ترغب بنشر مسار تعليمي؟ اضغط هنا

Path-Normalized Optimization of Recurrent Neural Networks with ReLU Activations

110   0   0.0 ( 0 )
 نشر من قبل Behnam Neyshabur
 تاريخ النشر 2016
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English




اسأل ChatGPT حول البحث

We investigate the parameter-space geometry of recurrent neural networks (RNNs), and develop an adaptation of path-SGD optimization method, attuned to this geometry, that can learn plain RNNs with ReLU activations. On several datasets that require capturing long-term dependency structure, we show that path-SGD can significantly improve trainability of ReLU RNNs compared to RNNs trained with SGD, even with various recently suggested initialization schemes.



قيم البحث

اقرأ أيضاً

We revisit the choice of SGD for training deep neural networks by reconsidering the appropriate geometry in which to optimize the weights. We argue for a geometry invariant to rescaling of weights that does not affect the output of the network, and s uggest Path-SGD, which is an approximate steepest descent method with respect to a path-wise regularizer related to max-norm regularization. Path-SGD is easy and efficient to implement and leads to empirical gains over SGD and AdaGrad.
Neural networks have been widely used to solve complex real-world problems. Due to the complicate, nonlinear, non-convex nature of neural networks, formal safety guarantees for the output behaviors of neural networks will be crucial for their applica tions in safety-critical systems.In this paper, the output reachable set computation and safety verification problems for a class of neural networks consisting of Rectified Linear Unit (ReLU) activation functions are addressed. A layer-by-layer approach is developed to compute output reachable set. The computation is formulated in the form of a set of manipulations for a union of polyhedra, which can be efficiently applied with the aid of polyhedron computation tools. Based on the output reachable set computation results, the safety verification for a ReLU neural network can be performed by checking the intersections of unsafe regions and output reachable set described by a union of polyhedra. A numerical example of a randomly generated ReLU neural network is provided to show the effectiveness of the approach developed in this paper.
We present polynomial time and sample efficient algorithms for learning an unknown depth-2 feedforward neural network with general ReLU activations, under mild non-degeneracy assumptions. In particular, we consider learning an unknown network of the form $f(x) = {a}^{mathsf{T}}sigma({W}^mathsf{T}x+b)$, where $x$ is drawn from the Gaussian distribution, and $sigma(t) := max(t,0)$ is the ReLU activation. Prior works for learning networks with ReLU activations assume that the bias $b$ is zero. In order to deal with the presence of the bias terms, our proposed algorithm consists of robustly decomposing multiple higher order tensors arising from the Hermite expansion of the function $f(x)$. Using these ideas we also establish identifiability of the network parameters under minimal assumptions.
Recurrent neural networks (RNNs) are notoriously difficult to train. When the eigenvalues of the hidden to hidden weight matrix deviate from absolute value 1, optimization becomes difficult due to the well studied issue of vanishing and exploding gra dients, especially when trying to learn long-term dependencies. To circumvent this problem, we propose a new architecture that learns a unitary weight matrix, with eigenvalues of absolute value exactly 1. The challenge we address is that of parametrizing unitary matrices in a way that does not require expensive computations (such as eigendecomposition) after each weight update. We construct an expressive unitary weight matrix by composing several structured matrices that act as building blocks with parameters to be learned. Optimization with this parameterization becomes feasible only when considering hidden states in the complex domain. We demonstrate the potential of this architecture by achieving state of the art results in several hard tasks involving very long-term dependencies.
In this paper, we construct neural networks with ReLU, sine and $2^x$ as activation functions. For general continuous $f$ defined on $[0,1]^d$ with continuity modulus $omega_f(cdot)$, we construct ReLU-sine-$2^x$ networks that enjoy an approximation rate $mathcal{O}(omega_f(sqrt{d})cdot2^{-M}+omega_{f}left(frac{sqrt{d}}{N}right))$, where $M,Nin mathbb{N}^{+}$ denote the hyperparameters related to widths of the networks. As a consequence, we can construct ReLU-sine-$2^x$ network with the depth $5$ and width $maxleft{leftlceil2d^{3/2}left(frac{3mu}{epsilon}right)^{1/{alpha}}rightrceil,2leftlceillog_2frac{3mu d^{alpha/2}}{2epsilon}rightrceil+2right}$ that approximates $fin mathcal{H}_{mu}^{alpha}([0,1]^d)$ within a given tolerance $epsilon >0$ measured in $L^p$ norm $pin[1,infty)$, where $mathcal{H}_{mu}^{alpha}([0,1]^d)$ denotes the Holder continuous function class defined on $[0,1]^d$ with order $alpha in (0,1]$ and constant $mu > 0$. Therefore, the ReLU-sine-$2^x$ networks overcome the curse of dimensionality on $mathcal{H}_{mu}^{alpha}([0,1]^d)$. In addition to its supper expressive power, functions implemented by ReLU-sine-$2^x$ networks are (generalized) differentiable, enabling us to apply SGD to train.

الأسئلة المقترحة

التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا