67 0 obj <> endobj \begin{split} Nov 2018: I organized a session on non-convex optimization for machine learning at INFORMS annual meeting; Oct 2018: our paper that says adding a neuron can eliminate all bad local minima will appear at NIPS 2018; Professional Services \begin{align} Other research interests: Information theory and wireless communications, such as interference alignment and base station association. 269, pp. Jan 2019: our paper on Adam-type methods (joint with Xiangyi Chen, Sijia Liu and Mingyi Hong) is accepted by ICLR 2019. Mixture of Gaussians. at work. Batch normalization [30] reestablishes these normalizations for every mini-batch and changes are back-propagated through the operation as well. We can now plug this into the Adam update equation by replacing \(\sqrt{\hat{v}_t} + \epsilon\) with \(u_t\) to obtain the AdaMax update rule: \(\theta_{t+1} = \theta_{t} - \dfrac{\eta}{u_t} \hat{m}_t\). This demonstrates again that momentum involves taking a step in the direction of the previous momentum vector and a step in the direction of the current gradient. Machine Learning 10-725 Instructor: Ryan Tibshirani (ryantibs at cmu dot edu) First-order nonconvex optimization (Yuanzhi Li) Slides (Scribed notes) Wed Nov 20: Bregman proximal methods (Javier Pena) Slides (Scribed notes) Mon Nov 25 (Thanksgiving break, no class) Support Vector Machines. , Qian, N. (1999). However, the open source version of TensorFlow currently does not support distributed functionality (see here). In contrast, running SGD asynchronously is faster, but suboptimal communication between workers can lead to poor convergence. Adaptive Subgradient Methods for Online Learning and Stochastic Optimization. \). Finally, we will consider additional strategies that are helpful for optimizing gradient descent. Advice on applying machine learning: Slides from Andrew's lecture on getting machine learning algorithms to work in practice can be found here. Make clear in your answer how you reach the final result; the road to the answer is very important. m_t &= \gamma m_{t-1} + \eta g_t\\ arXiv Preprint arXiv:1502.03167v3. \). hbbd```b``"A$Vs,"cCAa XDL6|`? For a great overview of some other common tricks, refer to [27]. Supervised Learning Setup. areas where the surface curves much more steeply in one dimension than in another [4], which are common around local optima. the update should have the same hypothetical units as the parameter. Large Scale Distributed Deep Networks. hb```f``*c`e`Jdb@ ! E]/8j00Ja@N*?\@"m} G0or8Lx|3TOu8 Ri*?(f 9pYDhR{zAA%c 7 1`R scPLkebUV~=k2Rg{7Fpr8!c; Value function approximation. 400407, 1951. http://doi.org/10.1007/3-540-49430-8_2 , Bengio, Y., Louradour, J., Collobert, R., & Weston, J. In case you found it helpful, consider citing the corresponding arXiv article as: The study of neural networks is an extension of my research on non-convex optimization for machine learning since PhD. Answers to many frequently asked questions for learners prior to the Lagunita retirement were available on our FAQ page. If it is neither of these, then CVX is not the correct tool for the task. \begin{split} Signal processing and information theory: IEEE Transaction on Information Theory, IEEE Transaction on Signal Processing, SPAWC, ICASSP. Batch gradient descent is guaranteed to converge to the global minimum for convex error surfaces and to a local minimum for non-convex surfaces. Proceedings of the 26th Annual International Conference on Machine Learning, 4148. The same thing happens to our parameter updates: The momentum term increases for dimensions whose gradients point in the same directions and reduces updates for dimensions whose gradients change directions. hj1_EodBJ- qiYag%KZ$` G gQ$A)!xX*PJ&@dKKPN ,aQ J #P,%,q*WWu.nSINwo/Ve=XV^r}xo/-Uz@M&P5doCtwL;~"u?1((eDTe?pDm_Pz &C$w{V[_}':R>}S8#3[j@6j-n}$z]s>wg6@ $~ Niu et al. On the other hand, for some cases where we aim to solve progressively harder problems, supplying the training examples in a meaningful order may actually lead to improved performance and better convergence. K-means. As adaptive learning rate methods have become the norm in training neural networks, practitioners noticed that in some cases, e.g. As we can see, the adaptive learning-rate methods, i.e. The authors note that the units in this update (as well as in SGD, Momentum, or Adagrad) do not match, i.e. (i.e. Retrieved from http://arxiv.org/abs/1410.4615 , Ioffe, S., & Szegedy, C. (2015). GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium. Also have a look here for a description of the same images by Karpathy and another concise overview of the algorithms discussed. Good default values are again \(\eta = 0.002\), \(\beta_1 = 0.9\), and \(\beta_2 = 0.999\). This post explores how many of the most popular gradient-based optimization algorithms such as Momentum, Adagrad, and Adam actually work. Running it provides good convergence but can be slow particularly