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Cyclades

This repository contains code for Cyclades, a general framework for parallelizing stochastic optimization algorithms in a shared memory setting. See https://arxiv.org/abs/1605.09721 for more information.

Here we implement SGD, SVRG and SAGA for sparse stochastic gradient descent methods applied to problems including matrix completion, graph eigenvalues, word embeddings and least squares.

Overview

Cyclades is a general framework for parallelizing stochastic optimization algorithms in a shared memory setting. By partitioning the conflict graph of datapoints into batches of non-conflicting updates, serializability can be maintained under execution of multiple cores.

Cyclades carefully samples updates, then finds conflict-groups, and allocates them across cores. Then, each core asynchronously updates the shared model, without incurring any read/write conflicts. This is possible by processing all the conflicting updates within the same core. After the processing of a batch is completed, the above is repeated, for as many iterations as required.

Experiments

Maintaining serializability confers numerous benefits, and the additional overhead of partitioning the conflict graph does not hinder performance too much. In fact, in some cases the avoidance of conflicts and the slightly better cache behavior of Cyclades leads to better performance.

Cyclades initially starts slower than Hogwild due to the overhead of partitioning the conflict graph. But by having better locality and avoiding conflicts Cyclades ends up slightly faster in terms of running time. In the plots both training methods were run for the same number of epochs, with the same learning rate. Note this graph was generated using the "custom" updater to optimize for performance.

Additionally, for various variance reduction algorithms we find that Cyclades' serial equivalance allows it to outperform Hogwild in terms of convergence.

On multithread SAGA, the serializability of Cyclades allows it to use a larger stepsize than Hogwild. With higher stepsizes, Hogwild diverges due to conflicts.

On multithread SVRG, by avoiding conflicts Cyclades achieves a lower objective loss value much faster than Hogwild.

For full experiment details please refer to the paper.

Building

Note that compilation requires git, make, cmake, OpenMP, and wget. Additionally, on a Mac, Xcode Command Line Tools and ClangOMP++ are required.

After cloning the repository, cd into the project directory

cd cyclades

Fetch the gflags submodule with

git submodule init && git submodule update

After the submodule fetches use cmake to generate a build file.

On Linux do

cmake .

While on Mac OS X do

cmake -DCMAKE_CXX_COMPILER=clang-omp++ .

Then make to compile

make

Fetching data

To fetch all experiment data, from the project home directory, run

cd data && sh fetch_all_data.sh && cd ..

Running

After compilation, a single executable called cyclades will be built. There are numerous flags that control the specifics of execution, such as learning rate, training type, number of epochs to run, etc.

To see a list of flags that can be set, run

./cyclades --help

A quick example to run after compiling and fetching the data is (run from the home directory)

./cyclades   --print_loss_per_epoch  --print_partition_time  --n_threads=2 --learning_rate=1e-2  -matrix_completion  -cyclades_trainer  -cyclades_batch_size=800 -n_epochs=20 -sparse_sgd --data_file="data/movielens/ml-1m/movielens_1m.data"

Guide On Writing Custom Models

Writing a model that can be optimized using Hogwild and Cyclades is straightforward. The two main classes that need to be overridden by the user are the Model and Datapoint classes, which capture all necessary information required for optimization.

The Model class is a wrapper around the user-defined model data, specifying methods that operate on the model (such as computing gradients and loss). The Datapoint class is a wrapper around the individual data elements used to train the model.

After defining the Datapoint and Model subclasses, the user can run Cyclades/Hogwild by including "run.h" and calling Run<CustomModel, CustomDatapoint>().

Data File Reading / Data File Format

The data file specified by the --data_file flag should contain information to initialize the model, as well as the individual data points that are used for training.

The first line of the data file is fed to the constructor of the model, and each subsequent line is used to instantiate separate instances of the Datapoint class.

For example, suppose we are writing the custom model class MyCustomModel and the custom data point class MyCustomDatapoint and the data file contains

1 2
1
2
3
4
5

This would result in the model being instantiated as MyCustomModel("1 2") and the creation of five separate instances of the data point class: MyCustomDatapoint("1"), MyCustomDatapoint("2"), ... , MyCustomDatapoint("5"). Note that the inputs are strings.

