ICLR 2026 Papers

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.

Recent large vision-language models (LVLMs) demonstrate remarkable capabilities in processing extended multi-modal sequences, yet the resulting key-value (KV) cache expansion creates a critical memory bottleneck that fundamentally limits deployment scalability. While existing KV cache compression methods focus on retaining high-importance KV pairs to minimize storage, they often overlook the modality-specific semantic redundancy patterns that emerge distinctively in multi-modal KV caches. In this work, we first analyze how, beyond simple importance, the KV cache in LVLMs exhibits varying levels of redundancy across attention heads. We show that relying solely on importance can only cover a subset of the full KV cache information distribution, leading to potential loss of semantic coverage. To address this, we propose MixKV, a novel method that mixes importance with diversity for optimized KV cache compression in LVLMs. MixKV adapts to head-wise semantic redundancy, selectively balancing diversity and importance when compressing KV pairs. Extensive experiments demonstrate that MixKV consistently enhances existing methods across multiple LVLMs. Under extreme compression (budget=64), MixKV improves baseline methods by an average of 5.1% across five multi-modal understanding benchmarks and achieves remarkable gains of 8.0% and 9.0% for SnapKV and AdaKV on GUI grounding tasks, all while maintaining comparable inference efficiency. Furthermore, MixKV extends seamlessly to LLMs with comparable performance gains.

Model editing techniques, particularly task arithmetic with task vectors, offer an efficient alternative to full fine-tuning by enabling direct parameter updates through simple arithmetic operations. While this approach promises substantial computational savings, its impact on fairness has remained largely unexplored---despite growing concern over biased outcomes in high-stakes applications such as hate speech detection. In this work, we present the first systematic study of group fairness in task arithmetic within this binary text and image classification regime, comparing it against full fine-tuning (FFT) and Low-Rank Adaptation (LoRA). We evaluate across multiple language models and datasets using standard group fairness metrics, including Demographic Parity and Equalized Odds. Our analysis shows that task vectors can be tuned to achieve competitive accuracy while reducing disparities, and that merging subgroup-specific task vectors provides a practical mechanism for steering fairness outcomes. We further provide a theoretical bound linking task vector scaling to fairness metrics, offering insight into the observed trade-offs. Together, these findings establish task arithmetic not only as a cost-efficient editing method but also as a fairness-aware alternative to existing adaptation techniques, within the standard group-fair classification setting, laying the groundwork for responsible deployment of large language models.

Heavy-tailed noise in nonconvex stochastic optimization has garnered increasing research interest, as empirical studies, including those on training attention models, suggest it is a more realistic gradient noise condition. This paper studies first-order nonconvex stochastic optimization under heavy-tailed gradient noise in a decentralized setup, where each node can only communicate with its direct neighbors in a predefined graph. Specifically, we consider a class of heavy-tailed gradient noise that is zero-mean and has only $p$-th moment for $p \in (1, 2]$. We propose GT-NSGDm, Gradient Tracking based Normalized Stochastic Gradient Descent with momentum, that utilizes normalization, in conjunction with gradient tracking and momentum, to cope with heavy-tailed noise on distributed nodes. We show that, when the communication graph admits primitive and doubly stochastic weights, GT-NSGDm guarantees, for the \textit{first} time in the literature, that the expected gradient norm converges at an optimal non-asymptotic rate $O\big(1/T^{(p-1)/(3p-2)}\big)$, which matches the lower bound in the centralized setup. When tail index $p$ is unknown, GT-NSGDm attains a non-asymptotic rate $O\big( 1/T^{(p-1)/(2p)} \big)$ that is, for $p < 2$, topology independent and has a speedup factor $n^{1-1/p}$ in terms of the number of nodes $n$. Finally, experiments on nonconvex linear regression with tokenized synthetic data and decentralized training of language models on a real-world corpus demonstrate that GT-NSGDm is more robust and efficient than baselines.

