Gradient-Free Optimization

Weight perturbation (WP) estimates learning directions from random parameter perturbations and a scalar change in loss, without backpropagating derivatives. We examine how batch size, training duration, and estimator stabilization affect its attainable accuracy in fully connected MNIST networks. In the shallow configuration, large-batch training reaches approximately 97% test accuracy, compared with an early backpropagation (BP) reference near 97.5--98%. The same comparison exposes a substantial efficiency gap: under a ten-epoch budget, shallow WP reaches 90.2%, whereas the reported BP reference reaches 97.4%. Epoch-indexed curves show diminishing returns from increasing batch size beyond 4,000. Learning-rate reduction and averaging independent perturbation-based updates also stabilize deeper networks, although their recorded results do not establish BP-level accuracy. We distinguish these empirical findings from a finite-horizon approximation argument: under stated regularity conditions, population WP can approximate a fixed gradient-descent trajectory arbitrarily closely at any fixed finite depth. Positive classification margins then suffice to preserve the reference predictions on a finite test set with arbitrarily high probability. Together, the results identify near-BP accuracy as a configuration- and budget-dependent possibility, rather than evidence of equal efficiency or an unconditional accuracy guarantee across depth.

Photoacoustic tomography (PAT) is a hybrid imaging modality that combines high optical contrast with high ultrasonic resolution for biomedical imaging applications. In this work, we investigate the inverse problem of recovering the initial pressure distribution from boundary measurements in the presence of nonlinear acoustic propagation and viscous attenuation effects. To model these phenomena more accurately, we consider a nonlinear damped viscoelastic wave equation incorporating spatially varying sound speed, temporal attenuation, and nonlinear propagation mechanisms. We first establish the well-posedness of the corresponding forward problem using a Galerkin approximation combined with energy estimates and a fixed-point argument. For the inverse problem, we derive existence, uniqueness, and local uniqueness results under suitable assumptions through a harmonic extension reduction, spectral Laplace transform techniques, and observability estimates. To numerically reconstruct the initial pressure field, we develop a hybrid reconstruction framework that combines a convolutional neural network (CNN) with a gradient-free optimization strategy based on the sequential quadratic Hamiltonian (SQH) method derived from Pontryagin's maximum principle. The CNN is used to generate an informative initial guess, while the SQH framework enforces the governing PDE dynamics during the reconstruction process. Numerical experiments demonstrate that the proposed hybrid strategy significantly improves reconstruction quality, contrast, and robustness compared to standalone time-reversal and CNN-based approaches.

Diffusion policies generate multimodal robot action sequences from demonstrations, but steering them toward deployment-time constraints typically relies on differentiable guidance costs. This excludes many practical safety constraints, such as binary collision checks, joint limits, and black-box rollout costs that are nondifferentiable. We propose Gradient-free Robot Action generation via Combined diffusion-MPPI posterior mean Estimation (GRACE), which guides a pretrained diffusion policy with Model Predictive Path Integral (MPPI) control using only forward cost evaluations. Building on the common score-ascent structure of diffusion and MPPI, GRACE constructs a cost-conditioned guidance posterior at each reverse step and estimates its mean with a single MPPI update centered at the diffusion reverse mean. For differentiable costs, GRACE recovers conventional gradient guidance under a first-order, matched-covariance approximation. GRACE attains higher success rates than diffusion-based and sampling-based baselines in simulation. On a real 7-DoF manipulator, GRACE avoids a deployment-time obstacle that the unguided prior collides with in every trial. Code and experiment videos are available at https://anonymous.4open.science/w/grace-70BB/.

