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

We consider the problem of learning to follow a desired trajectory when given a small number of demonstrations from a sub-optimal expert. We present an algorithm that (i) extracts the---initially unknown---desired trajectory from the sub-optimal expert's demonstrations and (ii) learns a local model suitable for control along the learned trajectory. We apply our algorithm to the problem of autonomous helicopter flight. In all cases, the autonomous helicopter's performance exceeds that of our expert helicopter pilot's demonstrations. Even stronger, our results significantly extend the state-of-the-art in autonomous helicopter aerobatics. In particular, our results include the first autonomous tic-tocs, loops and hurricane, vastly superior performance on previously performed aerobatic maneuvers (such as in-place flips and rolls), and a complete airshow, which requires autonomous transitions between these and various other maneuvers.

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Learning for control from multiple demonstrations

When we look at a picture, our prior knowledge about the world allows us to resolve some of the ambiguities that are inherent to monocular vision, and thereby infer 3d information about the scene. We also recognize different objects, decide on their orientations, and identify how they are connected to their environment. Focusing on the problem of autonomous 3d reconstruction of indoor scenes, in this paper we present a dynamic Bayesian network model capable of resolving some of these ambiguities and recovering 3d information for many images. Our model assumes a "floorwall" geometry on the scene and is trained to recognize the floor-wall boundary in each column of the image. When the image is produced under perspective geometry, we show that this model can be used for 3d reconstruction from a single image. To our knowledge, this was the first monocular approach to automatically recover 3d reconstructions from single indoor images.

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A Dynamic Bayesian Network Model for Autonomous 3D Reconstruction from a Single Indoor Image

In the model-based policy search approach to reinforcement learning (RL), policies are found using a model (or "simulator") of the Markov decision process. However, for high-dimensional continuous-state tasks, it can be extremely difficult to build an accurate model, and thus often the algorithm returns a policy that works in simulation but not in real-life. The other extreme, model-free RL, tends to require infeasibly large numbers of real-life trials. In this paper, we present a hybrid algorithm that requires only an approximate model, and only a small number of real-life trials. The key idea is to successively "ground" the policy evaluations using real-life trials, but to rely on the approximate model to suggest local changes. Our theoretical results show that this algorithm achieves near-optimal performance in the real system, even when the model is only approximate. Empirical results also demonstrate that---when given only a crude model and a small number of real-life trials---our algorithm can obtain near-optimal performance in the real system.

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Using inaccurate models in reinforcement learning

Although discriminatively trained classifiers are usually more accurate when labeled training data is abundant, previous work has shown that when training data is limited, generative classifiers can out-perform them. This paper describes a hybrid model in which a high-dimensional subset of the parameters are trained to maximize generative likelihood, and another, small, subset of parameters are discriminatively trained to maximize conditional likelihood. We give a sample complexity bound showing that in order to fit the discriminative parameters well, the number of training examples required depends only on the logarithm of the number of feature occurrences and feature set size. Experimental results show that hybrid models can provide lower test error and can produce better accuracy/coverage curves than either their purely generative or purely discriminative counterparts. We also discuss several advantages of hybrid models, and advocate further work in this area.

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Classification with Hybrid Generative/Discriminative Models

The HITS and the PageRank algorithms are eigenvector methods for identifying "authoritative" or "influential" articles, given hyperlink or citation information That such algorithms should give consistent answers is surely a desideratum, and in this paper, we address the question of when they can be expected to give stable rankings under small perturbations to the hyperlink patterns Using tools from matrix perturbation theory and Markov chain theory, we provide conditions under which these methods are stable, and give specific examples of instability when these conditions are violated We also briefly describe a modification to HITS that improves its stability

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Andrew Y. Ng

Papers

Learning for control from multiple demonstrations

A Dynamic Bayesian Network Model for Autonomous 3D Reconstruction from a Single Indoor Image

Using inaccurate models in reinforcement learning

Classification with Hybrid Generative/Discriminative Models

Link analysis, eigenvectors and stability