on large datasets. hWMoA+;SrRNTtHREHTr3oD9dK9yc{2%V%R)VJ)5*\)' q`%@T+TgJ3q)!n \end{align} Finally, with extensive numerical evaluations on convex optimization problems, we illustrate that our designed schemes achieve state-of-the-art communication complexity compared to several key baselines using second-order information. Recently, I have been studying optimization in deep learning, such as landscape of neural-nets, GANs and Adam. Are there any obvious algorithms to improve SGD that I've missed? More information about CVX can be found in the CVX Users Guide, which can be found online in a searchable format, or downloaded as a PDF. Assistant Professor at University of Illinois at Urbana-Champaign. It remains to be seen whether AMSGrad is able to consistently outperform Adam in practice. ECE 273. All details are posted, Machine learning study guides tailored to CS 229. , Dozat, T. (2016). Dec 2019: my survey optimization for deep learning: theory and algorithms is available at arxiv https://arxiv.org/abs/1912.08957 Comments are welcome. In Proceedings of ICLR 2019. (2018) [19] formalize this issue and pinpoint the exponential moving average of past squared gradients as a reason for the poor generalization behaviour of adaptive learning rate methods. A learning rate that is too small leads to painfully slow convergence, while a learning rate that is too large can hinder convergence and cause the loss function to fluctuate around the minimum or even to diverge. As \(m_t\) and \(v_t\) are initialized as vectors of 0's, the authors of Adam observe that they are biased towards zero, especially during the initial time steps, and especially when the decay rates are small (i.e. Berkeley Assured Autonomy Seminar (TBD, Spring 2022). Curriculum learning. Logical Interfaces to Z3 (2015). Weighted Least Squares. Netwon's Method. In Proceedings of ICLR 2019. "q Z \(g_{t, i}\) is then the partial derivative of the objective function w.r.t. Convex Optimization Overview, Part II ; Hidden Markov Models ; The Multivariate Gaussian Distribution ; More on Gaussian Distribution ; Gaussian Processes ; Other Resources. A MOOC on convex optimization, CVX101, was run from 1/21/14 to 3/14/14. \end{split} Constrained optimization So far: Projected gradient descent Conditional gradient method Barrier and Interior Point methods Convex problem . [, Functional after implementing stump_booster.m in PS2. However, this short-term memory of the gradients becomes an obstacle in other scenarios. - Bring Your Own Algorithm for Optimal Differentially Private The running average \(E[g^2]_t\) at time step \(t\) then depends (as a fraction \(\gamma \) similarly to the Momentum term) only on the previous average and the current gradient: \(E[g^2]_t = \gamma E[g^2]_{t-1} + (1 - \gamma) g^2_t \). points where one dimension slopes up and another slopes down. Not all solvers support MIDCPs, and those that do cannot guarantee a successful solution in reasonable time for all models. As training progresses and we update parameters to different extents, we lose this normalization, which slows down training and amplifies changes as the network becomes deeper. %PDF-1.5 % Neural Information Processing Systems Conference (NIPS 2015), 124. We will then briefly summarize challenges during training. \begin{split} some parameters. Quasi-hyperbolic momentum and Adam for deep learning. Expanding the second equation with the definitions of \(\hat{m}_t\) and \(m_t\) in turn gives us: \(\theta_{t+1} = \theta_{t} - \dfrac{\eta}{\sqrt{\hat{v}_t} + \epsilon} (\dfrac{\beta_1 m_{t-1}}{1 - \beta^t_1} + \dfrac{(1 - \beta_1) g_t}{1 - \beta^t_1})\). LQG. 94305. TensorFlow [25] is Google's recently open-sourced framework for the implementation and deployment of large-scale machine learning models. v_t &= \beta_2 v_{t-1} + (1 - \beta_2) g_t^2 Aug 30 Lecture 13 Descent Methods and Convex Optimization - Recording CEG Sec. Portfolio optimization (tigher control) Returns Ri [i ui,i +ui] with ERi = i guarantee return with probability 1 maximize ,t t subject to Prob Xn i=1 Rixi t 1 valueatrisk is non-convex in x, approximate it? annealing, i.e. In Advances in Neural Information Processing Systems 30 (NIPS 2017). Nevertheless, we believe that MIDCP support is a powerful addition to CVX and we look forward to seeing how our users take advantage of it. }!z##>/ohVmz(;L7FrF41E.