arXiv subjects

Haichen Hu

Publications and source records attributed to Haichen Hu.

11 recordsLinked to original sources

arXiv2026

Sharp Structure-Agnostic Minimax Risk for Partial Linear Models

We characterize the sharp structure-agnostic minimax risk for coefficient estimation in the partial linear model when the outcome and treatment nuisances are learned by two distinct black-box learners, which resolves the open problem in double machine learning posed by Gu (2025). For each nuisance $q\in\{\mu,\pi\}$, we characterize the available learner by an approximation-error budget $a_q$ and a stochastic-error budget $s_q$, with the latter controlled through localized Rademacher complexity. Writing $\mathcal E_n$ for the minimax mean-squared error, we show that \[\mathcal E_n\asymp1\wedge\left\{\frac1n+\left(a_\mu a_\pi+\min\left\{a_\pi s_\mu+s_\pi^2,\,a_\mu s_\pi+s_\mu^2\right\}\right)^2\right\}.\] The main new ingredient is a novel lower bound for the general two-learner problem. Our proof constructs four finite-mixture testing experiments using orthogonal code functions. Across these experiments, the hidden perturbations are placed outside both learner classes, outside only the treatment learner class, outside only the outcome learner class, or inside both learner classes. These four configurations capture, respectively, the interaction between the two approximation errors, the two asymmetric interactions between one learner's approximation error and the other learner's learning error, and the joint estimation difficulty of learning both nuisances. Combining the four resulting lower bounds yields the displayed rate, which matches the latest upper bound in Gu (2026). Our result shows that standard double machine learning can overstate the intrinsic difficulty of target estimation and provides a target-specific principle for learner selection: approximation error and stochastic complexity must be jointly balanced across the two nuisance learners rather than optimized separately.

cs.LG↗

arXiv2026

Simultaneous Pointwise Majorization for Mixed Tail Processes with Applications in Gaussian Chaos and Ergodic Diffusions

Classical chaining controls an indexed stochastic process through a single worst-case bound and can therefore obscure substantial variation across the index set. We develop the first simultaneous pointwise majorization theory for Banach-valued processes with finite-metric mixed-tail increments. Suppose that an anchored process $(Z_t)_{t\in T}$ satisfies, for some integer $m\ge1$, pseudo-metrics $d_1,\ldots,d_m$, and orders $\alpha_1,\ldots,\alpha_m>0$, \begin{align*} \mathbb{P}\{\|Z_t-Z_s\|>\sum_{j=1}^m u^{1/\alpha_j}d_j(s,t)\}\le 2e^{-u},s,t\in T. \end{align*} For ambient priors $\mu_1,\ldots,\mu_m$, let $v_j(t):=d_j(t,t_0), \Phi_j(t):=\int_0^{4v_j(t)}(\log\frac{1}{\mu_j(B_{d_j}(t,r))})^{1/\alpha_j}dr$. We prove that, $\forall \delta\in(0,1)$, with probability at least $1-\delta$, simultaneously for all $t\in T$, \begin{align*} \|Z_t\|\le C_{m,\boldsymbol\alpha}\sum_{j=1}^m\{\Phi_j(t)+v_j(t)(\log(e/\delta))^{1/\alpha_j}\}. \end{align*} Here $\boldsymbol\alpha:=(\alpha_1,\ldots,\alpha_m)$ and $C_{m,\boldsymbol\alpha}$ depend only on $m$ and these tail orders. The result subsumes single-metric sub-Weibull processes of every positive order as the case $m=1$. In the Gaussian setting, it sharpens the pointwise upper bound of \citet{xu2026} by eliminating the logarithmic terms generated by dyadic peeling. The proof retains the index-wise costs of measure-generated admissible chains and synchronizes the regimes through a nested common refinement. Finally, we apply our theorems to stationary diffusion empirical processes and decoupled Gaussian chaos to obtain simultaneous pointwise envelope bounds, which can further be applied to other statistics problems.

