ICLR 2026 Papers

We study the problem of posterior sampling in the context of score based generative models. We have a trained score network for a prior $p(x)$, a measurement model $p(y|x)$, and are tasked with sampling from the posterior $p(x|y)$. Prior work has shown this to be intractable in KL (in the worst case) under well-accepted computational hardness assumptions. Despite this, popular algorithms for tasks such as image super-resolution, stylization, and reconstruction enjoy empirical success. Rather than establishing distributional assumptions or restricted settings under which exact posterior sampling is tractable, we view this as a more general "tilting" problem of biasing a distribution towards a measurement. Under minimal assumptions, we show that one can tractably sample from a distribution that is simultaneously close to the posterior of a noised prior in KL divergence and the true posterior in Fisher divergence. Intuitively, this combination ensures that the resulting sample is consistent with both the measurement and the prior. To the best of our knowledge these are the first formal results for (approximate) posterior sampling in polynomial time.

We revisit Proximal Policy Optimization (PPO) from a function-space perspective. Our analysis decouples policy evaluation and improvement in a reproducing kernel Hilbert space (RKHS): (i) A kernelized temporal-difference (TD) critic performs efficient RKHS-gradient updates using only one-step state–action transition samples. (ii) a KL-regularized, natural-gradient policy step exponentiates the evaluated action-value, recovering a PPO/TRPO-style proximal update in continuous state-action spaces. We provide non-asymptotic, instance-adaptive guarantees whose rates depend on RKHS entropy, unifying tabular, linear, Sobolev, Gaussian, and Neural Tangent Kernel (NTK) regimes, and we derive a sampling rule for the proximal update that ensures the optimal $k^{-1/2}$ convergence rate for stochastic optimization. Empirically, the theory-aligned schedule improves stability and sample efficiency on common control tasks (e.g., CartPole, Acrobot), while our TD-based critic attains favorable throughput versus a GAE baseline. Altogether, our results place PPO on a firmer theoretical footing beyond finite-dimensional assumptions and clarify when RKHS-proximal updates with kernel-TD critics yield global policy improvement with practical efficiency.

Developing efficient multi-objective optimization methods to compute the Pareto set of optimal compromises between conflicting objectives remains a key challenge, especially for large-scale and expensive problems. To bridge this gap, we introduce SPREAD, a generative framework based on Denoising Diffusion Probabilistic Models (DDPMs). SPREAD first learns a conditional diffusion process over points sampled from the decision space and then, at each reverse diffusion step, refines candidates via a sampling scheme that uses an adaptive multiple gradient descent-inspired update for fast convergence alongside a Gaussian RBF–based repulsion term for diversity. Empirical results on multi-objective optimization benchmarks, including offline and Bayesian surrogate-based settings, show that SPREAD matches or exceeds leading baselines in efficiency, scalability, and Pareto front coverage. Code is available at https://github.com/safe-autonomous-systems/moo-spread .

Score-based methods, such as diffusion models and Bayesian inverse problems, are often interpreted as learning the data distribution in the low-noise limit ($\sigma \to 0$). In this work, we propose an alternative perspective: their success arises from implicitly learning the data manifold rather than the full distribution. Our claim is based on a novel analysis of scores in the small-$\sigma$ regime that reveals a sharp separation of scales: information about the data manifold is $\Theta(\sigma^{-2})$ stronger than information about the distribution. We argue that this insight suggests a paradigm shift from the less practical goal of distributional learning to the more attainable task of geometric learning, which provably tolerates $O(\sigma^{-2})$ larger errors in score approximation. We illustrate this perspective through three consequences: i) in diffusion models, concentration on data support can be achieved with a score error of $o(\sigma^{-2})$, whereas recovering the specific data distribution requires a much stricter $o(1)$ error; ii) more surprisingly, learning the uniform distribution on the manifold—an especially structured and useful object—is also $O(\sigma^{-2})$ easier; and iii) in Bayesian inverse problems, the maximum entropy prior is $O(\sigma^{-2})$ more robust to score errors than generic priors. Finally, we validate our theoretical findings with preliminary experiments on large-scale models, including Stable Diffusion.

We introduce a new, flexible, and theoretically justified framework for dimension reduction in high-dimensional regression, based on an ensemble of random projections. Specifically, we consider disjoint groups of independent random projections, retain the best projection in each group according to the empirical regression performance on the projected covariates, and then aggregate the selected projections via singular value decomposition. The singular values quantify the relative importance of corresponding projection directions and guide the dimension selection process. We investigate various aspects of our framework, including the choice of projection distribution and the number of projections used. Our theoretical results show that the expected estimation error decreases as the number of groups of projections increases. Finally, we demonstrate that our proposal consistently matches or outperforms state-of-the-art methods through extensive numerical studies on simulated and real data.