Related papers: Statistical-Computational Trade-offs for Greedy Recursive Partitioning Estimators

Statistical-Computational Trade-offs for Greedy Recursive Partitioning Estimators

URL: http://arxiv.org/abs/2411.04394v1
Date: Thu, 07 Nov 2024 03:11:53 GMT
Title: Statistical-Computational Trade-offs for Greedy Recursive Partitioning Estimators
Authors: Yan Shuo Tan, Jason M. Klusowski, Krishnakumar Balasubramanian,
Abstract summary: We show that greedy algorithms for high-dimensional regression get stuck at local optima. We show that greedy training requires $exp(Omega(d))$ to achieve low estimation error. We also show that greedy training can attain small estimation error with only $O(log d)$ samples.
Score: 23.056208049082134
License:
Abstract: Models based on recursive partitioning such as decision trees and their ensembles are popular for high-dimensional regression as they can potentially avoid the curse of dimensionality. Because empirical risk minimization (ERM) is computationally infeasible, these models are typically trained using greedy algorithms. Although effective in many cases, these algorithms have been empirically observed to get stuck at local optima. We explore this phenomenon in the context of learning sparse regression functions over $d$ binary features, showing that when the true regression function $f^*$ does not satisfy the so-called Merged Staircase Property (MSP), greedy training requires $\exp(\Omega(d))$ to achieve low estimation error. Conversely, when $f^*$ does satisfy MSP, greedy training can attain small estimation error with only $O(\log d)$ samples. This performance mirrors that of two-layer neural networks trained with stochastic gradient descent (SGD) in the mean-field regime, thereby establishing a head-to-head comparison between SGD-trained neural networks and greedy recursive partitioning estimators. Furthermore, ERM-trained recursive partitioning estimators achieve low estimation error with $O(\log d)$ samples irrespective of whether $f^*$ satisfies MSP, thereby demonstrating a statistical-computational trade-off for greedy training. Our proofs are based on a novel interpretation of greedy recursive partitioning using stochastic process theory and a coupling technique that may be of independent interest.

Related papers

MGDA Converges under Generalized Smoothness, Provably [27.87166415148172]
Multi-objective optimization (MOO) is receiving more attention in various fields such as multi-task learning. Recent works provide some effective algorithms with theoretical analysis but they are limited by the standard $L$-smooth or bounded-gradient assumptions. We study a more general and realistic class of generalized $ell$-smooth loss functions, where $ell$ is a general non-decreasing function of gradient norm.
arXiv Detail & Related papers (2024-05-29T18:36:59Z)
Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs [56.237917407785545]
We consider the problem of learning an $varepsilon$-optimal policy in a general class of continuous-space Markov decision processes (MDPs) having smooth Bellman operators. Key to our solution is a novel projection technique based on ideas from harmonic analysis. Our result bridges the gap between two popular but conflicting perspectives on continuous-space MDPs.
arXiv Detail & Related papers (2024-05-10T09:58:47Z)
Computational-Statistical Gaps in Gaussian Single-Index Models [77.1473134227844]
Single-Index Models are high-dimensional regression problems with planted structure. We show that computationally efficient algorithms, both within the Statistical Query (SQ) and the Low-Degree Polynomial (LDP) framework, necessarily require $Omega(dkstar/2)$ samples.
arXiv Detail & Related papers (2024-03-08T18:50:19Z)
Effective Minkowski Dimension of Deep Nonparametric Regression: Function Approximation and Statistical Theories [70.90012822736988]
Existing theories on deep nonparametric regression have shown that when the input data lie on a low-dimensional manifold, deep neural networks can adapt to intrinsic data structures. This paper introduces a relaxed assumption that input data are concentrated around a subset of $mathbbRd$ denoted by $mathcalS$, and the intrinsic dimension $mathcalS$ can be characterized by a new complexity notation -- effective Minkowski dimension.
arXiv Detail & Related papers (2023-06-26T17:13:31Z)
Distributional Reinforcement Learning with Dual Expectile-Quantile Regression [51.87411935256015]
quantile regression approach to distributional RL provides flexible and effective way of learning arbitrary return distributions. We show that distributional guarantees vanish, and we empirically observe that the estimated distribution rapidly collapses to its mean estimation. Motivated by the efficiency of $L$-based learning, we propose to jointly learn expectiles and quantiles of the return distribution in a way that allows efficient learning while keeping an estimate of the full distribution of returns.
arXiv Detail & Related papers (2023-05-26T12:30:05Z)
Generalization and Stability of Interpolating Neural Networks with Minimal Width [37.908159361149835]
We investigate the generalization and optimization of shallow neural-networks trained by gradient in the interpolating regime. We prove the training loss number minimizations $m=Omega(log4 (n))$ neurons and neurons $Tapprox n$. With $m=Omega(log4 (n))$ neurons and $Tapprox n$, we bound the test loss training by $tildeO (1/)$.
arXiv Detail & Related papers (2023-02-18T05:06:15Z)
Provably Efficient Offline Reinforcement Learning with Trajectory-Wise Reward [66.81579829897392]
We propose a novel offline reinforcement learning algorithm called Pessimistic vAlue iteRaTion with rEward Decomposition (PARTED) PARTED decomposes the trajectory return into per-step proxy rewards via least-squares-based reward redistribution, and then performs pessimistic value based on the learned proxy reward. To the best of our knowledge, PARTED is the first offline RL algorithm that is provably efficient in general MDP with trajectory-wise reward.
arXiv Detail & Related papers (2022-06-13T19:11:22Z)
An Improved Analysis of Gradient Tracking for Decentralized Machine Learning [34.144764431505486]
We consider decentralized machine learning over a network where the training data is distributed across $n$ agents. The agent's common goal is to find a model that minimizes the average of all local loss functions. We improve the dependency on $p$ from $mathcalO(p-1)$ to $mathcalO(p-1)$ in the noiseless case.
arXiv Detail & Related papers (2022-02-08T12:58:14Z)
Towards an Understanding of Benign Overfitting in Neural Networks [104.2956323934544]
Modern machine learning models often employ a huge number of parameters and are typically optimized to have zero training loss. We examine how these benign overfitting phenomena occur in a two-layer neural network setting. We show that it is possible for the two-layer ReLU network interpolator to achieve a near minimax-optimal learning rate.
arXiv Detail & Related papers (2021-06-06T19:08:53Z)
An efficient projection neural network for $\ell_1$-regularized logistic regression [10.517079029721257]
This paper presents a simple projection neural network for $ell_$-regularized logistics regression. The proposed neural network does not require any extra auxiliary variable nor any smooth approximation. We also investigate the convergence of the proposed neural network by using the Lyapunov theory and show that it converges to a solution of the problem with any arbitrary initial value.
arXiv Detail & Related papers (2021-05-12T06:13:44Z)
Convergence of Online Adaptive and Recurrent Optimization Algorithms [0.0]
We prove local convergence of several notable descent algorithms used in machine learning. We adopt an "ergodic" rather than probabilistic viewpoint, working with empirical time averages instead of probability distributions.
arXiv Detail & Related papers (2020-05-12T09:48:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.