ICLR 2026 Papers

Generating physically stable and novel metal-organic frameworks (MOFs) for inverse design that meet specific performance targets is a significant challenge. Existing generative models often struggle to explore the vast chemical and structural space effectively, leading to suboptimal solutions or mode collapse. To address this, we propose PRO-MOF, a hierarchical reinforcement learning (HRL) framework for controllable MOF generation. Our approach decouples the MOF design process into two policies: a high-level policy for proposing chemical building blocks and a low-level policy for assembling their 3D structures. By converting the deterministic Flow Matching model into a Stochastic Differential Equation (SDE), we enable the low-level policy to perform compelling exploration. The framework is optimized in a closed loop with high-fidelity physical reward signals provided by a pre-trained universal atomistic model (UMA). Furthermore, we introduce a Pass@K Group Relative Policy Optimization (GRPO) scheme that effectively balances exploration and exploitation by rewarding in-group diversity. Experiments on multiple inverse design tasks, such as maximizing CO2 working capacity and targeting specific pore diameters, show that PRO-MOF significantly outperforms existing baselines, including diffusion-based methods and genetic algorithms, in both success rate and the discovery of top-performing materials. Our work demonstrates that hierarchical reinforcement learning combined with a high-fidelity physical environment is a powerful paradigm for solving complex material discovery problems.

Detecting AI risks becomes more challenging as stronger models emerge and find novel methods such as Alignment Faking to circumvent these detection attempts. Inspired by how risky behaviors in humans (i.e., illegal activities that may hurt others) are sometimes guided by strongly-held values, we believe that identifying values within AI models can be an early warning system for AI's risky behaviors. We create LitmusValues, an evaluation pipeline to reveal AI models' priorities on a range of AI value classes. Then, we collect AIRiskDilemmas, a diverse collection of dilemmas that pit values against one another in scenarios relevant to AI safety risks such as Power Seeking. By measuring an AI model's value prioritization using its aggregate choices, we obtain a self-consistent set of predicted value priorities that uncover potential risks. We show that values in LitmusValues (including seemingly innocuous ones like Care) can predict for both seen risky behaviors in AIRiskDilemmas and unseen risky behaviors in HarmBench.

Achieving zero constraint violations in safe reinforcement learning poses a significant challenge. We discover a key obstacle called the safety paradox, where improving policy safety reduces the frequency of constraint-violating samples, thereby impairing feasibility function estimation and ultimately undermining policy safety. We theoretically prove that the estimation error bound of the feasibility function increases as the proportion of violating samples decreases. To overcome the safety paradox, we propose an algorithm called feasible dual policy iteration (FDPI), which employs an additional policy to strategically maximize constraint violations while staying close to the original policy. Samples from both policies are combined for training, with data distribution corrected by importance sampling. Extensive experiments show FDPI's state-of-the-art performance on the Safety-Gymnasium benchmark, achieving the lowest violation and competitive-to-best return simultaneously.

The current success of diffusion transformers are built on the compressed latent space shaped by the pre-trained variational autoencoder(VAE). However, this two-stage training paradigm inevitably introduces accumulated errors and decoding artifacts. To avoid these problems, researchers return to pixel space modeling but at the cost of complicated cascade pipelines and increased token complexity. Motivated by the simple yet effective diffusion transformer architectures on the latent space, we propose to model pixel space diffusion using a large-patch diffusion transformer and employ neural fields to decode these large patches, leading to a single-stage streamlined end-to-end solution, which we coin as pixel neural field diffusion transformer (**PixNerd**). Thanks to the efficient neural field representation in PixNerd, we achieve **1.93 FID** on ImageNet 256x256 and nearly **8x lower latency** without any complex cascade pipeline or VAE. We also extend our PixNerd framework to text-to-image applications. Our PixNerd-XXL/16 achieves a competitive 0.73 overall score on the GenEval benchmark and 80.9 overall score on the DPG benchmark.

The importance of mobile phone GPS trajectory data is widely recognized across many fields, yet the use of real data is often hindered by privacy concerns, limited accessibility, and high acquisition costs. As a result, generating pseudo–GPS trajectory data has become an active area of research. Recent diffusion-based approaches have achieved strong fidelity but remain limited in spatial scale (small urban areas), transportation-mode diversity, and efficiency (requiring numerous sampling steps). To address these challenges, we introduce TrajFlow, the first flow-matching–based generative model for GPS trajectory generation. TrajFlow leverages the flow-matching paradigm to improve robustness across multiple geospatial scales and incorporates a trajectory harmonization & reconstruction strategy to jointly address scalability, diversity, and efficiency. Using a nationwide mobile phone GPS dataset with millions of trajectories across Japan, we show that TrajFlow consistently outperforms diffusion-based and deep generative baselines at urban, metropolitan, and nationwide levels. As the first nationwide, multi-scale GPS trajectory generation model, TrajFlow demonstrates strong potential to support inter-region urban planning, traffic management, and disaster response, thereby advancing the resilience and intelligence of future mobility systems.