Backpropagation (BP) dominates deep learning training, but its reliance on gradients brings inherent troubles -- vanishing and exploding gradients. The pursuit of gradient-free methods has long been a goal in the field of artificial intelligence. This paper shows that indeed the simplest Monte Carlo algorithm implemented on a single GPU -- randomly mutate a parameter, keep it if the loss decreases, otherwise retry -- can practically train deep networks. This gradient-free method does not even need common techniques such as batch normalization or residual connections to directly train sufficiently deep networks. More remarkably, its flexibility extends to several nontrivial scenarios: it enables pure pruning training, supports discrete weights, accommodates unconventional transfer functions such as Gaussian, and reveals the substantial redundancy of deep networks. We have demonstrated its feasibility on deep networks with more than 20 layers, single-hidden-layer wide networks with up to 16,384 hidden neurons, and even a simple Transformer architecture trained on both image classification (MNIST) and character-level language modeling (Tiny Shakespeare). This simple gradient-free method may offer a complementary perspective for understanding the self-organization and learning mechanisms of neural networks, and also provides an alternative route for building physically inspired deep learning systems.

Continual learning that is gradient-free, local, online, and append-only is attractive for edge and streaming deployment, but its value is usually argued informally. We give a provable account on recurring-regime streams. Given segmentation, a warm-start library learner attains amortized recovery cost $O\!\big(KD/\varepsilon^2+(R-K)\logK/Δ^2\big)$ versus a memoryless re-estimator's $Θ(RD/\varepsilon^2)$, an advantage $(R-K)\,Θ(D/\varepsilon^2)$ growing with dimension $D$ and recurrence density. The mechanism is a decoupling: recognizing which of $K$ seen regimes is active costs $O(\log K/Δ^2)$, independent of $D$, whereas estimating a regime costs $Θ(D/\varepsilon^2)$. We prove this is tight: matching lower bounds give recognition $Θ(\log K/Δ^2)$ and a memoryless-class bound $Ω(RD/\varepsilon^2)$, so each term is individually minimax-tight (the joint statement is conditional). The separation is born-immune (a memoryless learner's advantage is identically zero) and paradigm-level: it matches, and does not beat, a fair spawn-capable Bayesian baseline; the contribution is attaining this cost structure without end-to-end backprop and with zero forgetting by construction. A count-calibrated variant ties the baseline's leading constant up to a bounded, never-negative per-recurrence overshoot, hyperparameter-free and with no per-step transcendentals. We bound the scope: recognizable regimes are capped by simplex packing (walls $e^{Θ(D)}$); autonomous segmentation is impossible at the packing wall (no detector escapes the false-alarm/delay frontier as regimes overlap); the advantage vanishes under overlap. The dimension-dependent separation is corroborated on synthetic streams and real $k$-mer genome distributions (memoryless cost $\propto D^{1.04}$, recognition $D$-independent); the one real sequential stream sits in the $D{=}1$ near-null corner.

Zeroth-order (ZO) optimization is a memory-efficient alternative to backpropagation for fine-tuning large language models, but its deployment is limited by the high variance of gradient estimation. We propose GRZO, a Group-Relative Zeroth-Order optimizer that draws one pseudo-independent perturbation per mini-batch example and aggregates the per-example losses through group-relative normalization, raising the effective gradient-direction count from one to the batch size at no additional forward cost while preserving inference-level memory. We prove that GRZO is directionally unbiased with variance shrinking proportionally to the batch size, yielding a tighter nonconvex convergence bound than MeZO. Across RoBERTa-large, Llama3-8B, and OPT-13B over multiple tasks, GRZO improves average accuracy on Llama3-8B by $+3.0$ over MeZO at $23\%$ lower peak GPU memory; as a drop-in replacement for the MeZO core, it lifts sparse, low-rank, and quantized ZO variants by $+6.0$ on average.

