Random Function Descent (RFD)

• Random Function Descent (RFD) is a collection of optimization methods that incorporate randomness into descent directions and step-size schedules to achieve average-case optimality.
• It employs random search directions, Bayesian-inspired step size schedules, random coordinate descent, and swarm-based heuristics to improve performance in high-dimensional and large-scale scenarios.
• RFD offers robust theoretical convergence guarantees and computational efficiency by blending probabilistic modeling with adaptive step size selection, bridging deterministic and Bayesian optimization approaches.

Random Function Descent (RFD) is a collection of optimization frameworks that generalize classical deterministic descent methods by incorporating randomness, either in the modeling of the objective function or in the selection of update directions. RFD encompasses random search direction stochastic gradient approaches, Bayesian-inspired step size schedules, random subspace and coordinate descent, and swarm-based heuristics. These methods are designed to handle high-dimensional or large-scale optimization problems efficiently, often providing robust theoretical convergence guarantees and improved empirical performance in settings where worst-case optimization is computationally prohibitive or overly pessimistic.

1. Conceptual Foundations and Frameworks

RFD departs from worst-case analysis by optimizing under a distributional or stochastic view. Instead of assuming the objective function is arbitrary within a regularity class (e.g., L-smooth), some RFD variants treat it as a realization from a probability distribution (often a Gaussian process). The update at each step seeks to minimize the conditional expectation of the cost, leading to an average-case-optimal progression. This contrasts with the harsh regularization of standard first-order methods derived from worst-case smoothness or convexity assumptions (Benning et al., 2023, Benning, 22 Jul 2025).

The general RFD update can be written as: $$\theta_{n+1} := \argmin_{\theta} \; \mathbb{E}\left[\, \mathbf{J}(\theta) \mid \mathbf{J}(\theta_n),\, \nabla \mathbf{J}(\theta_n) \right]$$ where $\mathbf{J}$ represents the random cost function. This stochastic Taylor approximation makes RFD scale-invariant and forms the foundation for theoretically derived, adaptive step size selection.

2. Algorithmic Realizations

Several algorithmic paradigms realize the RFD framework:

• Random Search Direction SGD: At each step, the descent is made along a random direction, not necessarily aligned with the negative gradient, but chosen such that the expectation (over random directions) recovers the standard gradient direction in aggregate. The stochastic update is given by:

$$X_{n+1} = X_n - \gamma_n D(V_{n+1}) \nabla f_{U_{n+1}}(X_n)$$

where $V_{n+1}$ is a random direction and $D(v) = v v^\top$ ensures expected directionality (Gbaguidi, 25 Mar 2025).

• Bayesian-Inspired Step Size Schedules: RFD uses a "stochastic Taylor" expansion to derive a step size $\Delta^*$ that solves:

$\Delta_n^* := \argmin_{\Delta} \left\{ \frac{\ikernel(\Delta^2/2)}{\ikernel(0)} (J(\theta_n)-\mu) - \Delta\, \frac{\ikernel'(\Delta^2/2)}{\ikernel'(0)} \|\nabla J(\theta_n)\| \right\}$

This underpins heuristic practices such as learning rate warmup and gradient clipping (Benning et al., 2023, Benning, 22 Jul 2025).

• Random Coordinate/Subspace Descent: Iteratively updating a random block or subspace (instead of the entire parameter vector) reduces per-iteration cost. Notably, random coordinate methods for linearly constrained convex problems achieve $\mathcal{O}(N^2/\epsilon)$ iteration complexity, with $N$ block coordinates, and strong numerical efficiency in high dimensions (Necoara et al., 2013, Chen et al., 2020).
• Swarm-Based Random Descent: Agents explore the parameter space with directions randomized within a cone around the negative gradient, and with individual step sizes and mass transfers according to their performance, balancing exploitation and exploration (Tadmor et al., 2023).

3. Theoretical Properties

RFD frameworks provide both average-case optimality and strong convergence results under various structural assumptions:

• Convergence Rates:
  • For smooth convex problems, random search direction SGD achieves $O(1/\sqrt{n})$ or better, and with an exact line-search or variance reduction, linear convergence under strong convexity (Gbaguidi, 25 Mar 2025, Lorenz et al., 2023).
  • Bayesian-inspired RFD yields an asymptotically optimal and scale-invariant step size schedule that adapts to the local curvature and uncertainty (Benning et al., 2023, Benning, 22 Jul 2025).
  • Random Coordinate/Subspace Descent achieves strong complexity bounds for composite convex objectives with linear coupling, outperforming full gradient methods when per-coordinate computation is cheaper (Necoara et al., 2013, Chen et al., 2020).
• Central Limit Theorem and Noise Control:
  • Under mild smoothness, the distribution of iterates around the optimum is asymptotically normal with an explicitly characterizable covariance matrix that reflects the choice of random direction distribution (Gbaguidi, 25 Mar 2025).
• Non-Asymptotic Guarantees:
  • Explicit $\mathbb{L}^p$ rates—e.g., $\mathbb{E}\|X_n - x^*\|^2 \leq K / n^\alpha$ for step size $\gamma_n = c/n^\alpha$, $1/2 < \alpha \leq 1$—hold for random search direction SGD (Gbaguidi, 25 Mar 2025).
• Scale Invariance:
  • Updates derived from the stochastic Taylor expansion are invariant under affine transformations of parameters and costs, increasing robustness across various problem rescalings (Benning et al., 2023).

