// Databricks notebook source exported at Sat, 18 Jun 2016 23:40:46 UTC
prepared by Raazesh Sainudiin and Sivanand Sivaram
The html source url of this databricks notebook and its recorded Uji :
- read: http://arxiv.org/pdf/1509.02256.pdf (also see References and Appendix A).
- and go through the notebooks here: Data Types - MLlib Programming Guide
Distributed Machine Learning: Computation and Storage by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning
(watch now/later? 8:48):
This is a transcript of Ameet’s lecture in the video above:
HOMEWORK: Make sure you are understanding everything he is trying to convey!
Recall from week 1’s lectures that we are weaving over Ameet’s course here to get to advanced applications “in a hurry” and it is assumed you have already done the edX Big Data Series from 2015 or will be doing the 2016 version, ideally with certification :)
In this segment, we’ll begin our discussion of distributed machine learning principles related to computation and storage. We’ll use linear regression as a running example to illustrate these ideas. As we previously discussed, the size of datasets is rapidly growing. And this has led to scalability issues for many standard machine learning methods. For instance, consider the least squares regression model. The fact that a closed-form solution exists is an appealing property of this method. But what happens when we try to solve this closed-form solution at scale? In this segment, we’ll focus on computational issues associated with linear regression. Though I should note that these same ideas apply in the context of ridge regression, as it has a very similar computational profile. So let’s figure out the time and space complexity of solving for this closed-form solution. We’ll start with time complexity, considering arithmetic operations as the basic units of computation when discussing big O complexity. Looking at the expression for w, if we perform each of the required operations separately, we see that computing X transpose X takes O of nd squared time, and inverting this resulting matrix takes O of d cubed time. Since matrix multiplication is an associative operation, we can multiply X transpose and y in O of nd time to get some intermediate d dimensional vector, and then perform the multiplication between the inverse matrix and this intermediate d dimensional vector in O of d squared time. Hence, in summary, the two computational bottlenecks in this process involve computing X transpose X, and subsequently computing its inverse. There are other methods to solve this equation that can be faster in practice, but they are only faster in terms of constants, and therefore they all have the same overall time complexity. In summary, we say that computing the closed-form solution for linear regression takes O of nd squared plus d cubed time. Now let’s consider the space complexity. And recall that our basic unit of storage here is the storage required to store a single float, which is typically 8 bytes. In order to compute w, we must first store the data matrix, which requires O of nd floats. Additionally, we must compute X transpose X and its inverse. In order to solve for w, each of these matrices are d by d and thus require O of d squared floats. These are the two bottlenecks storage-wise. And thus our space complexity is O of nd plus d squared. So now that we’ve considered the time and space complexity required to solve for the closed-form solution, let’s consider what happens as our data grows large. The first situation we’ll consider is one where n, or the number of observations, is large, while d, or the number of features, is relatively small. Specifically, we’ll assume that we’re in a setting where d is small enough such that O of d cubed operate computation and O of d squared storage is feasible on a single machine. In this scenario, the terms in our big O complexity involving n are the ones that dominate, and thus storing X and computing X transpose X are the bottlenecks. It turns out that in this scenario, we’re well suited for a distributed computation. First, we can store the data points or rows of X across several machines, thus reducing the storage burden. Second, we can compute X transpose X in parallel across the machines by treating this multiplication as a sum of outer products. To understand this alternative interpretation of matrix multiplication in terms of outer products, let’s first recall our typical definition of matrix multiplication. We usually think about each entry of the output matrix being computed via an inner product between rows and columns of the input matrices. So, for instance, in the example on the slide, to compute the top left entry, we compute the inner product between the first row of the left input matrix and the first column of the right input matrix. Similarly, to compute the top right entry, we compute the inner product between the first row of the left input matrix and the second column of the right input matrix. We perform additional inner products to compute the two remaining entries of the output matrix. There is, however, an alternative interpretation of matrix multiplication as the sum of outer products between corresponding rows and columns of the input matrices. Let’s look at the same example from the last slide to get a better sense of what this means. First consider the first column of the left input matrix and the first row of the right input matrix. We can compute their outer product with the result being the 2 by 2 matrix on the bottom of the slide. Next, we can consider the second column of the left input matrix and the second row of the right input matrix, and again compute their outer product, resulting in another 2 by 2 matrix. We can repeat this process a third time to generate a third outer product or 2 by 2 matrix. The sum of these outer products matches the result we obtained in the previous slide using the traditional definition of matrix multiplication. And more generally, taking a sum of outer products of corresponding rows and columns of the input matrices, always returns the desired matrix multiplication result. Now we can use this new interpretation of matrix multiplication to our benefit when distributing the computation of X transpose X for linear regression. Let’s first represent X visually by its