2005년 대한 생화학·분자생물학회 하계학술대회를 다녀와서

We show that passing input points through a simple Fourier feature mapping enables a multilayer perceptron (MLP) to learn high-frequency functions in low-dimensional problem domains These results shed light on recent advances in computer vision and graphics that achieve state-of-the-art results by using MLPs to represent complex 3D objects and scenes Using tools from the neural tangent kernel (NTK) literature, we show that a standard MLP fails to learn high frequencies both in theory and in practice To overcome this spectral bias, we use a Fourier feature mapping to transform the effective NTK into a stationary kernel with a tunable bandwidth We suggest an approach for selecting problem-specific Fourier features that greatly improves the performance of MLPs for low-dimensional regression tasks relevant to the computer vision and graphics communities

Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains

Gradient descent finds a global minimum in training deep neural networks despite the objective function being non-convex. The current paper proves gradient descent achieves zero training loss in polynomial time for a deep over-parameterized neural network with residual connections (ResNet). Our analysis relies on the particular structure of the Gram matrix induced by the neural network architecture. This structure allows us to show the Gram matrix is stable throughout the training process and this stability implies the global optimality of the gradient descent algorithm. We further extend our analysis to deep residual convolutional neural networks and obtain a similar convergence result.

/pdf/gradient-descent-finds-global-minima-of-deep-neural-networks-u4z03tf3zz.pdf

Gradient descent finds global minima of deep neural networks

The offline reinforcement learning (RL) problem, also known as batch RL, refers to the setting where a policy must be learned from a static dataset, without additional online data collection. This setting is compelling as it potentially allows RL methods to take advantage of large, pre-collected datasets, much like how the rise of large datasets has fueled results in supervised learning in recent years. However, existing online RL benchmarks are not tailored towards the offline setting, making progress in offline RL difficult to measure. In this work, we introduce benchmarks specifically designed for the offline setting, guided by key properties of datasets relevant to real-world applications of offline RL. Examples of such properties include: datasets generated via hand-designed controllers and human demonstrators, multi-objective datasets where an agent can perform different tasks in the same environment, and datasets consisting of a mixtures of policies. To facilitate research, we release our benchmark tasks and datasets with a comprehensive evaluation of existing algorithms and an evaluation protocol together with an open-source codebase. We hope that our benchmark will focus research effort on methods that drive improvements not just on simulated tasks, but ultimately on the kinds of real-world problems where offline RL will have the largest impact.

/pdf/d4rl-datasets-for-deep-data-driven-reinforcement-learning-162ppxtve5.pdf

D4RL: Datasets for Deep Data-Driven Reinforcement Learning

The phenomenon of benign overfitting is one of the key mysteries uncovered by deep learning methodology: deep neural networks seem to predict well, even with a perfect fit to noisy training data. Motivated by this phenomenon, we consider when a perfect fit to training data in linear regression is compatible with accurate prediction. We give a characterization of linear regression problems for which the minimum norm interpolating prediction rule has near-optimal prediction accuracy. The characterization is in terms of two notions of the effective rank of the data covariance. It shows that overparameterization is essential for benign overfitting in this setting: the number of directions in parameter space that are unimportant for prediction must significantly exceed the sample size. By studying examples of data covariance properties that this characterization shows are required for benign overfitting, we find an important role for finite-dimensional data: the accuracy of the minimum norm interpolating prediction rule approaches the best possible accuracy for a much narrower range of properties of the data distribution when the data lie in an infinite-dimensional space vs. when the data lie in a finite-dimensional space with dimension that grows faster than the sample size.

https://www.pnas.org/content/pnas/117/48/30063.full.pdf

Benign overfitting in linear regression

How well does a classic deep net architecture like AlexNet or VGG19 classify on a standard dataset such as CIFAR-10 when its “width”— namely, number of channels in convolutional layers, and number of nodes in fully-connected internal layers — is allowed to increase to infinity? Such questions have come to the forefront in the quest to theoretically understand deep learning and its mysteries about optimization and generalization. They also connect deep learning to notions such as Gaussian processes and kernels. A recent paper [Jacot et al., 2018] introduced the Neural Tangent Kernel (NTK) which captures the behavior of fully-connected deep nets in the infinite width limit trained by gradient descent; this object was implicit in some other recent papers. An attraction of such ideas is that a pure kernel-based method is used to capture the power of a fully-trained deep net of infinite width. The current paper gives the first efficient exact algorithm for computing the extension of NTK to convolutional neural nets, which we call Convolutional NTK (CNTK), as well as an efficient GPU implementation of this algorithm. This results in a significant new benchmark for performance of a pure kernel-based method on CIFAR-10, being 10% higher than the methods reported in [Novak et al., 2019], and only 6% lower than the performance of the corresponding finite deep net architecture (once batch normalization etc. are turned off). Theoretically, we also give the first non-asymptotic proof showing that a fully-trained sufficiently wide net is indeed equivalent to the kernel regression predictor using NTK.

