Optimization Landscape

UAVs have emerged as highly flexible platforms for data sensing in Wireless Sensor Networks (WSNs). Path planning for UAVs in such tasks plays a key role to assure remote sensing effectiveness and friendly energy consumption. However, existing approaches show two key limitations: i) they are primarily hand-crafted with certain design biases that harm adaptation on unseen tasks. ii) they predominantly assume idealized spatial complexities of actual environments through simplified simulation, causing them to underperform during real-world deployment. In this paper, we propose a novel learning-assisted planning framework, termed Landscape-Aware Meta Differential Evolution (LAMDE), to tackle the mentioned limitations. The major contributions come from the following aspects. We first re-formulate such UAV path planning problem to embrace challenging constraints. To efficiently navigate this highly constrained space, we propose a bi-level learning to optimize approach, where the meta-level is a trainable algorithm configuration policy that meta-learns an adaptable planning strategy for low-level planning algorithm. To address the potential training data scarcity and distribution shift in real-world environments, we introduce a landscape-aware automatic augmentation scheme that enriches training data. At the low-level, a Differential Evolution algorithm is deployed for solving the path planning tasks. To enhance the solving flexibility, we further design a variable-length encoding strategy that dynamically prunes redundant hover points and optimizes continuous flight parameters concurrently within a unified search space. Based on all proposed designs, we meta-train LAMDE and compare it with representative baselines. Comprehensive experiments demonstrate that LAMDE achieves state-of-the-art performance on the tested complex UAV path planning tasks in WSN data collection scenarios.

We study zeroth-order optimization of non-convex functions with the aid of directional hints, which are cheap but potentially inaccurate approximations of the true gradient direction, given by linear subspaces at each iteration. To leverage these hints adaptively while maintaining robustness to their quality, we introduce Control-Variate Zeroth-Order Descent (CV-ZOD), a new framework that refines the classical zeroth-order gradient estimator with a control variate that can be set based on the directional hints. We first show that the oracle algorithm that optimally sets the reference vector and step size at each iteration achieves a convergence rate that interpolates between the first-order $O(1/T)$ rate and the zeroth-order $O(d/T)$ rate, depending on the quality of the hints along the trajectory. We then develop a practical variant of CV-ZOD that achieves the same oracle guarantee up to logarithmic factors, without any prior knowledge of the hint quality. We validate the method empirically on simulation-based scientific optimization tasks, demonstrating sustained progress on non-convex landscapes where zeroth-order descent is slower and existing guided methods stall as guidance deteriorates.

Permutation optimization arises whenever the components of a system are fixed but their ordering affects performance. We introduce PRISM, a predictive protocol for permutation optimization that measures a fitness landscape before selecting a search strategy. PRISM uses inexpensive landscape diagnostics, including one-step move autocorrelation and fitness-distance correlation, to predict useful mutation operators, identify when structured search is likely to outperform random sampling, and detect regimes in which search provides little advantage. Across synthetic permutation landscapes, neural architecture benchmarks, scientific machine learning pipelines, and large-language-model instruction ordering, the protocol makes testable predictions about search behavior before optimization begins. Exhaustive instruction-ordering experiments reveal substantial performance variation induced solely by permutation, while cross-model experiments show that useful ordering structure can transfer across model families and task difficulty. Additional experiments demonstrate that instruction ordering remains consequential after prompt wording is optimized, indicating that content optimization and ordering optimization are complementary. The results position PRISM not as a universally superior optimizer, but as a framework for determining when permutation search is useful, which representation and operator should be used, and when simpler alternatives are preferable.

This work revealed a relationship between a multi-objective optimization problem and single-objective optimization problems that exist in the multi-objective problem. This work focused on combinatorial problems and investigated the relations between the local optima networks of the single-objective problems and the Pareto optima network of the multi-objective problem. Each of their networks has a graph structure. We divided the entire network into subgraphs. Each subgraph was called a component and characterized by overlapping relations between the single-objective local optima networks and the multi-objective Pareto optima network. Results on multi-objective landscape problems showed that most Pareto optimal solutions were reachable from the single-objective local optimal solutions. This tendency was emphasized by increasing the number of objectives and the objective correlation. The number of co-variables impacted the number of cross-link relations between the single-objective local optima networks and the multi-objective Pareto optima network. The results suggested that searching for single-objective problems is a clue to multi-objective optimization.

An ensemble of surrogate models helps improve the prediction quality and robustness of surrogate models, and in turn, the search performance of surrogate-assisted evolutionary algorithms (SAEAs). Although different degrees of smoothness of the approximated fitness landscapes need to be carefully designed for an effective ensemble, little attention has been paid to the explicit tuning of the degree of smoothness derived by surrogate models. This study proposes an adaptive ensemble SAEA, which automatically constructs plausible ensemble models by optimizing their parameter settings. Unlike existing adaptive/ensemble SAEAs, which consider prediction accuracy alone, the proposed algorithm optimizes the structure of radial basis function networks (RBFNs) by solving bi-objective minimization problems of approximation error and model complexity, resulting in robust ensemble models of accurate surrogate models with different degrees of smoothness of the approximated fitness landscapes. As a result, the over/under-fittings are reduced. Additionally, an infill criterion is designed so that surrogate models with different degrees of smoothness can contribute to the solution prescreening. The experimental results demonstrated the statistical superiority of our algorithm over state-of-the-art SAEAs on a single-objective benchmark and real-world problem sets under an expensive optimization scenario. The source code of the proposed algorithm is available at https://github.com/haranychan/EPOS

Anomaly detection is inherently characterised by severe class imbalance, making the interpretation of evaluation metrics challenging. Although metrics such as AUROC, AUPR, F1-score, and MCC are widely used, their values convey different meanings depending on the anomaly ratio. In this work, we analyse the behaviour of those four common anomaly detection metrics under varying levels of imbalance. We focus on the study of metric landscapes, visualisations that relate metric values to true positive and true negative rates, providing an intuitive view of metric preferences and stability. Our analysis offers practical guidance for interpreting and comparing anomaly detection results across datasets with different imbalance ratios.

