Noise Reduction Methods for Large-Scale Machine Learning

I have two posts remaining in my series on “Optimization Methods for Large-Scale Machine Learning” by Bottou, Curtis, and Nocedal. You can find the entire series here. These last two posts will discuss improvements on the base stochastic gradient method. Below I have reproduced Figure 3.3, which suggests two general approaches. I will cover noise reduction in this post, and second-order methods in the next.

SGDirections

The left-to-right direction on the diagram signifies noise reduction techniques. We say that the SG search direction is “noisy” because it includes information from only one (randomly generated) sample per iteration. We use a noisy direction, of course, because it’s too expensive to use the entire gradient. But we can consider using a small batch of samples per iteration (a “minibatch”), or using information from previous iterations. The idea here is to find a happy medium between the far left of the diagram, which represents one sample per iteration, and the right, which represents using the full gradient.

Section 5 describes several noise reduction techniques. Dynamic sample size methods vary the number of samples in a minibatch per iteration, for example by increasing the batch size geometrically with the iteration count. Gradient aggregation, as the name suggests, involves the use of gradient information from past iterations. The SVRG method involves starting with a full batch gradient, then for subsequent iterations updating the gradient using gradient information at a single sample. The SAGA method involves “taking the average of stochastic gradients evaluated at previous iterates”. Finally, iterate averaging methods use the iterates from multiple previous steps to update the current iterate.

The motivations behind these various noise reduction methods are more or less the same: make more progress on a single step without paying too much of a computational cost. The primary tradeoff, in addition to increased computational cost per iteration, of course, is the extra storage associated with keeping extra state to compute search direction. Section 5 of the paper discusses these tradeoffs in light of convergence criteria.

In our final post, we will move to second-order methods, which involve looking at the curvature of the objective function to compute search directions that will lead to optimal solutions in fewer iterations.

Analyses of Stochastic Gradient Methods

I am continuing my series on “Optimization Methods for Large-Scale Machine Learning” by Bottou, Curtis, and Nocedal. You can find the entire series here.

Last time we discussed stochastic and batch methods for optimization in machine learning. In both cases we’re trying to optimize a loss function that will give us good learning parameters for a deep learning model. We do this iteratively. A pure stochastic gradient (SG) approach picks one sample per iteration to define a search direction, whereas a batch method will pick multiple samples.

In this post I want to cover most of Section 4, which concerns the convergence and complexity of SG methods. For completeness I have reproduced the authors’ summary of a general purpose SG algorithm below:

Choose an iterate w_1
for k = 1, 2, … do
   generate a random number s_k
   compute a stochastic vector g(w_k, s_k)
   choose a stepsize a_k > 0.
   w_k+1 <- w_k – a_k g(w_k, s_k)

In this algorithm, g is the search direction. In this series, we’ve already discussed three different choices for g:

  • g is the gradient. This is conventional gradient descent. In this case the random number is ignored.
  • g is the gradient for a single sample. This is conventional stochastic gradient.
  • g is the gradient over a subset of the samples. This is “mini-batch” SG.

As the authors show, it’s not easy to make definitive statements on comparison between mini batch and SG. The gist of what they show is that SG has better convergence properties in theory, but batch methods can provide certain practical advantages. That is: “one can, however, realize benefits of mini-batching in practice since it offers important opportunities for software optimization and parallelization; e.g., using sizeable mini-batches is often the only way to fully leverage a GPU processor.” The rest of Section 4 spells this out in more detail through convergence results and complexity analysis.

As a practitioner, I honestly don’t place huge weight on the convergence theorems presented in Section 4. The key result for me is Theorem 4.9, which states that for a nonconvex objective (which we have in most deep learning scenarios) the SG method converges when the stepsize diminishes according to a somewhat loosely defined schedule.

More interesting (for me) is Section 4.4, which discusses the overall work complexity for applying SG to deep learning scenarios. In many real-world big data scenarios, we’re optimizing a loss function using previously trained model parameters under a particular computational time limit. This is different from many traditional optimization scenarios, where we let the code run until we achieve a particular solution accuracy. In our situation, the total expected error in the model is the sum of three components: the expected risk using optimal parameters, the estimated error in the expected and empirical risk, and the optimization accuracy. Minimizing this error involves tradeoffs, for example “if one decides to make the optimization more accurate … one might need to make up for the additional computing time by: (i) reducing the sample size n, potentially increasing the estimation error; or (ii) simplifying the function family H, potentially increasing the approximation error.” These are familiar techniques for machine learning practitioners; the benefit provided here is a more formal characterization of how these techniques impact the overall solution error.

