NAdam

Gradient-Based

Nesterov-accelerated Adaptive Moment Estimation (Nadam) optimization algorithm.

Algorithm Overview

This module implements the Nadam (Nesterov-accelerated Adaptive Moment Estimation) optimization algorithm. Nadam combines Adam with Nesterov momentum, incorporating lookahead into the gradient computation which can lead to faster convergence.

Nadam 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) m_bar = beta1 * m_hat + (1 - beta1) * gradient / (1 - beta1^t) x = x - learning_rate * m_bar / (sqrt(v_hat) + epsilon)

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

Usage

from opt.gradient_based.nadam import Nadam
from opt.benchmark.functions import sphere

optimizer = Nadam(
    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	NADAM_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.
seed	int | None	None	Random seed for reproducibility.

Algorithm Metadata

Property	Value
Algorithm Name	Nesterov-accelerated Adaptive Moment
Acronym	NADAM
Year Introduced	2016
Authors	Dozat, Timothy
Algorithm Class	Gradient-Based
Complexity	O(dim)
Properties	Gradient-based, Stochastic
Implementation	Python 3.10+
COCO Compatible	Yes

Mathematical Formulation

Core update equations:

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

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
• $ϵ$ is a small constant for numerical stability
• $m_t,v_t$ are biased first and second moment estimates
• ${\overline{m}}_t$ is the Nesterov-corrected first moment

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	0.002	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 constant

Sensitivity Analysis:

• learning_rate: High impact on convergence
• beta1, beta2: Medium impact
• Recommended tuning ranges: $\alpha \in [0.0001,0.01]$, ${\beta }_1\in [0.8,0.95]$

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 updates
• Space complexity: $O(dim)$ for storing moment estimates
• BBOB budget usage: Typically uses 50-65% of dim*10000 budget for convergence

BBOB Performance Characteristics:

• Best function classes: Unimodal, moderately multimodal 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): Often faster than Adam, competitive with best

Convergence Properties:

• Convergence rate: Faster than Adam due to Nesterov momentum
• Local vs Global: Tends toward local optima (gradient-based)
• Premature convergence risk: Low - momentum helps exploration

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 Nesterov lookahead

Known Limitations:

• Learning rate requires tuning
• Gradient approximation via finite differences less accurate
• May overshoot in some scenarios

Version History:

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

References

[1] Dozat, T. (2016). "Incorporating Nesterov Momentum into Adam." ICLR Workshop. http://cs229.stanford.edu/proj2015/054_report.pdf

[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: Dozat (2016) - Stanford report
• This implementation: Nadam with BBOB compliance

See Also

Adam: Base algorithm without Nesterov momentum BBOB Comparison: Nadam often converges faster than Adam

NesterovAcceleratedGradient: Classical Nesterov momentum BBOB Comparison: Nadam combines this with adaptive learning rates

AdamW: Adam with decoupled weight decay BBOB Comparison: Different optimization approaches for similar goals

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

Related BBOB Algorithm Classes:

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

Benchmark Performance

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

Run-based charts

Convergence, distribution and ECDF charts appear here once this optimizer is included in the benchmark suite.

Related Pages

• Gradient-Based Algorithms
• All Algorithms
• Benchmark Functions

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Source Code

View the implementation: nadam.py