Discovering governing partial differential equations (PDEs) from noisy observational data is a fundamental challenge in scientific machine learning. Traditional symbolic regression (SR) methods often struggle to identify accurate equations within vast combinatorial search spaces, largely due to their inability to incorporate essential domain-specific prior knowledge. Furthermore, reliance on pointwise evaluations and discrete finite differences inherently amplifies high-frequency noise, creating deceptive fitness landscapes that derail the optimization process. To resolve these bottlenecks, we propose LLM-PDESR, a framework that integrates the structural hypothesis generation of Large Language Models (LLMs) with a mathematically rigorous evaluation environment. By employing C^4-continuous quintic splines for robust differentiation and subdomain weighted residuals as natural low-pass filters, our approach effectively mitigates the fitness landscape distortion that plagues existing methods. A Pareto-driven feedback loop then enables the LLM to iteratively refine candidate equations, balancing predictive accuracy with structural parsimony. We evaluate LLM-PDESR on 23 canonical PDEs and five structurally novel equations (including a multivariate system) specifically designed to preclude dataset memorization and test true discovery capabilities. Demonstrating real-world applicability, the framework successfully extracts a consistent structural skeleton for an interpretable 1D dynamical surrogate (1D-CACE) directly from noisy ERA5 reanalysis data. Extensive experiments and out-of-distribution testing confirm that LLM-PDESR significantly outperforms state-of-the-art methodologies in structural recovery, noise resilience, and the avoidance of spurious complexity and equation bloat.

Classical space-filling designs often fail to provide reliable statistical results for Exploratory Landscape Analysis (ELA) when only limited evaluation budgets are available, as commonly occurs in high-dimensional problems or other resource-constrained settings, resulting in noisy and unstable landscape descriptors. To address this challenge, we propose an alternative sampling strategy for ELA based on random linear embeddings. Rather than sampling uniformly in the full decision space, we allocate the budget to randomly oriented low-dimensional subspaces and investigate whether this improves the robustness of the resulting landscape descriptors. We compare full-space and embedding-based sampling strategies across several classical ELA feature sets on the noiseless Black-Box Optimization Benchmarking (BBOB) test suite from the COmparing Continuous Optimizers (COCO) environment, in a 20-dimensional setting. Our results suggest that random linear embeddings constitute a promising alternative for budget-constrained ELA, although their effectiveness remains dependent on the feature class and the underlying problem.

The Self-Improving Alignment (SAIL) algorithm addresses distribution shift by reducing a bilevel formulation of the problem to an efficient, single-level method. Empirically, SAIL has demonstrated strong performance on this task. However, a formal analysis of its convergence properties has been lacking. We identify a key theoretical challenge: the standard SAIL objective function is not guaranteed to be strongly concave due to unfavorable properties of its Hessian. To address this limitation, we propose a regularized objective, SAIL-RevKL, which incorporates a reverse Kullback-Leibler (KL) divergence penalty to improve the optimization landscape. Our central theoretical contribution is to prove that this regularized objective satisfies the Polyak-Lojasiewicz (PL) condition within a bounded parameter space. We establish global convergence guarantees, achieving a near-linear sample complexity. We further validate the effectiveness and stability of SAIL-RevKL through empirical evaluations, demonstrating that it outperforms the vanilla SAIL on both MuJoCo benchmarks and LLM alignment tasks.

In this work, Neural ODEs equipped with a curated collection of equilibrium points have been successfully employed for classification tasks. The planted attractors serve as indicators for the target classes, while the velocity field leveraging the universal approximation capabilities of the architecture shapes the dynamical landscape. This process defines the basins of attraction of the trained model, effectively directing each input (provided as an initial condition) toward its corresponding destination target.

We introduce a new simple model to study the fitness progress of Evolution Strategies (ES) in generic problems. In this model, we bypass the underlying fitness landscape and assume that the mutation of any individual produces an offspring whose fitness relative to the parent is given by an invariant distribution $Z$, such as a mean-shifted Gaussian. This serves as a prototypical model for the optimisation landscape when an evolution algorithm operates far from the global optimum. This simple model can be used to approximate the optimisation process for problems where it is intractable to model the exact fitness function, including tasks such as hyperparameter tuning in machine learning models. We rigorously analyse the expected growth rate $\mathcal{R}_μ$ of the continuous steady-state $(μ+1)$-ES in this model. Unlike comma-selection strategies, the steady-state $(μ+1)$-ES maintains overlapping generations, introducing complex mathematical dependencies among surviving parents that make it harder to analyse. We give a general technique to analyse the the $(μ+ 1)$-ES by constructing modified processes whose growth rates provably sandwich that of the original process. These modified processes are then easier to analyse but still close enough to the true process to give a tight bound on the expected growth rate. When $Z = \mathcal{N}(-δ, 1)$ and $μ\le e^δ$, we show that $\mathcal{R}_μ = \frac{\log^{1 + o(1)} μ}μ \mathcal{R}_1$.

