اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn the Connection Between Learning Two-Layers Neural Networks and Tensor Decomposition
112
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We establish connections between the problem of learning a two-layer neural network and tensor decomposition. We consider a model with feature vectors $boldsymbol x in mathbb R^d$, $r$ hidden units with weights ${boldsymbol w_i}_{1le i le r}$ and output $yin mathbb R$, i.e., $y=sum_{i=1}^r sigma( boldsymbol w_i^{mathsf T}boldsymbol x)$, with activation functions given by low-degree polynomials. In particular, if $sigma(x) = a_0+a_1x+a_3x^3$, we prove that no polynomial-time learning algorithm can outperform the trivial predictor that assigns to each example the response variable $mathbb E(y)$, when $d^{3/2}ll rll d^2$. Our conclusion holds for a `natural data distribution, namely standard Gaussian feature vectors $boldsymbol x$, and output distributed according to a two-layer neural network with random isotropic weights, and under a certain complexity-theoretic assumption on tensor decomposition. Roughly speaking, we assume that no polynomial-time algorithm can substantially outperform current methods for tensor decomposition based on the sum-of-squares hierarchy. We also prove generalizations of this statement for higher degree polynomial activations, and non-random weight vectors. Remarkably, several existing algorithms for learning two-layer networks with rigorous guarantees are based on tensor decomposition. Our results support the idea that this is indeed the core computational difficulty in learning such networks, under the stated generative model for the data. As a side result, we show that under this model learning the network requires accurate learning of its weights, a property that does not hold in a more general setting.
قيم البحث
اقرأ أيضاً
The fundamental learning theory behind neural networks remains largely open. What classes of functions can neural networks actually learn? Why doesnt the trained network overfit when it is overparameterized? In this work, we prove that overparamete
rized neural networks can learn some notable concept classes, including two and three-layer networks with fewer parameters and smooth activations. Moreover, the learning can be simply done by SGD (stochastic gradient descent) or its variants in polynomial time using polynomially many samples. The sample complexity can also be almost independent of the number of parameters in the network. On the technique side, our analysis goes beyond the so-called NTK (neural tangent kernel) linearization of neural networks in prior works. We establish a new notion of quadratic approximation of the neural network (that can be viewed as a second-order variant of NTK), and connect it to the SGD theory of escaping saddle points.
We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy. Our algorithms can
decompose a 4-tensor with $n$-dimensional orthonormal components in the presence of error with constant spectral norm (when viewed as an $n^2$-by-$n^2$ matrix). The running time is $n^5$ which is close to linear in the input size $n^4$. We also obtain algorithms with similar running time to learn sparsely-used orthogonal dictionaries even when feature representations have constant relative sparsity and non-independent coordinates. The only previous polynomial-time algorithms to solve these problem are based on solving large semidefinite programs. In contrast, our algorithms are easy to implement directly and are based on spectral projections and tensor-mode rearrangements. Or work is inspired by recent of Hopkins, Schramm, Shi, and Steurer (STOC16) that shows how fast spectral algorithms can achieve the guarantees of SOS for average-case problems. In this work, we introduce general techniques to capture the guarantees of SOS for worst-case problems.
Several recent works have shown separation results between deep neural networks, and hypothesis classes with inferior approximation capacity such as shallow networks or kernel classes. On the other hand, the fact that deep networks can efficiently ex
press a target function does not mean that this target function can be learned efficiently by deep neural networks. In this work we study the intricate connection between learnability and approximation capacity. We show that learnability with deep networks of a target function depends on the ability of simpler classes to approximate the target. Specifically, we show that a necessary condition for a function to be learnable by gradient descent on deep neural networks is to be able to approximate the function, at least in a weak sense, with shallow neural networks. We also show that a class of functions can be learned by an efficient statistical query algorithm if and only if it can be approximated in a weak sense by some kernel class. We give several examples of functions which demonstrate depth separation, and conclude that they cannot be efficiently learned, even by a hypothesis class that can efficiently approximate them.
On-chip edge intelligence has necessitated the exploration of algorithmic techniques to reduce the compute requirements of current machine learning frameworks. This work aims to bridge the recent algorithmic progress in training Binary Neural Network
s and Spiking Neural Networks - both of which are driven by the same motivation and yet synergies between the two have not been fully explored. We show that training Spiking Neural Networks in the extreme quantization regime results in near full precision accuracies on large-scale datasets like CIFAR-$100$ and ImageNet. An important implication of this work is that Binary Spiking Neural Networks can be enabled by In-Memory hardware accelerators catered for Binary Neural Networks without suffering any accuracy degradation due to binarization. We utilize standard training techniques for non-spiking networks to generate our spiking networks by conversion process and also perform an extensive empirical analysis and explore simple design-time and run-time optimization techniques for reducing inference latency of spiking networks (both for binary and full-precision models) by an order of magnitude over prior work.
Daniely and Schacham recently showed that gradient descent finds adversarial examples on random undercomplete two-layers ReLU neural networks. The term undercomplete refers to the fact that their proof only holds when the number of neurons is a vanis
hing fraction of the ambient dimension. We extend their result to the overcomplete case, where the number of neurons is larger than the dimension (yet also subexponential in the dimension). In fact we prove that a single step of gradient descent suffices. We also show this result for any subexponential width random neural network with smooth activation function.
الأسئلة المقترحة
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