Returning to the choice of conventional versus batch approaches, the discussion in Section 4.5 shows that “Even though a batch approach possesses a better dependency on epsilon, this advantage does not make up for its dependence on n. […] In conclusion, we have found that a stochastic optimization algorithm performs better in terms of expected error, and, hence, makes a better learning algorithm in the sense considered here.” Again, even this statement could be mitigated somewhat by the computational benefits associated with batching on a particular computational infrastructure, that is, the benefits of GPUs and parallelism.

Section 4.5 is a commentary on some of the remaining challenges and questions that must be confronted when using SG for large-scale machine learning. Since the issues raised in Section 4.5 pair nicely with the discussion in Section 5, I’ll save both for next time.

Stochastic and Batch Methods for Machine Learning

I am continuing my series on “Optimization Methods for Large-Scale Machine Learning” by Bottou, Curtis, and Nocedal. You can find the entire series here.

In previous posts we discussed the use of deep neural networks (DNNs) in machine learning, and the pivotal role of the optimization of carefully selected prediction functions in training such models. These topics roughly correspond to the first two sections of the paper.

Section 3 provides an overview of optimization methods appropriate for DNNs. We begin where we left off in Section 2 by assuming a family of prediction functions parameterized by w. In other words, w represents the learning parameters. We also assume a loss function that depends on predicted and actual values, for example misclassification rate. Typically we’re minimizing the loss function f over a set of samples – the empirical risk. We call this R_n(w), which in turn is the average of the loss for each sample: (1/n) Ʃ_i∇f_i(). Given all of this, there are two fundamental paths for minimizing risk. In each case we iteratively improve the learning parameters step by step, where we write the parameters at step k as w_k.

  • stochastic: w_k+1 ← w_k − α_k ∇f_ik (w_k) where the index ik is chosen randomly.
  • batch: w_k+1 ← w_k − (α_k / n) Ʃ_i∇f_i(w_k)

The difference between the two paths is how many samples we consider on each step. The tradeoff is between per iteration cost (where stochastic wins) versus per iteration improvement (where batch wins). Stochastic algorithms a good choice for DNNs because they employ information more efficiently. If training samples are the same, or similar, than the added value of per step improvement from batch is not going to be worth it, because some (or most) of the added value turns out to be redundant. “If one believes that working with only, say, half of the data in the training set is sufficient to make good predictions on unseen data, then one may argue against working with the entire training set in every optimization iteration. Repeating this argument, working with only a quarter of the training set may be useful at the start, or even with only an eighth of the data, and so on. In this manner, we arrive at motivation for the idea that working with small samples, at least initially, can be quite appealing.”

The authors summarize: “there are intuitive, practical, and theoretical arguments in favor of stochastic over batch approaches in optimization methods for large-scale machine learning”. I will omit a summary of the theoretical motivations, but they are found in Section 3.3.

Next the authors consider improving on SG. One path involves trying to realize the best of both worlds between batch and stochastic methods, namely preserving the low per-iteration cost of stochastic methods while improving the per-iteration improvement. Another path is to “attempt to overcome the adverse effects of high nonlinearity and ill-conditioning”.  This involves trying to employ information beyond just the gradient. We’ll examine these alternatives in future posts in this series, when we get to later sections in the paper.

Model Building for Large-Scale Machine Learning

In this post on my series on “Optimization Methods for Large-Scale Machine Learning” by Bottou, Curtis, and Nocedal, I want to focus on model building in machine learning.

Section 2 of the paper describes several case studies, with the purpose of showing how “the process of machine learning leads to the selection of a prediction function through solving an optimization problem.” A prediction function is a mathematical function that links the model inputs to the quantity we wish to predict. From the practitioner’s point of view, a prediction function is implicitly specified by the technique the data scientist has chosen (for example, regression or neural networks) and trained model parameters (what is actually learned when the technique is applied to data).

For example, the structure of a neural network amounts to a description of a family of related functions. In the diagram below I have given two simple neural networks with corresponding prediction functions. The first simply adds the two inputs together. The second specifies a linear function involving a vector of inputs and training parameters W and b.