Auto-regressive models have emerged as powerful tools for sequential data, from language to video. Understanding how and why these models learn latent representations remains an open theoretical question. In this work, we demonstrate that when trained by empirical risk minimization on data from partially observed linear dynamical systems, two-layer linear auto-regressive models naturally learn to approximate Kalman filtering. In particular, we show that the learned hidden representation coincides, up to a similarity transformation, with the state estimates produced by the optimal (Kalman) filter, even though the model has no explicit knowledge of the underlying dynamics or state. The result follows from three main insights. First, we establish that the Kalman filter is well approximated by an auto-regressive model with bounded truncation error. Second, we show that despite non-convexity, the two-layer optimization landscape is benign, i.e., all stationary points are either strict saddles or global minima. Finally, as our main contributions, we provide finite-sample guarantees on prediction error, parameter estimation error, and latent state recovery. Numerical simulations support the theoretical results and demonstrate that the latent representations of auto-regressive models recover state estimates.

Knowledge graph completion (KGC) models are commonly evaluated using rank-based metrics such as MRR and Hits@K, despite different users often requiring different evaluation perspectives. In this demo, we present PROBE-Web, an interactive system for probing diverse evaluation landscapes for KGC models. PROBE-Web enables users to flexibly evaluate KGC models by adjusting two critical perspectives: (P1) predictive sharpness and (P2) popularity-bias robustness. Through a user-friendly GUI, users easily evaluate multiple KGC models and analyze their strengths and weaknesses. PROBE-Web provides four key functionalities: (1) conventional evaluation toolkit, (2) flexible perspective-aware evaluation, (3) explainable case studies, and (4) evaluation landscape exploration. We believe that PROBE-Web can help users better understand KGC models aligning with their objectives.

We analyze the two combinatorial problems of Dominating Set and Vertex Coloring regarding what kind of local optima are present for various instances. For a variety of graph classes each, we determine whether the induced landscapes are unimodal, plateau-unimodal (all optima are just one plateau), equimodal (all local optima are global) or truly multimodal. We do this for two different neighborhood operators, one based on making only a single change and one also allowing swaps (interchanging two parts of the solution).

AI-Driven Research Systems (ADRS) -- systems coupling LLMs with automated evaluation to discover algorithms, proofs, and designs -- are being optimized and adopted across domains, but the tools to analyze them have not kept pace. ADRS performance depends on component interactions that are poorly understood, expensive to explore, and (as we show) not well captured by standard convergence guarantees. These guarantees rely on structural assumptions that do not hold under the ADRS process we formalize. We introduce GAMBLe, a framework that decomposes ADRS behavior into four parameters (generator $G$, assessor $\mathcal{A}$, discovery mechanism $\mathcal{M}$, budget $B$) and one compositional object, the effective landscape $L_{\text{eff}} = \mathcal{A} \circ G$, which reveals that distinct generator-assessor pairs induce structurally different per-problem optimization landscapes. We exercise the framework on 760+ replicated runs (>46,000 iterations) spanning generators from single LLMs to dynamically-adaptive ensembles, mechanisms from greedy selection to co-evolutionary meta-search, and three NP-hard problems whose assessors range from continuous scoring to cliff functions. The experiments reveal no total ordering of generators or mechanisms: frontier models can underperform open-source alternatives and the simplest mechanism sometimes outperforms state-of-the-art meta-search. Results show that even under limited budgets (60 iterations per run), the right component choices can improve performance by 13-67% and search efficiency by 6-39x.

Algorithm Selection (AS) aims to automatically identify the most suitable optimization algorithm for a given problem instance by leveraging measurable problem characteristics and historical performance data. In this study, we investigate the generalization ability of AS models across both synthetic and real-world optimization landscapes. We consider two widely used academic benchmark suites (BBOB and CEC) and two real-world problem sets (robotics trajectory optimization tasks and unmanned aerial vehicle path-planning problems). Through a systematic cross-benchmark evaluation, we analyze how AS models transfer between domains, identify where generalization succeeds or breaks down, and highlight the challenges that arise when applying AS in realistic, domain-specific contexts. Our findings provide insights into the robustness of current AS approaches and inform the development of more reliable, broadly applicable AS systems for real-world optimization.

This paper investigates the optimization mechanisms of non-convex Model Predictive Control (MPC) using the Maximum Entropy Differential Dynamic Programming (ME-DDP) framework. Navigating non-convex cost landscapes induced by nonlinear dynamics, multiple obstacles, etc. remains a fundamental challenge in robotics, where gradient-based methods frequently converge to suboptimal local minima. We demonstrate a dual-step optimization mechanism designed to overcome these traps. (1) an initial phase of using DDP to exploit the gradient of the cost landscape, followed by (2) disruption of the optimization via sampling from policies characterized by the inverse Hessian of the action-value function. We provide a rigorous analysis of this sampling mechanism of three ME-DDP variants: Unimodal Gaussian ME-DDP, Multimodal Gaussian ME-DDP, and Stein Variational DDP. Furthermore, with navigation tasks of four robotic systems under cluttered environments, we conduct extensive benchmarking of three variants of the ME-DDP, against deterministic DDP, and one of the most successful sampling-based schemes, Model Predictive Path Integral (MPPI) control with three policy parameterizations and update laws that correspond to those of ME-DDPs. The results show that in low-dimensional systems where the cost landscapes are relatively simple and local information is sufficiently representative, our framework consistently outperforms MPPIs. In high-dimensional systems, MPPI can occasionally discover aggressive maneuvers that enable it to steer the systems faster than DDP-based methods, whereas our method maintains a higher, more stable success rate. Finally, we validate the practical efficacy of the framework through hardware experiments with a quadrotor navigating a dense, non-convex obstacle field, confirming the robustness of the proposed framework for real-world deployment.