rows or data points. Then we can express this matrix multiplication as a sum of outer products where each outer product involves only a single row of X or a single data point. Let’s see how we can use this insight to effectively distribute our computation. Consider a toy example where we have a cluster of three workers and a data set with six data points. We can distribute the storage of our six data points across the three workers so that each worker is storing two data points. Now we can express matrix multiplication as a simple MapReduce operation. In the map step, we take each point and compute its outer product with itself. And in the subsequent reduce step, we simply sum over all of these outer products. We can then solve for the final linear regression model locally, which includes computing the inverse of this resulting matrix. Now let’s look at the storage and computation involved at each step. In the first step, we’re not doing any computation, but we need to store the input data, which requires O of nd storage. This is a bottleneck in our setting since n is large. However, the storage can be distributed over several machines. Next, during the map step, we perform an outer product for each data point. Each outer product takes O of d squared time, and we have to compute n of these outer products. This is the computational bottleneck in our setting, but again, it is distributed across multiple workers. In terms of storage, we must store the outer products computed on each machine. Note that although we may be computing several outer products per machine, we can keep a running sum of these outer products, so the local storage required for each machine is O of d squared. Finally, in the reduce step, we must take the sum of these outer products, though the computational bottleneck is, in fact, inverting the resulting matrix, which is cubic nd. However, we’re assuming that d is small enough for this computation to be feasible on a single machine. Similarly, the O of d squared storage required to store X transpose X and its inverse is also feasible on a single machine by assumption. This entire process can be concisely summarized via the following Spark code snippet. In this code, train data is an RDD of rows of X. In the map step, we compute an outer product for each row. And in the reduce step, we sum these outer products and invert the resulting matrix. In the final reduce step, we can also perform the remaining steps required to obtain our final regression model.
Distributed Machine Learning: Computation and Storage (Part 2) by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning
(watch now/later? 4:02):
This is a transcript of Ameet’s lecture in the video above:
Homework: Make sure you are understanding everything he is trying to convey!
In this segment, we’ll continue our discussion of distributed machine learning principles related to computation and storage. We’ll focus on the problem when D, the number of features, grows large. In the previous segment, we discussed the big N small D setting. In this setting, we can naturally use a distributed computing environment to solve for the linear regression closed form solution. To do this, we store our data across multiple machines and we compute X transpose X as a sum of outer products. This strategy can be written as a simple MapReduce operation, expressed very concisely in Spark. Now, let’s consider what happens when D grows large. As before, storing X and computing X transpose X are bottlenecks. However, storing and operating on X transpose X is now also a bottleneck. And we can no longer use our previous strategy. So let’s see what goes wrong. Here’s what our strategy looks like in the small D setting with data stored across workers, outer products computed in the map step, and sum of these outer products performed in the reduced step. However, we can no longer perform D cubed operations locally or store D squared floats locally in our new setting. This issue leads to a more general rule of thumb, which is that when N and D are large, we need the computation and storage complexity to be at most linear in N and D. So how do we devise methods that are linear in space and time complexity? One idea is to exploit sparsity. Sparse data is quite prevalent in practice. Some data is inherently sparse, such as rating information and collaborative filtering problems or social networking or other grafted. Additionally, we often generate sparse features during a process of feature extraction, such as when we represent text documents via a bag-of-words features or when we convert categorical features into numerical representations. Accounting for sparsity can lead to orders of magnitudes of savings in terms of storage and computation. A second idea is to make a late and sparsity assumption, whereby we make the assumption that our high dimensional data can in fact be represented in a more succinct fashion, either exactly or approximately. For example, we can make a low rank modeling assumption where we might assume that our data matrix can in fact be represented by the product of two skinny matrices, where the skinny dimension R is much smaller than either N or D. Exploiting this assumption can also yield significant computational and storage gigs. A third option is to use different algorithms. For instance, instead of learning a linear regression model via the closed form solution, we could alternatively use gradient descent. Gradient descent is an iterative algorithm that requires layer computation and storage at each iteration thus making it attractive in the big N and big D setting. So let’s see how gradient descent stacks up with a closed form solution in our toy example on a cluster with three machines. As before, we can store the data across the worker machines. Now in the map step, we require O of ND computation, and this computation is distributed across workers. And we also require O of D storage locally. In the reduced step, we require O of D local computation as well as O of D local storage. Moreover, unlike the closed form case, we need to repeat this process several times since gradient descent is an iterative algorithm. At this point, I haven’t really told you how these question marks work. And in the next segment, we’ll talk about what actually is going on with gradient decent.
