اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA Convergence Theory for Deep Learning via Over-Parameterization
376
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. On the theoretical side, a long line of works has been focusing on training neural networks with one hidden layer. The theory of multi-layer networks remains largely unsettled. In this work, we prove why stochastic gradient descent (SGD) can find $textit{global minima}$ on the training objective of DNNs in $textit{polynomial time}$. We only make two assumptions: the inputs are non-degenerate and the network is over-parameterized. The latter means the network width is sufficiently large: $textit{polynomial}$ in $L$, the number of layers and in $n$, the number of samples. Our key technique is to derive that, in a sufficiently large neighborhood of the random initialization, the optimization landscape is almost-convex and semi-smooth even with ReLU activations. This implies an equivalence between over-parameterized neural networks and neural tangent kernel (NTK) in the finite (and polynomial) width setting. As concrete examples, starting from randomly initialized weights, we prove that SGD can attain 100% training accuracy in classification tasks, or minimize regression loss in linear convergence speed, with running time polynomial in $n,L$. Our theory applies to the widely-used but non-smooth ReLU activation, and to any smooth and possibly non-convex loss functions. In terms of network architectures, our theory at least applies to fully-connected neural networks, convolutional neural networks (CNN), and residual neural networks (ResNet).
قيم البحث
اقرأ أيضاً
How does a 110-layer ResNet learn a high-complexity classifier using relatively few training examples and short training time? We present a theory towards explaining this in terms of Hierarchical Learning. We refer hierarchical learning as the learne
r learns to represent a complicated target function by decomposing it into a sequence of simpler functions to reduce sample and time complexity. We formally analyze how multi-layer neural networks can perform such hierarchical learning efficiently and automatically by applying SGD. On the conceptual side, we present, to the best of our knowledge, the FIRST theory result indicating how deep neural networks can still be sample and time efficient using SGD on certain hierarchical learning tasks, when NO KNOWN existing algorithm is efficient. We establish a new principle called backward feature correction, where training higher-level layers in the network can improve the features of lower-level ones. We believe this is the key to understand the deep learning process in multi-layer neural networks. On the technical side, we show for regression and even binary classification, for every input dimension $d>0$, there is a concept class of degree $omega(1)$ polynomials so that, using $omega(1)$-layer neural networks as learners, SGD can learn any function from this class in $mathsf{poly}(d)$ time and sample complexity to any $frac{1}{mathsf{poly}(d)}$ error, through learning to represent it as a composition of $omega(1)$ layers of quadratic functions. In contrast, we do not know any other simple algorithm (including layer-wise training or applying kernel method sequentially) that can learn this concept class in $mathsf{poly}(d)$ time even to any $d^{-0.01}$ error. As a side result, we prove $d^{omega(1)}$ lower bounds for several non-hierarchical learners, including any kernel methods, neural tangent or neural compositional kernels.
There has been an increased interest in discovering heuristics for combinatorial problems on graphs through machine learning. While existing techniques have primarily focused on obtaining high-quality solutions, scalability to billion-sized graphs ha
s not been adequately addressed. In addition, the impact of budget-constraint, which is necessary for many practical scenarios, remains to be studied. In this paper, we propose a framework called GCOMB to bridge these gaps. GCOMB trains a Graph Convolutional Network (GCN) using a novel probabilistic greedy mechanism to predict the quality of a node. To further facilitate the combinatorial nature of the problem, GCOMB utilizes a Q-learning framework, which is made efficient through importance sampling. We perform extensive experiments on real graphs to benchmark the efficiency and efficacy of GCOMB. Our results establish that GCOMB is 100 times faster and marginally better in quality than state-of-the-art algorithms for learning combinatorial algorithms. Additionally, a case-study on the practical combinatorial problem of Influence Maximization (IM) shows GCOMB is 150 times faster than the specialized IM algorithm IMM with similar quality.
It has been empirically observed that, in deep neural networks, the solutions found by stochastic gradient descent from different random initializations can be often connected by a path with low loss. Recent works have shed light on this intriguing p
henomenon by assuming either the over-parameterization of the network or the dropout stability of the solutions. In this paper, we reconcile these two views and present a novel condition for ensuring the connectivity of two arbitrary points in parameter space. This condition is provably milder than dropout stability, and it provides a connection between the problem of finding low-loss paths and the memorization capacity of neural nets. This last point brings about a trade-off between the quality of features at each layer and the over-parameterization of the network. As an extreme example of this trade-off, we show that (i) if subsets of features at each layer are linearly separable, then almost no over-parameterization is needed, and (ii) under generic assumptions on the features at each layer, it suffices that the last two hidden layers have $Omega(sqrt{N})$ neurons, $N$ being the number of samples. Finally, we provide experimental evidence demonstrating that the presented condition is satisfied in practical settings even when dropout stability does not hold.
How quickly can a given class of concepts be learned from examples? It is common to measure the performance of a supervised machine learning algorithm by plotting its learning curve, that is, the decay of the error rate as a function of the number of
training examples. However, the classical theoretical framework for understanding learnability, the PAC model of Vapnik-Chervonenkis and Valiant, does not explain the behavior of learning curves: the distribution-free PAC model of learning can only bound the upper envelope of the learning curves over all possible data distributions. This does not match the practice of machine learning, where the data source is typically fixed in any given scenario, while the learner may choose the number of training examples on the basis of factors such as computational resources and desired accuracy. In this paper, we study an alternative learning model that better captures such practical aspects of machine learning, but still gives rise to a complete theory of the learnable in the spirit of the PAC model. More precisely, we consider the problem of universal learning, which aims to understand the performance of learning algorithms on every data distribution, but without requiring uniformity over the distribution. The main result of this paper is a remarkable trichotomy: there are only three possible rates of universal learning. More precisely, we show that the learning curves of any given concept class decay either at an exponential, linear, or arbitrarily slow rates. Moreover, each of these cases is completely characterized by appropriate combinatorial parameters, and we exhibit optimal learning algorithms that achieve the best possible rate in each case. For concreteness, we consider in this paper only the realizable case, though analogous results are expected to extend to more general learning scenarios.
While over-parameterization is widely believed to be crucial for the success of optimization for the neural networks, most existing theories on over-parameterization do not fully explain the reason -- they either work in the Neural Tangent Kernel reg
ime where neurons dont move much, or require an enormous number of neurons. In practice, when the data is generated using a teacher neural network, even mildly over-parameterized neural networks can achieve 0 loss and recover the directions of teacher neurons. In this paper we develop a local convergence theory for mildly over-parameterized two-layer neural net. We show that as long as the loss is already lower than a threshold (polynomial in relevant parameters), all student neurons in an over-parameterized two-layer neural network will converge to one of teacher neurons, and the loss will go to 0. Our result holds for any number of student neurons as long as it is at least as large as the number of teacher neurons, and our convergence rate is independent of the number of student neurons. A key component of our analysis is the new characterization of local optimization landscape -- we show the gradient satisfies a special case of Lojasiewicz property which is different from local strong convexity or PL conditions used in previous work.
الأسئلة المقترحة
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