We prove that a minimal Transformer with frozen weights emulates a broad class of algorithms by in-context prompting. We formalize two modes of in-context algorithm emulation. In the *task-specific mode*, for any continuous function $f: \mathbb{R} \to \mathbb{R}$, we show the existence of a single-head softmax attention layer whose forward pass reproduces reproduces functions of the form $f(w^\top x - y)$ to arbitrary precision. This general template subsumes many popular machine learning algorithms (e.g., gradient descent, linear regression, ridge regression). In the *prompt-programmable mode*, we prove universality: a single fixed-weight two-layer softmax attention module emulates all algorithms from the task-specific class (i.e., each implementable by a single softmax attention) via only prompting. Our key idea is to construct prompts that encode an algorithm’s parameters into token representations, creating sharp dot-product gaps that force the softmax attention to follow the intended computation. This construction requires no feed-forward layers and no parameter updates. All adaptation happens through the prompt alone. Numerical results corroborate our theory. These findings forge a direct link between in-context learning and algorithmic emulation, and offer a simple mechanism for large Transformers to serve as prompt-programmable interpreters of algorithms. They illuminate how GPT-style foundation models may swap algorithms via prompts alone, and establish a form of algorithmic universality in modern Transformer models.

While test-time scaling with verification has shown promise in improving the performance of large language models (LLMs), role of the verifier and its imperfections remain underexplored. The effect of verification manifests through interactions of three quantities: (i) the generator’s *coverage*, (ii) the verifier’s *region of convergence* (ROC), and (iii) the sampling algorithm’s *sub-optimality*. Though recent studies capture subsets of these factors, a unified framework quantifying the geometry of their interplay is missing. We frame verifiable test-time scaling as a transport problem. This characterizes the interaction of coverage, ROC, and sub-optimality, and uncovers that the sub-optimality–coverage curve exhibits three regimes. A *transport regime* -- where sub-optimality increases with coverage, a *policy improvement regime* -- where sub-optimality may decrease with coverage, depending on the verifier’s ROC, and a *saturation regime* -- where sub-optimality plateaus, unaffected by coverage. We further propose and analyze two classes of sampling algorithms -- *sequential* and *batched*, and examine how their computational complexities shape these trade-offs. Empirical results with \texttt{Qwen}, \texttt{Llama}, and \texttt{Gemma} models corroborate our theoretical findings.

The training of neural networks has been extensively studied from both algorithmic and complexity-theoretic perspectives, yet recent results in this direction almost exclusively concern real-valued networks. In contrast, advances in machine learning practice highlight the benefits of quantization, where network parameters and data are restricted to finite integer domains, yielding significant improvements in speed and energy efficiency. Motivated by this gap, we initiate a systematic complexity-theoretic study of ReLU Neural Network Training in the full quantization mode. We establish strong lower bounds by showing that hardness already arises in the binary setting and under highly restrictive structural assumptions on the architecture, thereby excluding parameterized tractability for natural measures such as depth and width. On the positive side, we identify nontrivial fixed-parameter tractable cases when parameterizing by input dimensionality in combination with width and either output dimensionality or error bound, and further strengthen these results by replacing width with the more general treewidth.

Multi-agent imitation learning (MA-IL) aims to learn optimal policies from expert demonstrations in multi-agent interactive domains. Despite existing guarantees on the performance of the extracted policy, characterizations of its distance to a Nash equilibrium are missing for offline MA-IL. In this paper, we demonstrate impossibility and hardness results of learning low-exploitable policies in general $n$-player Markov Games. We do so by providing examples where even exact measure matching fails, and present challenges associated with the practical case of approximation errors. We then show how these challenges can be overcome using strategic dominance assumptions on the expert equilibrium, assuming BC error $\epsilon_{\text{BC}}$. Specifically, for the case of dominant strategy expert equilibria, this provides a Nash imitation gap of $\mathcal{O}\left(n\epsilon_{\text{BC}}/(1-\gamma)^2\right)$ for a discount factor $\gamma$. We generalize this result with a new notion of best-response continuity, and argue that this is implicitly encouraged by standard regularization techniques.

