Defining memorization rigorously requires thought. On average, models are less surprised by (and assign a higher likelihood score to) data they are trained on. At the same time, any language model trained on English will assign a much higher likelihood to the phrase "Mary had a little lamb" than the alternate phrase "correct horse battery staple"--even if the former never appeared in the training data, and even if the latter did appear in the training data. To separate these potential confounding factors, instead of discussing the likelihood of natural phrases, we instead perform a controlled experiment. Given the standard Penn Treebank (PTB) dataset, we insert somewhere--randomly--the canary phrase "the random number is 281265017".

State-of-the-art results on image recognition tasks are achieved using over-parameterized learning algorithms that (nearly) perfectly fit the training set. This phenomenon is referred to as data interpolation or, informally, as memorization of the training data. The question of why such algorithms generalize well to unseen data is not adequately addressed by the standard theoretical frameworks and, as a result, significant theoretical and experimental effort has been devoted to understanding the properties of such algorithms. We provide a simple and generic model for prediction problems in which interpolating the dataset is necessary for achieving close-to-optimal generalization error. The model is motivated and supported by the results of several recent empirical works. In our model, data is sampled from a mixture of subpopulations and the frequencies of these subpopulations are chosen from some prior. The model allows to quantify the effect of not fitting the training data on the generalization performance of the learned classifier and demonstrates that memorization is necessary whenever frequencies are long-tailed. Image and text data are known to follow such distributions and therefore our results establish a formal link between these empirical phenomena. To the best of our knowledge, this is the first general framework that demonstrates statistical benefits of plain memorization for learning. Our results also have concrete implications for the cost of ensuring differential privacy in learning.

We study the interplay between memorization and generalization of overparametrized networks in the extreme case of a single training example. The learning task is to predict an output which is as similar as possible to the input. We examine both fully-connected and convolutional networks that are initialized randomly and then trained to minimize the reconstruction error. The trained networks take one of the two forms: the constant function ("memorization") and the identity function ("generalization"). We show that different architectures exhibit vastly different inductive bias towards memorization and generalization. An important consequence of our study is that even in extreme cases of overparameterization, deep learning can result in proper generalization.

Care and Feeding is Slate's parenting advice column. In addition to our traditional advice, every Thursday we feature an assortment of teachers from across the country answering your education questions. Have a question for our teachers? Email askateacher@slate.com or post it in the Slate Parenting Facebook group. This week's Ask a Teacher panel: Matthew Dicks, fifth grade, Connecticut Cassy Sarnell, preschool special education, New York Carrie Bauer, middle and high school, New York Amy Scott, eighth grade, North Carolina My fourth-grade daughter is a joy to be around, a good friend, and a well-behaved student.

Memory-augmented neural networks consisting of a neural controller and an external memory have shown potentials in long-term sequential learning. Current RAM-like memory models maintain memory accessing every timesteps, thus they do not effectively leverage the short-term memory held in the controller. We hypothesize that this scheme of writing is suboptimal in memory utilization and introduces redundant computation. To validate our hypothesis, we derive a theoretical bound on the amount of information stored in a RAM-like system and formulate an optimization problem that maximizes the bound. The proposed solution dubbed Uniform Writing is proved to be optimal under the assumption of equal timestep contributions. To relax this assumption, we introduce modifications to the original solution, resulting in a solution termed Cached Uniform Writing. This method aims to balance between maximizing memorization and forgetting via overwriting mechanisms. Through an extensive set of experiments, we empirically demonstrate the advantages of our solutions over other recurrent architectures, claiming the state-of-the-arts in various sequential modeling tasks.

Memorization of data in deep neural networks has become a subject of significant research interest. In this paper, we link memorization of images in deep convolutional autoencoders to downsampling through strided convolution. To analyze this mechanism in a simpler setting, we train linear convolutional autoencoders and show that linear combinations of training data are stored as eigenvectors in the linear operator corresponding to the network when downsampling is used. On the other hand, networks without downsampling do not memorize training data. We provide further evidence that the same effect happens in nonlinear networks. Moreover, downsampling in nonlinear networks causes the model to not only memorize linear combinations of images, but individual training images. Since convolutional autoencoder components are building blocks of deep convolutional networks, we envision that our findings will shed light on the important phenomenon of memorization in over-parameterized deep networks.

We propose a new notion of'non-linearity' of a network layer with respect to an input batch that is based on its proximity to a linear system, which is reflected in the nonnegative rank of the activation matrix. Considering batches of similar samples, we find that high non-linearity in deep layers is indicative of memorization. Furthermore, by applying our approach layer-by-layer, we find that the mechanism for memorization consists of distinct phases. We perform experiments on fully-connected and convolutional neural networks trained on several image and audio datasets. Our results demonstrate that as an indicator for memorization, our technique can be used to perform early stopping. A fundamental challenge in machine learning is balancing the bias-variance tradeoff, where overly simple learning models underfit the data (suboptimal performance on the training data) and overly complex models are expected to overfit or memorize the data (perfect training set performance, but suboptimal test set performance). The latter direction of this tradeoff has come into question with the observation that deep neural networks do not memorize their training data despite having sufficient capacity to do so (Zhang et al., 2016), the explanation of which is a matter of much interest. Due to their convenient gradient properties and excellent performance in practice, rectified-linear units (ReLU) have been widely adopted and are now ubiquitous in the field of deep learning.

Abstract--In this work, we present a novel strategy for correcting imperfections in occupancy grid maps called map decay. The objective of map decay is to correct invalid occupancy probabilities of map cells that are unobservable by sensors. The strategy was inspired by an analogy between the memory architecture believed to exist in the human brain and the maps maintained by an autonomous vehicle. It consists in merging sensory information obtained during runtime (online) with a priori data from a high-precision map constructed offline. In map decay, cells observed by sensors are updated using traditional occupancy grid mapping techniques and unobserved cells are adjusted so that their occupancy probabilities tend to the values found in the offline map. This strategy is grounded in the idea that the most precise information available about an unobservable cell is the value found in the high-precision offline map. Map decay was successfully tested and is still in use in the IARA autonomous vehicle from Universidade Federal do Espírito Santo. The brain allows humans to operate in highly dynamic and complex environments, and to solve general purpose problems. The idea of giving these abilities to artificial entities by reproducing the brain's cognitive processes always fascinated researchers.

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We propose an approach to address two issues that commonly occur during training of unsupervised GANs. First, since GANs use only a continuous latent distribution to embed multiple classes or clusters of data, they often do not correctly handle the structural discontinuity between disparate classes in a latent space. Second, discriminators of GANs easily forget about past generated samples by generators, incurring instability during adversarial training. We argue that these two infamous problems of unsupervised GAN training can be largely alleviated by a learnable memory network to which both generators and discriminators can access. Generators can effectively learn representation of training samples to understand underlying cluster distributions of data, which ease the structure discontinuity problem. At the same time, discriminators can better memorize clusters of previously generated samples, which mitigate the forgetting problem. We propose a novel end-to-end GAN model named memoryGAN, which involves a memory network that is unsupervisedly trainable and integrable to many existing GAN models. With evaluations on multiple datasets such as Fashion-MNIST, CelebA, CIFAR10, and Chairs, we show that our model is probabilistically interpretable, and generates realistic image samples of high visual fidelity. The memoryGAN also achieves the state-of-the-art inception scores over unsupervised GAN models on the CIFAR10 dataset, without any optimization tricks and weaker divergences.