Inference-time reward alignment steers pretrained diffusion and flow-based generative models to satisfy user-specified rewards without retraining. Recently, Sequential Monte Carlo (SMC) has emerged as a powerful framework for this task by iteratively filtering and propagating multiple particles. However, we show that standard SMC-based methods often suffer from poor performance because they initialize particles from a standard prior, whereas high-reward regions in complex reward landscapes are extremely rare. Further, we show that even recent reward-aware initial sampling approaches remain vulnerable to getting trapped in local modes, as complex reward landscapes are often multi-modal. To overcome these limitations, we propose PATHS (PArallel Tempering for High-complexity reward Sampling), a novel initialization method that couples multiple sampling chains through parallel tempering. PATHS maintains a ladder of reward-tempered chains and periodically performs Metropolis swaps, enabling efficient exploration across flattened reward landscapes, thereby mitigating the mode-trapping issues. Our analysis reveals that this mechanism substantially enhances the finite-budget exploration of rare, high-reward regions that are typically challenging to sample. Experiments on layout-to-image and quantity-aware generation show that PATHS achieves consistent gains in alignment quality, particularly on complex prompts.

Landscape feature representations play a central role in automated algorithm selection and meta-learning for black-box optimization, yet little is known about how different representations agree (or disagree) in the structures they impose on problem spaces. This paper presents a systematic unsupervised evaluation of four state-of-the-art representations (ELA, DeepELA, TransOptAS, and DoE2Vec) using a diverse set of affine combinations of BBOB functions (MA-BBOB). By applying extensive clustering analyses, coverage-based stability measures, and cross-representation similarity assessments, we show that each representation organizes the same problems in markedly different ways: ELA and TransOptAS form compact geometric structures, DeepELA provides a balanced intermediate view, and DoE2Vec achieves strong semantic alignment but with substantial fragmentation. Our results reveal that no single representation dominates; rather, they capture complementary aspects of the underlying landscapes. These findings highlight the importance of multi-view analyses for understanding representation behavior and offer guidance on selecting or combining representations in downstream meta-learning and algorithm selection tasks. In addition, across two different algorithm families (Differential Evolution and Particle Swarm Optimization), we show that landscape representations face an inherent trade-off in how well they align structural landscape descriptions with observed performance, indicating that no single representation can fully capture algorithm performance.

Protein sequence optimization under tight oracle budgets requires methods that explore vast combinatorial spaces while making each evaluation informative. Existing reinforcement learning and off-policy generative approaches often degrade under surrogate noise, and position-agnostic mutation proposals risk disrupting functionally critical residues. We introduce SILO, a trajectory-level self-improvement imitation framework for oracle-budgeted protein design. SILO uses a hierarchical edit policy that decomposes each mutation into a position choice followed by a residue choice. In each active-learning round, the policy samples candidate trajectories via incremental stochastic beam search without replacement (SBS), and a UCB-based proxy ensemble, combined with an alanine-scan fitness score (AFS), selects candidates with functionally relevant edits for in silico oracle evaluation. The policy is then updated by next-action cross-entropy imitation on the round's best oracle-labeled trajectories, avoiding value-function estimation. Across eight reproduced protein fitness landscapes and five strong baselines from prior work, SILO achieves the highest maximum and top-100 mean fitness on 8 of 8 landscapes within our evaluations, often exhibiting faster early-stage improvement. In low-data and noisy-proxy stress tests on two landscapes per setting, SILO remains competitive or best when several baselines degrade. Ablations show that SBS with AFS account for much of the gains, with iterative imitation providing additional improvement. Code is available at: https://github.com/grimmlab/SILO.git

Across machine learning, biology, and physics, independently evolving systems often converge toward strikingly similar high-level structures despite radically different microscopic details. Grokking circuits converge across random seeds, evolutionary lineages rediscover similar metabolic solutions, and renormalization flows approach common fixed points. We propose the Hierarchical Emergence Framework (HEF) as a candidate universality framework for such convergence phenomena. HEF models emergence as a phase transition in a mechanism landscape constrained by thermodynamic and information-theoretic laws. The framework introduces a critical energy threshold Ec separating an exploration regime with competing mechanisms from a convergence regime governed by a unique minimum-cost mechanism. Under structural assumptions, we prove physical feasibility, derive strict metric contraction, and establish convergence toward a unique fixed-point representation independent of initial conditions. We further connect this convergence structure to causal emergence through Effective Information and mechanism competition entropy. To test the framework, we study delayed generalization ("grokking") in modular arithmetic transformers across 111 experiments. We identify a reproducible empirical fingerprint of the Ec transition: the weight norm peaks systematically before grokking in 92% of runs. Normalized accuracy curves collapse onto a tanh kink (R^2=0.93) consistent with a Landau-Ginzburg universality class, and all grokked models converge to 0.9745+/-0.014 regardless of initialization, weight decay, or training fraction (ANOVA p>0.13). HEF is not presented as a universal theory of emergence, but as a falsifiable mathematical scaffold for studying convergence phenomena across complex systems.

