COMS6998 Homework 3 Solved

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Problem 1 – Adaptive Learning Rate Methods, CIFAR-10

We will consider five methods, AdaGrad, RMSProp, RMSProp+Nesterov, AdaDelta, Adam, and study their convergence using CIFAR-10 dataset. We will use multi-layer neural network model with two fully connected hidden layers with 1000 hidden units each and ReLU activation with minibatch size of 128.

  1. Write the weight update equations for the five adaptive learning rate methods. Explain each term clearly. What are the hyperparameters in each policy ? Explain how AdaDelta and Adam are different from RMSProp. (5+1)
  2. Train the neural network using all the five methods with L2-regularization for 200 epochs each and plot the training loss vs number of epochs. Which method performs best (lowest training loss) ? (5)
  3. Add dropout (probability 0.2 for input layer and 0.5 for hidden layers) and train the neural network again using all the five methods for 200 epochs. Compare the training loss with that in part 2. Which method performs the best ? For the five methods, compare their training time (to finish 200 epochs with dropout) to the training time in part 2 (to finish 200 epochs without dropout). (5)
  4. Compare test accuracy of trained model for all the five methods from part 2 and part 3. Note that to calculate test accuracy of model trained using dropout you need to appropriately scale the weights (by the dropout probability). (4)


Problem 2 – TF 2.0, tensorflow.distribute.strategy, Strong and Weak Scaling

In this problem we will compare strong scaling and weak scaling in distributed training using tensorflow.distribute.strategy in Tensorflow 2.0. tf.distribute.Strategy is a TensorFlow API to distribute training across multiple GPUs, multiple machines or TPUs. In strong scaling, each worker computes with (batch size/# workers) training examples whereas in weak scaling, the effective batch size of SGD grows as the number of workers increases. For example, in strong scaling, if the batch size with 1 worker is 256, with 2 workers it will be 128 per worker, with 4 workers it will be 64 per worker, thus keeping the effective batch size at 256. In weak scaling, if the batch size with 1 worker is 64, with 2 workers it will be still be 64 per worker (with an effective batch size of 128 with 2 workers), thus the effective batch size increases linearly with number of workers. So the amount of compute per worker decreases in strong scaling whereas with weak scaling it remains constant.

Using FashionMNIST dataset and Resnet50 you will run distributed training using tensorflow.distribute.strategy and compare strong and weak scaling scenarios. Using an effective batch size of 256 you will run training jobs with 1,2,4,8,16 learners (each learner is a K80 GPU). For 8 or less number of learners, all the GPUs can be allocated on the same node (GCP povides 8 K80s on one node). You will run each training job for 10 epochs and measure average throughout, training time, and training cost. In total, you will be running 10 training jobs, 5 (with 1,2,4,8,16 GPUs) for weak scaling and 5 for strong scaling. For single node (worker) training using multiple GPUs you will use tf.distribute.MirroredStrategy with default all-reduce. For training with two or more workers you will use tf.distribute.experimental.MultiWorkerMirroredStrategy with CollectiveCommunication.AUTO.

  1. Plot throughput vs number of learners for weak and strong scaling. (5)
  2. Plot training time vs number of learners for weak and strong scaling. (5)
  3. Plot training cost vs number of learners for weak and strong scaling. The training cost can be estimated using GPU per unit hour cost and the training time. (2)
  4. For weak scaling, calculate scaling efficiency defined as the increase in time to finish one iteration at a learner as the number of learners increases. Show the plot of scaling efficiency vs number of learners for weak scaling. (5)
  5. MirroredStrategy uses NVIDIA NCCL (tf.distribute.NcclAllReduce) as the default all-reduce. Change this to tf.distribute.HierarchicalCopyAllReduce and tf.distribute.ReductionToOneDevice and compare throughput of the three all-reduce implementations. You will be doing this for 1,2,4,8 GPUs single-node training. So you will be running 8 new training jobs (4 with HierarchicalCopyAllReduce and 4 with ReductionToOneDevice). For NcclAllReduce you can reuse results from part 1 of the question. (8)
  6. Change MultiWorkerMirroredStrategy to use CollectiveCommunication.NCCL and CollectiveCommunication.RING and repeat the experiment with 2 nodes. Yow will be running two new training jobs (one with RING and one with NCCL). For AUTO you can reuse throughput from part 1 of the question. Compare the throughput of the three all-reduce methods (AUTO, NCCL, RING) ? Does AUTO gives the best throughput ? (5)


