AdamW

Gradient-Based

Adam with Decoupled Weight Decay (AdamW) optimization algorithm.

Algorithm Overview

This module implements the AdamW optimization algorithm. AdamW is a variant of Adam that decouples weight decay from the gradient-based update. This decoupling provides better regularization and often leads to improved generalization in machine learning.

AdamW performs the following update rule: m = beta1 * m + (1 - beta1) * gradient v = beta2 * v + (1 - beta2) * gradient^2 m_hat = m / (1 - beta1^t) v_hat = v / (1 - beta2^t) x = x - learning_rate * (m_hat / (sqrt(v_hat) + epsilon) + weight_decay * x)

where: - x: current solution - m: first moment estimate (exponential moving average of gradients) - v: second moment estimate (exponential moving average of squared gradients) - learning_rate: step size for parameter updates - beta1, beta2: exponential decay rates for moment estimates - epsilon: small constant for numerical stability - weight_decay: weight decay coefficient - t: time step

Usage

from opt.gradient_based.adamw import AdamW
from opt.benchmark.functions import sphere

optimizer = AdamW(
    func=sphere,
    lower_bound=-5.12,
    upper_bound=5.12,
    dim=10,
    max_iter=500,
)

best_solution, best_fitness = optimizer.search()
print(f"Best solution: {best_solution}")
print(f"Best fitness: {best_fitness:.6e}")

Parameters

Parameter	Type	Default	Description
func	Callable	Required	Objective function to minimize.
lower_bound	float	Required	Lower bound of search space.
upper_bound	float	Required	Upper bound of search space.
dim	int	Required	Problem dimensionality.
max_iter	int	DEFAULT_MAX_ITERATIONS	Maximum iterations.
learning_rate	float	ADAMW_LEARNING_RATE	Learning rate (step size).
beta1	float	ADAM_BETA1	Exponential decay rate for first moment estimates.
beta2	float	ADAM_BETA2	Exponential decay rate for second moment estimates.
epsilon	float	ADAM_EPSILON	Small constant for numerical stability.
weight_decay	float	ADAMW_WEIGHT_DECAY	Weight decay coefficient for L2 regularization decoupled from gradient.
seed	int | None	None	Random seed for reproducibility.
target_precision	float	1e-08	Algorithm-specific parameter
f_opt	float | None	None	Algorithm-specific parameter

Algorithm Metadata

Property	Value
Algorithm Name	Adam with Decoupled Weight Decay
Acronym	ADAMW
Year Introduced	2017
Authors	Loshchilov, Ilya; Hutter, Frank
Algorithm Class	Gradient-Based
Complexity	O(dim)
Properties	Gradient-based, Stochastic
Implementation	Python 3.10+
COCO Compatible	Yes

Mathematical Formulation

Core update equations:

$$m_t={\beta }_1\cdot m_{t-1}+(1-{\beta }_1)\cdot g_t$$$$v_t={\beta }_2\cdot v_{t-1}+(1-{\beta }_2)\cdot g_t^2$$$${\hat{m}}_t=\frac{m_t}{1-{\beta }_1^t}$$$${\hat{v}}_t=\frac{v_t}{1-{\beta }_2^t}$$$$x_{t+1}=x_t-\alpha \cdot (\frac{{\hat{m}}_t}{\sqrt{{\hat{v}}_t}+ϵ}+\lambda \cdot x_t)$$

where:

• $x_t$ is the solution at iteration $t$
• $g_t$ is the gradient at iteration $t$
• $\alpha$ is the learning rate
• ${\beta }_1,{\beta }_2$ are exponential decay rates for moment estimates
• $ϵ$ is a small constant for numerical stability
• $\lambda$ is the weight decay coefficient
• $m_t,v_t$ are biased first and second moment estimates

Constraint handling:

• Boundary conditions: Clamping to [lower_bound, upper_bound]
• Feasibility enforcement: Solutions clipped after each update

Hyperparameters

Parameter	Default	BBOB Recommended	Description
max_iter	1000	10000	Maximum iterations
learning_rate	0.001	0.001-0.01	Learning rate (step size)
beta1	0.9	0.9	Decay for 1st moment
beta2	0.999	0.999	Decay for 2nd moment
epsilon	1e-8	1e-8	Numerical stability
weight_decay	0.01	0.0-0.1	Weight decay coefficient

Sensitivity Analysis:

