Styblinski–Tang Function in Optimization

• Styblinski–Tang function is a non-convex benchmark defined by quartic, quadratic, and linear terms, exhibiting multiple local minima with a global minimum near -2.903534.
• Its multimodal landscape and analytic gradients make it ideal for testing classical optimization methods and quantum heuristics like QAOA.
• The function enhances gradient-guided data selection strategies, improving training efficiency and robustness in machine learning applications.

The Styblinski–Tang function is a widely used non-convex benchmark for optimization and machine learning, particularly valued for its multimodal landscape and controllable analytic properties. It is defined for $N$ continuous variables as

$$f(\mathbf{x}) = \frac{1}{2} \sum_{k=1}^N [x_k^4 - 16x_k^2 + 5x_k]$$

and exhibits multiple local minima, with a global minimum located symmetrically in all coordinates near $x_k \approx -2.903534$. Its sharp wells and plateaus make it suitable for thoroughly probing optimization algorithms, data selection procedures, and quantum computing heuristics.

1. Mathematical Formulation and Properties

The $N$-dimensional Styblinski–Tang function is given by

$$f(\mathbf{x}) = \frac{1}{2} \sum_{i=1}^N (x_i^4 - 16x_i^2 + 5x_i)$$

For the $d=2$ case, this simplifies to

$$f(x, y) = \frac{1}{2}\left[ x^4 - 16x^2 + 5x + y^4 - 16y^2 + 5y \right]$$

The function’s primary features include:

• Multimodality: Several distinct local minima (“wells” or “basins”).
• Global Minimum: Located at $\mathbf{x}^* \approx [-2.903534,\ldots,-2.903534]$ with

$f(\mathbf{x}^*) \approx -39.16617 \cdot N$

• Non-uniform curvature: Regions of steep slopes between wells separated by relatively flat plateaus.
• Analytic gradients: The gradient is

$$\nabla f(\mathbf{x}) = 2x_k^3 - 16x_k + \frac{5}{2}$$

These gradients are crucial for both numerical optimization and gradient-guided sampling.

The combination of quartic, quadratic, and linear terms ensures rich behavior, including regions with rapid energy changes and regions with minimal variation.

2. Use in Benchmarking Optimization Algorithms

The Styblinski–Tang function is a standard benchmark for evaluating optimization algorithms due to its challenging landscape. It has been employed in testing both classical methods (stochastic gradient descent variants, global optimization heuristics) and quantum algorithms.

A notable algorithmic application is in the Quantum Approximate Optimization Algorithm (QAOA) for continuous variables (Verdon et al., 2019). In this setting, the function is encoded into a cost Hamiltonian $\hat{H}_C = f(\hat{\mathbf{x}})$, and the system is evolved under alternating cost and mixer (kinetic) Hamiltonians:

$\hat{H}_M = \frac{1}{2}\sum_j \hat{p}_j^2$

where $\hat{p}_j$ are the conjugate momenta to the position operators $\hat{x}_j$. For optimization, the QAOA protocol applies the unitary,

$$U(\eta, \gamma) = \prod_{j=1}^P \exp(-i \gamma_j \hat{H}_M) \exp(-i\eta_j \hat{H}_C)$$

which, in the Heisenberg picture, yields an update

$$\hat{\mathbf{x}} \rightarrow \hat{\mathbf{x}} + \gamma \hat{\mathbf{p}} - \eta \gamma \nabla f(\hat{\mathbf{x}})$$

This is directly analogous to gradient descent with momentum, but acting simultaneously on all components in superposition.

3. Role in Gradient-Guided Data Selection

The Styblinski–Tang function serves as an instructive example in training set selection studies for machine learning, as demonstrated in the evaluation of Gradient Guided Furthest Point Sampling (GGFPS) (Trestman et al., 10 Oct 2025). The function’s multidimensional energy surface—characterized by wells and regions of sharp energy change—enables the paper of data efficiency and robustness in learning contexts.

GGFPS leverages the local norm of the gradient (the "force") to adjust sampling. Each candidate point is scored as

$s_j = (g(\mathbf{x}_j))^\beta d_j$

where

• $g(\mathbf{x}) = \|\nabla f(\mathbf{x})\|$ is the gradient norm,
• $d_j$ is the minimum Euclidean distance to prior samples,
• $\beta$ controls the bias toward high-gradient regions.

Empirically, for the two-dimensional Styblinski–Tang function, GGFPS achieves up to twofold reduction in required training set size for equivalent accuracy compared to FPS, with lower prediction errors and variance. Standard FPS, which ignores the energy landscape’s gradients, may oversample flat regions and miss steep, informative transitions.

4. Numerical Simulation and Quantum Algorithm Performance

In the QAOA context, numerical studies of Styblinski–Tang optimization reveal the characteristic transformation of the output probability distribution:

• Initial state: Momentum-squeezed vacuum yields a broad Gaussian superposition.
• First layer: The state develops interference fringes and starts localizing near multiple minima.
• Second layer: Density accumulates at local minima.
• Third (final) layer: The distribution is sharply peaked around the global minimum at $(x, y) \approx (-2.90353, -2.90353)$.

Simulations in the Strawberry Fields framework (with quartic gates) using $P = 3$ layers and $T = 0.1$ per layer gave best observed values $f(x^*,y^*) \approx -78.33216$ over $1000$ samples, closely matching $f_{\text{global}} \approx -78.3323$. On average, nearly optimal solutions required about $50$ samples.

5. Implications for Robust Learning and Sampling

The explicit use of the Styblinski–Tang function in GGFPS demonstrates that incorporating gradient information into training set selection:

• Improves data efficiency: Fewer points are needed for the same model accuracy.
• Enhances robustness: Prediction errors and variances decrease, especially in regions of sharp curvature such as the steep “walls” between wells.
• Mitigates FPS limitations: While classic Furthest Point Sampling (FPS) can spread samples only over the descriptor space, it may under-cover high-gradient regions not at the geometric periphery. GGFPS ensures that energetically important regions are preferentially sampled.

In molecular and chemical machine learning applications, this strategy is directly applicable when the target property (e.g., potential energy) varies nonlinearly with the system’s geometric descriptors.

6. Complexity-Theoretic Aspects and Quantum Information Perspective

Implementing the Styblinski–Tang function as a cost Hamiltonian in QAOA enables examination of fundamental complexity aspects. Specifically:

• Quadratic speedup: Variants of continuous-variable QAOA recover Grover-like speedup by encoding the search objective as an oracle Hamiltonian.
• CV-IQP simulation: Single-step evolution under the QAOA protocol can reproduce continuous-variable instantaneous quantum polynomial circuits, believed to be classically hard to simulate.

While demonstrated empirically in two dimensions, these features suggest that algorithmic primitives tested on the Styblinski–Tang function may generalize to higher-dimensional or more intricate continuous-variable optimization settings.

7. Context within Optimization Landscape Research

The Styblinski–Tang function functions as an archetype for rugged, multimodal optimization landscapes—valuable for testing numerical solvers, quantum heuristics, and data selection procedures. Both the functional form and its analytic gradient are leveraged to explicitly expose the behavior of algorithms under non-trivial landscape conditions.

Its application as a cost Hamiltonian in variational quantum algorithms provides a tangible instantiation of kinetic and potential energy alternation, enabling direct mapping between quantum operator dynamics and classical optimization analogues, such as gradient descent with momentum. Similarly, its variable curvature enables rigorous assessment of sample selection criteria in data-driven modeling, highlighting both algorithmic efficiency and robustness metrics.