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

This is the Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence, which was held in Montreal, QC, Canada, June 18 - 21 2009.

Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence (2009)

Medical image segmentation models are typically supervised by expert annotations at the pixel-level, which can be expensive to acquire. In this work, we propose a method that combines the high quality of pixel-level expert annotations with the scale of coarse DNN-generated saliency maps for training multi-label semantic segmentation models. We demonstrate the application of our semi-supervised method, which we call CheXseg, on multi-label chest X-ray interpretation. We find that CheXseg improves upon the performance (mIoU) of fully-supervised methods that use only pixel-level expert annotations by 9.7% and weakly-supervised methods that use only DNN-generated saliency maps by 73.1%. Our best method is able to match radiologist agreement on three out of ten pathologies and reduces the overall performance gap by 57.2% as compared to weakly-supervised methods.

CheXseg: Combining Expert Annotations with DNN-generated Saliency Maps for X-ray Segmentation

We study computational and sample complexity of parameter and structure learning in graphical models. Our main result shows that the class of factor graphs with bounded factor size and bounded connectivity can be learned in polynomial time and polynomial number of samples, assuming that the data is generated by a network in this class. This result covers both parameter estimation for a known network structure and structure learning. It implies as a corollary that we can learn factor graphs for both Bayesian networks and Markov networks of bounded degree, in polynomial time and sample complexity. Unlike maximum likelihood estimation, our method does not require inference in the underlying network, and so applies to networks where inference is intractable. We also show that the error of our learned model degrades gracefully when the generating distribution is not a member of the target class of networks.

Learning Factor Graphs in Polynomial Time & Sample Complexity

The application of deep learning to pathology assumes the existence of digital whole slide images of pathology slides. However, slide digitization is bottlenecked by the high cost of precise motor stages in slide scanners that are needed for position information used for slide stitching. We propose GloFlow, a two-stage method for creating a whole slide image using optical flow-based image registration with global alignment using a computationally tractable graph-pruning approach. In the first stage, we train an optical flow predictor to predict pairwise translations between successive video frames to approximate a stitch. In the second stage, this approximate stitch is used to create a neighborhood graph to produce a corrected stitch. On a simulated dataset of video scans of WSIs, we find that our method outperforms known approaches to slide-stitching, and stitches WSIs resembling those produced by slide scanners.

GloFlow: Global Image Alignment for Creation of Whole Slide Images for Pathology from Video.

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Andrew Y. Ng

Papers

Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence (2009)

CheXseg: Combining Expert Annotations with DNN-generated Saliency Maps for X-ray Segmentation

Learning Factor Graphs in Polynomial Time & Sample Complexity

GloFlow: Global Image Alignment for Creation of Whole Slide Images for Pathology from Video.

CheXseg: Combining Expert Annotations with DNN-generated Saliency Maps for X-ray Segmentation