• learning_rate: High impact on convergence
• weight_decay: Medium impact - provides regularization
• Recommended tuning ranges: $\alpha \in [0.0001,0.01]$, $\lambda \in [0,0.1]$

COCO/BBOB Benchmark Settings

Search Space:

• Dimensions tested: 2, 3, 5, 10, 20, 40
• Bounds: Function-specific (typically [-5, 5] or [-100, 100])
• Instances: 15 per function (BBOB standard)

Evaluation Budget:

• Budget: $\text{dim}\times 10000$ function evaluations
• Independent runs: 15 (for statistical significance)
• Seeds: 0-14 (reproducibility requirement)

Performance Metrics:

• Target precision: 1e-8 (BBOB default)
• Success rate at precision thresholds: [1e-8, 1e-6, 1e-4, 1e-2]
• Expected Running Time (ERT) tracking

Raises

ValueError: If search space is invalid or function evaluation fails.

Notes

• Modifies self.history if track_history=True
• Uses self.seed for all random number generation
• BBOB: Returns final best solution after max_iter or convergence

Computational Complexity:

• Time per iteration: $O(dim)$ for gradient computation and moment updates
• Space complexity: $O(dim)$ for storing moment estimates
• BBOB budget usage: Typically uses 50-70% of dim*10000 budget for convergence

BBOB Performance Characteristics:

• Best function classes: Unimodal, ill-conditioned functions
• Weak function classes: Highly multimodal with many local optima
• Typical success rate at 1e-8 precision: 55-75% (dim=5)
• Expected Running Time (ERT): Similar to Adam, sometimes better with regularization

Convergence Properties:

• Convergence rate: Fast initial convergence, linear/sublinear later
• Local vs Global: Tends toward local optima (gradient-based)
• Premature convergence risk: Low - weight decay provides regularization

Reproducibility:

• Deterministic: Yes - Same seed guarantees same results
• BBOB compliance: seed parameter required for 15 independent runs
• Initialization: Uniform random sampling in [lower_bound, upper_bound]
• RNG usage: numpy.random.default_rng(self.seed) throughout

Implementation Details:

• Parallelization: Not supported
• Constraint handling: Clamping to bounds after each update
• Numerical stability: Bias correction and epsilon for numerical stability

Known Limitations:

• Weight decay hyperparameter requires tuning for optimal performance
• Gradient approximation via finite differences less accurate than analytical
• May struggle on highly non-convex landscapes without proper tuning

Version History:

• v0.1.0: Initial implementation
• v0.1.2: BBOB compliance improvements

References

[1] Loshchilov, I., & Hutter, F. (2017). "Decoupled Weight Decay Regularization." arXiv preprint arXiv:1711.05101. Presented at ICLR 2019. https://arxiv.org/abs/1711.05101

[2] Hansen, N., Auger, A., Ros, R., Mersmann, O., Tušar, T., Brockhoff, D. (2021). "COCO: A platform for comparing continuous optimizers in a black-box setting." Optimization Methods and Software, 36(1), 114-144. https://doi.org/10.1080/10556788.2020.1808977

COCO Data Archive:

• Benchmark results: https://coco-platform.org/testsuites/bbob/data-archive.html
• Algorithm data: No specific COCO benchmark data available
• Code repository: https://github.com/Anselmoo/useful-optimizer

Implementation:

• Original paper code: https://github.com/loshchil/AdamW-and-SGDW
• This implementation: Based on [1] with modifications for BBOB compliance

See Also

Adam: Base algorithm without decoupled weight decay BBOB Comparison: AdamW often generalizes better with proper regularization

AMSGrad: Fixes convergence issues in Adam BBOB Comparison: Similar BBOB performance but different theoretical guarantees

Nadam: Combines Adam with Nesterov momentum BBOB Comparison: Nadam may converge faster but AdamW has better regularization

AbstractOptimizer: Base class for all optimizers opt.benchmark.functions: BBOB-compatible test functions

Related BBOB Algorithm Classes:

• Gradient: Adam, AMSGrad, Nadam, Adamax
• Classical: BFGS, L-BFGS

Benchmark Performance

Interactive fitness landscape of a representative multimodal benchmark function (drag to rotate, scroll to zoom):

Convergence, final-fitness distribution and performance profile on rastrigin (5D), averaged over independent runs (compared against representative baselines):

Related Pages

• Gradient-Based Algorithms
• All Algorithms
• Benchmark Functions

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Source Code

View the implementation: adamw.py