We introduce a physics-inspired continuous relaxation framework that yields substantially improved solutions for NP-hard combinatorial optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), binary sparse coding, and planted-solution Ising models. By parameterizing discrete binary variables as continuous wave-like states on the complex unit circle, we inherently smooth highly non-convex energy landscapes. We show that representing binary variables as complex phases reveals an implicit regularization mechanism that promotes convergence toward discrete states. Extracting this mechanism yields significant improvements even within standard real-valued optimization frameworks, using this regularizer explicitly. Empirically, this regularization yields vastly higher ground-state convergence rates than standard real-valued alternatives. Our models achieved zero error in large-scale 160x160 QUBO tasks under severe noise (sigma=0.25), and outperformed traditional algorithms (OMP and LASSO) in underdefined sparse coding with perfect recovery at sigma=0.15. The solver's robustness was further validated by recovering exact ground-state configurations in 8 out of 11 rigorously engineered planted-solution benchmarks.

The robustness of synchronization is typically characterized by scalar, per-node stability indices whose dependence on topology is studied via network science or graph neural networks (GNNs). We propose a novel upstream task, learning stability landscapes, which provide deeper insights into synchronization behavior and from which many such scalar indices can be derived. Crucially, we pioneer a graph-to-image prediction paradigm: learning image-like landscapes as per-node targets directly from graph topology, a formulation we are not aware of having been established elsewhere in the literature. To support this task, we release two datasets of 10,000 graphs each at 20 and 100 nodes with per-node landscape labels, based on a conceptual oscillator model, capturing power grid synchronization behavior. A GNN encodes topology and a CNN decoder renders per-node images, learned end-to-end with good in-distribution accuracy, generalizing across graph sizes and to realistic power grid topologies. This demonstrates that stability landscapes, while beyond the reach of conventional network science, are learnable from topology and open new avenues for moving beyond scalar stability indices in biology, neuroscience, and power grids.

Nonnegative matrix factorization (NMF) seeks a low-rank approximation $X \approx UV^T$ with nonnegative factors and is commonly solved using interior methods that enforce feasibility throughout optimization. We show that such constraint-driven approaches can impede progress in the nonconvex landscape, leading to slow convergence or convergence to suboptimal stationary points. We propose an exterior framework for NMF (eNMF) that separates low-rank approximation from nonnegativity enforcement. Our method initializes from the optimal unconstrained factorization and introduces a rotation procedure that maps unconstrained factors to an exterior point closest to the nonnegative orthant. This viewpoint yields an algorithmic framework in which simple iterative updates converge to KKT-satisfying stationary points on the boundary of the positive orthant. The exterior formulation also enables a geometric interpretation of NMF solutions, clarifying equivalence classes of factorizations under permutation and orthogonal transformations. An intriguing numerical result, involving 400 NMF experiments across both real and synthetic datasets, show that in 99% of the cases, different algorithms tend to converge towards equivalent factor matrices. We benchmark eNMF against 9 state-of-the-art NMF algorithms with 9 initialization schemes across 3 real-world and 2 synthetic datasets. eNMF consistently outperforms all 81 competitors, achieving up to 30% lower reconstruction error under equal-time settings and up to 150% speedup under equal-error settings. The downstream experiments further demonstrate substantial performance gains in audio processing and recommendation tasks, corroborating the practical benefits of the proposed exterior optimization framework. Code is available at https://github.com/roychowdhuryresearch/eNMF

Multimodal optimization requires finding many optima rather than merely keeping a diverse population. Yet most niching-based evolutionary algorithms rely on distances or density estimators without explicitly recovering the underlying peak--basin organization in the decision space, which can lead to pseudo-multimodality: many distinct individuals ultimately collapse into only a few basins. We introduce Chaotic Landscape-Decoding Evolution (CLDE), a decision-space-centric framework that turns multimodal search into a closed loop of decode--value--allocate--refine. CLDE injects controlled global exploration via a logistic chaotic map with a decaying step size, then builds a $k$-nearest-neighbor graph on a decoding canvas and performs persistence-guided basin growing that merges peaks only when they are not separated by deep valleys. An adaptive persistence threshold continuously tunes the decoding resolution online to avoid over-fragmentation and over-merging. Guided by the decoded structure, CLDE carries out basin-wise selection and refinement to improve solution quality while preserving basin coverage. We instantiate CLDE as CLDE-S and CLDE-M for single- and multi-objective multimodal optimization. Experiments on 20 CEC2013 functions show that CLDE-S achieves strong peak ratio under the same evaluation budget, while on DTLZ and MMMOP suites CLDE-M attains competitive IGD/IGDx, with pronounced gains on strongly multimodal problems.

The integration of Large Language Models (LLMs) into evolutionary frameworks has established a new paradigm for automated heuristic discovery. Despite their promise, these methods typically search in the discrete space of program syntax, relying on stochastic sampling to navigate a highly non-convex optimization landscape. This work proposes a continuous heuristic discovery framework that shifts optimization to a learned latent manifold. We employ an encoder to map discrete programs into continuous embeddings and train a differentiable surrogate model to predict performance, enabling gradient-based search. To regularize the optimization trajectory, an invertible normalizing flow maps these embeddings to a structured Gaussian prior, where we perform gradient ascent. The resulting optimized latent vectors are projected through a learned mapper into soft prompts, which condition a frozen LLM to synthesize novel executable heuristics. We evaluate the proposed method on the Traveling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP), the Knapsack Problem (KSP), and Online Bin Packing (OBP). Empirical results demonstrate that continuous latent-space optimization achieves performance competitive with state-of-the-art discrete evolutionary baselines while offering a complementary methodological alternative for automated algorithm design. The implementation code is available at \url{https://github.com/cheikh025/LHS}.