4. Implementation and Computational Considerations

A major motivation for RFD is computational scalability in high-dimensional and large-sample regimes:

• Complexity per Iteration:
  • Bayesian optimization with full covariance updates scales as $\mathcal{O}(n^3d^3)$, whereas RFD-style updates, by reducing trust region computations to local conditional expectations, achieve $\mathcal{O}(nd)$ complexity (Benning et al., 2023).
  • Random coordinate and subspace methods further reduce cost by only updating a fraction of parameters per iteration, and, when properly parallelized, can operate at very large scales (Necoara et al., 2013, Chen et al., 2020).
• Adjoint-Free and Transpose-Free Variants:
  • In settings where only forward operator evaluations are possible (e.g., ill-posed inverse problems), random descent with exact line search along random directions eliminates the need for adjoint computations, outperforming established transpose-free methods such as TFQMR and CGS in certain cases (Lorenz et al., 2023).
• Search Distribution Choice:
  • The choice of random direction distribution influences both the convergence rate and the asymptotic noise (variance). Spherical, Gaussian, or coordinate sampling can be optimized depending on the problem structure, with explicit formulas linking the search distribution to the asymptotic covariance (Gbaguidi, 25 Mar 2025).

5. Applications and Practical Performance

RFD and its variants have demonstrated strong empirical and theoretical performance across a spectrum of domains:

• Large-Scale Machine Learning:
  • Numerical studies on support vector machines, convex regression, and logistic regression illustrate RFD’s speed and robustness, especially for massive datasets where full gradient methods are infeasible (Necoara et al., 2013, Gbaguidi, 25 Mar 2025).
  • In deep learning, the RFD step size schedule explains heuristic tools such as learning rate warmup and gradient clipping in convolutional network training scenarios (e.g., MNIST benchmarks), and matches or surpasses Adam/SGD in stability and convergence (Benning et al., 2023).
• Ill-posed Inverse Problems:
  • Random descent with line search effectively handles rough signals and semi-convergence in regularized least squares without access to operator adjoints, often beating classical solvers in both speed and accuracy for inverse integration and other problems (Lorenz et al., 2023).
• Nonconvex and Global Optimization:
  • Swarm-based random descent finds global minima in high-dimensional multimodal landscapes, outperforming deterministic swarm-based gradient descent in standard benchmarks (Ackley, Rosenbrock, Rastrigin, etc.) due to improved exploration properties (Tadmor et al., 2023).

6. Relation to Other Approaches and Methodological Comparisons

RFD both generalizes and complements existing method classes:

Method Class	Key Feature	Relationship to RFD
Gradient Descent (GD)	Full-gradient, conservative step-size via $1/L$	RFD recovers GD as special case under deterministic functions
Bayesian Optimization (BO)	Average-case, high complexity	RFD bridges BO and GD with local, scalable average-case steps
Coordinate/Subspace Descent	Low per-iteration cost, random subsampling	RFD encompasses random direction and coordinate updates
Swarm-Based Algorithms	Multi-agent, mass/energy transfer, global search	RFD includes swarm-randomized descent as a subset

Key distinctions arise in adaptivity (RFD’s step size is theoretically derived per step), scalability (RFD avoids cubic matrix inversions in high dimensions), and average-case optimality (leveraging distributional assumptions on the cost).

7. Future Directions

Several open research avenues and methodological enhancements are suggested in the literature:

• Beyond Gaussian Assumptions:

Adapting the stochastic Taylor expansion to non-Gaussian processes, possibly via best linear unbiased estimators or other conditional expectation models (Benning et al., 2023).

• Momentum and Memory:

RFD, in its basic form, only exploits information from the latest step. Incorporating momentum or adaptive smoothing may improve performance and stability in risk-affine or noisy regimes.

• Adaptive and Problem-Specific Search Distributions:

Investigating search direction distributions or subspace decompositions suited to particular problem structures, including anisotropic or non-stationary objectives (Gbaguidi, 25 Mar 2025, Benning, 22 Jul 2025).

• Variance Reduction and Hybridization:

Combining RFD updates with variance reduction for stochastic gradients or integrating with coordinate selection heuristics to accelerate convergence.

• Theoretical Analysis in Overparameterized Regimes:

Understanding double descent and generalization in random features models trained by SGD, with explicit non-asymptotic error bounds influenced by step sizes and randomness (Liu et al., 2021).

RFD's probabilistic, scalable, and flexible framework positions it as a unifying principle for future developments in optimization, particularly in high-dimensional and data-intensive applications.