ICML 2026 Papers

We study gradient testing and gradient estimation of smooth functions using only a comparison oracle that, given two points, indicates which one has the larger function value. For any smooth $f:\mathbb R^n\to\mathbb R$, $\mathbf{x}\in\mathbb R^n$, and $\varepsilon>0$, we design a gradient testing algorithm that determines whether the normalized gradient $\nabla f(\mathbf{x})/\lVert\nabla f(\mathbf{x})\rVert$ is $\varepsilon$-close or $2\varepsilon$-far from a given unit vector $\mathbf{v}$ using $O(1)$ queries, as well as a gradient estimation algorithm that outputs an $\varepsilon$-estimate of $\nabla f(\mathbf{x})/\lVert\nabla f(\mathbf{x})\rVert$ using $O(n\log(1/\varepsilon))$ queries which we prove to be optimal. Furthermore, we study gradient estimation in the quantum comparison oracle model where queries can be made in superpositions, and develop a quantum algorithm using $O(\log (n/\varepsilon))$ queries.

We study simultaneous confidence bounds for aggregated effects over downward-closed subset families of independent statistical tests. The bounds are obtained by bootstrap calibration of the maximum normalized aggregated effect over the relevant subset family, yielding valid post-hoc inference for data-selected subsets and tighter bounds than classical methods that protect all linear contrasts. A central challenge is that the required maximization is a nonlinear combinatorial optimization problem. We cast it as a weighted densest-subgraph problem and derive exact linear and mixed integer linear program reformulations, and we further develop a fully polynomial-time approximation scheme that exploits the rank-1 structure of the objective to scale to large families. On the statistical side, we establish a finite-sample coverage guarantee with a lighter dependence on the subset family size than high-dimensional central limit theorems provide. We illustrate the method on synthetic and real machine learning applications.

We consider minimizing an objective function subject to constraints defined by the intersection of lower-level sets of convex functions. We study two cases: (i) strongly convex and Lipschitz-smooth objective function and (ii) convex but possibly nonsmooth objective function. To deal with the constraints that are not easy to project on, we use a randomized feasibility algorithm with Polyak steps and a random number of sampled constraints per iteration, while taking (sub)gradient steps to minimize the objective function. For case (i), we prove linear convergence in expectation of the objective function values to any prescribed tolerance using an adaptive stepsize. For case (ii), we develop a fully problem parameter-free and adaptive stepsize scheme that yields an $O(1/\sqrt{T})$ worst-case rate in expectation. The infeasibility of the iterates decreases geometrically with the number of feasibility updates almost surely, while for the averaged iterates, we establish an expected lower bound on the function values relative to the optimal value that depends on the distribution for the random number of sampled constraints. For certain choices of sample-size growth, optimal rates are achieved. Finally, simulations on a Quadratically Constrained Quadratic Programming (QCQP) problem, Support Vector Machines (SVM), and logistic regression with group fairness constraints demonstrate the computational efficiency of our algorithm compared to other state-of-the-art methods.

Optimization modeling underpins decision-making in logistics, manufacturing, energy, and finance, yet translating natural-language requirements into correct optimization formulations and solver-executable code remains labor-intensive. Although large language models (LLMs) have been explored for this task, evaluation is still dominated by toy-sized or synthetic benchmarks, masking the difficulty of industrial problems with $10^{3}$--$10^{6}$ (or more) variables and constraints. A key bottleneck is the lack of benchmarks that align natural-language specifications with reference formulations/solver code grounded in real optimization models. To fill in this gap, we introduce MIPLIB-NL, built via a structure-aware reverse construction methodology from real mixed-integer linear programs in MIPLIB~2017. Our pipeline (i) recovers compact, reusable model structure from flat solver formulations, (ii) reverse-generates natural-language specifications explicitly tied to this recovered structure under a unified model--data separation format, and (iii) performs iterative semantic validation through expert review and human--LLM interaction with independent reconstruction checks. This yields 223 one-to-one reconstructions that preserve the mathematical content of the original instances while enabling realistic natural-language-to-optimization evaluation. Experiments show substantial performance degradation on MIPLIB-NL for systems that perform strongly on existing benchmarks, exposing failure modes invisible at toy scale.

Category-level 6D object pose estimation is typically formulated as a multi-category joint learning problem with fully shared model parameters. However, pronounced geometric heterogeneity across categories entangles incompatible optimization signals in shared modules, resulting in gradient conflicts and negative transfer during training. To address this challenge, we first introduce gradient-based diagnostics to quantify module-level cross-category contention. Building on results of diagnostics, we propose DecomPose, a difficulty-aware decomposition framework that mitigates optimization contention via: (1) difficulty-aware gradient decoupling, which groups categories using a data-driven difficulty proxy and routes each instance to a group-specific correspondence branch to isolate incompatible updates; and (2) stability-driven asymmetric branching, which assigns higher-capacity branches to structurally simple categories as stable optimization anchors while constraining complex categories with lightweight branches to suppress noisy updates and alleviate negative transfer. Extensive experiments on REAL275, CAMERA25, and HouseCat6D demonstrate that DecomPose effectively reduces cross-category optimization contention and delivers superior pose estimation performance across multiple benchmarks.