NeuralNetwork

Training the neural network amounts to choosing a particular function from the family corresponding to the nodes. Neural networks are interesting because they yield “large-scale, highly nonlinear, and nonconvex optimization problems”. For optimization practitioners, the “nonconvex” part of this statement is important because nonconvex optimization problems are particularly challenging. Here is a snippet from Stephen Boyd’s Convex Optimization I class that makes the point well.

With this in mind we may be tempted to avoid neural networks, and deep learning, altogether. However, as section 2.2 points out, certain classification tasks, like those involving speech and images, are “not well performed in an automated manner using computer programs based on sets of prescribed rules.” Deep neural networks (DNNs) involve many internal layers of manipulations and transformations, which lead to very flexible, highly parameterized models. Therefore while the corresponding optimization models for DNN are really damn hard, the potential payoff is worth it.

When a machine learning application is trying to classify data, for example in handwriting recognition, it is typical to minimize a function that relates to the misclassification rate. There are various choices for the specific function, as noted here and here (for empirical risk minimization). While we want to minimize a loss function relating to the misclassification rate, we also want classifiers that are general. In other words, if they work great on the data that we have at the time the classifier is learned, but poorly on data that comes in after that, our classifier is not very useful. For this we often divide our data into training, validation, and testing sets. Read here for more.

Section 2.3 considers the determination of a prediction function that accurately predicts outputs given inputs. We want this function to work well over the set of inputs that we will see in the real world, not just the training set. Therefore “one should choose the prediction function h by attempting to minimize a risk measure over an adequately selected family of prediction functions”. A family of functions can be described in many ways, for example as a particular functional form with parameters in it as in m x + b for parameters (m, b). Adequately selected means:

  • Able to achieve low empirical risk by choosing a rich family of functions or by using knowledge about the problem domain.
  • The gap between expected and empirical risk should be small, that is, they should not be biased towards or underfit the input data.
  • Chosen so the resulting optimization problem can be solved efficiently.

These considerations are at odds with one another as some point towards broader, more complicated families of functions and others simpler. With regards to the first consideration, increasing the number of training samples is helpful. So is choosing a function family with a high “capacity”, which can be loosely described as a function’s “complexity, expressive power, richness, or flexibility.

Having considered what makes a good prediction function, the authors next consider procedures for finding them. The approach considered in Section 2.3 is called structural risk minimization – here is a good overview. A nice visual representation is given in Figure 2.5, but the point is to avoid both underfitting and overfitting. Underfitting happens which happens when the observed empirical risk (the frequency of observed misclassification) is high. This happens when the prediction function is insufficiently expressive to link inputs to outputs, which can happen when the network structure doesn’t make sense or is too simplistic. Overfitting happens when increasing the number or complexity of the model parameters begins to increase the misclassification rate in real-world data. This can happen even as the misclassification rate on our training data decreases. In other words, the model no longer effectively generalizes to the real world – it is too highly tuned to the data at hand. All of this implies that picking functional families that give good empirical risk may be counterproductive. The remedy to this situation is to split input data into training, validation, and testing sets, as alluded to in the first post in this series.

In the next post in this series, we’ll cover Section 3, which describes the optimization methods used to train these models.

Optimization Methods for Large-Scale Machine Learning

Hey, so I mostly read a 93 page paper. The topic is a worthy one: optimization methods for large-scale machine learning. Deep learning powers best in class speech, image, and text intelligence on the web today, and deep learning is in turn powered by optimization. I will summarize “Optimization Methods for Large-Scale Machine Learning” by Bottou, Curtis, and Nocedal over the next few posts because it provides a useful operations research-centered evaluation of an important area in machine learning. In general, machine learning practitioners don’t know shit about operations research, and vice versa. This paper, along with work of Stephen Wright at Wisconsin (check out this talk), will certainly help to remedy this situation. I also predict that this paper will spur new advances in deep learning.

Here goes, and remember that I’m trying to summarize 93 pages!

The title of the paper is quite broad, but the focus is primarily on the use of the stochastic gradient descent (SGD) method (and variants) in deep learning applications. If you don’t have any previous experience with these topics, this series may not be for you, but I will try to summarize anyway. The term “deep learning” describes a range of machine learning algorithms that are used to classify or predict. Deep learning is primarily distinguished by:

  1. The use of much more input data than is typical for machine learning,
  2. Models that have many internal layers of data manipulation and transformation,
  3. A reliance on parallel and GPU processing.