The MyCustomDatapoint(const std::string &input_line) and MyCustomModel(const std::string &input_line) constructors can then be defined by the user to specify how to initialize the objects using the given data file inputs. For example the model data input may specify the dimension of the model, and the constructor may use this information to pre-allocate enough memory to hold it.

It is important to note that the user must manage the underlying data behind their custom model / datapoint classes. For the model, the underlying raw model data should be captured by a std::vector<double>.

Defining the Datapoint Subclass

The following virtual methods of Datapoint are required to be overridden.

Datapoint(const std::string &input_line, int order)

The constructor for the subclass of Datapoint. The order argument should be passed in to the superclass constructor call. For example: CustomDatapoint(const std::string &input_line, int order) : Datapoint(input_line, order) { ... }

Args:
  • input_line - The line of input from the data file.
  • order - The order in which this data point appears in the shuffled permutation of data points.

virtual std::vector<double> & GetWeights()

Return a vector of weights where the i'th weight in the returned vector corresponds to the i'th coordinate of GetCoordinates().

virtual std::vector<int> & GetCoordinates()

Return a vector of coordinates where the i'th coordinate of the returned vector corresponds to the i'th weight of GetWeights().

Defining the Model Subclass

The following virtual methods of Model are required to be overridden.

Model(const std::string &input_line)

The constructor for the subclass of Model.

Args:
  • input_line - first line of the data file.
virtual double ComputeLoss(const std::vector<Datapoint *> &datapoints)

Compute loss value given a list of data points.

Args:
  • datapoints - vector of pointers to the user defined data point structure. If the custom data point structure has methods not listed in the Model parent class you may need to cast the pointer to a pointer of the custom class.
virtual int NumParameters()

Return the number of coordinates of the model.

virtual int CoordinateSize()

Return the size of the coordinate vectors of the model. For scalar coordinates, return 1.

virtual std::vector<double> & ModelData()

Return a reference to the underlying data. ModelData().size() should be NumParameters() * CoordinateSize().

For the following gradient related methods, we formulate the gradient at a datapoint x, model coordinate j as [∇f(x)]_j = λ_j * x_j − κ_j + h_bar_j(x).

virtual void PrecomputeCoefficients(Datapoint *datapoint, Gradient *g, std::vector<double> &local_model)

Do any sort of precomputation (E.G: computing dot product) on a datapoint before calling methods for computing lambda, kappa and h_bar. Note that PrecomputeCoefficients is called by multiple threads.

Args:
  • datapoint - Data point to precompute gradient information.
  • g - Gradient object for storing any precomputed data. This is passed to the h_bar method afterwards. The relevant Gradient attribute is g->coeffs, a vector to store arbitrary data. Note that g->coeffs is initially size 0, so in PrecomputeCoefficients the user needs to resize this vector according to their needs. Gradient objects are thread local objects that are re-used. Thus, g->coeffs may contain junk precompute info from a previous iteration.
  • cur_model - a vector of doubles that contains the raw data of the model to precompute gradient information.
virtual void H_bar(int coordinate, std::vector<double> &out, Gradient *g, std::vector<double> &local_model)

Write to output h_bar_j of [∇f(x)]_j = λ_j * x_j − κ_j + h_bar_j(x). Note that this function is called by multiple threads.

Args:
  • coordinate - The model coordinate j for which h_bar_j should be computed.
  • out - Reference to vector to which the value of h_bar should be written to.
  • g - Gradient object which contains the precomputed data previously set by PrecomputeCoefficients. Further note that g->datapoint is a pointer to the data point whose gradient is being computed (which is the data point that was used by PrecomputeCoefficients to precompute gradient information).
  • local_model - The raw data of the model for which lambda should be computed for.
virtual void Lambda(int coordinate, double &out, std::vector<double> &local_model)

Write to output the λ_j coefficient of the gradient equation [∇f(x)]_j = λ_j * x_j − κ_j + h_bar_j(x). Note that this function is called by multiple threads.