math.PR↗

arXiv2026

Optimizing the Preconditioner: A Black-box Online-to-Nonconvex Conversion with Static Regret Minimization Oracles

We study whether stochastic nonconvex optimization can be reduced to ordinary static regret minimization in online convex optimization in a black-box manner. For smooth nonconvex objectives, our reduction maintains a predictable gradient tracker, while a black-box online learner selects a preconditioner that determines how this tracker is transformed into the update direction. The learner receives linear convex losses and is evaluated against a single fixed comparator over one undiscounted online game. For a $\beta$-smooth objective with range bounded by $M$ and an unbiased stochastic-gradient oracle with variance bounded by $\sigma^2$, we establish $$\frac{1}{T}\sum_{t=1}^T \mathbb E\!\left[\|\nabla f(x_t)\|_2^2\right] \lesssim \frac{\sigma\sqrt{M\beta}}{\sqrt T} + \frac{\sqrt{M\beta}\, \mathscr R_T(\mathcal A,I_d)}{T} + \frac{M\beta}{T}.$$ Consequently, any black-box OCO algorithm with $\mathscr R_T(\mathcal A,I_d)=O(\sqrt T)$ recovers the classical $O(\frac{1}{\sqrt{T}})$ convergence rate. We further show that the same black-box framework extends beyond the smooth setting to Lipschitz nonconvex objectives without Lipschitz continuous gradients. Importantly, this extension continues to rely only on an ordinary static-regret guarantee and requires no stronger notion of online regret. When the OCO oracle admits square-root static regret, the resulting conversion achieves the optimal $O(T^{-2/7})$ convergence rate for the corresponding Goldstein stationary point. These results resolve the open problem posed by Chen and Hazan (2024). More broadly, our framework separates optimizer design into gradient prediction and online preconditioner selection, providing a principled perspective on how adaptive optimization methods may be understood through static regret and applied in nonconvex optimization.

cs.LG↗

arXiv2026

Model-Based Reinforcement Learning with Double Oracle Efficiency in Policy Optimization and Offline Estimation

Reinforcement learning (RL) in large environments often suffers from severe computational bottlenecks, as conventional regret minimization algorithms require repeated, costly calls to planning and statistical estimation oracles. While recent advances have explored offline oracle-efficient algorithms, their computational complexity typically scales with the cardinality of the state and action spaces, rendering them intractable for large-scale or continuous environments. In this paper, we address this fundamental limitation by studying offline oracle-efficient episodic RL through the lens of log-barrier and log-determinant regularization. Specifically, for tabular Markov Decision Processes (MDPs), we propose a novel algorithm that achieves the optimal $\tilde{O}(\sqrt{T})$ regret bound while requiring only $O(H\log\log T)$ calls to both the offline statistical estimation and planning oracles when $T$ is known and $O(H\log T)$ calls when $T$ is unknown. Crucially, this oracle complexity is entirely independent of the size of the state and action spaces. This strict independence drastically reduces the planning oracle complexity, representing a substantial improvement over existing offline oracle-efficient algorithms (Qian et al., 2024). Furthermore, we demonstrate the versatility of our framework by generalizing the algorithm to linear MDPs featuring infinite state spaces and arbitrary action spaces. We prove that this generalized approach successfully attains meaningful sub-linear regret. Consequently, our work yields the first doubly oracle-efficient (i.e., efficient with respect to both statistical estimation and policy optimization) regret minimization algorithm capable of solving MDPs with infinite state and action spaces, significantly expanding the boundaries of computationally tractable RL.

cs.LG↗

arXiv2026

Interleaved Resampling and Refitting: Data and Compute-Efficient Evaluation of Black-Box Predictors