Training a deep learning algorithm involves finding model parameters that produce effective predictors or classifiers. Finding the values of variables that produce the best results for a particular objective (or “goal”) is the job of optimization. The stochastic gradient descent method is so-named because it repeatedly takes steps in the direction of steepest descent, which is defined by the gradient of the objective we want to optimize. If we think of the objective function as a hilly field, then the gradient always points in the steepest direction down, when we examine the immediate area around where we stand. The “stochastic” part of SGD applies because rather than looking at all of the samples over which the objective function is defined, we only look one (or a few) randomly determined sample. As compared to using the full gradient, this approach takes less time to take a single step, but the step is possibly less effective in improving the value of our objective function. In theory and practice we can establish that often the tradeoff is worth it. Characterizing these tradeoffs more concretely is one of the objectives of the paper. As as supplement to the paper and this post, check out this great post by Sebastian Ruder for an overview of gradient descent algorithms for machine learning.

In my next post in this series, I will cover Section 2 which describes the selection of a prediction function that is useful for modeling but practical for model training at scale.

Updated 8/2/2016 to correctly summarize SGD. Thanks J-F!

Notes on Mesos: A Platform for Fine-Grained Resource Sharing in the Data Center

Here are my notes on the influential paper “Mesos: A Platform for Fine-Grained Resource Sharing in the Data Center”. My notes pertain only to the original the paper itself, and not improvements or changes in the theory or implementation of Apache Mesos since 2010.

Mesos is “a platform for sharing commodity clusters between multiple diverse cluster computing frameworks”. A framework is a “software system that manages and executes one or more jobs on a cluster”, for example Hadoop, Spark, or MPI. Frameworks are responsible for running tasks, for example running a machine learning algorithm. When multiple frameworks run on a cluster without a platform like Mesos, there are often unintended consequences, for example one framework may grab resources for a job that would gave been better suited for another framework’s job.

Multiple frameworks often run on a single cluster because different frameworks are best suited for different kinds of computational workloads. For example, Spark for iterative workloads on shared data, or Flink for streaming workloads. Mesos shares cluster resources across frameworks with the goals of high utilization and efficient data sharing. Cluster resources can be shared without a framework, for example by simply partitioning the nodes in cluster to frameworks, or by allocating virtual machines to each framework, but utilization and efficiency suffer.

Determining how to share cluster resources between frameworks is especially challenging because individual frameworks already manage resources themselves. Cluster managers must either work with, augment, or replace these framework capabilities. For example, Hadoop’s Fair Scheduler assigns cluster nodes to jobs so that all jobs “get, on average, an equal share of resources”. Mesos does not seek to replace framework schedulers; it seeks to harmonize them so that total cluster utilization and efficiency is maximized, even though framework schedulers are unaware of each other’s existence. Mesos does this in a non-intrusive way by adopting a two phase approach:

  1. Mesos decides how many resources to offer each framework,
  2. Frameworks decide which resources to accept and which tasks should run on them (using their own scheduler).

There are several advantages to this approach:

  • Frameworks can continue to use their own schedulers.
  • Mesos can accommodate newly developed frameworks.
  • The Mesos implementation itself can be kept simple (since concerns are separated).
  • Mesos is scalable, because Mesos does not attempt to compute a global schedule for all tasks across all frameworks.

The primary disadvantage is that Mesos is denied the ability to globally optimize task allocation across frameworks.

Figures 2 and 3 in the paper are useful visual depictions of the Mesos offer process. Here is a simplified architectural diagram:

Mesos

There are two components to the Mesos architecture: masters and workers. Masters are responsible for issuing offers to workers and interacting with workers and framework schedulers. Workers are responsible for running tasks on cluster resources.