Args:
  • coordinate - The model coordinate j for which λ_j should be computed.
  • out - Reference to scalar double to which the value of lambda should be written to.
  • local_model - The raw data of the model for which lambda should be computed for.
virtual void Kappa(int coordinate, std::vector<double> &out, std::vector<double> &local_model)

Write to output the κ_j coefficient of the gradient equation [∇f(x)]_j = λ_j * x_j − κ_j + h_bar_j(x). Note that Kappa is called by multiple threads.

Args:
  • coordinate - The model coordinate j for which κ_j should be computed.
  • out - Reference to vector to which the value of kappa should be written to.
  • local_model - The raw data of the model for which lambda should be computed for.

Example: Sparse Least Squares

Here we show how to define a custom model and datapoint class to solve the classic least squares problem.

In the least squares problem we are interested in minimizing the function ||Ax-b||^2. A data point in this sense is a row of A which we refer to as a_i, and the model we are optimizing is x. Note that minimizing ||Ax-b||^2 is equivalent to minimizing sum (dot(a_i, x) - b_i)^2.

For the purposes of this example, we will name our least squares model SimpleLSModel and our least squares datapoint SimpleLSDatapoint. The full source code is in the examples directory.

Starting off

To begin using Cyclades, we must include the "run.h" and "defines.h" files in the src directory. Furthermore, to make use of gflags, we must also initialize gflags.

#include "../src/run.h"
#include "../src/defines.h"

int main(int argc, char **argv) {
    // Initialize gflags
    gflags::ParseCommandLineFlags(&argc, &argv, true);
}

After implementing SimpleLSDatapoint and SimpleLSModel we can call

Run<SimpleLSModel, SimpleLSDatapoint>();

to solve the least squares problem with the passed in command line flags.

Data File Format / Data File Reading

To store the A matrix and b label vector, we will use the following format.

line 1 (input to model constructor) : the dimension of the x model vector
line 2...n (input to datapoint constructor) : m index_of_nnz_1 value_of_nnz_1 ... index_of_nnz_m value_of_nnz_m label

For the purposes of this example we will use the following data input:

10
1 0 1 1
1 1 1 2
1 2 1 3
1 3 1 4
1 4 1 5
1 5 1 6
1 6 1 7
1 7 1 8
1 8 1 9
1 9 1 10

This is equivalent to solving the problem with

A = 1 0 0 0 0 0 0 0 0 0       b = 1
    0 1 0 0 0 0 0 0 0 0           2
    0 0 1 0 0 0 0 0 0 0           3
    0 0 0 1 0 0 0 0 0 0           4
    0 0 0 0 1 0 0 0 0 0           5
    0 0 0 0 0 1 0 0 0 0           6
    0 0 0 0 0 0 1 0 0 0           7
    0 0 0 0 0 0 0 1 0 0           8
    0 0 0 0 0 0 0 0 1 0           9
    0 0 0 0 0 0 0 0 0 1           10

Defining SimpleLSDatapoint

First we subclass Datapoint, and keep a weights vector, a coordinates vector, and a label double.

class SimpleLSDatapoint : public Datapoint {
public:
    std::vector<double> weights;
    std::vector<int> coordinates;
    double label;
};

weights[i] will contain the value of the row of A at the index specified coordinates[i]. This is a sparse representation of a row of a matrix. label will be the corresponding b_i of the row specified by the data point.

Note that the Cyclades framework does not require a label variable, but it will be used by the model when computing gradients.

To read the datapoint values from a line of the input file, we implement the constructor to read according to the format.