\oD2)PK*RBoQ|. While batch gradient descent converges to the minimum of the basin the parameters are placed in, SGD's fluctuation, on the one hand, enables it to jump to new and potentially better local minima. Simultaneous Interior and Boundary Optimization of Volumetric Domain Parameterizations for IGA Hao Liu, Yang Yang, Yuan Liu, Xiao-Ming Fu Computer-Aided Geometric Design (GMP), 2020. This way, it a) reduces the variance of the parameter updates, which can lead to more stable convergence; and b) can make use of highly optimized matrix optimizations common to state-of-the-art deep learning libraries that make computing the gradient w.r.t. Value Iteration and Policy Iteration. Finally, we've considered other strategies to improve SGD such as shuffling and curriculum learning, batch normalization, and early stopping. While the Lagunita platform has been retired, we offer many other platforms for extended education. The root mean squared error of parameter updates is thus: \(RMS[\Delta \theta]_{t} = \sqrt{E[\Delta \theta^2]_t + \epsilon} \). An overview of gradient descent optimisation algorithms. STANFORD COURSES ON THE LAGUNITA LEARNING PLATFORM, Stanford Center for Professional Development, Entrepreneurial Leadership Graduate Certificate, Energy Innovation and Emerging Technologies. m_t &= \gamma m_{t-1} + \eta g_t\\ With Adadelta, we do not even need to set a default learning rate, as it has been eliminated from the update rule. Consequently, if you care about fast convergence and train a deep or complex neural network, you should choose one of the adaptive learning rate methods. Vanilla gradient descent, aka batch gradient descent, computes the gradient of the cost function w.r.t. \begin{split} It is based on their experience with DistBelief and is already used internally to perform computations on a large range of mobile devices as well as on large-scale distributed systems. Retrieved from http://jmlr.org/papers/v12/duchi11a.html , Dean, J., Corrado, G. S., Monga, R., Chen, K., Devin, M., Le, Q. V, Ng, A. Y. Given the ubiquity of large-scale data solutions and the availability of low-commodity clusters, distributing SGD to speed it up further is an obvious choice. Principal Component Analysis. As has been shown, SGD usually achieves to find a minimum, but it might take significantly longer than with some of the optimizers, is much more reliant on a robust initialization and annealing schedule, and may get stuck in saddle points rather than local minima. Microsofts Activision Blizzard deal is key to the companys mobile gaming efforts. ). Essentially, when using momentum, we push a ball down a hill. \). This way, AMSGrad results in a non-increasing step size, which avoids the problems suffered by Adam. \end{align} \begin{align} Momentum [5] is a method that helps accelerate SGD in the relevant direction and dampens oscillations as can be seen in Image 3. h bHp)0A>"0` Ry"X,,,r`m i.g`b F ). Update 13.04.16: A distributed version of TensorFlow has been released. We thus only need to modify the gradient \(g_t\) to arrive at NAG: \( Convex Optimization and Applications (4) This course covers some convex optimization theory and algorithms. Proceedings of ICLR 2018. 2012 CVX Research, Inc. All rights reserved. RMSprop and Adadelta have both been developed independently around the same time stemming from the need to resolve Adagrad's radically diminishing learning rates. First, let us recall the momentum update rule using our current notation : \( Hinton suggests \(\gamma\) to be set to 0.9, while a good default value for the learning rate \(\eta\) is 0.001. arXiv Preprint arXiv:1611.0455. Incorporating Nesterov Momentum into Adam. Note: Some implementations exchange the signs in the equations. These algorithms, however, are often used as black-box optimizers, as practical explanations of their strengths and weaknesses are hard to come by. Other experiments, however, show similar or worse performance than Adam. [23] introduce an update scheme called Hogwild! bco f{Alkp6_ ~ \uTz9Dqzw@uFVlz&xvcW.7s)j K c:*`/sV_!i -d F69>x.t1 `-5a hgo~fLF(T;CJ4%TA 99i},Onx@mYQ;oHC4zKpatn21vQYL|5`<{{?[>,YJeZmMXmvlK~}"8)xGNq}=gdd+?uAxuu==zy|puvO]P6"FFRzlTB\g(rD. The sum of two convex functions (for example, L 2 loss + L 1 regularization) is a convex function. With the right learning algorithm, we can start to fit by minimizing J() as a function of to find optimal parameters. Stanford University. Note: If you are looking for a review paper, this blog post is also available as an article on arXiv. Delay-Tolerant Algorithms for Asynchronous Distributed Online Learning. In code, instead of iterating over examples, we now iterate over mini-batches of size 50: Vanilla mini-batch gradient descent, however, does not guarantee good convergence, but offers a few challenges that need to be addressed: Choosing a proper learning rate can be difficult. Looking for your Lagunita course? As the denominator is just the root mean squared (RMS) error criterion of the gradient, we can replace it with the criterion short-hand: \( \Delta \theta_t = - \dfrac{\eta}{RMS[g]_{t}} g_t\). In settings where Adam converges to a suboptimal solution, it has been observed that some minibatches provide large and informative gradients, but as these minibatches only occur rarely, exponential averaging diminishes their influence, which leads to poor convergence. Based on the book "Convex Optimization Theory," Athena Scientific, 2009, and the book "Convex Optimization Algorithms," Athena Scientific, 2014.