By specifying behaviour across multiple agents, social norms are a coordination approach to resolving social dilemmas. Decentralized and wide adoption can be achieved by norms whose prescription involves interpreting stochastic signals in the environment. Such signals must have enough correlation to orchestrate mutually beneficial coordination and enough disincentivizing uncertainty about the benefits of exploiting that coordination. Evolutionary game theory of matrix games has been used to describe how, by rational agents comparing and adopting norms, a norm can evolve to become dominant in a population. Morsky & Akçay (2019) classify norms according to a set of rationality criteria. Joint player strategies that adopt norms that are consistent with optimal single-player strategies with respect to expected reward naturally satisfy a correlated, rather than Nash game theoretic equilibrium condition. Here, we present a version of this theory that clarifies the basic ingredients. We formulate it in the more general Markov game setting more commonly used in reinforcement learning theory. We illustrate the theory by mapping norms over the signal and reward space, while also giving a detailed exposition of the underlying mechanics of the approach. Finally, we give a general solution and analysis of replicator dynamics, which Morsky & Akçay (2019) propose as a means by which these norms could emerge.

Differentiable planning enables gradient-based optimization of decision-making problems by leveraging differentiable models of system dynamics. However, in highly nonlinear and hybrid discrete-continuous domains, the resulting optimization landscapes are often ill-conditioned, with flat regions and sharp transitions that hinder effective optimization. We propose Model-Driven Policy Optimization (MDPO), a framework that introduces stochastic exploration into differentiable planning by injecting noise into the action space during optimization. Leveraging access to the model, MDPO further adapts the noise magnitude based on gradient-derived sensitivity of the trajectory objective, yielding a time-dependent exploration profile. This enables improved exploration of the objective landscape and helps escape poor local optima via dynamic allocation of exploration across timesteps and iterations. Experiments on benchmark domains demonstrate that MDPO consistently outperforms deterministic differentiable planning, including both the noise-free variant of our method and available state-of-the-art implementations, as well as model-free baselines such as PPO, significantly improving solution quality across challenging nonlinear and hybrid settings. We further analyze the evolution of the adaptive noise magnitude across both time steps and optimization iterations, providing insight into how exploration is allocated during learning.

Many scientific and combinatorial problems admit multiple correct solutions, not a single label. Standard supervised learning resolves this ambiguity by choosing one solution as the target, but this hidden selector can be arbitrary, discontinuous, and harder to learn than the underlying solution set. We study bifurcation models, a weight-tied dynamical view in which different initializations can converge to different stable equilibria, so the model represents an attractor landscape rather than one chosen branch. We prove that broad set-valued maps with locally Lipschitz branches can be represented by regular equilibrium dynamics and that the induced selectors are almost everywhere regular, while manual selectors can be arbitrarily irregular. Experiments on frustrated Ising models show that such dynamics can discover multiple valid equilibria without branch labels and outperform single-branch supervision. Allen--Cahn experiments further show that diversity is not automatic: it can be encouraged explicitly, but with an accuracy--diversity tradeoff.

Large language models have become drivers of evolutionary search, but most systems rely on a fixed, prompt-elicited policy to sample next candidates. This limits adaptation in practical engineering and research tasks, where evaluations are expensive, and progress depends on learning task-specific search dynamics. We introduce PACEvolve++, an advisor-model reinforcement learning framework for test-time policy adaptation in evolutionary search agents. PACEvolve++ decouples strategic search decisions from implementation: a trainable advisor generates, assesses, and selects hypotheses, while a stronger frontier model translates selected hypotheses into executable candidates. To train the advisor under non-stationary feedback, we propose a phase-adaptive approach that adapts its optimization strategy to different phases of the evolutionary process. Early in evolution, it uses group-relative feedback to learn broad search preferences; later, as reward gaps compress, it emphasizes best-of-$k$ frontier contribution to support stable refinement. Across expert-parallel load balancing, sequential recommendation, and protein fitness extrapolation, PACEvolve++ outperforms the state-of-the-art evolutionary search framework with frontier models, achieving faster convergence and stabilizing test-time training during evolutionary search.

Attacking semantic segmentation models is significantly harder than image classification models because an attacker must flip thousands of pixel predictions simultaneously. Standard pixel-wise cross-entropy (CE) is ill-suited to this setting: it tends to overemphasize already-misclassified pixels, which slows optimization and overstates model robustness. To address these issues, we introduce TsallisPGD, an adversarial attack built on the Tsallis cross-entropy, a generalization of CE parameterized by $q$, which adaptively reshapes the gradient landscape by controlling gradient concentration across pixels. By varying $q$, we steer the attack toward pixels at different confidence levels. We first show that no single fixed-$q$ is universally optimal, as its effectiveness depends on the dataset, model architecture, and perturbation budget. Motivated by this, we propose a dynamic $q$-schedule that sweeps $q$ during optimization. Extensive experiments on Cityscapes, Pascal VOC, and ADE20K show that TsallisPGD, using a single validation-selected schedule, achieves the best average attack rank across all evaluated settings and improves over CEPGD, SegPGD, CosPGD, JSPGD, and MaskedPGD in reducing accuracy and mIoU on both standard and robust models.