Problem 3 – Convolutional Neural Networks Architectures

In this problem we will study and compare different convolutional neural network architectures. We will calculate number of parameters (weights, to be learned) and memory requirement of each network. We will also analyze inception modules and understand their design.

  1. Calculate the number of parameters in Alexnet. You will have to show calculations for each layer and then sum it to obtain the total number of parameters in Alexnet. When calculating you will need to account for all the filters (size, strides, padding) at each layer. Look at Sec. 3.5 and Figure 2 in Alexnet paper (see reference). Points will only be given when explicit calculations are shown for each layer. (5)
  2. VGG (Simonyan et al.) has an extremely homogeneous architecture that only performs 3×3 convolutions with stride 1 and pad 1 and 2×2 max pooling with stride 2 (and no padding) from the beginning to the end. However VGGNet is very expensive to evaluate and uses a lot more memory and parameters. Refer to VGG19 architecture on page 3 in Table 1 of the paper by Simonyan et al. You need to complete Table 1 below for calculating activation units and parameters at each layer in VGG19 (without counting biases). Its been partially filled for you. (6)
  3. VGG architectures have smaller filters but deeper networks compared to Alexnet (3×3 compared to 11×11 or 5×5). Show that a stack of N convolution layers each of filter size F × F has the same receptive field as one convolution layer with filter of size (NF N + 1) × (NF N + 1). Use this to calculate the receptive field of 3 filters of size 5×5. (4)
  4. The original Googlenet paper (Szegedy et al.) proposes two architectures for Inception module, shown in Figure 2 on page 5 of the paper, referred to as naive and dimensionality reduction respectively.

(a) What is the general idea behind designing an inception module (parallel convolutional filters of different sizes with a pooling followed by concatenation) in a convolutional neural network ? (3)

Layer Number of Activations (Memory) Parameters (Compute)
Input 224*224*3=150K 0
CONV3-64 224*224*64=3.2M (3*3*3)*64 = 1,728
CONV3-64 224*224*64=3.2M (3*3*64)*64 = 36,864
POOL2 112*112*64=800K 0
POOL2 56*56*128=400K 0
CONV3-256 56*56*256=800K (3*3*256)*256 = 589,824
POOL2   0
CONV3-512 28*28*512=400K (3*3*256)*512 = 1,179,648
CONV3-512 28*28*512=400K  
POOL2   0
POOL2   0
FC 4096  
FC 4096 4096*4096 = 16,777,216
FC 1000  

Table 1: VGG19 memory and weights (b) Assuming the input to inception module (referred to as ”previous layer” in Figure 2 of the paper) has size 32x32x256, calculate the output size after filter concatenation for the naive and dimensionality reduction inception architectures with number of filters given in Figure 1. (4)

  • Next calculate the total number of convolutional operations for each of the two inception architecture again assuming the input to the module has dimensions 32x32x256 and number of filters given in Figure 1. (4)
  • Based on the calculations in part (c) explain the problem with naive architecture and how dimensionality reduction architecture helps (Hint: compare computational complexity). How much is the computational saving ? (2+2)

Figure 1: Two types of inception module with number of filters and input size for calculation in Question

3.4(b) and 3.4(c).


Problem 4 – Batch Augmentation, Cutout Regularization

In this problem we will be achieving large-batch SGD using batch augmentation techniques. In batch augmentation instances of samples within the same batch are generated with different data augmentations. Batch augmentation acts as a regularizer and an accelerator, increasing both generalization and performance scaling. One such augmentation scheme is using Cutout regularization, where additional samples are generated by occluding random portions of an image.