We study the problem of evaluating the excess risk of large-scale empirical risk minimization under the square loss. Leveraging the idea of wild refitting and resampling, we assume only black-box access to the training algorithm and develop an efficient procedure for estimating the excess risk. Our evaluation algorithm is both computationally and data efficient. In particular, it requires access to only a single dataset and does not rely on any additional validation data. Computationally, it only requires refitting the model on several much smaller datasets obtained through sequential resampling, in contrast to previous wild refitting methods that require full-scale retraining and might therefore be unsuitable for large-scale trained predictors. Our algorithm has an interleaved sequential resampling-and-refitting structure. We first construct pseudo-responses through a randomized residual symmetrization procedure. At each round, we thus resample two sub-datasets from the resulting covariate pseudo-response pairs. Finally, we retrain the model separately on these two small artificial datasets. We establish high probability excess risk guarantees under both fixed design and random design settings, showing that with a suitably chosen noise scale, our interleaved resampling and refitting algorithm yields an upper bound on the prediction error. Our theoretical analysis draws on tools from empirical process theory, harmonic analysis, Toeplitz operator theory, and sharp tensor concentration inequalities.

cs.LG↗

arXiv2025

Perturbing the Derivative: Doubly Wild Refitting for Model-Free Evaluation of Opaque Machine Learning Predictors

We study the problem of excess risk evaluation for empirical risk minimization (ERM) under convex losses. We show that by leveraging the idea of wild refitting, one can upper bound the excess risk through the so-called "wild optimism," without relying on the global structure of the underlying function class but only assuming black box access to the training algorithm and a single dataset. We begin by generating two sets of artificially modified pseudo-outcomes created by stochastically perturbing the derivatives with carefully chosen scaling. Using these pseudo-labeled datasets, we refit the black-box procedure twice to obtain two wild predictors and derive an efficient excess risk upper bound under the fixed design setting. Requiring no prior knowledge of the complexity of the underlying function class, our method is essentially model-free and holds significant promise for theoretically evaluating modern opaque deep neural networks and generative models, where traditional learning theory could be infeasible due to the extreme complexity of the hypothesis class.

cs.LG↗

arXiv2025

Perturbing the Derivative: Wild Refitting for Model-Free Evaluation of Machine Learning Models under Bregman Losses

We study the excess risk evaluation of classical penalized empirical risk minimization (ERM) with Bregman losses. We show that by leveraging the idea of wild refitting, one can efficiently upper bound the excess risk through the so-called "wild optimism," without relying on the global structure of the underlying function class. This property makes our approach inherently model-free. Unlike conventional analysis, our framework operates with just one dataset and black-box access to the training procedure. The method involves randomized Rademacher symmetrization and constructing artificially modified outputs by perturbation in the derivative space with appropriate scaling, upon which we retrain a second predictor for excess risk estimation. We establish high-probability performance guarantee under the fixed design setting, demonstrating that wild refitting under Bregman losses, with an appropriately chosen wild noise scale, yields a valid upper bound on the excess risk. Thus, our work is promising for theoretically evaluating modern opaque ML models, such as deep neural networks and generative models, where the function class is too complex for classical learning theory and empirical process techniques.

stat.ML↗

arXiv2025

Pre-Trained AI Model Assisted Online Decision-Making under Missing Covariates: A Theoretical Perspective

We study a sequential contextual decision-making problem in which certain covariates are missing but can be imputed using a pre-trained AI model. From a theoretical perspective, we analyze how the presence of such a model influences the regret of the decision-making process. We introduce a novel notion called "model elasticity", which quantifies the sensitivity of the reward function to the discrepancy between the true covariate and its imputed counterpart. This concept provides a unified way to characterize the regret incurred due to model imputation, regardless of the underlying missingness mechanism. More surprisingly, we show that under the missing at random (MAR) setting, it is possible to sequentially calibrate the pre-trained model using tools from orthogonal statistical learning and doubly robust regression. This calibration significantly improves the quality of the imputed covariates, leading to much better regret guarantees. Our analysis highlights the practical value of having an accurate pre-trained model in sequential decision-making tasks and suggests that model elasticity may serve as a fundamental metric for understanding and improving the integration of pre-trained models in a wide range of data-driven decision-making problems.