Bridge diffusion models have shown great promise in image restoration by constructing a direct path from degraded to clean images. However, they often rely on predefined, high-action trajectories, which limits both sampling efficiency and final restoration quality. To address this, we propose a Consistency Geodesic Bridge (CGB) framework to construct a lower-action, geodesic trajectory. We achieve this by designing a novel bridge process that evolves over a shorter time horizon and makes the reverse process start from an entropy-regularized point that mixes the degraded image and Gaussian noise, which theoretically reduces the required trajectory action. To ensure this path approximates a geodesic on the data manifold, we innovatively leverage a pretrained denoiser as a dynamic geodesic guidance field. To solve this process efficiently, we draw inspiration from consistency models to learn a single-step mapping function, optimized via a continuous-time consistency objective tailored for our trajectory, so as to analytically map any state on the path to the target image. Notably, the path length in our framework becomes a tunable task-adaptive knob, allowing the model to adaptively balance information preservation against generative power for tasks of varying degradation, such as denoising versus super-resolution. Extensive experiments demonstrate that CGB achieves state-of-the-art performance across various image restoration tasks, while enabling high-quality recovery with a single or fewer sampling steps. Our project page is \url{https://cgbridge.github.io/}.

We study the classic problem of prediction with expert advice under the constraint of local differential privacy (LDP). In this context, we first show that a classical algorithm naturally satisfies LDP and then design two new algorithms that improve it: RW-AdaBatch and RW-Meta. For RW-AdaBatch, we exploit the limited-switching behavior induced by LDP to provide a novel form of privacy amplification that grows stronger on easier data, analogous to the shuffle model in offline learning. Drawing on the theory of random walks, we prove that this improvement carries essentially no utility cost. For RW-Meta, we develop a general method for privately selecting between experts that are themselves non-trivial learning algorithms, and we show that in the context of LDP this carries no extra privacy cost. In contrast, prior work has only considered data-independent experts. We also derive formal regret bounds that scale inversely with the degree of independence between experts. Our analysis is supplemented by evaluation on real-world data reported by hospitals during the COVID-19 pandemic; RW-Meta outperforms both the classical baseline and a state-of-the-art \textit{central} DP algorithm by 1.5-3$\times$ on the task of predicting which hospital will report the highest density of COVID patients each week.

Prompt-based methods have recently gained prominence in Continual Learning (CL) due to their strong performance and memory efficiency. A prevalent strategy in this paradigm assigns a dedicated subset of prompts to each task, which, while effective, incurs substantial computational overhead and causes memory requirements to scale linearly with the number of tasks. Conversely, approaches employing a single shared prompt across tasks offer greater efficiency but often suffer from degraded performance due to knowledge interference. To reconcile this trade-off, we propose **SMoPE**, a novel framework that integrates the benefits of both task-specific and shared prompt strategies. Inspired by recent findings on the relationship between Prefix Tuning and Mixture of Experts (MoE), SMoPE organizes a shared prompt into multiple "prompt experts" within a sparse MoE architecture. For each input, only a select subset of relevant experts is activated, effectively mitigating interference. To facilitate expert selection, we introduce a prompt-attention score aggregation mechanism that computes a unified proxy score for each expert, enabling dynamic and sparse activation. Additionally, we propose an adaptive noise mechanism to encourage balanced expert utilization while preserving knowledge from prior tasks. To further enhance expert specialization, we design a prototype-based loss function that leverages prefix keys as implicit memory representations. Extensive experiments across multiple CL benchmarks demonstrate that SMoPE consistently outperforms task-specific prompt methods and achieves performance competitive with state-of-the-art approaches, all while significantly reducing parameter counts and computational costs.

Protein foundation models have advanced rapidly, with most approaches falling into two dominant paradigms. Sequence-only language models (e.g., ESM-2) capture sequence semantics at scale but lack structural grounding. MSA-based predictors (e.g., AlphaFold 2/3) achieve accurate folding by exploiting evolutionary couplings, but their reliance on homologous sequences makes them less reliable in highly mutated or alignment-sparse regimes. We present FlexProtein, a pretrained protein model that jointly learns from amino acid sequences and three-dimensional structures. Our pretraining strategy combines masked language modeling with diffusion-based denoising, enabling bidirectional sequence-structure learning without requiring MSAs. Trained on both experimentally resolved structures and AlphaFold 2 predictions, FlexProtein captures global folds as well as flexible conformations critical for biological function. Evaluated across diverse tasks spanning interface design, intermolecular interaction prediction, and protein function prediction, FlexProtein establishes new state-of-the-art performance on 12 different tasks, with particularly strong gains in mutation-rich settings where MSA-based methods often struggle.