/pdf/on-exact-computation-with-an-infinitely-wide-neural-net-5bwgiyxlk5.pdf

On Exact Computation with an Infinitely Wide Neural Net

Recent works have cast some light on the mystery of why deep nets fit any data and generalize despite being very overparametrized. This paper analyzes training and generalization for a simple 2-layer ReLU net with random initialization, and provides the following improvements over recent works: 
(i) Using a tighter characterization of training speed than recent papers, an explanation for why training a neural net with random labels leads to slower training, as originally observed in [Zhang et al. ICLR'17]. 
(ii) Generalization bound independent of network size, using a data-dependent complexity measure. Our measure distinguishes clearly between random labels and true labels on MNIST and CIFAR, as shown by experiments. Moreover, recent papers require sample complexity to increase (slowly) with the size, while our sample complexity is completely independent of the network size. 
(iii) Learnability of a broad class of smooth functions by 2-layer ReLU nets trained via gradient descent. 
The key idea is to track dynamics of training and generalization via properties of a related kernel.

Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks

/pdf/fine-grained-analysis-of-optimization-and-generalization-for-2yxelvm9r0.pdf

Fine-grained analysis of optimization and generalization for overparameterized two-layer neural networks

How well does a classic deep net architecture like AlexNet or VGG19 classify on a standard dataset such as CIFAR-10 when its width --- namely, number of channels in convolutional layers, and number of nodes in fully-connected internal layers --- is allowed to increase to infinity? Such questions have come to the forefront in the quest to theoretically understand deep learning and its mysteries about optimization and generalization. They also connect deep learning to notions such as Gaussian processes and kernels. A recent paper [Jacot et al., 2018] introduced the Neural Tangent Kernel (NTK) which captures the behavior of fully-connected deep nets in the infinite width limit trained by gradient descent; this object was implicit in some other recent papers. An attraction of such ideas is that a pure kernel-based method is used to capture the power of a fully-trained deep net of infinite width. 
The current paper gives the first efficient exact algorithm for computing the extension of NTK to convolutional neural nets, which we call Convolutional NTK (CNTK), as well as an efficient GPU implementation of this algorithm. This results in a significant new benchmark for the performance of a pure kernel-based method on CIFAR-10, being $10\%$ higher than the methods reported in [Novak et al., 2019], and only $6\%$ lower than the performance of the corresponding finite deep net architecture (once batch normalization, etc. are turned off). Theoretically, we also give the first non-asymptotic proof showing that a fully-trained sufficiently wide net is indeed equivalent to the kernel regression predictor using NTK.

Modern deep learning methods provide effective means to learn good representations. However, is a good representation itself sufficient for efficient reinforcement learning? This question is largely unexplored, and the extant body of literature mainly focuses on conditions which permit efficient reinforcement learning with little understanding of what are necessary conditions for efficient reinforcement learning. This work provides strong negative results for reinforcement learning methods with function approximation for which a good representation (feature extractor) is known to the agent, focusing on natural representational conditions relevant to value-based learning and policy-based learning. For value-based learning, we show that even if the agent has a highly accurate linear representation, the agent still needs to sample an exponential number trajectories in order to find a near-optimal policy. For policy-based learning, we show even if the agent's linear representation is capable of perfectly predicting the optimal action at any state, the agent still needs to sample an exponential number of trajectories in order to find a near-optimal policy. These lower bounds highlight the fact that having a good (value-based or policy-based) representation in and of itself is insufficient for efficient reinforcement learning and that additional assumptions are needed. In particular, these results provide new insights into why the analysis of existing provably efficient reinforcement learning methods make assumptions which are partly model-based in nature. Furthermore, our lower bounds also imply exponential separations on the sample complexity between 1) value-based learning with perfect representation and value-based learning with a good-but-not-perfect representation, 2) value-based learning and policy-based learning, 3) policy-based learning and supervised learning and 4) reinforcement learning and imitation learning.

/pdf/is-a-good-representation-sufficient-for-sample-efficient-4c9zq50h33.pdf

Ruosong Wang

Papers

On Exact Computation with an Infinitely Wide Neural Net

Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks

Fine-grained analysis of optimization and generalization for overparameterized two-layer neural networks

On Exact Computation with an Infinitely Wide Neural Net

Is a Good Representation Sufficient for Sample Efficient Reinforcement Learning