A Compositional Operation Toolbox for Gradient-based Bi-Level Optimization

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BOAT (OperAtion-level Toolbox for gradient-based BLO) is a compositional, operation-level framework designed to bridge the gap between theoretical modeling and practical implementation in Bi-Level Optimization (BLO).

Unlike existing libraries that typically encapsulate fixed solver routines, BOAT factorizes the BLO workflow into atomic, reusable primitives. Through a unified constraint reconstruction perspective, it empowers researchers to automatically compose over 85+ solver variants from a compact set of 17 gradient operations.

This is the PyTorch-based version of BOAT, designed for efficiency and wide compatibility. BOAT also supports other backends via separate branches:

• Jittor-based: Accelerated version with meta-operators.
• MindSpore-based: Optimized for ascending chips and industry scenarios.

🔑 Key Features

• 🧩 Compositional Operation-Level Abstraction: Deconstructs BLO solvers into three modular stages: Gradient Mapping (GM), Numerical Approximation (NA), and First-Order (FO).
• 🏭 Generative Solver Construction: Supports dynamic serialization of operations. Users can recover classical algorithms (e.g., DARTS, MAML) or discover novel hybrid solvers (e.g., NGD + DI + PTT) simply by changing configurations.
• 🛠 Configuration-Driven: Define complex optimization strategies via simple JSON configurations (boat_config & loss_config), decoupling algorithmic logic from model definitions.
• 🚀 Unified Computational Analysis: Offers a comprehensive complexity analysis of gradient-based BLO techniques to guide users in selecting optimal configurations for efficiency and accuracy.
• ✅ Comprehensive Testing: Achieves 99% code coverage through rigorous testing with pytest, ensuring software robustness and reliability.

📚 Supported Operation Libraries

BOAT implements 17 atomic gradient operations organized into three modular libraries. These primitives can be dynamically serialized to generate over 85+ solver variants, covering the full spectrum of BLO methodologies.

Library	Functional Role	Supported Atomic Operations
GM-OL (Gradient Mapping)	Reconstructs the LL iterative trajectory. Customizes the dynamic mapping rules ($\mathcal{T}_k$) to shape the optimization path and variable coupling.	• NGD (Naive Gradient Descent) • GDA (Gradient Descent Aggregation) • DI (Dynamic Initialization) • DM (Dual Multiplier / KKT)
NA-OL (Numerical Approx.)	Resolves the auxiliary gradient bottleneck. Approximates the implicit gradients or hyper-gradients via automatic differentiation, numerical inversion, or truncation.	• RAD (Reverse-AD / Unrolled) • RGT (Reverse Gradient Truncation) • PTT (Pessimistic Trajectory Truncation) • FD (Finite Difference / DARTS) • CG (Conjugate Gradient) • NS (Neumann Series) • IGA (Implicit Gradient Approximation) • IAD (Init-based AD / MAML) • FOA (First-Order Approx. / Reptile)
FO-OL (First-Order)	Constructs single-level surrogates. Reformulates the nested problem into first-order objectives using value-functions or penalties, avoiding Hessian computations.	• VSO (Value-Function Sequential) • VFO (Value-Function First-Order) • MESO (Moreau Envelope) • PGDO (Penalty Gradient Descent)

🔨 Installation

To install BOAT (PyTorch version), we recommend using a virtual environment.

1. Create Environment

conda create -n boat python=3.12
conda activate boat

2. Create Environment

You can install the latest stable version from PyPI or the latest development version from GitHub:

# Install from PyPI
pip install boat-torch

# Or install from Source
git clone [https://github.com/callous-youth/BOAT.git](https://github.com/callous-youth/BOAT.git)
cd BOAT
pip install -e .

⚡ How to Use BOAT

BOAT separates the problem definition from the solver configuration, allowing you to switch algorithms without changing your model code.

1. Load Configuration

Define your optimization strategy in boat_config.jsonand your objectives in loss_config.json.

import json
import boat_torch as boat

# boat_config defines the operations (e.g., NGD + CG)
with open("configs/boat_config.json", "r") as f:
    boat_config = json.load(f)

# loss_config defines the Upper/Lower objectives
with open("configs/loss_config.json", "r") as f:
    loss_config = json.load(f)

2. Define Models and Optimizers

You need to specify both the upper-level and lower-level PyTorch models along with their respective optimizers.

import torch

# Define models
upper_model = UpperModel(*args, **kwargs)  # Replace with your upper-level model
lower_model = LowerModel(*args, **kwargs)  # Replace with your lower-level model

# Define optimizers
upper_opt = torch.optim.Adam(upper_model.parameters(), lr=1e-3)
lower_opt = torch.optim.SGD(lower_model.parameters(), lr=1e-2)

3. Customize & Initialize Problem

Inject your runtime objects (models, optimizers) into the configuration and initialize the boat.Problem instance.

# Example gradient mapping and numerical approximation opreation Combination.
gm_op = ["NGD", "DI", "GDA"]  # Dynamic Methods (Demo Only)
na_op = ["RGT","RAD"]          # Hyper-Gradient Methods (Demo Only)

# Add methods and model details to the configuration
boat_config["gm_op"] = gm_op
boat_config["na_op"] = na_op
boat_config["lower_level_model"] = lower_model
boat_config["upper_level_model"] = upper_model
boat_config["lower_level_opt"] = lower_opt
boat_config["upper_level_opt"] = upper_opt
boat_config["lower_level_var"] = list(lower_model.parameters())
boat_config["upper_level_var"] = list(upper_model.parameters())

# Initialize the BOAT core
b_optimizer = boat.Problem(boat_config, loss_config)

4. Build Solvers

This step automatically composes the solver based on the operations defined in boat_config (e.g., constructing the hyper-gradient graph).

# Build solvers for lower and upper levels
b_optimizer.build_ll_solver()  # Build Lower-Level Solver
b_optimizer.build_ul_solver()  # Build Upper-Level Solver

5. Define Data Feeds

Execute the optimization. run_iter handles the forward pass, inner-loop optimization, and hyper-gradient calculation automatically.

# Training loop
for x_itr in range(1000):
    # Prepare data batches
    ul_feed_dict = {"data": ul_data, "target": ul_target}
    ll_feed_dict = {"data": ll_data, "target": ll_target}
    
    # Run one step of Bilevel Optimization
    loss, run_time = b_optimizer.run_iter(ll_feed_dict, ul_feed_dict, current_iter=x_itr)
    
    if x_itr % 100 == 0:
        print(f"Iter {x_itr}: UL Loss {loss:.4f}")

🌍 Applications

BOAT covers a wide spectrum of BLO applications, categorized by the optimization target:

• Data-Centric: Data Hyper-Cleaning, Synthetic Data Reweighting, Diffusion Model Guidance.

• Model-Centric: Neural Architecture Search (NAS), LLM Prompt Optimization, Parameter Efficient Fine-Tuning (PEFT).

• Strategy-Centric: Meta-Learning, Hyperparameter Optimization (HO), Reinforcement Learning from Human Feedback (RLHF).

📝 Citation

If you find BOAT useful in your research, please consider citing our paper:

@article{liu2025boat,
  title={BOAT: A Compositional Operation Toolbox for Gradient-based Bi-Level Optimization},
  author={Liu, Yaohua and Pan, Jibao and Jiao, Xianghao and Gao, Jiaxin and Liu, Zhu and Liu, Risheng},
  journal={Submitted to Journal of Machine Learning Research (JMLR)},
  year={2025}
}

License

MIT License

Copyright (c) 2024 Yaohua Liu

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.