Computing the Bias of Constant-step Stochastic Approximation with Markovian Noise

• URL: http://arxiv.org/abs/2405.14285v2
• Date: Fri, 25 Oct 2024 08:25:02 GMT
• Title: Computing the Bias of Constant-step Stochastic Approximation with Markovian Noise
• Authors: Sebastian Allmeier, Nicolas Gast,
• Abstract summary: We study approximation algorithms with Markovian noise and constant step-size $alpha$. We show that the time-averaged bias is equal to $alpha V + O(alpha2)$, where $V$ is a constant characterized by a Lyapunov equation. We also show that $bartheta_n$ converges with high probability around $theta*+alpha V$.
• Score: 1.068128849363198
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: We study stochastic approximation algorithms with Markovian noise and constant step-size $\alpha$. We develop a method based on infinitesimal generator comparisons to study the bias of the algorithm, which is the expected difference between $\theta_n$ -- the value at iteration $n$ -- and $\theta^*$ -- the unique equilibrium of the corresponding ODE. We show that, under some smoothness conditions, this bias is of order $O(\alpha)$. Furthermore, we show that the time-averaged bias is equal to $\alpha V + O(\alpha^2)$, where $V$ is a constant characterized by a Lyapunov equation, showing that $\mathbb{E}[\bar{\theta}_n] \approx \theta^*+V\alpha + O(\alpha^2)$, where $\bar{\theta}_n=(1/n)\sum_{k=1}^n\theta_k$ is the Polyak-Ruppert average. We also show that $\bar{\theta}_n$ converges with high probability around $\theta^*+\alpha V$. We illustrate how to combine this with Richardson-Romberg extrapolation to derive an iterative scheme with a bias of order $O(\alpha^2)$.

Related papers

• On the $O(\rac{\sqrt{d}}{K^{1/4}})$ Convergence Rate of AdamW Measured by $\ell_1$ Norm [54.28350823319057]
  This paper establishes the convergence rate $frac1Ksum_k=1KEleft[|nabla f(xk)|_1right]leq O(fracsqrtdCK1/4) for AdamW measured by $ell_$ norm, where $K$ represents the iteration number, $d denotes the model dimension, and $C$ matches the constant in the optimal convergence rate of SGD.
  arXiv  Detail & Related papers  (2025-05-17T05:02:52Z)
• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• Revisiting Step-Size Assumptions in Stochastic Approximation [1.3654846342364308]
  It is shown for the first time that this assumption is not required for convergence and finer results.<n>Rates of convergence are obtained for the standard algorithm and for estimates obtained via the averaging technique of Polyak and Ruppert.<n>Results from numerical experiments illustrate that $beta_theta$ may be large due to a combination of multiplicative noise and Markovian memory.
  arXiv  Detail & Related papers  (2024-05-28T05:11:05Z)
• Sample-Efficient Linear Regression with Self-Selection Bias [7.605563562103568]
  We consider the problem of linear regression with self-selection bias in the unknown-index setting. We provide a novel and near optimally sample-efficient (in terms of $k$) algorithm to recover $mathbfw_1,ldots,mathbfw_kin. Our algorithm succeeds under significantly relaxed noise assumptions, and therefore also succeeds in the related setting of max-linear regression.
  arXiv  Detail & Related papers  (2024-02-22T02:20:24Z)
• Provably learning a multi-head attention layer [55.2904547651831]
  Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. In this work, we initiate the study of provably learning a multi-head attention layer from random examples. We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
  arXiv  Detail & Related papers  (2024-02-06T15:39:09Z)
• Solving Dense Linear Systems Faster Than via Preconditioning [1.8854491183340518]
  We show that our algorithm has an $tilde O(n2)$ when $k=O(n0.729)$. In particular, our algorithm has an $tilde O(n2)$ when $k=O(n0.729)$. Our main algorithm can be viewed as a randomized block coordinate descent method.
  arXiv  Detail & Related papers  (2023-12-14T12:53:34Z)
• Average R\'{e}nyi Entropy of a Subsystem in Random Pure State [0.0]
  We plot the $ln m$-dependence of the quantum information derived from $widetildeS_alpha (m,n)$. It is remarkable to note that the nearly vanishing region of the information becomes shorten with increasing $alpha$, and eventually disappears in the limit of $alpha rightarrow infty$.
  arXiv  Detail & Related papers  (2023-01-22T08:08:51Z)
• Faster Sampling from Log-Concave Distributions over Polytopes via a Soft-Threshold Dikin Walk [28.431572772564518]
  We consider the problem of sampling from a $d$-dimensional log-concave distribution $pi(theta) propto e-f(theta)$ constrained to a polytope $K$ defined by $m$ inequalities. Our main result is a "soft-warm'' variant of the Dikin walk Markov chain that requires at most $O((md + d L2 R2) times MDomega-1) log(fracwdelta)$ arithmetic operations to sample from $pi$
  arXiv  Detail & Related papers  (2022-06-19T11:33:07Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Mean Estimation in High-Dimensional Binary Markov Gaussian Mixture Models [12.746888269949407]
  We consider a high-dimensional mean estimation problem over a binary hidden Markov model. We establish a nearly minimax optimal (up to logarithmic factors) estimation error rate, as a function of $|theta_*|,delta,d,n$.
  arXiv  Detail & Related papers  (2022-06-06T09:34:04Z)
• Optimal Mean Estimation without a Variance [103.26777953032537]
  We study the problem of heavy-tailed mean estimation in settings where the variance of the data-generating distribution does not exist. We design an estimator which attains the smallest possible confidence interval as a function of $n,d,delta$.
  arXiv  Detail & Related papers  (2020-11-24T22:39:21Z)
• Sparse sketches with small inversion bias [79.77110958547695]
  Inversion bias arises when averaging estimates of quantities that depend on the inverse covariance. We develop a framework for analyzing inversion bias, based on our proposed concept of an $(epsilon,delta)$-unbiased estimator for random matrices. We show that when the sketching matrix $S$ is dense and has i.i.d. sub-gaussian entries, the estimator $(epsilon,delta)$-unbiased for $(Atop A)-1$ with a sketch of size $m=O(d+sqrt d/
  arXiv  Detail & Related papers  (2020-11-21T01:33:15Z)
• Accelerating Optimization and Reinforcement Learning with Quasi-Stochastic Approximation [2.294014185517203]
  This paper sets out to extend convergence theory to quasi-stochastic approximations. It is illustrated with applications to gradient-free optimization and policy gradient algorithms for reinforcement learning.
  arXiv  Detail & Related papers  (2020-09-30T04:44:45Z)
• Model-Free Reinforcement Learning: from Clipped Pseudo-Regret to Sample Complexity [59.34067736545355]
  Given an MDP with $S$ states, $A$ actions, the discount factor $gamma in (0,1)$, and an approximation threshold $epsilon > 0$, we provide a model-free algorithm to learn an $epsilon$-optimal policy. For small enough $epsilon$, we show an improved algorithm with sample complexity.
  arXiv  Detail & Related papers  (2020-06-06T13:34:41Z)

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

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.