Local Optima Networks (LONs) represent the global structure of search spaces as graphs, but their construction requires iterative execution of a search algorithm to find local optima and approximate transitions between Basins of Attraction (BoAs). In continuous optimization, this high computational cost prevents systematic investigation of the relationship between LON features and evolutionary algorithm performance. To address this issue, we propose an alternative definition of BoAs for Max-Set of Gaussians (MSG) landscapes with explicitly tunable multimodality. This bypasses search-based BoA identification, enabling low-cost LON construction. Moreover, we leverage Novelty Search (NS) to explore the parameter space of the MSG landscape generator, producing instances with diverse graph topologies. Our experiments show that the proposed BoAs closely align with gradient-based BoAs, and that NS successfully generates instances with varied search difficulty and connectivity patterns among optima. Finally, over the instances generated by NS, we predict the success rate of two well-established evolutionary algorithms from LON features. While our LON construction is specific to MSG landscapes, the proposed framework provides a dataset that serves as a foundation for landscape-aware optimization.

Reinforcement Learning (RL) enhances LLM reasoning, yet a paradox emerges as models scale: strong base models saturate standard benchmarks (e.g., MATH), yielding correct but homogeneous solutions. In such environments, the lack of failure cases causes the advantage signal in group-relative algorithms (e.g., GRPO) to vanish, driving policies into mode collapse. To address this, we propose Constrained Uniform Top-K Sampling (CUTS), a parameter-free decoding strategy enforcing structure-preserving exploration. Unlike standard sampling that follows model biases, CUTS flattens the local optimization landscape by sampling uniformly from constrained high-confidence candidates. We integrate this into Mixed-CUTS, a training framework synergizing exploitative and exploratory rollouts to amplify intra-group advantage variance. Experiments on Qwen3 models demonstrate that our approach prevents policy degeneration and significantly boosts out-of-domain generalization. Notably, Mixed-CUTS improves Pass@1 accuracy on the challenging AIME25 benchmark by up to 15.1% over standard GRPO, validating that maintaining diversity within the semantic manifold is critical for rigorous reasoning.

Exploratory Landscape Analysis (ELA) provides numerical features for characterizing black-box optimization problems. In high-dimensional settings, however, ELA suffers from sparsity effects, high estimator variance, and the prohibitive cost of computing several feature classes. Dimensionality reduction has therefore been proposed as a way to make ELA applicable in such settings, but it remains unclear whether features computed in reduced spaces still reflect intrinsic properties of the original landscape. In this work, we investigate the robustness of ELA features under dimensionality reduction via Random Gaussian Embeddings (RGEs). Starting from the same sampled points and objective values, we compute ELA features in projected spaces and compare them to those obtained in the original search space across multiple sample budgets and embedding dimensions. Our results show that linear random projections often alter the geometric and topological structure relevant to ELA, yielding feature values that are no longer representative of the original problem. While a small subset of features remains comparatively stable, most are highly sensitive to the embedding. Moreover, robustness under projection does not necessarily imply informativeness, as apparently robust features may still reflect projection-induced artifacts rather than intrinsic landscape characteristics.

Protein fitness optimization is a discrete search problem, and the representation used for prediction also determines the neighborhood graph traversed by an optimizer. We introduce Q-BioLat, a framework that maps pretrained protein-language-model embeddings to compact binary codes and fits a quadratic unconstrained binary optimization (QUBO) surrogate with unary and pairwise latent interactions. Our central contribution is an optimization-aware view of representation: binary encodings that are similar in pointwise predictive accuracy can induce different Hamming neighborhoods, local optima, and search trajectories. We formalize when a recoding is only a Hamming-isometric reparameterization and give a constructive example showing that exact pointwise agreement does not imply optimization equivalence. We study experimentally measured GFP and AAV fitness landscapes from ProteinGym. The internal QUBO surrogate is evaluated against labels withheld from QUBO fitting. A conservative retrieval analysis maps optimized codes to measured variants and reports their experimental fitness, while neural decoding of potentially unmeasured sequences is evaluated separately with an experiment-trained sequence surrogate and is interpreted only as model-based candidate prioritization. Across the reported comparisons, PCA followed by per-coordinate median thresholding yields a more balanced and decodable binary space than the post-hoc-zero-threshold AE/VAE baselines, despite the latter's low continuous reconstruction error. In the measured-library retrieval analysis, simulated annealing, genetic algorithms, and greedy hill climbing frequently return high-percentile variants; decoded candidates are reported separately using surrogate-predicted scores.

