Online Setting

In the online setting, we assume unrestricted access to the true objective functions. This allows us to generate as many training samples as needed before the diffusion training phase. Here, we consider the ZDT2 benchmark problem as an example.

import numpy as np
import torch

# Import the SPREAD solver
from moospread import SPREAD

# Import a test problem
from moospread.tasks import ZDT2

# Define the problem
problem = ZDT2(n_var=30)

When initializing the SPREAD solver, the parameter mode must be set to "online" and the size of the training dataset is specified using the data_size argument. If no value is provided, a default of 10000 is used. Setting timesteps to values greater than or equal to 1000 typically leads to better performance.

# Initialize the SPREAD solver
solver = SPREAD(
    problem,
    data_size=10000,
    num_blocks=2,
    timesteps=5000,
    num_epochs=1000,
    train_tol=100,
    mode="online",
    seed=2026,
    verbose=True
)

To solve the problem, specify the number of solutions to generate using the num_points_sample argument. If the solver is reused with the same initialization, retraining can be avoided by setting load_models=True.

In the online setting, we recommend:

• setting rho_scale_gamma = 0.9 for bi-objective problems, and

• setting rho_scale_gamma = 0.001 for problems with more than two objectives.

If intermediate Pareto fronts are not needed, simply set iterative_plot=False.

# Solve the problem
res_x, res_y = solver.solve(
    num_points_sample=200,
    rho_scale_gamma=0.9,
    iterative_plot=True,
    plot_period=10,
    max_backtracks=25,
    save_results=True,
    samples_store_path="./samples_dir/",
    images_store_path="./images_dir/"
)

Since in this example we enable iterative_plot, we can visualize the generative process using the images saved in images_store_path (see Visualization).

solver.create_video(
    image_folder="images_dir/ZDT2_online",
    output_video="videos_dir/ZDT2_online.mp4",
    total_duration_s=20.0,
    first_transition_s=2.0,
    fps=30,
)

Generative process

Final generated Pareto front