Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1512.09170

Statistical Query Algorithms for Mean Vector Estimation and Stochastic Convex Optimization

Vitaly Feldman·Cristobal Guzman·Santosh Vempala
View original source ↗Read source document ↗Export metadata ↓

Abstract

Stochastic convex optimization, where the objective is the expectation of a random convex function, is an important and widely used method with numerous applications in machine learning, statistics, operations research and other areas. We study the complexity of stochastic convex optimization given only statistical query (SQ) access to the objective function. We show that well-known and popular first-order iterative methods can be implemented using only statistical queries. For many cases of interest we derive nearly matching upper and lower bounds on the estimation (sample) complexity including linear optimization in the most general setting. We then present several consequences for machine learning, differential privacy and proving concrete lower bounds on the power of convex optimization based methods. The key ingredient of our work is SQ algorithms and lower bounds for estimating the mean vector of a distribution over vectors supported on a convex body in $\mathbb{R}^d$. This natural problem has not been previously studied and we show that our solutions can be used to get substantially improved SQ versions of Perceptron and other online algorithms for learning halfspaces.

Explore related subjects

cs.LGcs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vitaly Feldman, Cristobal Guzman, Santosh Vempala. 2016-11-22. Statistical Query Algorithms for Mean Vector Estimation and Stochastic Convex Optimization. https://arxiv.org/abs/1512.09170

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Online Regularized Statistical Learning in Reproducing Kernel Hilbert Space With Non-Stationary Data

We study recursive regularized learning algorithms in the reproducing kernel Hilbert space (RKHS) with non-stationary online data streams. We introduce the concept of a random Tikhonov regularization path and decompose the tracking error of the algorithm's output for the regularization path into random difference equations in RKHS. We show that the tracking error vanishes in mean square and almost surely if the regularization path is slowly time-varying. Then, leveraging the monotonicity of inverse operators and the spectral decomposition of compact operators, and introducing the RKHS persistence of excitation condition, we develop a dominated convergence method to prove the mean square and almost sure consistency between the regularization path and the unknown function to be learned. Especially, for independent and non-identically distributed data streams, the mean square and almost sure consistency between the algorithm's output and the unknown function is achieved if the input data's marginal probability measures are slowly time-varying and the average measure over each fixed-length time period is uniformly above a strictly positive finite Borel measure.

cs.LG↗
arXiv2026

Reflective Policy Optimization

On-policy reinforcement learning methods, like Trust Region Policy Optimization (TRPO) and Proximal Policy Optimization (PPO), often demand extensive data per update, leading to sample inefficiency. This paper introduces Reflective Policy Optimization (RPO), a novel on-policy extension that amalgamates past and future state-action information for policy optimization. This approach empowers the agent for introspection, allowing modifications to its actions within the current state. Theoretical analysis confirms that policy performance is monotonically improved and contracts the solution space, consequently expediting the convergence procedure. Empirical results demonstrate RPO's feasibility and efficacy in two reinforcement learning benchmarks, culminating in superior sample efficiency. The source code of this work is available at https://github.com/Edgargan/RPO.

cs.LG↗
arXiv2026

Transductive Off-policy Proximal Policy Optimization

Proximal Policy Optimization (PPO) is a popular model-free reinforcement learning algorithm, esteemed for its simplicity and efficacy. However, due to its inherent on-policy nature, its proficiency in harnessing data from disparate policies is constrained. This paper introduces a novel off-policy extension to the original PPO method, christened Transductive Off-policy PPO (ToPPO). Herein, we provide theoretical justification for incorporating off-policy data in PPO training and prudent guidelines for its safe application. Our contribution includes a novel formulation of the policy improvement lower bound for prospective policies derived from off-policy data, accompanied by a computationally efficient mechanism to optimize this bound, underpinned by assurances of monotonic improvement. Comprehensive experimental results across six representative tasks underscore ToPPO's promising performance.

cs.LG↗