Many networks not only support but also rely on transient non-normal amplification, an orders-of-magnitude increase in the activity of an otherwise stable system. Constructing such networks under hard sign/sparsity/diagonal constraints -- the regime relevant for biological connectomes and structured RNN initializations -- has so far required either gradient-based local search with thousands of inner-loop eigendecompositions or Schur-form direct construction in an abstract basis that breaks the constraints under projection. Here we introduce DAGGER (Directed Acyclic Graph Guided Edge Reweighting), a gradient-free single-pass algorithm. Given a stable signed sparse matrix, DAGGER produces an output with the same sign, sparsity, and diagonal. A single scalar $β$ controls a Wasserstein-2 budget that smoothly trades exact multiset preservation ($β= 0$) for amplification; peak amplification grows essentially without bound with $β$, empirically reaching $10^{10}$ before numerical overflow. DAGGER matches or exceeds gradient-based methods at multiset preservation in a single forward pass -- 30-100$\times$ fewer eigendecompositions than a typical gradient inner loop -- and at moderate $β$ beats them by orders of magnitude with connectivity exactly preserved. We develop the algorithm, compare it to the existing methods and on a downstream signal-detection task, and examine the diagnostics that show why DAGGER is structurally different from other amplifying networks.

Text-agent environments are typically modeled as partially observable Markov decision processes (POMDPs), assuming that the simulator's latent state and transition dynamics are hidden from the agent. Yet little work has examined whether executable code can be induced to serve as a world model for prediction and planning under partial observability. We introduce PatchWorld, a gradient-free framework that turns offline trajectories into executable Python world models through counterexample-guided code repair. Instead of predicting the next observation with a black-box model, PatchWorld induces symbolic belief-state programs whose action updates can be inspected, replayed, and locally patched. Across seven AgentGym environments, PatchWorld-Simple achieves the highest code-based planning score among evaluated methods, reaching 76.4% macro success in live one-step lookahead while invoking no LLM calls inside the world-model prediction module itself. We further find that a human-specified residual-memory bias improves surface observation fidelity but weakens decision utility. This exposes a tradeoff in executable world models, since improving observation fidelity can come at the expense of action-discriminative dynamics, and vice versa. Code is available at https://github.com/HKBU-KnowComp/PatchWorld.

Spiking Neural Networks (SNNs) offer compelling energy efficiency on neuromorphic hardware, yet their training remains challenging because the discrete spike threshold is non-differentiable. Surrogate-gradient methods sidestep this by approximating the derivative, but they impose backpropagation infrastructure that is incompatible with on-chip learning. Evolution Strategies (\es) are a natural gradient-free alternative, yet their computational cost scales with the number of parameters, making them impractical for large weight matrices. We present a method for training SNNs using EGGROLL, a low-rank factorisation of ES perturbations that reduces per-generation memory from $\mathcal{O}(mn)$ to $\mathcal{O}(r(m{+}n))$. Combining EGGROLL with a Leaky Integrate-and-Fire SNN on N-MNIST, we demonstrate that gradient-free training achieves 79.21% test accuracy while reducing per-generation wall-clock time by 2.23$\times$ relative to full-rank ES. Our results demonstrate EGGROLL is viable for SNN training, with a clear accuracy-speed tradeoff, compatible with training on neuromorphic hardware without surrogate gradients.

Existing reward alignment methods for diffusion and flow models rely on multi-step stochastic trajectories, making them difficult to extend to deterministic generators. A natural alternative is noise-space optimization, but existing approaches require backpropagation through the generator and reward pipeline, limiting applicability to differentiable settings. To address this, here we present ZeNO (Zeroth-order Noise Optimization), a gradient-free framework that formulates noise optimization as a path-integral control problem, estimable from zeroth-order reward evaluations. When instantiated with an Ornstein--Uhlenbeck reference process and a one-step receding horizon, the repeated update connects to Langevin dynamics targeting a reward-tilted distribution. ZeNO enables effective inference-time scaling and demonstrates strong performance across diverse generators and reward functions, including a protein structure generation task where backpropagation is infeasible.