SimpleLSDatapoint(const std::string &input_line, int order) : Datapoint(input_line, order) {
    // Create a string stream from the input line.
    // This lets us read from the line as if reading via cin.
    std::stringstream in(input_line);

    // We expect a sparse row from the A matrix, a_i, as well as the
    // corresponding value of b_i. The first value of the line is the number
    // of nnz values in the row of the matrix.
    // E.g:
    // n index_1 value_1 index_2 value_2 .... index_n value_n label
    int n;
    in >> n;
    weights.resize(n);
    coordinates.resize(n);
    for (int i = 0; i < n; i++) {
        in >> coordinates[i];
        in >> weights[i];
    }
    in >> label;
}

Finally, we fill in the required GetWeights, GetCoordinates() and GetNumCoordinateTouches() methods.

std::vector<double> & GetWeights() override {
    return weights;
}

std::vector<int> & GetCoordinates() override {
    return coordinates;
}

int GetNumCoordinateTouches() override {
    return coordinates.size();
}

Defining SimpleLSModel

First we subclass Model, and define x, the raw data containing the model we are trying to optimize.

class SimpleLSModel : public Model {
public:
    std::vector<double> x;
};

Initialization with the data input line from the constructor follows a similar process to that in SimpleLSDatapoint.

SimpleLSModel(const std::string &input_line) {
    // Create a string stream from the input line.
    // This lets us read from the line as if reading via cin.
    std::stringstream in(input_line);

    // We expect a single integer describing the number
    // of coordinates of the model.
    int num_coordinates;
    in >> num_coordinates;

    // Preallocate the x model and randomly initialize.
    x.resize(num_coordinates);
    for (int i = 0; i < x.size(); i++) {
        x[i] = ((double)rand()/(double)RAND_MAX);
    }
}

Here we additionally allocate enough memory to hold the model x and furthermore randomly initialize its values.

For this problem, it is also convenient to define a dot product operation which computes the dot product between a row specified by SimpleLSDatapoint and the raw model.

double dot(SimpleLSDatapoint *a_i, std::vector<double> &x) {
    double product = 0;
    for (int i = 0; i < a_i->GetNumCoordinateTouches(); i++) {
        int index = a_i->GetCoordinates()[i];
        double value = a_i->GetWeights()[i];
        product += value * x[index];
    }
    return product;
}

Computing the loss requires iterating through the data points and summing up (dot(a_i, x) - b_i)^2.

// Minimize loss = sum (a_i * x - b)^2.
double ComputeLoss(const std::vector<Datapoint *> &datapoints) override {
    double loss = 0;

    // Note that ComputeLoss is called by a SINGLE thread.
    // It is possible to parallelize this via
    // #pragma omp parallel for
    for (int i = 0; i < datapoints.size(); i++) {
        SimpleLSDatapoint *a_i = (SimpleLSDatapoint *)datapoints[i];
        double b_i = a_i->label;
        double dot_product = dot(a_i, x);
        loss += (dot_product - b_i) * (dot_product - b_i);
    }

    // Let's also print out the model to see if we are on the right track.
    std::cout << "Model Parameters: " << std::endl;
    for (int i = 0; i < NumParameters(); i++) {
        std::cout << x[i] << " ";
    }
    std::cout << std::endl;

    return loss / datapoints.size();
}

Since each coordinate of x is a scalar value, CoordinateSize should return 1.

int CoordinateSize() override {
    // Each coordinate in the model is a single scalar.
    return 1;
}

Additionally we indicate the size of the model, and allow the updaters a reference to the raw data.

int NumParameters() override {
    return x.size();
}

std::vector<double> &ModelData() override {
    return x;
}

To define Lambda, H_bar, Kappa, recall for least squares the gradient at a datapoint is

d/dx fx = d/dx (dot(a_i, x) - b_i)^2 = 2(dot(a_i, x) - b_i) a_i

Casting this in terms of

[∇f(x)]_j = λ_j * x_j − κ_j + h_bar_j(x)

indicates that λ_j = 0, x_j = 0 and h_bar_j(x) = 2(dot(a_i, x) - b_i) a_i.

void Lambda(int coordinate, double &out, std::vector<double> &local_model) override {
    out = 0;
}

void Kappa(int coordinate, std::vector<double> &out, std::vector<double> &local_model) override {
    // out.size() == local_model.size()
    out[0] = 0;
}