The two phase approach for task scheduling and execution is summarized in Figure 3 in the original paper. The process begins with workers reporting available resources. You can think of these “reports” as tuples (w_i, r_1, r_2, …r_n) where w_i identifies the worker, and r_1, …, r_n represent resource attributes. For example, r_1 may represent the number of CPUs, r_2 may represent memory, r_3 the presence or absence of a GPU, and so on. Armed with the knowledge of the capabilities of the cluster, the master can begin issuing offers to framework schedulers. An offer is also a tuple (w_i, r_1, …, r_n) – it’s a record that represents resources that a scheduler can choose to use. At this point, the framework scheduler can choose to either accept or reject the offer. Frameworks decide to accept or reject based on the pending list of tasks that need to be executed by the framework. There are legitimate reasons for rejecting offers even if tasks are pending; for example pending tasks may require a GPU but the offer does not include one. When an offer is accepted, the framework scheduler sends back a list of tuples (t_i, u_1, …, u_n), with t_i identifying a task to be executed, and u_i representing the resources that will be utilized by the task when it is executed. The master can then send the tasks to workers for execution. It also “adjusts the books” so that future resource offers will account for the running tasks. When tasks are completed, the master is notified so that it can then account for these newly available resources.

It might be helpful to compare this process to home mortgages. In this world, Mesos plays the role of a mortgage broker. A Mesos offer represents the terms of a mortgage, offered to lenders (schedulers). An approval constitutes an agreement by a lender to fund the loan.

As the paper notes, the ability for frameworks to reject offers is an important extensibility point that allows for frameworks to account for its own considerations, without burdening Mesos with the details.

The process of brokering offers and launching tasks is the heart of Mesos. There are a number of important additional considerations, of course: how to handle long running or “zombie” tasks, task isolation, robustness, and fault tolerance. Mesos relies on existing framework or cluster node mechanisms to handle these considerations when possible, and adds simple policies to Mesos itself when this is not possible. This all falls under the general design principle of keeping Mesos simple. These mechanisms are described in Section 3 of the paper. The details are interesting but are not fundamental to understanding the design.

As noted earlier, Mesos takes a decentralized approach: offers are made to frameworks, and the frameworks schedule accepted offers. Frameworks are (implicitly) incented by Mesos to adopt certain policies in order to improve throughput. These incentives are given in Section 4.4:

  • Uses short tasks,
  • Uses resources as soon as they are allocated,
  • Ability to scale down,
  • Does not accept unknown resources.

Frameworks that follow these guidelines yield high utilization when managed by Mesos.

Mesos does not claim to be the only viable solution for cluster resource management. For example, in a traditional HPC-style cluster environment with specialized, largely homogeneous hardware and fixed-size jobs, centralized scheduling may be more appropriate. In a grid computing environment where geographically separate and separately administered resources are marshaled together for a computation (like me and my colleagues did for the famed “nug30” problem back in 2000), additional layers may need to sit on top of a framework such as Mesos.

Nonetheless for many modern cluster workloads, especially those for large scale machine learning, Mesos is an excellent choice.

Finding Optimal State Capitol Tours on the Cloud with NEOS

My last article showed you how to find an optimal tour of all 48 continental US state capitols using operations research. I used the Python API of the popular Gurobi solver to create and solve a traveling salesman problem (TSP) model in a few seconds.

In this post I want to show you how to use Concorde, the world’s best TSP solver for free on the cloud using the NEOS optimization service. In less than 100 lines of Python code, you can find the best tour. Here it is:

TSP_Tour48_Bokeh

Using NEOS is pretty easy. You need to do three things to solve an optimization problem:

  1. Create a NEOS account.
  2. Create an input file for the problem you want to solve.
  3. Give the input file to NEOS, either through their web interface, or by calling an API.

Let’s walk through those steps for the state capitol problem. If you just want to skip to the punchline, here is my code.

Concorde requires a problem specification in the TSPLIB format. This is a text based format where we specify the distances between all pairs of cities. Recall that Randy Olson found the distances between all state capitols using the Google Maps API in this post. Here is a file with this information. Using the distances, I created a TSPLIB input file with the distance matrix – here it is.

The next step is to submit the file to NEOS. Using the xmlrpc Python module, I wrote a simple wrapper to submit TSPLIB files to NEOS. The NEOS submission is an XML file that wraps the contents of the TSPLIB data, and also tells NEOS that we want to use the Concorde solver. The XML file is given to NEOS via an XML-RPC call. NEOS returns the results as a string – the end of the string contains the optimal tour. Here is the body of the primary Python function that carries out these steps:

def solve_tsp_neos_concorde(dist):
xml = make_neos_concorde(dist)
neos = NeosClient()
result = neos.run(xml)
return tour_from_neos_concorde_result(result)

When I run this code, I obtain the same tour as in my initial post. Hooray! You can also extend my code (which is based on NEOS documentation) to solve many other kinds of optimization models.