#01Nov 3, 2026
stat.ME
Post-stratified importance sampling
Yuling Yao, Sergei Tikhonov
Importance sampling is a building block of modern statistical computing, but the weighted estimate is prone to high or infinite variance. Moreover, its empirical standard error is often unreliable, leaving importance sampling without practical uncertainty quantification. Motivated by multilevel regression and poststratification (MRP) in survey sampling, we propose post-stratified importance sampling (PIS) for estimating target expectations from proposal draws. PIS reformulates arbitrarily high-dimensional weighting into two one-dimensional probabilistic modeling problems: regression for the conditional mean given the importance ratio and marginal density estimation for the importance ratio. Different modeling choices yield a family of PIS estimators, encompassing ordinary importance sampling as a special case. We develop a model-based implementation that combines one-dimensional Gaussian process regression for the conditional mean and score-informed heavy-tailed density estimation for the marginal law, using both the observed weights and the available density evaluations. We study the oracle gains of PIS and establish consistency under general and model-specific conditions. Simulations and applications to cross-validation demonstrate improvements in estimation accuracy and uncertainty quantification.
#02Nov 3, 2026
cs.LG
Universal Hardness From Masking Channels
Parikshit Gopalan, Manoj Prabhakaran, Pranay Tankala
We prove a broad learning-theoretic generalization of the celebrated hardcore lemma concerning the average-case complexity of computing a Boolean function (Impagliazzo, 1995). Our starting point is the observation that the classic notion of a hardcore measure, which reweights a function's input distribution, is equivalent to an input-dependent masking channel that injects randomness into the function's output distribution until it appears fully pseudorandom. The analogue of density of the measure is the fidelity of the channel. From this viewpoint, we develop a general framework for certifying hardness that allows for non-Boolean functions $f$ and a wide range of notions of hardness besides the classic 0-1 loss, by choosing the right notion of fidelity via convex duality. We show the difficulty of learning $f$ using a class $\mathcal{C}$ to minimize a convex loss is tightly characterized by the maximum fidelity possible for a masking channel, which measures the least amount of noise the channel must add to hide the structure of $f$ and make it $\mathcal{C}$-indistinguishable from random noise. Using our framework, we derive a number of multiclass generalizations of the hardcore lemma, and reveal some unexpected subtleties in going to larger alphabets. We construct a universal masking channel that applies to many natural notions of hardness simultaneously. Such channels are dual to omnipredictors, which optimize multiple losses at once (Gopalan, Kalai, Reingold, Sharan, and Wieder, 2022). We observe a symmetry between masking channels and generalizations of the hardcore lemma using pseudoentropy. We present a tight characterization of pseudoentropy using the MaxEnt algorithm from machine learning. A long line of research has shown that learning and hardness are two sides of the same coin; our framework gives a crisp and general formulation of this principle via convex duality.
#03Nov 3, 2026
cs.LG
Agnostic Learning by Imitating
Sílvia Casacuberta, Parikshit Gopalan, Varun Kanade and 3 more
Given a class of machine learning models and sample access to a data distribution with binary labels, which computational task is harder: training a discriminative model that distinguishes positive and negative labels as well as the best member of the class, or training a generative model whose outputs are indistinguishable from real data by all members of the class? We prove that these two fundamental tasks in computational learning theory are not just closely related, but actually equivalent to each other in complexity. One direction of this equivalence was previously known. Indeed, the powerful regularity lemma in complexity theory gives an efficient oracle reduction from the indistinguishability task (a.k.a. multiaccuracy) to the distinguishing task (a.k.a. agnostic learning) in the style of boosting algorithms. Our main contribution is an efficient oracle reduction in the other direction. Taking this idea further, we show that a multiaccuracy oracle is a remarkably powerful primitive, achieving not just agnostic learning but also a wide range of seemingly more demanding learning objectives via similarly efficient and direct constructions. Our results crucially use the power of making adaptive queries to the oracle, and stand in stark contrast to prior work, which showed that post-processing a single multiaccurate predictor is insufficient for even weak forms of learning. On a technical level, we draw a connection between our (offline) problem and a version of no-regret learning in the online setting, which allows us to use follow-the-leader type algorithms to modify the label distribution and aggregate the predictions of the multiaccuracy oracle.
#04Nov 3, 2026
cs.LG
De Rham cohomological architectures for PINNs and PDEs with topological holes
Andrew Gracyk
We attempt to alleviate failure modes of classical physics-informed neural networks (PINNs) in scenarios of topological pathologies corresponding to punctures, holes, poles, and voids by introducing de Rham cohomological augmentations, or Hodge-de Rham PINNs. Vanilla PINNs often encounter pitfalls in hole-based settings due to discontinuous interfaces, nonzero closed path integration, path dependence, correspondence to exact forms via the exterior derivative, etc., therefore we absolve traditional failures via parameterizing a 1-form by the Hodge-Morrey-Friedrichs (HMF) decomposition for manifolds with boundary, drawing a connection to de Rham cohomology such as through the HMF isomorphism theorem. 1-forms can act as representations of scalar-valued PDE solutions through integral operator mappings. We propose performing a PINN-based loss on the parameterized 1-form through an associated PDE, which subsequently solves the potential-based PINN problem and moreover remedies the failures of vanilla PINNs via the Sobolev-regular operator under a Fréchet differentiation interconnection. The operator relationship often has a nontrivial closed-form and often requires integration or a case-by-case analysis, which we address thoroughly in our experiments. When an integrability condition is not met on our losses, we can equip exhaustion functions to train the networks. We demonstrate our methods on the Aharonov-Bohm effect, which, to our understanding, is largely nonexistent in PINN literature. We develop theoretical guarantees of our methods and demonstrate our pathological safeguards on a broad selection of topologically-adverse problems, including scenarios of inexact forms, and those of nontrivial harmonics and coexact terms.
#05Nov 3, 2026
cs.RO
GSBR: Geometric Support Behavior Regularization for Online RL
Charles Xu, Jacob Berg, Liyiming Ke and 1 more
Regularizing online reinforcement learning (RL) toward a pretrained base policy can improve sample efficiency and learning stability, but regularization that is too restrictive limits how much the policy can improve. Base policies -- typically diffusion or flow-matching models trained by imitation learning on large, diverse datasets -- capture many behaviors that are reasonable but suboptimal. Existing regularizers using such base policies are too restrictive in one of two ways: behavioral cloning regularizers pull the actor toward the full base-policy distribution, penalizing it for discarding suboptimal modes, while hard support constraints forbid improvements that lie just outside the base policy's support. We propose Geometric Support Behavior Regularization (GSBR), which penalizes actions by their distance from the base policy's support. Actions within the support incur no penalty, so the actor can concentrate on high-value modes; outside the support, where value estimates are less reliable, the penalty grows smoothly with distance while still permitting improvement when the value gained outweighs the penalty. GSBR estimates this distance from a finite set of base-policy samples, so it requires only sample access to the base policy. Across 19 tasks on four robotic benchmarks, GSBR raises the average success rate from 33% for the base policy to 95%, achieving the highest average success rate and throughput among all compared methods, with about 23% higher normalized throughput than the next-best RL method. Project website and open-source code at https://charlesxu0124.github.io/gsbr/ .