// h_bar_j(x) = 2(a_i * x - b_i) a_i
// We can just precompute the each h_bar_j directly.
void PrecomputeCoefficients(Datapoint *datapoint, Gradient *g, std::vector<double> &local_model) override {
    // We need to make sure g->coeffs can store the gradient to the model.
    if (g->coeffs.size() != 1) g->coeffs.resize(NumParameters());

    // Compute 2(a_i * x - b_i).
    SimpleLSDatapoint *a_i = (SimpleLSDatapoint *)datapoint;
    double b_i = a_i->label;
    double coefficient = 2 * (dot(a_i, local_model) - b_i);

    // For each nnz weight of the data point, set g->coeffs appropriately.
    for (int i = 0; i < datapoint->GetNumCoordinateTouches(); i++) {
        int index = datapoint->GetCoordinates()[i];
        double weight = datapoint->GetWeights()[i];
        g->coeffs[index] = coefficient * weight;
    }
}

// Since g->coeffs[coordinate] = 2(a_i * x - b_i) a_i, we can set out[0] to be g->coeffs[coordinate]
void H_bar(int coordinate, std::vector<double> &out, Gradient *g, std::vector<double> &local_model) override {
    out[0] = g->coeffs[coordinate];
}

Note that in this example, although we precompute the whole gradient directly in PrecomputeCoefficients, there are cases where it is necessary / more efficient to do part of the computation in PrecomputeCoefficients and the rest in H_bar.

Furthermore note that g->coeffs is not zeroed out between gradient computations, so it may be filled with junk value. This also means that the resizing of g->coeffs is only done once per thread.

Putting It All Together

Finally, we can add a call to Run in the main function to trigger optimization.

int main(int argc, char **argv) {
    // Initialize gflags
    gflags::ParseCommandLineFlags(&argc, &argv, true);

    Run<SimpleLSModel, SimpleLSDatapoint>();
}

Here is our final source code.

#include <iostream>
#include <vector>
#include "../src/run.h"
#include "../src/defines.h"

/* Minimize the equation sum (a_i x - b_i)^2
 * Each row of the A matrix (a_i) is represented by a single data point.
 * Also store b_i for each corresponding a_i.
 */
class SimpleLSDatapoint : public Datapoint {
public:
    std::vector<double> weights;
    std::vector<int> coordinates;
    double label;

    SimpleLSDatapoint(const std::string &input_line, int order) : Datapoint(input_line, order) {
        // Create a string stream from the input line.
        // This lets us read from the line as if reading via cin.
        std::stringstream in(input_line);

        // We expect a sparse row from the A matrix, a_i, as well as the
        // corresponding value of b_i. The first value of the line is the number
        // of nnz values in the row of the matrix.
        // E.g:
        // n index_1 value_1 index_2 value_2 .... index_n value_n label
        int n;
        in >> n;
        weights.resize(n);
        coordinates.resize(n);
        for (int i = 0; i < n; i++) {
            in >> coordinates[i];
            in >> weights[i];
        }
        in >> label;
    }

    std::vector<double> & GetWeights() override {
        return weights;
    }

    std::vector<int> & GetCoordinates() override {
        return coordinates;
    }

    int GetNumCoordinateTouches() override {
        return coordinates.size();
    }
};

/* Minimize the equation sum (a_i x - b_i)^2
 * Represents the x model.
 *
 * The gradient at a datapoint a_i is
 * d/dx f(x) = 2(a_i * x - b_i) x.
 * Therefore, in terms of [∇f(x)]_j = λ_j * x_j − κ_j + h_bar_j(x),
 * which defines the gradient at a datapoint at a coordinate j of the model,
 * we have λ_j = 0, x_j = 0, h_bar_j(x) = 2(a_i * x - b_i) a_i.
 *
 */
class SimpleLSModel : public Model {
private:
    double dot(SimpleLSDatapoint *a_i, std::vector<double> &x) {
        double product = 0;
        for (int i = 0; i < a_i->GetNumCoordinateTouches(); i++) {
            int index = a_i->GetCoordinates()[i];
            double value = a_i->GetWeights()[i];
            product += value * x[index];
        }
        return product;
    }

public:
    std::vector<double> x;

