TNO PET Lab - Machine Learning - Cox Regression

Implementation of a Federated Learning scheme for Cox Regression. It is based on the library for Logistic Regression in the TNO PET lab. This library was designed to facilitate both developers that are new to federated learning and developers that are more familiar the technique.

Supports:

• Any number of clients and one server.
• Horizontal fragmentation
• Both fixed learning rate or second-order methods (Hessian)

This software implementation was financed via EUREKA ITEA labeling under Project reference number 20050.

PET Lab

The TNO PET Lab consists of generic software components, procedures, and functionalities developed and maintained on a regular basis to facilitate and aid in the development of PET solutions. The lab is a cross-project initiative allowing us to integrate and reuse previously developed PET functionalities to boost the development of new protocols and solutions.

The package tno.fl.protocols.cox_regression is part of the TNO Python Toolbox.

Limitations in (end-)use: the content of this software package may solely be used for applications that comply with international export control laws.
This implementation of cryptographic software has not been audited. Use at your own risk.

Documentation

Documentation of the tno.fl.protocols.cox_regression package can be found here.

Install

Easily install the tno.fl.protocols.cox_regression package using pip:

console $ python -m pip install tno.fl.protocols.cox_regression

Note: If you are cloning the repository and wish to edit the source code, be sure to install the package in editable mode:

console $ python -m pip install -e 'tno.fl.protocols.cox_regression'

If you wish to run the tests you can use:

console $ python -m pip install 'tno.fl.protocols.cox_regression[tests]'

The package has two groups of optional dependencies:

• schoenfeld_residuals: The packages required to run the Schoenfeld residual functionality
• examples: The packages required to run the example script in ./examples/schoenfeld
• tests: Required packages for running the tests included in this package

Usage

In Federated Cox, several clients, each with their own data, wish to fit a Cox model on their combined data. Each client computes a local update on their model and sends this update to a central server. This server combines these updates, updates the global model from this aggregated update and sends this new model back to the clients. Then the process repeats: the clients compute the local updates on this new model, send this to the server, which combines it and so on. This is done until the server notices that the model has converged. Afterwards, the results can be verified in a distributed privacy-preserving manner using a novel variant of the Schoenfeld residuals.

This package uses a combination of data preprocessing and federated logistic regression to perform this federated Cox modelling. Each client inputs their data. Next, the package performs a procedure called survival stacking. This procedure modifies the data in such a way that if we fit logistic regression on the new data set, we obtain the coefficients of a Cox model. Therefore, the next steps performs federated logistic regression and returns its coefficients.

Implementation

Communication

The client and the servers must be given a communication pool during initialization. This is a Pool object from the tno.mpc.communication package, which is also part of the PET lab. It is used for the communication amongst the server and the clients. We refer to this package for more information about this. The example file also gives an example of how to set up a simple communication pool.

Since the communication package uses asyncio for asynchronous handling, this federated learning package depends on it as well. For more information about this, we refer to the tno.mpc.communication documentation

Input data

For each subject, the data consists of a number of covariates, a failure/censoring time and a binary indicator whether the subject experienced failure or was censored. The data must be passed as numpy arrays. Each row in the covariates, should correspond to the same row in the times array and the events array.

Survival Stacking

As mentioned above, we use a method to transform fitting a cox model to fitting a logistic regression model. This method is called 'Survival Stacking' and described in this paper. Basically, it transforms the data such a way that we can obtain the Cox model parameters by performing logistic regression. See the paper for more details. Note that best results are achieved for large risk sets and will err the most for events that occur near the end of the time period.

Time bins

Time binning is used to limit the size of the stacked data set. When using time bins, the subjects are grouped together based on their censoring/failure time. A maximum number of bins can be set, or custom bins can be supplied in the run method. The number of time bins is passed as a property on the server.

Federated Logistic Regression

For the implementation of federated logistic regression, we rely on this logistic regression package. This implementation uses Newton-Raphson approximation by default.

Schoenfeld Residuals

After the model is trained completely, Schoenfeld residuals are often used to verify the quality of the model. We use a novel method to perform this calculation without leaking additional information. You can find a more detailed description as well as a usage example in the /examples folder.

Structure

The implementation of the federated Cox model consists of two classes with the suggestive names Client and Server. Each client is an instance of Client and the orchestrating server is an instance of the Server class. These classes are passed their own communication pool, maximum number of iterations, and the number of time bins.

Next, the clients and the servers call their respective run() methods. The clients supply the covariates, times and events as arguments. The server has one optional argument: custom time bins. If no time bins are supplied, it will space them evenly over the failure times based on the n_time_bins property.

Example code

Below is a minimal example of how to use the library. It consists of two clients, Alice and Bob, who want to fit a cox model based on a data set. The code is also supplied under examples/rotterdam