Latent-space optimization methods for counterfactual explanations—framed as minimal semantic perturbations that change model predictions—inherit the ambiguity of Wachter et al.’s objective: the choice of distance metric dictates whether perturbations are meaningful or adversarial. Existing approaches adopt flat or misaligned geometries, leading to off-manifold artifacts, semantic drift, or adversarial collapse. We introduce Perceptual Counterfactual Geodesics (PCG), a method that constructs counterfactuals by tracing geodesics under a perceptually Riemannian metric induced from robust vision features. This geometry aligns with human perception and penalizes brittle directions, enabling smooth, on-manifold, semantically valid transitions. Experiments on three vision datasets show that PCG outperforms baselines and reveals failure modes hidden under standard metrics.

When a deep ReLU network is initialized with small weights, gradient descent (GD) is at first dominated by the saddle at the origin in parameter space. We study the so-called escape directions along which GD leaves the origin, which play a similar role as the eigenvectors of the Hessian for strict saddles. We show that the optimal escape direction features a \textit{low-rank bias} in its deeper layers: the first singular value of the $\ell$-th layer weight matrix is at least $\ell^{\frac{1}{4}}$ larger than any other singular value. We also prove a number of related results about these escape directions. We suggest that deep ReLU networks exhibit saddle-to-saddle dynamics, with GD visiting a sequence of saddles with increasing bottleneck rank.

We propose a sample-efficient Q-learning algorithm for reinforcement learning with the Conditional Value-at-Risk (CVaR) objective. Our algorithm is built upon predictive tail value function, a novel formulation of risk-sensitive action value, that admits a recursive structure as in the conventional risk-neutral Bellman equation. This structure enables the Q-learning algorithm to utilize the entire set of sample trajectories rather than relying only on worst-case outcomes, enhancing the sample efficiency. We further derive a Bellman optimality equation and a policy improvement theorem, which provide theoretical foundations of our algorithm and remedy inconsistencies that have existed in the literature. Empirical results demonstrate that our method consistently improves CVaR performance while maintaining stable and interpretable learning dynamics.

We consider the inverse problem of reconstructing the spatial layout of a place, a home floorplan for example, from a user’s movements inside that layout. Direct inversion is ill-posed since many floorplans can explain the same movement trajectories. We adopt a diffusion-based posterior sampler to generate layouts consistent with the measurements. While active research is in progress on generative inverse solvers, we find that the forward operator in our problem poses new challenges. The path planning process inside a floorplan is a non-invertible, non-differentiable function, and causes instability while optimizing using the likelihood score. We break-away from existing approaches and reformulate the likelihood score in a smoother embedding space. The embedding space is trained with a contrastive loss which brings compatible floorplans and trajectories close to each other, while pushing mismatched pairs far apart. We show that a surrogate form of the likelihood score in this embedding space is a valid approximation of the true likelihood score, making it possible to steer the denoising process towards the posterior. Across extensive experiments, our model CoGuide produces more consistent floorplans from trajectories, and is more robust than differentiable-planner baselines and guided-diffusion methods.

Autoregressive (AR) models have emerged as a powerful framework for image generation, yet they remain bound by a fundamental limitation: once a prediction is made, it cannot be revised. Each step marches forward in a strict left-to-right sequence, causing small errors to accumulate and compromise the final image. In this work, we reimagine this process with TensorAR, a decoder-only AR model that shifts from predicting discrete tokens to predicting overlapping tensor windows. This simple change transforms image synthesis into a process of next-tensor prediction, enabling the model to refine earlier outputs while preserving the causal structure that defines autoregression. To guard against information leakage during training, we introduce a discrete tensor noising mechanism inspired by discrete diffusion theory, which injects categorical noise into input tensors. TensorAR is designed to be plug-and-play: unlike masked AR methods, it requires no architectural modifications, and unlike autoregressive diffusion, it preserves the familiar AR training paradigm. We evaluate TensorAR across both class-to-image and text-to-image tasks, showing consistent gains in generation quality and instruction-following ability, while achieving a superior balance between quality and latency. In doing so, TensorAR offers a new path forward for autoregressive generation---one where predictions are not just produced, but continually refined.

Generative Pre-trained Transformer (GPT) architectures are the most popular design for language modeling. Energy-based modeling is a different paradigm that views inference as a dynamical process operating on an energy landscape. We propose a minimal modification of the GPT setting to unify it with the EBM framework. The inference step of our model, which we call eNeRgy-GPT (NRGPT), is conceptualized as an exploration of the tokens on the energy landscape. We prove, and verify empirically, that under certain circumstances this exploration becomes gradient descent, although they don’t necessarily lead to the best performing models. We demonstrate that our model performs well for simple language (Shakespeare dataset), algebraic ListOPS tasks, and richer settings such as OpenWebText language modeling. We also observe that our models may be more resistant to overfitting, doing so only during very long training.