Communication Hierarchy by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning
(watch later 2:32):
SUMMARY: Access rates fall sharply with distance.
- roughly 50 x gap between reading from memory and reading from either disk or the network.
We must take this communication hierarchy into consideration when developing parallel and distributed algorithms.
Distributed Machine Learning: Communication Principles by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning
(watch later 11:28):
Focusing on strategies to reduce communication costs.
- access rates fall sharply with distance.
- so this communication hierarchy needs to be accounted for when developing parallel and distributed algorithms.
- parallelism makes our computation faster
but network communication slows us down
- BINGO: perform parallel and in-memory computation.
Persisting in memory is a particularly attractive option when working with iterative algorithms that read the same data multiple times, as is the case in gradient descent.
Several machine learning algorithms are iterative!
- Limits of multi-core scaling (powerful multicore machine with several CPUs,
and a huge amount of RAM).
- sidestep any network communication when working with a single multicore machine
- can indeed handle fairly large data sets, and they’re an attractive option in many settings.
- can be quite expensive (due to specialized hardware),
- not as widely accessible as commodity computing nodes.
- this approach does have scalability limitations, as we’ll eventually hit a wall when the data grows large enough! This is not the case for a distributed environment (like the AWS EC2 cloud under the hood here).
Simple strategies for algorithms in a distributed setting: to reduce network communication, simply keep large objects local
- In the big n, small d case for linear regression
- we can solve the problem via a closed form solution.
- And this requires us to communicate \(O(d)^2\) intermediate data.
- the largest object in this example is our initial data, which we store in a distributed fashion and never communicate! This is a data parallel setting.
- In the big n, big d case:
- for linear regression.
- we use gradient descent to iteratively train our model and are again in a data parallel setting.
- At each iteration we communicate the current parameter vector \(w_i\) and the required \(O(d)\) communication is feasible even for fairly large d.
- for linear regression.
- In the small n, small d case:
- for ridge regression
- we can communicate the small data to all of the workers.
- this is an example of a model parallel setting where we can train the model for each hyper-parameter in parallel.
- for ridge regression
- Linear regression with big n and huge d is an example of both data and model parallelism.
HOMEWORK: Watch the video and find out why Linear regression with big n and huge d is an example of both data and model parallelism.
In this setting, since our data is large, we must still store it across multiple machines. We can still use gradient descent, or stochastic variants of gradient descent to train our model, but we may not want to communicate the entire d dimensional parameter vector at each iteration, when we have 10s, or hundreds of millions of features. In this setting we often rely on sparsity to reduce the communication. So far we discussed how we can reduce communication by keeping large data local.
Simple strategies for algorithms in a distributed setting: compute more and communicate less per iteration
HOMEWORK: watch the video and understand why it is important at each iteration of an iterative algorithm to compute more and communicate less.
Recall from week 1’s lecture that the ideal mathematical preparation to fully digest this material requires a set of self-tutorials from Reza Zadeh’s course in Distributed Algorithms and Optimization from Stanford:
This is a minimal pre-requisite for designing new algorithms or improving exixting ones!!!