main.py

```python """ This module runs the Federated Cox regression protocol on the Rotterdam data set. See https://rdrr.io/cran/survival/man/rotterdam.html for more information. By running the script three times with command line argument 'server', 'alice' and 'bob' respectively, you can get a demonstration of how it works. """

from future import annotations

import logging import sys

import pandas as pd

from tno.mpc.communication import Pool

from tno.fl.protocols.coxregression.client import Client from tno.fl.protocols.coxregression.server import Server

logger = logging.getLogger(name) logger.setLevel(logging.DEBUG)

async def run_client(name: str, port: int) -> None: """ Run the client.

:param name: The client's name, either alice or bob.
:param port: The port the client is running.
"""
# Create Pool
pool = Pool()
pool.add_http_server(addr="localhost", port=port)
pool.add_http_client(name="server", addr="localhost", port=8080)
# Get Data
csv_data = pd.read_csv(f"data_{name}.csv")
covariates = csv_data[
    ["age", "grade", "nodes", "pgr", "er", "meno", "hormon"]
].to_numpy()
times = csv_data["dtime"].to_numpy()
events = csv_data["death"].to_numpy()
# Create Client
client = Client(pool)
logger.info(await client.run(covariates, times, events))
logger.info(await client.compute_statistics())
await pool.shutdown()

async def runserver() -> None: """ Run the server. """ # Create Pool pool = Pool() pool.addhttpserver(addr="localhost", port=8080) pool.addhttpclient(name="alice", addr="localhost", port=8081) pool.addhttpclient(name="bob", addr="localhost", port=8082) # Create Client server = Server(pool, ntimebins=100) await server.run() await server.computestatistics() await pool.shutdown()

async def asyncmain() -> None: """ The asynchronous main function """ if len(sys.argv) < 2: raise ValueError("Player name must be provided.") if sys.argv[1].lower() == "server": await runserver() elif sys.argv[1].lower() == "alice": await runclient("alice", 8081) elif sys.argv[1].lower() == "bob": await runclient("bob", 8082) else: raise ValueError( "This player has not been implemented. Possible values are: server, alice, bob" ) ```

To run the scripts in the examples folder, call main.py from the folder with the data files. As command line argument, pass it the name of the party running the app: 'Alice', 'Bob', or 'Server'. To run in on a single computer, run the following three commands, each in a different terminal. Note that if a client is started prior to the server, it will throw a ClientConnectorError. The client tries to send a message to the server, which has not started yet. After starting the server, the error disappears.

console python main.py alice python main.py bob python main.py server

For the rotterdam example (which uses the R Rotterdam data set), the output for the clients will be something similar to:

```console

python main.py alice 20xx-yy-zz 16:11:28,676 - tno.mpc.communication.httphandlers - INFO - Serving on localhost:8002 20xx-yy-zz 16:11:28,691 - tno.mpc.communication.httphandlers - INFO - Received message from 127.0.0.1:8000 20xx-yy-zz 16:11:28,691 - tno.mpc.communication.httphandlers - INFO - Received message from 127.0.0.1:8000 20xx-yy-zz 16:11:28,707 - tno.mpc.communication.httphandlers - INFO - Received message from 127.0.0.1:8000 20xx-yy-zz 16:11:28,707 - tno.mpc.communication.httphandlers - INFO - Received message from 127.0.0.1:8000 20xx-yy-zz 16:11:28,722 - tno.mpc.communication.httphandlers - INFO - Received message from 127.0.0.1:8000 20xx-yy-zz 16:11:28,722 - tno.mpc.communication.httphandlers - INFO - Received message from 127.0.0.1:8000 20xx-yy-zz 16:11:28,734 - tno.mpc.communication.httphandlers - INFO - Received message from 127.0.0.1:8000 20xx-yy-zz 16:11:28,734 - tno.mpc.communication.httphandlers - INFO - Received message from 127.0.0.1:8000 20xx-yy-zz 16:11:28,734 - tno.mpc.communication.httphandlers - INFO - Received message from 127.0.0.1:8000 20xx-yy-zz 16:11:28,750 - tno.mpc.communication.httphandlers - INFO - Received message from 127.0.0.1:8000 20xx-yy-zz 16:11:28,750 - tno.mpc.communication.httphandlers - INFO - Received message from 127.0.0.1:8000 2024-10-30 17:16:27,514 - main - INFO - [ 1.84107500e-02 3.80357702e-01 9.09068022e-02 -4.09343915e-04 -4.61119239e-05 -3.42392644e-02 -5.30740663e-02 -9.70355122e+00 -8.76459672e+00 -8.07314057e+00 -7.47193459e+00 -7.35069891e+00 -7.11330304e+00 -6.69510433e+00 ... 20xx-yy-zz 16:11:28,754 - tno.mpc.communication.httphandlers - INFO - Received message from 127.0.0.1:8080 20xx-yy-zz 16:11:29,783 - main - INFO - [{'se': 0.0062415538530286715, 'z': 2.7883563969710345, 'p': 0.005297622817553194}, {'se': 0.11341501923706382, 'z': 3.427426052562356, 'p': 0.0006093322338218687}, {'se': 0.01122222728224323, 'z': 12.132338463949338, 'p': 0.0}, ... 20xx-yy-zz 16:11:29,784 - tno.mpc.communication.httphandlers - INFO - HTTPServer: Shutting down server task 20xx-yy-zz 16:11:29,784 - tno.mpc.communication.httphandlers - INFO - Server localhost:8082 shutdown ```

We first see the client setting up the connection with the server. Then we have ten rounds of training, as passed to the server. Finally, we print the resulting model. We obtain the following coefficients for the covariates and the coefficients of the risk set indicators. Next, it gives the corresponding statistics. Here we parse the results of the first three covariates.

| Parameter | Coefficient | standard error | p-value | | --------- | -------------- | -------------- | -------- | | age | 1.84107500e-02 | 0.00624 | 0.005297 | | grade | 3.80357702e-01 | 0.11341 | 0.000609 | | nodes | 9.09068022e-02 | 0.01122 | 0 |