Heterogeneity is a defining feature of deployed multi-robot teams: platforms often differ in sensing modalities, ranges, fields of view, and failure patterns. Controllers trained under nominal sensing can degrade sharply when deployed on robots with missing or mismatched sensors, even when the task and action interface are unchanged. We present DC-Ada, a reward-only decentralized adaptation method that keeps a pretrained shared policy frozen and instead adapts compact per-robot observation transforms to map heterogeneous sensing into a fixed inference interface. DC-Ada is gradient-free and communication-minimal: it uses budgeted accept/reject random search with short common-random-number rollouts under a strict step budget. We evaluate DC-Ada against four baselines in a deterministic 2D multi-robot simulator covering warehouse logistics, search and rescue, and collaborative mapping, across four heterogeneity regimes (H0--H3) and five seeds with a matched budget of $200{,}000$ joint environment steps per run. Results show that heterogeneity can substantially degrade a frozen shared policy and that no single mitigation dominates across all tasks and metrics. Observation normalization is strongest for reward robustness in warehouse logistics and competitive in search and rescue, while the frozen shared policy is strongest for reward in collaborative mapping. DC-Ada offers a useful complementary operating point: it improves completion most clearly in severe coverage-based mapping while requiring only scalar team returns and no policy fine-tuning or persistent communication. These results position DC-Ada as a practical deploy-time adaptation method for heterogeneous teams.

Evolution Strategies (ES) have emerged as a scalable gradient-free alternative to reinforcement learning based LLM fine-tuning, but it remains unclear whether comparable task performance implies comparable solutions in parameter space. We compare ES and Group Relative Policy Optimization (GRPO) across four tasks in both single-task and sequential continual-learning settings. ES matches or exceeds GRPO in single-task accuracy and remains competitive sequentially when its iteration budget is controlled. Despite this similarity in task performance, the two methods produce markedly different model updates: ES makes much larger changes and induces broader off-task KL drift, whereas GRPO makes smaller, more localized updates. Strikingly, the ES and GRPO solutions are linearly connected with no loss barrier, even though their update directions are nearly orthogonal. We develop an analytical theory of ES that explains all these phenomena within a unified framework, showing how ES can accumulate large off-task movement on weakly informative directions while still making enough progress on the task to match gradient-based RL in downstream accuracy. These results show that gradient-free and gradient-based fine-tuning can reach similarly accurate yet geometrically distinct solutions, with important consequences for forgetting and knowledge preservation. The source code is publicly available: https://github.com/Bhoy1/ESvsGRPO.

Backpropagation dominates modern machine learning, yet it is not the only principled method for optimizing dynamical systems. We propose Kalman World Models (KWM), a class of learned state-space models trained via recursive Bayesian filtering rather than reverse-mode automatic differentiation. Instead of gradient descent updates, we replace parameter learning with Kalman-style gain adaptation. Training becomes online filtering; error signals become innovations. We further extend this framework to transformer-based large language models (LLMs), where internal activations are treated as latent dynamical states corrected via innovation terms. This yields a gradient-free training and adaptation paradigm grounded in control theory. We derive stability conditions, analyze computational complexity, and provide empirical results on sequence modeling tasks demonstrating competitive performance with improved robustness and continual adaptation properties.

Federated Learning (FL) is a popular distributed learning paradigm to break down data silo. Traditional FL approaches largely rely on gradient-based updates, facing significant issues about heterogeneity, scalability, convergence, and overhead, etc. Recently, some analytic-learning-based work has attempted to handle these issues by eliminating gradient-based updates via analytical (i.e., closed-form) solutions. Despite achieving superior invariance to data heterogeneity, these approaches are fundamentally limited by their single-layer linear model with a frozen pre-trained backbone. As a result, they can only achieve suboptimal performance due to their lack of representation learning capabilities. In this paper, to enable representable analytic models while preserving the ideal invariance to data heterogeneity for FL, we propose our Deep Analytic Federated Learning approach, named DeepAFL. Drawing inspiration from the great success of ResNet in gradient-based learning, we design gradient-free residual blocks in our DeepAFL with analytical solutions. We introduce an efficient layer-wise protocol for training our deep analytic models layer by layer in FL through least squares. Both theoretical analyses and empirical evaluations validate our DeepAFL's superior performance with its dual advantages in heterogeneity invariance and representation learning, outperforming state-of-the-art baselines by up to 5.68%-8.42% across three benchmark datasets.