Transformers are remarkably versatile and their design is largely consistent across a variety of applications. But are they optimal for any given task or dataset? The answer may be key for pushing AI beyond merely scaling current designs. **Method.** We present a method to optimize a transformer architecture for a given dataset, which we use as a tool to study optimal task-specific inductive biases. This method replaces the most important non-linearities (GeLUs,;softmax) with functions learned on held-out data. We then train the resulting architectures on other datasets, as a way to evaluate the compatibility between pairs of tasks. **Findings.** On algorithmic toy tasks, we identify new architectures with dramatic improvements in learning speed, in- and out-of-distribution generalization, and stability across seeds. The new designs prove very task-specific however, and indicate that these tasks require inductive biases very different from those of standard transformers. On code and language modeling datasets, we also find architectures with consistent, yet smaller improvements. These designs transfer much better across datasets and domains (English & computer code). **Implications.** Our results show that standard transformers are rarely a local optimum in the space of architectures. Simple alternatives can perform much better but sacrifice universality. This suggests that there may be room for improved architectures that better support multiple capabilities simultaneously, such as fluency and robust reasoning.

The geometry of dynamical systems estimated from trajectory data is a major challenge for machine learning applications. Koopman and transfer operators provide a linear representation of nonlinear dynamics through their spectral decomposition, offering a natural framework for comparison. We propose a novel approach representing each system as a distribution of its joint operator eigenvalues and spectral projectors and defining a metric between systems leveraging optimal transport. The proposed metric is invariant to the sampling frequency of trajectories. It is also computationally efficient, supported by finite-sample convergence guarantees, and enables the computation of Fréchet means, providing interpolation between dynamical systems. Experiments on simulated and real-world datasets show that our approach consistently outperforms standard operator-based distances in machine learning applications, including dimensionality reduction and classification, and provides meaningful interpolation between dynamical systems.

Large language models (LLMs) have achieved remarkable performance across diverse reasoning tasks, yet their deployment is hindered by prohibitive computational and memory costs. Quantization-aware training (QAT) enables ultra-low-bit compression (<4 bits per weight), but existing QAT methods often degrade reasoning capability, partly because complex knowledge structures are introduced during the post-training process in LLMs. In this paper, through a systematic investigation of how quantization affects different data domains, we find that its impact on pre-training and reasoning capabilities differs. Building on this insight, we propose a novel two-stage QAT pipeline specifically designed for reasoning LLMs. In the first stage, we quantize the model using mixed-domain calibration data to preserve essential capabilities across domains; in the second stage, we fine-tune the quantized model with a teacher-guided reward-rectification loss to restore reasoning capability. We first demonstrate that mixed-domain calibration outperforms single-domain calibration by up to 2.74% improvement on average over six tasks, including reasoning and pre-trained tasks. Following experiments on five reasoning benchmarks show that our 2-bit-quantized Qwen3-8B outperforms post-training quantization (PTQ) baselines by 50.45% on average. Moreover, compared to ultra-low-bit-specialized models such as BitNet-2B4T, our pipeline achieves about 2% higher mathematical-reasoning accuracy using only 40K training sequences.

The growing demand for high-quality 3D mesh models has fueled the need for efficient 3D mesh compression techniques. However, existing methods often exhibit suboptimal compression performance due to the inefficient representation of mesh data. To address this issue, we propose a novel neural mesh compression method based on Sparse Implicit Representation (SIR). Specifically, SIR records signed distance field (SDF) values only on regular grids near the surface, enabling high-resolution structured representation of arbitrary geometric data with a significantly lower memory cost, while still supporting precise surface recovery. Building on this representation, we construct a lightweight Sparse Neural Compression (SNC) network to extract compact embedded features from the SIR and encode them into a bitstream. Extensive experiments and ablation studies demonstrate that our method outperforms state-of-the-art mesh and point cloud compression approaches in both compression performance and computational efficiency across a variety of mesh models. The source code is available at https://github.com/yydlmzyz1/SIR-SNC.