  1. Explain cutout regularization and its advantages compared to simple dropout (as argued in the paper by DeVries et al) in your own words. Select any 2 images from CIFAR10 and show how does these images look after applying cutout. Use a square-shaped fixed size zero-mask to a random location of each image and generate its cutout version. Refer to the paper by DeVries et al (Section 3) and associated github repository. (2+4)
  2. Using CIFAR10 datasest and Resnet-44 we will first apply simple data augmentation as in He et al. (look at Section 4.2 of He et al.) and train the model with batch size 64. Note that testing is always done with original images. Plot validation error vs number of training epochs. (4)
  3. Next use cutout for data augmentation in Resnet-44 as in Hoffer et al. and train the model and use the same set-up in your experiments. Plot validation error vs number of epochs for different values of M (2,4,8,16,32) where M is the number of instances generated from an input sample after applying cutout M times effectively increasing the batch size to M B, where B is the original batch size (before applying cutout augmentation). You will obtain a figure similar to Figure 3(a) in the paper by Hoffer et al. Also compare the number of epochs and wallclock time to reach 94% accuracy for different values of M. Do not run any experiment for more than 100 epochs. If even after 100 epochs of training you did not achieve 94% then just report the accuracy you obtain and the corresponding wallclock time to train for 100 epochs. Before attempting this question it is advisable to read paper by Hoffer et al. and especially Section 4.1. (5+5)

You may reuse code from github repository associated with Hoffer et al. work for answering part 2 and 3 of this question. References: • DeVries et al. Improved Regularization of Convolutional Neural Networks with Cutout.

Paper available at

Code available at

  • Hoffer et al. Augment your batch: better training with larger batches. 2019

Paper available at

Code available at • He et al. Deep residual learning for image recognition. Paper available at

Problem 5 – Universal Approximators: Depth Vs. Width

Multilayer layer feedforward network, with as little as two layers and sufficiently large hidden units can approximate any arbitrary function. Thus one can tradeoff between deep and shallow networks for the same problem. In this problem we will study this tradeoff using the Eggholder function defined as:

Let y(x1,x2) = f(x1,x2) + N(0,0.3) be the function that we want to learn from a neural network through regression with −512 ≤ x1 ≤ 512 and −512 ≤ x2 ≤ 512. Draw a dataset of 100K points from this function (uniformly sampling in the range of x1 and x2) and do a 80/20 training/test split.

  1. Assume that total budget for number of hidden units we can have in the network is 512. Train a 1, 2, and 3 hidden layers feedforward neural network to learn the regression function. For each neural network you can consider a different number of hidden units per hidden layer so that the total number of hidden units does not exceed 512. We would recommend to work with 16, 32, 64, 128, 256, 512, hidden units per layer. So if there is only one hidden layer you can have at most 512 units in that layer.

If there are two hidden layers, you can have any combination of hidden units in each layer, e.g., 16 and 256, 64 and 128, etc. such that the total is less than 512. Plot the RMSE (Root Mean Square Error) on test set for networks with different number of hidden layers as a function of total number of hidden units. If there are more than one network with the same number of hidden units (say a two hidden layer with 16 in first layer and 128 in second layer and another network with 128 in first layer and 16 in second) you will use the average RMSE. So you will have a figure with three curves, one each for 1, 2, and 3 layer networks, with x-axis being the total number of hidden units. Also plot another curve but with the x-axis being the number of parameters (weights) that you need to learn in the network. (20)

  1. Comment on the tradeoff between number of parameters and RMSE as you go from deeper (3 hidden layers) to shallow networks (1 hidden layer). Also measure the wall clock time for training each configuration and plot training time vs number of parameters. Do you see a similar tradeoff in training time ? (10)

For networks with 2 and 3 layers you will use batch normalization as regularization. For hidden layers use ReLU activation and for training use SGD with Nesterov momentum. Take a batch size of 1000 and train for 2000 epochs. You can pick other hyperparameter values (momentum, learning rate schedule) or use the default values in the framework implementation.