This work considers a number of optimization problems and reductive relations between them. The two main problems we are interested in are the Optimal Decision Tree and Set Cover. We study these two fundamental tasks under precedence constraints, that is, if a test (or set) $X$ is a predecessor of $Y$, then in any feasible decision tree $X$ needs to be an ancestor of $Y$ (or respectively, if $Y$ is added to set cover, then so must be $X$). For the Optimal Decision Tree we consider two optimization criteria: worst case identification time (height of the tree) or the average identification time. Similarly, for the Set Cover we study two cost measures: the size of the cover or the average cover time. Our approach is to develop a number of algorithmic reductions, where an approximation algorithm for one problem provides an approximation for another via a black-box usage of a procedure for the former. En route we introduce other optimization problems either to complete the `reduction landscape' or because they hold the essence of combinatorial structure of our problems. The latter is brought by a problem of finding a Maximum Density Precedence-Closed Subfamily, where the density is defined as the ratio of the number of items the family covers to its size. We provide $\mathcal{O}^*(\sqrt{m})$-approximation polynomial-time algorithms for all aforementioned problems. The picture is complemented by a number of hardness reductions that provide $\mathcal{O}(m^{1/12-ε})$-inapproximability results for the decision tree and covering problems. Besides giving a complete set of results for general precedence constraints, we also provide polylogarithmic approximation guarantees for two most typically studied and applicable graph types, outforests and inforests. By providing corresponding hardness results, we show most of these results to be tight.

Toeplitz covariance estimation is a classical problem in statistical signal processing, yet the geometry of the Gaussian maximum-likelihood objective remains only partially understood. Recent algorithms, including Newton-type, majorization-minimization, and gradient-based methods, indicate that the nonconvex problem can often be globally solved when the number of samples is sufficiently large, but they also reveal a difficult computational landscape. In this work, we study this phenomenon through an overparameterized Caratheodory representation of positive definite Toeplitz covariance matrices. The Caratheodory decomposition parameterizes the covariance using a combination of steering vectors with different frequencies and amplitudes. Our first result shows that fixed-grid amplitude optimization is fundamentally insufficient. Even in the population setting, and even with arbitrarily many fixed frequency grid points, amplitude-only optimization can have a strictly positive error floor under grid mismatch. This motivates optimizing both amplitudes and frequencies. In this case, our main theoretical result proves that the joint optimization has a benign population landscape: every stationary point that produces a positive definite covariance matrix recovers the true Toeplitz covariance. These findings suggest a simple interpretation of the Toeplitz covariance problem: the population landscape is globally benign, but may be highly ill-conditioned. In our numerical experiments, overparameterization improves convergence speed and finite-sample accuracy. In particular, it allows simple gradient descent to approach the Cramer Rao bound while keeping the implementation simple.

In this work, we study the nonlinear dynamics of a shallow neural network trained with mean-squared loss and leaky ReLU activation. Under Gaussian inputs and equal layer width k, (1) we establish, based on the equivariant gradient degree, a theoretical framework, applicable to any number of neurons k>= 4, to detect bifurcation of critical points with associated symmetries from global minimum as leaky parameter $α$ varies. Typically, our analysis reveals that a multi-mode degeneracy consistently occurs at the critical number 0, independent of k. (2) As a by-product, we further show that such bifurcations are width-independent, arise only for nonnegative $α$ and that the global minimum undergoes no further symmetry-breaking instability throughout the engineering regime $α$ in range (0,1). An explicit example with k=5 is presented to illustrate the framework and exhibit the resulting bifurcation together with their symmetries.

Swarm-based optimization algorithms have demonstrated remarkable success in solving complex problems, yet their widespread adoption remains limited due to poor transparency in how algorithmic components influence performance. This work presents a multi-faceted explainability framework for Particle Swarm Optimization (PSO) through two complementary perspectives: landscape-based and algorithmic explainability. From the landscape-based perspective, we develop a comprehensive characterization framework using Exploratory Landscape Analysis (ELA) to quantify problem difficulty, multimodality, and ruggedness, extracting ELA meta-features, dispersion measures, and information content statistics, while a machine learning approach employing Decision Tree and Random Forest classifiers enables prediction of optimal topology-specific hyperparameter configurations for unseen problems. From the algorithmic explainability perspective, we integrate IOHxplainer for temporal convergence profiling and Search Trajectory Networks (STN) for spatial navigation mapping, proposing three novel STN metrics-Connectivity Density, Fragmentation Score, and Search Efficiency-that enhance visual explainability by quantifying topology-specific search organization and transition effectiveness. Through systematic experimentation across 24 benchmark functions in multiple dimensions with Star, Ring, and Von Neumann topologies, we establish practical guidelines for topology selection and parameter configuration. Our findings uncover the black-box nature of PSO, providing greater transparency and interpretability to swarm intelligence systems. The source code is available at https://github.com/GitNitin02/ioh_pso.

Diffusion models generate conditional samples by progressively denoising Gaussian noise, yet the denoising trajectory can stall at visually plausible but low-quality outcomes with conditional misalignment or structural artifacts. We interpret this behavior as local optima in a surrogate quality landscape: Once early denoising commits to a suboptimal global structure, later steps mainly sharpen details and seldom correct the underlying mistake. While existing inference-time approaches explore alternative diffusion states via re-noising with fixed strength or direction, they exhibit limited capacity to escape steep quality plateaus. We propose Controlled Random Zigzag Sampling (Ctrl-Z Sampling),a scalable sampling strategy that detects plateaus in quality landscape via a surrogate score, and allocates exploration only when a plateau is detected. Upon detection, Ctrl-Z Sampling rolls back to noisier states, samples a set of alternative continuations, and updates the trajectory when a candidate improves the score, otherwise escalating the exploration depth to escape the current plateau. The proposed method is model-agnostic and broadly compatible with existing diffusion frameworks. Experiments show that Ctrl-Z Sampling consistently improves generation quality over other inference-time scaling samplers across different NFE budgets, offering a scalable compute-quality trade-off. Code available at: https://github.com/ShunqiM/Ctrl-Z-Sampling.