MICE: Multi-Iteration stochastiC Estimator

MICE is a gradient estimator for stochastic optimization that uses successive control variates along the optimization path to reduce variance. By adaptively selecting which iterates to include in its index set and optimally distributing samples, MICE achieves accurate mean gradient estimation at minimal computational cost. The article can be found here: Applied Mathematics & Optimization, 93, 93 (2026) (DOI)

Key Features

• Adaptive variance reduction: Controls relative L² error with user-specified tolerance ε
• Efficient sample allocation: Minimizes gradient sampling cost subject to error constraints
• Index-set operators: Add, Drop, Restart, and Clip operations for optimal hierarchy management
• Flexible integration: Non-intrusive design couples seamlessly with SGD, Adam, and other optimizers
• Dual problem support: Handles both expectation minimization and finite-sum problems
• Robust stopping: Resampling-based gradient norm estimation for stable termination criteria

Theoretical Performance

For smooth, strongly convex problems, SGD-MICE achieves a gradient evaluation complexity of O(tol⁻¹) to reach tolerance tol, compared to O(tol⁻¹ log(tol⁻¹)) for standard adaptive batch-size SGD.

Installation

pip install mice

For development or to run experiments:

git clone https://github.com/agcarlon/mice.git
cd mice
pip install -e .

Quick Start

import numpy as np
from mice import MICE
from mice.policy import DropRestartClipPolicy

# Define gradient function: grad(x, thetas) -> gradients array
def gradient(x, thetas):
    """Compute gradients for batch of samples."""
    return x - thetas  # Simple example: minimize E[(x - θ)²]

# Define sampler: sampler(n) -> batch of n samples
def sampler(n):
    return np.random.randn(n, 1)

# Create MICE estimator
estimator = MICE(
    grad=gradient,
    sampler=sampler,
    eps=0.577,              # Relative error tolerance (1/√3)
    min_batch=10,
    policy=DropRestartClipPolicy(
        drop_param=0.5,
        restart_param=0.0,
        max_hierarchy_size=100
    ),
    max_cost=10000,         # Maximum gradient evaluations
    stop_crit_norm=1e-6,    # Stopping criterion
)

# Use in optimization loop
x = np.array([10.0])
for iteration in range(100):
    grad_estimate = estimator(x)
    if estimator.terminate:
        print(f"Terminated early: {estimator.terminate_reason}")
        break
    x = x - 0.1 * grad_estimate  # Gradient descent step
    print(f"Iteration {iteration}: x = {x[0]:.6f}")

Advanced Features

Finite-Sum Problems

For finite datasets (empirical risk minimization):

# Load your dataset
X_train = ...  # Training features
y_train = ...  # Training labels
data = np.column_stack([y_train, X_train])

# MICE automatically handles finite sampling
estimator = MICE(
    grad=your_gradient_function,
    sampler=data,  # Pass data directly
    eps=0.577,
    # ... other parameters
)

Policy Configuration

Control index-set management with DropRestartClipPolicy:

from mice.policy import DropRestartClipPolicy

policy = DropRestartClipPolicy(
    drop_param=0.5,           # Threshold for dropping last iterate
    restart_param=0.0,        # Threshold for restarting hierarchy
    max_hierarchy_size=100,   # Maximum |L_k|
    clip_type="full",         # Clipping strategy ("full", "all", or None)
    aggr_cost=0.1,           # Aggregation cost factor
)

estimator = MICE(grad=..., sampler=..., policy=policy)

Resampling-Based Norm Estimation

Enable robust norm estimation for sizing and stopping:

estimator = MICE(
    grad=gradient,
    sampler=sampler,
    use_resampling=True,
    re_part=5,              # Number of jackknife partitions
    re_quantile=0.05,       # Quantile for tolerance
    re_tot_cost=0.2,        # Resampling cost budget
    # ... other parameters
)

API Reference

MICE

Main estimator class.

Parameters:

• grad (callable): Gradient function with signature grad(x: ndarray, thetas: Any) -> ndarray
• sampler (callable or array): Sampler function sampler(n: int) -> Any or finite dataset
• eps (float): Relative error tolerance parameter (default: 0.577)
• min_batch (int): Minimum batch size (default: 10)
• restart_factor (int): Restart batch multiplier (default: 10)
• max_cost (float): Maximum gradient evaluations (default: inf)
• stop_crit_norm (float): Stopping criterion for gradient norm (default: 0.0)
• stop_crit_prob (float): Stopping criterion probability (default: 0.05)
• convex (bool): Whether problem is convex (default: False)
• policy (DropRestartClipPolicy): Index-set management policy
• use_resampling (bool): Enable resampling-based norm estimation (default: True)
• recorder (Recorder): Optional event recorder for diagnostics

Methods:

• evaluate(x: ndarray) -> ndarray: Evaluate MICE gradient estimate at point x
• __call__(x: ndarray) -> ndarray: Alias for evaluate
• get_log() -> list: Return recorded events (if recorder enabled)

Reproducible Experiments

The repository includes all numerical experiments from the manuscript "Multi-Iteration Stochastic Optimizers". See experiments/README.md for detailed instructions on:

• Running operator ablations and sensitivity sweeps (quadratic benchmarks)
• Training logistic regression on mushrooms, gisette, and HIGGS datasets
• Generating all figures and tables from the paper

Citation

If you use MICE in your research, please cite:

@article{carlon2026multi,
  title={Multi-iteration Stochastic Optimizers},
  author={Carlon, Andr{\'e} and Espath, Luis and Holdorf, Rafael and Tempone, Ra{\'u}l},
  journal={Applied Mathematics \& Optimization},
  volume={93},
  number={93},
  year={2026},
  doi={10.1007/s00245-026-10451-x}
}

Published article: Applied Mathematics & Optimization, 93, 93 (2026) (DOI)

Documentation

Full documentation available at mice.readthedocs.io

Build docs locally:

python -m pip install -r docs/requirements.txt
python -m pip install -e .
sphinx-build -b html docs docs/_build/html

License

This project is licensed under the GNU General Public License v3.0 - see the LICENSE file for details.

Authors

• André Carlon (RWTH Aachen University)
• Luis Espath (University of Nottingham)
• Rafael Holdorf (Federal University of Santa Catarina)
• Raúl Tempone (KAUST & RWTH Aachen University)