We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little memory requirements, is invariant to diagonal rescaling of the gradients, and is well suited for problems that are large in terms of data and/or parameters. The method is also appropriate for non-stationary objectives and problems with very noisy and/or sparse gradients. The hyper-parameters have intuitive interpretations and typically require little tuning. Some connections to related algorithms, on which Adam was inspired, are discussed. We also analyze the theoretical convergence properties of the algorithm and provide a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. Finally, we discuss AdaMax, a variant of Adam based on the infinity norm.

Adam: A Method for Stochastic Optimization

We trained a large, deep convolutional neural network to classify the 1.2 million high-resolution images in the ImageNet LSVRC-2010 contest into the 1000 different classes. On the test data, we achieved top-1 and top-5 error rates of 37.5% and 17.0% which is considerably better than the previous state-of-the-art. The neural network, which has 60 million parameters and 650,000 neurons, consists of five convolutional layers, some of which are followed by max-pooling layers, and three fully-connected layers with a final 1000-way softmax. To make training faster, we used non-saturating neurons and a very efficient GPU implementation of the convolution operation. To reduce overriding in the fully-connected layers we employed a recently-developed regularization method called "dropout" that proved to be very effective. We also entered a variant of this model in the ILSVRC-2012 competition and achieved a winning top-5 test error rate of 15.3%, compared to 26.2% achieved by the second-best entry.

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ImageNet Classification with Deep Convolutional Neural Networks

In this work we investigate the effect of the convolutional network depth on its accuracy in the large-scale image recognition setting. Our main contribution is a thorough evaluation of networks of increasing depth using an architecture with very small (3x3) convolution filters, which shows that a significant improvement on the prior-art configurations can be achieved by pushing the depth to 16-19 weight layers. These findings were the basis of our ImageNet Challenge 2014 submission, where our team secured the first and the second places in the localisation and classification tracks respectively. We also show that our representations generalise well to other datasets, where they achieve state-of-the-art results. We have made our two best-performing ConvNet models publicly available to facilitate further research on the use of deep visual representations in computer vision.

Very Deep Convolutional Networks for Large-Scale Image Recognition

We propose a deep convolutional neural network architecture codenamed Inception that achieves the new state of the art for classification and detection in the ImageNet Large-Scale Visual Recognition Challenge 2014 (ILSVRC14). The main hallmark of this architecture is the improved utilization of the computing resources inside the network. By a carefully crafted design, we increased the depth and width of the network while keeping the computational budget constant. To optimize quality, the architectural decisions were based on the Hebbian principle and the intuition of multi-scale processing. One particular incarnation used in our submission for ILSVRC14 is called GoogLeNet, a 22 layers deep network, the quality of which is assessed in the context of classification and detection.

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Going deeper with convolutions

Deep neural networks (DNNs) are now a central component of nearly all state-of-the-art speech recognition systems. Building neural network acoustic models requires several design decisions including network architecture, size, and training loss function. This paper offers an empirical investigation on which aspects of DNN acoustic model design are most important for speech recognition system performance. We report DNN classifier performance and final speech recognizer word error rates, and compare DNNs using several metrics to quantify factors influencing differences in task performance. Our first set of experiments use the standard Switchboard benchmark corpus, which contains approximately 300 hours of conversational telephone speech. We compare standard DNNs to convolutional networks, and present the first experiments using locally-connected, untied neural networks for acoustic modeling. We additionally build systems on a corpus of 2,100 hours of training data by combining the Switchboard and Fisher corpora. This larger corpus allows us to more thoroughly examine performance of large DNN models -- with up to ten times more parameters than those typically used in speech recognition systems. Our results suggest that a relatively simple DNN architecture and optimization technique produces strong results. These findings, along with previous work, help establish a set of best practices for building DNN hybrid speech recognition systems with maximum likelihood training. Our experiments in DNN optimization additionally serve as a case study for training DNNs with discriminative loss functions for speech tasks, as well as DNN classifiers more generally.

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Building DNN Acoustic Models for Large Vocabulary Speech Recognition

We consider the problem of modeling a helicopter's dynamics based on state-action trajectories collected from it. The contribution of this paper is two-fold. First, we consider the linear models such as learned by CIFER (the industry standard in helicopter identification), and show that the linear parameterization makes certain properties of dynamical systems, such as inertia, fundamentally difficult to capture. We propose an alternative, acceleration based, parameterization that does not suffer from this deficiency, and that can be learned as efficiently from data. Second, a Markov decision process model of a helicopter's dynamics would explicitly model only the one-step transitions, but we are often interested in a model's predictive performance over longer timescales. In this paper, we present an efficient algorithm for (approximately) minimizing the prediction error over long time scales. We present empirical results on two different helicopters. Although this work was motivated by the problem of modeling helicopters, the ideas presented here are general, and can be applied to modeling large classes of vehicular dynamics.

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Learning vehicular dynamics, with application to modeling helicopters

Humans have an amazing ability to perceive depth from a single still image; however, it remains a challenging problem for current computer vision systems. In this paper, we will present algorithms for estimating depth from a single still image.

There are numerous monocular cues--such as texture variations and gradients, defocus, color/haze, etc.--that can be used for depth perception. Taking a supervised learning approach to this problem, in which we begin by collecting a training set of single images and their corresponding ground-truth depths, we learn the mapping from image features to the depths. We then apply these ideas to create 3-d models that are visually-pleasing as well as quantitatively accurate from individual images. We also discuss applications of our depth perception algorithm in robotic navigation, in improving the performance of stereovision, and in creating large-scale 3-d models given only a small number of images.

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Make3D: depth perception from a single still image

We consider the task of accurately controlling a complex system, such as autonomously sliding a car sideways into a parking spot. Although certain regions of this domain are extremely hard to model (i.e., the dynamics of the car while skidding), we observe that in practice such systems are often remarkably deterministic over short periods of time, even in difficult-to-model regions. Motivated by this intuition, we develop a probabilistic method for combining closed-loop control in the well-modeled regions and open-loop control in the difficult-to-model regions. In particular, we show that by combining 1) an inaccurate model of the system and 2) a demonstration of the desired behavior, our approach can accurately and robustly control highly challenging systems, without the need to explicitly model the dynamics in the most complex regions and without the need to hand-tune the switching control law. We apply our approach to the task of autonomous sideways sliding into a parking spot, and show that we can repeatedly and accurately control the system, placing the car within about 2 feet of the desired location; to the best of our knowledge, this represents the state of the art in terms of accurately controlling a vehicle in such a maneuver.

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A probabilistic approach to mixed open-loop and closed-loop control, with application to extreme autonomous driving

A computer-implemented method is provided for defining interesting portions of a workflow of a business or other type of process. Using a tracking profile editor, a portion of a given workflow is selected and associated with a named process part. Items of payload data to be used in reports are selected from a message schema associated with the given workflow. A profile is generated based on the given workflow and the selected items of payload data. A tracking profile compiler receives the created tracking profile and generates a star schema, a flattened view and OLAP cube based on the tracking profile. An interceptor extracts monitored workflow events and stores event information to a tracking stream, based on the tracking profile. A tracking service extracts information pertaining to the monitored events from the tracking stream and stores the information in the queryable database. The OLAP cube and the flattened view are updated based on the information written to the queryable database.

Andrew Y. Ng

Papers

Building DNN Acoustic Models for Large Vocabulary Speech Recognition

Learning vehicular dynamics, with application to modeling helicopters

Make3D: depth perception from a single still image

A probabilistic approach to mixed open-loop and closed-loop control, with application to extreme autonomous driving

Support for real-time queries concerning current state, data and history of a process