cs.LG↗

arXiv2025

Constrained Online Decision-Making: A Unified Framework

Contextual online decision-making problems with constraints appear in a wide range of real-world applications, such as adaptive experimental design under safety constraints, personalized recommendation with resource limits, and dynamic pricing under fairness requirements. In this paper, we investigate a general formulation of sequential decision-making with stage-wise feasibility constraints, where at each round, the learner must select an action based on observed context while ensuring that a problem-specific feasibility criterion is satisfied. We propose a unified algorithmic framework that captures many existing constrained learning problems, including constrained bandits, active learning with label budgets, online hypothesis testing with Type I error control, and model calibration. Central to our approach is the concept of upper counterfactual confidence bounds, which enables the design of practically efficient online algorithms with strong theoretical guarantees using any offline conditional density estimation oracle. To handle feasibility constraints in complex environments, we introduce a generalized notion of the eluder dimension, extending it from the classical setting based on square loss to a broader class of metric-like probability divergences. This allows us to capture the complexity of various density function classes and characterize the utility regret incurred due to feasibility constraint uncertainty. Our result offers a principled foundation for constrained sequential decision-making in both theory and practice.

stat.ML↗

arXiv2025

Contextual Online Decision Making with Infinite-Dimensional Functional Regression

Contextual sequential decision-making problems play a crucial role in machine learning, encompassing a wide range of downstream applications such as bandits, sequential hypothesis testing and online risk control. These applications often require different statistical measures, including expectation, variance and quantiles. In this paper, we provide a universal admissible algorithm framework for dealing with all kinds of contextual online decision-making problems that directly learns the whole underlying unknown distribution instead of focusing on individual statistics. This is much more difficult because the dimension of the regression is uncountably infinite, and any existing linear contextual bandits algorithm will result in infinite regret. To overcome this issue, we propose an efficient infinite-dimensional functional regression oracle for contextual cumulative distribution functions (CDFs), where each data point is modeled as a combination of context-dependent CDF basis functions. Our analysis reveals that the decay rate of the eigenvalue sequence of the design integral operator governs the regression error rate and, consequently, the utility regret rate. Specifically, when the eigenvalue sequence exhibits a polynomial decay of order $\frac{1}{\gamma}\ge 1$, the utility regret is bounded by $\tilde{\mathcal{O}}\Big(T^{\frac{3\gamma+2}{2(\gamma+2)}}\Big)$. By setting $\gamma=0$, this recovers the existing optimal regret rate for contextual bandits with finite-dimensional regression and is optimal under a stronger exponential decay assumption. Additionally, we provide a numerical method to compute the eigenvalue sequence of the integral operator, enabling the practical implementation of our framework.

stat.ML↗

arXiv2024

Offline Oracle-Efficient Learning for Contextual MDPs via Layerwise Exploration-Exploitation Tradeoff

Motivated by the recent discovery of a statistical and computational reduction from contextual bandits to offline regression (Simchi-Levi and Xu, 2021), we address the general (stochastic) Contextual Markov Decision Process (CMDP) problem with horizon H (as known as CMDP with H layers). In this paper, we introduce a reduction from CMDPs to offline density estimation under the realizability assumption, i.e., a model class M containing the true underlying CMDP is provided in advance. We develop an efficient, statistically near-optimal algorithm requiring only O(HlogT) calls to an offline density estimation algorithm (or oracle) across all T rounds of interaction. This number can be further reduced to O(HloglogT) if T is known in advance. Our results mark the first efficient and near-optimal reduction from CMDPs to offline density estimation without imposing any structural assumptions on the model class. A notable feature of our algorithm is the design of a layerwise exploration-exploitation tradeoff tailored to address the layerwise structure of CMDPs. Additionally, our algorithm is versatile and applicable to pure exploration tasks in reward-free reinforcement learning.

cs.LG↗