    SimpleLSModel(const std::string &input_line) {
        // Create a string stream from the input line.
        // This lets us read from the line as if reading via cin.
        std::stringstream in(input_line);

        // We expect a single integer describing the number
        // of coordinates of the model.
        int num_coordinates;
        in >> num_coordinates;

        // Preallocate the x model and randomly initialize.
        x.resize(num_coordinates);
        for (int i = 0; i < x.size(); i++) {
            x[i] = ((double)rand()/(double)RAND_MAX);
        }
    }

    // Minimize loss = sum (a_i * x - b)^2.
    double ComputeLoss(const std::vector<Datapoint *> &datapoints) override {
        double loss = 0;

        // Note that ComputeLoss is called by a SINGLE thread.
        // It is possible to parallelize this via
        // #pragma omp parallel for
        for (int i = 0; i < datapoints.size(); i++) {
            SimpleLSDatapoint *a_i = (SimpleLSDatapoint *)datapoints[i];
            double b_i = a_i->label;
            double dot_product = dot(a_i, x);
            loss += (dot_product - b_i) * (dot_product - b_i);
        }

        std::cout << "Model Parameters: " << std::endl;
        for (int i = 0; i < NumParameters(); i++) {
            std::cout << x[i] << " ";
        }
        std::cout << std::endl;

        return loss / datapoints.size();
    }

    int NumParameters() override {
        return x.size();
    }

    int CoordinateSize() override {
        // Each coordinate in the model is a single scalar.
        return 1;
    }

    std::vector<double> &ModelData() override {
        return x;
    }

    void Lambda(int coordinate, double &out, std::vector<double> &local_model) override {
        out = 0;
    }

    void Kappa(int coordinate, std::vector<double> &out, std::vector<double> &local_model) override {
        out[0] = 0;
    }

    // h_bar_j(x) = 2(a_i * x - b_i) a_i
    // We can just precompute the each h_bar_j directly.
    void PrecomputeCoefficients(Datapoint *datapoint, Gradient *g, std::vector<double> &local_model) override {
        // We need to make sure g->coeffs can store the gradient to the model.
        if (g->coeffs.size() != 1) g->coeffs.resize(NumParameters());

        // Compute 2(a_i * x - b_i).
        SimpleLSDatapoint *a_i = (SimpleLSDatapoint *)datapoint;
        double b_i = a_i->label;
        double coefficient = 2 * (dot(a_i, local_model) - b_i);

        // For each nnz weight of the data point, set g->coeffs appropriately.
        for (int i = 0; i < datapoint->GetNumCoordinateTouches(); i++) {
            int index = datapoint->GetCoordinates()[i];
            double weight = datapoint->GetWeights()[i];
            g->coeffs[index] = coefficient * weight;
        }
    }

    // Since g->coeffs[0] = 2(a_i * x - b_i),
    // The gradient is g->coeffs[0] * a_ij (i'th datapoint, j'th
    void H_bar(int coordinate, std::vector<double> &out, Gradient *g, std::vector<double> &local_model) override {
        out[0] = g->coeffs[coordinate];
    }
};

int main(int argc, char **argv) {
    std::cout << "Simple least squares custom optimization example:" << std::endl;
    std::cout << "Minimize the equation ||Ax-b||^2 = Minimize Sum (a_i x - b_i)^2" << std::endl;

    // Initialize gflags
    gflags::ParseCommandLineFlags(&argc, &argv, true);

    Run<SimpleLSModel, SimpleLSDatapoint>();
}

The github repository contains the simple_ls target which builds and compiles the above example. To compile, do

make simple_ls

To run do

./simple_ls --sparse_sgd --n_epochs=10  --learning_rate=1e-1 --print_loss_per_epoch --cyclades_trainer --n_threads=1 --data_file='./examples/simple_ls_data'