examples required by the semantic abstraction approach.

cs.LG↗

arXiv2016

Deep SimNets

We present a deep layered architecture that generalizes convolutional neural networks (ConvNets). The architecture, called SimNets, is driven by two operators: (i) a similarity function that generalizes inner-product, and (ii) a log-mean-exp function called MEX that generalizes maximum and average. The two operators applied in succession give rise to a standard neuron but in "feature space". The feature spaces realized by SimNets depend on the choice of the similarity operator. The simplest setting, which corresponds to a convolution, realizes the feature space of the Exponential kernel, while other settings realize feature spaces of more powerful kernels (Generalized Gaussian, which includes as special cases RBF and Laplacian), or even dynamically learned feature spaces (Generalized Multiple Kernel Learning). As a result, the SimNet contains a higher abstraction level compared to a traditional ConvNet. We argue that enhanced expressiveness is important when the networks are small due to run-time constraints (such as those imposed by mobile applications). Empirical evaluation validates the superior expressiveness of SimNets, showing a significant gain in accuracy over ConvNets when computational resources at run-time are limited. We also show that in large-scale settings, where computational complexity is less of a concern, the additional capacity of SimNets can be controlled with proper regularization, yielding accuracies comparable to state of the art ConvNets.

cs.NE↗

arXiv2016

Long-term Planning by Short-term Prediction

We consider planning problems, that often arise in autonomous driving applications, in which an agent should decide on immediate actions so as to optimize a long term objective. For example, when a car tries to merge in a roundabout it should decide on an immediate acceleration/braking command, while the long term effect of the command is the success/failure of the merge. Such problems are characterized by continuous state and action spaces, and by interaction with multiple agents, whose behavior can be adversarial. We argue that dual versions of the MDP framework (that depend on the value function and the $Q$ function) are problematic for autonomous driving applications due to the non Markovian of the natural state space representation, and due to the continuous state and action spaces. We propose to tackle the planning task by decomposing the problem into two phases: First, we apply supervised learning for predicting the near future based on the present. We require that the predictor will be differentiable with respect to the representation of the present. Second, we model a full trajectory of the agent using a recurrent neural network, where unexplained factors are modeled as (additive) input nodes. This allows us to solve the long-term planning problem using supervised learning techniques and direct optimization over the recurrent neural network. Our approach enables us to learn robust policies by incorporating adversarial elements to the environment.

cs.LG↗

arXiv2014

SimNets: A Generalization of Convolutional Networks

We present a deep layered architecture that generalizes classical convolutional neural networks (ConvNets). The architecture, called SimNets, is driven by two operators, one being a similarity function whose family contains the convolution operator used in ConvNets, and the other is a new soft max-min-mean operator called MEX that realizes classical operators like ReLU and max pooling, but has additional capabilities that make SimNets a powerful generalization of ConvNets. Three interesting properties emerge from the architecture: (i) the basic input to hidden layer to output machinery contains as special cases kernel machines with the Exponential and Generalized Gaussian kernels, the output units being "neurons in feature space" (ii) in its general form, the basic machinery has a higher abstraction level than kernel machines, and (iii) initializing networks using unsupervised learning is natural. Experiments demonstrate the capability of achieving state of the art accuracy with networks that are an order of magnitude smaller than comparable ConvNets.

cs.NE↗

arXiv2012

Tightening Fractional Covering Upper Bounds on the Partition Function for High-Order Region Graphs

In this paper we present a new approach for tightening upper bounds on the partition function. Our upper bounds are based on fractional covering bounds on the entropy function, and result in a concave program to compute these bounds and a convex program to tighten them. To solve these programs effectively for general region graphs we utilize the entropy barrier method, thus decomposing the original programs by their dual programs and solve them with dual block optimization scheme. The entropy barrier method provides an elegant framework to generalize the message-passing scheme to high-order region graph, as well as to solve the block dual steps in closed-form. This is a key for computational relevancy for large problems with thousands of regions.

