Often, we stop our iterations when the loss values haven't improved in a pre-decided number, say, 10, or 20 iterations. We would like to thank Leon Bottou, Yann LeCun, Yann Dauphin, Yuandong Tian, Mark Tygert, Levent Sagun, Lechao Xiao, Tengyu Ma, Jason Lee, Matus Telgarsky, Ju Sun, Wei Hu, Simon Du, Lei Wu, Quanquan Gu, Justin Sirignano, Tian Ding, Dawei Li, Shiyu Liang, R. Srikant, for discussions on various results reviewed in this article. The z axis represents the value of the loss function for a particular value of two weights. Consider the prototype convergence rate result in convex optimization: the epoch-complexityFootnote 14 is \(O( \kappa \log 1/\varepsilon )\) or \(O( \beta /\varepsilon )\). 1019–1028 (2017), Neyshabur, B., Salakhutdinov, R.R., Srebro, N.: Path-sgd: Path-normalized optimization in deep neural networks. : Generalization error in deep learning. A desirable property of a minima should be it that it should be on the flatter side.

4140–4149 (2017), Li, Y., Yuan, Y.: Convergence analysis of two-layer neural networks with relu activation. One could also consider a point that is a local minima for the "all-example-loss". I Image recognition Wright (UW-Madison) Optimization in Data Analysis Oct 2017 4 / 63.

INFORMS (2017), Goyal, P., Dollár, P., Girshick, R., Noordhuis, P., Wesolowski, L., Kyrola, A., Tulloch, A., Jia, Y., He, K.: Accurate, large minibatch sgd: Training imagenet in 1 hour (2017). In contrast to Stochastic Gradient Descent, where each example is stochastically chosen, our earlier approach processed all examples in one single batch, and therefore, is known as Batch Gradient Descent. In: International Conference on Learning Representations (2019c). Correspondence to �sGX���b��4@D�����*(�1���$��7ߧy�. arXiv:1901.10159, Salimans, T., Kingma, D.P. Article This direction is given by the gradient. 520–529 (2018), Arora, S., Cohen, N., Golowich, N., Hu, W.: A convergence analysis of gradient descent for deep linear neural networks (2018).

If the loss function is not quadratic, but a general loss function \( \ell ( y, h^L ) \), we only need to replace \(e = 2(h^L - y)\) by \(e = \frac{\partial \ell }{ \partial h^L } \). In: International Conference on Learning Representations (2019a). 770–778 (2016), Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale image recognition arXiv:1409.1556 (2014), Srivastava, R.K., Greff, K., Schmidhuber, J.: Highway networks (2015). However, there's still one missing block about Gradient descent that we haven't talked about in this post, and that is addressing the problem of pathological curvature. arXiv:1810.05270, Lee, N., Ajanthan, T., Torr, P.: SNIP: single-shot network pruning based on connection sensitivity. Therefore, we would like to understand the challenges and opportunities from a theoretical perspective and review the existing research in this field. 315–323 (2013), Defazio, A., Bach, F., Lacoste-Julien, S.: Saga: a fast incremental gradient method with support for non-strongly convex composite objectives. : Theoretical insights into the optimization landscape of over-parameterized shallow neural networks. 1026–1034 (2015). Here, we assume the dimension and the number of samples are both n, and a refined analysis can show the dependence on the two parameters. Go too slow, and the training might turn out to be too long to be feasible at all. In the loss landscape of a neural network, there are just way too many minimum, and a "good" local minima might perform just as well as a global minima. 6240–6249 (2017), Wei, C., Lee, J.D., Liu, Q., Ma, T.: On the margin theory of feedforward neural networks (2018). Now, once we have the direction we want to move in, we must decide the size of the step we must take.

: Recovery guarantees for one-hidden-layer neural networks.

arXiv:1811.03962, Zou, D., Cao, Y., Zhou, D., Gu, Q.: Stochastic gradient descent optimizes over-parameterized deep relu networks (2018a). 586–594 (2016), Lu, H., Kawaguchi, K.: Depth creates no bad local minima (2017). Since the neural network optimization problem is often of huge size (at least millions of optimization variables and millions of samples), a method that directly inverts a matrix in an iteration, such as Newton method, is often considered impractical. In general, parallel computation in numerical optimization is quite complicated, which is why the whole book [101] is devoted to this topic.

arXiv:1910.01663, Novak, R., Xiao, L., Hron, J., Lee, J., Alemi, A.A., Sohl-Dickstein, J., Schoenholz, S.S.: Neural tangents: Fast and easy infinite neural networks in python (2019b).

Sun, R. Optimization for Deep Learning: An Overview.

When we initialize our weights, we are at point A in the loss landscape. Ruo-Yu Sun. arXiv:1706.02677, Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, Ł., Polosukhin, I.: Attention is all you need. In practice, we might never exactly reach the minima, but we keep oscillating in a flat region in close vicinity of the minima. Multiple iterations in one epoch of CD or SGD may not be parallellizable in the worst case (e.g. Realise that even if we keep the learning rate constant, the size of step can change owing to changes in magnitude of the gradient, ot the steepness of the loss contour. Commun. Of course the contribution to optimization area cannot just be judged by the number of citations, but the attention Adam received is still quite remarkable. 2408–2417 (2015), Osawa, K., Tsuji, Y., Ueno, Y., Naruse, A., Yokota, R., Matsuoka, S.: Second-order optimization method for large mini-batch: training resnet-50 on imagenet in 35 epochs (2018). In: Proceedings of the IEEE International Conference on Computer Vision, pp. In: Advances in Neural Information Processing Systems, pp. MATH Bertsekas, D.P. arXiv:1809.10749, Li, Dawei, D., Tian, S., Ruoyu: Over-parameterized deep neural networks have no strict local minima for any continuous activations (2018a).

22(2), 341–362 (2012), Johnson, R., Zhang, T.: Accelerating stochastic gradient descent using predictive variance reduction.

8(1), 141–148 (1988), Becker, S., Le Cun, Y., et al. If we go too fast, we might overshoot the minima, and keep bouncing along the ridges of the "valley" without ever reaching the minima. 343–351 (2013), Tan, C., Ma, S., Dai, Y.-H., Qian, Y.: Barzilai-borwein step size for stochastic gradient descent. The direction of steepest descent is the direction exactly opposite to the gradient, and that is why we are subtracting the gradient vector from the weights vector.

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