cs.LG↗

arXiv2012

Convergent Message-Passing Algorithms for Inference over General Graphs with Convex Free Energies

Inference problems in graphical models can be represented as a constrained optimization of a free energy function. It is known that when the Bethe free energy is used, the fixedpoints of the belief propagation (BP) algorithm correspond to the local minima of the free energy. However BP fails to converge in many cases of interest. Moreover, the Bethe free energy is non-convex for graphical models with cycles thus introducing great difficulty in deriving efficient algorithms for finding local minima of the free energy for general graphs. In this paper we introduce two efficient BP-like algorithms, one sequential and the other parallel, that are guaranteed to converge to the global minimum, for any graph, over the class of energies known as "convex free energies". In addition, we propose an efficient heuristic for setting the parameters of the convex free energy based on the structure of the graph.

cs.LG↗

arXiv2011

ShareBoost: Efficient Multiclass Learning with Feature Sharing

Multiclass prediction is the problem of classifying an object into a relevant target class. We consider the problem of learning a multiclass predictor that uses only few features, and in particular, the number of used features should increase sub-linearly with the number of possible classes. This implies that features should be shared by several classes. We describe and analyze the ShareBoost algorithm for learning a multiclass predictor that uses few shared features. We prove that ShareBoost efficiently finds a predictor that uses few shared features (if such a predictor exists) and that it has a small generalization error. We also describe how to use ShareBoost for learning a non-linear predictor that has a fast evaluation time. In a series of experiments with natural data sets we demonstrate the benefits of ShareBoost and evaluate its success relatively to other state-of-the-art approaches.

cs.LG↗

arXiv2010

The Latent Bernoulli-Gauss Model for Data Analysis

We present a new latent-variable model employing a Gaussian mixture integrated with a feature selection procedure (the Bernoulli part of the model) which together form a "Latent Bernoulli-Gauss" distribution. The model is applied to MAP estimation, clustering, feature selection and collaborative filtering and fares favorably with the state-of-the-art latent-variable models.

cs.LG↗

arXiv2010

Norm-Product Belief Propagation: Primal-Dual Message-Passing for Approximate Inference

In this paper we treat both forms of probabilistic inference, estimating marginal probabilities of the joint distribution and finding the most probable assignment, through a unified message-passing algorithm architecture. We generalize the Belief Propagation (BP) algorithms of sum-product and max-product and tree-rewaighted (TRW) sum and max product algorithms (TRBP) and introduce a new set of convergent algorithms based on "convex-free-energy" and Linear-Programming (LP) relaxation as a zero-temprature of a convex-free-energy. The main idea of this work arises from taking a general perspective on the existing BP and TRBP algorithms while observing that they all are reductions from the basic optimization formula of $f + \sum_i h_i$ where the function $f$ is an extended-valued, strictly convex but non-smooth and the functions $h_i$ are extended-valued functions (not necessarily convex). We use tools from convex duality to present the "primal-dual ascent" algorithm which is an extension of the Bregman successive projection scheme and is designed to handle optimization of the general type $f + \sum_i h_i$. Mapping the fractional-free-energy variational principle to this framework introduces the "norm-product" message-passing. Special cases include sum-product and max-product (BP algorithms) and the TRBP algorithms. When the fractional-free-energy is set to be convex (convex-free-energy) the norm-product is globally convergent for estimating of marginal probabilities and for approximating the LP-relaxation. We also introduce another branch of the norm-product, the "convex-max-product". The convex-max-product is convergent (unlike max-product) and aims at solving the LP-relaxation.

cs.AI↗

arXiv2009

Introduction to Machine Learning: Class Notes 67577

Introduction to Machine learning covering Statistical Inference (Bayes, EM, ML/MaxEnt duality), algebraic and spectral methods (PCA, LDA, CCA, Clustering), and PAC learning (the Formal model, VC dimension, Double Sampling theorem).

cs.LG↗