The Variance of Thought: Policy Variance, Critical Forks, and Local Credit Assignment

• URL: http://arxiv.org/abs/2608.22467v1
• Date: Sun, 23 Aug 2026 15:46:56 GMT
• Title: The Variance of Thought: Policy Variance, Critical Forks, and Local Credit Assignment
• Abstract summary: Policy variance is the sole source of return variance and is injected in discrete pulses at states we call critical forks.<n> bootstrapping removes the Monte Carlo advantage estimate by converting a product of survival probabilities into a sum.
• Score: 2.6994604905555004
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Long-horizon language-model tasks --- multi-step reasoning and tool-using agents alike --- are limited by credit assignment. We analyze it through the policy variance $σ_π^2(s)=\operatorname{Var}_{a\simπ}[Q_π(s,a)]$, which in a deterministic MDP is the sole source of return variance and is injected in discrete pulses at states we call critical forks. Three results follow. (i) Policy variance is a discovery budget: observing an action of advantage $c$ requires $Ω(c^2/σ_π^2(s))$ draws, a bound that is exact on the canonical two-point fork. (ii) Policy variance is bounded by the policy's Gini dispersion, $σ_π^2(s)\le 1-\|π(\cdot|s)\|_2^2$, a rollout-free necessary condition for criticality computable from logits alone. (iii) The remaining horizon sets the estimation cost: at a fork whose downstream success probability is $P$, the Monte Carlo advantage estimate has signal-to-noise ratio of order $\sqrt{P}$, so its sample cost scales as $1/P$ --- a cost that branched sampling shares. Bootstrapping removes it by converting a product of survival probabilities into a sum, provided the value representation is multiplicatively accurate, which argues for log-value parameterization.

Related papers

• Privacy Amplification Without Independence: How Far Negative Dependence Carries the Guarantees of Poisson Subsampling [3.6238624372073254]
  We prove a linear law for the Rényi-difference criterion for random allocation with $k=1$.<n>We give a substitution map for privacy accounting: when Poisson-based computations remain sound for structured participation, where they fail, and what sound alternatives cost in deployment.
  arXiv  Detail & Related papers  (2026-09-01T23:20:11Z)
• Robust Average-Reward Markov Decision Processes: Minimax-Optimal Learning via Plug-in Reductions [51.50375419691955]
  Distributionally robust Markov decision processes provide a principled framework for sequential decision making under model uncertainty.<n>We study how many samples are necessary and sufficient to learn an $varepsilon$-optimal robust policy under the average-reward criterion.
  arXiv  Detail & Related papers  (2026-08-06T19:49:48Z)
• Online Covariance Estimation in Averaged SGD: Improved Batch-Mean Rates and Minimax Optimality via Trajectory Regression [12.805268849262243]
  We study online covariance matrix estimation for Polyak-Ruppert averaged gradient descent (SGD)<n>The construction reveals that the bottleneck is the sublinear accumulation of information about the Hessian from the SGD drift.
  arXiv  Detail & Related papers  (2026-04-12T20:49:33Z)
• Adaptive stable distribution and Hurst exponent by method of moments moving estimator for nonstationary time series [0.49728186750345144]
  We will focus on novel more agnostic approach: moving estimator, which estimates parameters separately for every time.<n>We will show its applications for alpha-Stable distribution, which also influences Hurst exponent, hence can be used for its adaptive estimation.
  arXiv  Detail & Related papers  (2025-05-20T08:36:49Z)
• Beyond likelihood ratio bias: Nested multi-time-scale stochastic approximation for likelihood-free parameter estimation [49.78792404811239]
  We study inference in simulation-based models where the analytical form of the likelihood is unknown.<n>We use a ratio-free nested multi-time-scale approximation (SA) method that simultaneously tracks the score and drives the parameter update.<n>We show that our algorithm can eliminate the original bias $Obig(sqrtfrac1Nbig)$ and accelerate the convergence rate from $Obig(beta_k+sqrtfracalpha_kNbig)$.
  arXiv  Detail & Related papers  (2024-11-20T02:46:15Z)
• Variance Reduction for the Independent Metropolis Sampler [11.074080383657453]
  We prove that if $pi$ is close enough under KL divergence to another density $q$, an independent sampler that obtains samples from $pi$ achieves smaller variance than i.i.d. sampling from $pi$. We propose an adaptive independent Metropolis algorithm that adapts the proposal density such that its KL divergence with the target is being reduced.
  arXiv  Detail & Related papers  (2024-06-25T16:38:53Z)
• Multiple-policy Evaluation via Density Estimation [30.914344538340412]
  We propose an algorithm named $mathrmCAESAR$ for this problem. Up to low order and logarithmic terms $mathrmCAESAR$ achieves a sample complexity $tildeOleft(fracH4epsilon2sum_h=1Hmax_kin[K]sum_s,afrac(d_hpik(s,a))2mu*_h(s,a)right)$, where $dpi
  arXiv  Detail & Related papers  (2024-03-29T23:55:25Z)
• TIC-TAC: A Framework for Improved Covariance Estimation in Deep Heteroscedastic Regression [109.69084997173196]
  Deepscedastic regression involves jointly optimizing the mean and covariance of the predicted distribution using the negative log-likelihood. Recent works show that this may result in sub-optimal convergence due to the challenges associated with covariance estimation. We study two questions: (1) Does the predicted covariance truly capture the randomness of the predicted mean? Our results show that not only does TIC accurately learn the covariance, it additionally facilitates an improved convergence of the negative log-likelihood.
  arXiv  Detail & Related papers  (2023-10-29T09:54:03Z)
• Adaptive Stochastic Variance Reduction for Non-convex Finite-Sum Minimization [52.25843977506935]
  We propose an adaptive variance method, called AdaSpider, for $L$-smooth, non-reduction functions with a finitesum structure. In doing so, we are able to compute an $epsilon-stationary point with $tildeOleft + st/epsilon calls.
  arXiv  Detail & Related papers  (2022-11-03T14:41:46Z)
• Horizon-Free and Variance-Dependent Reinforcement Learning for Latent Markov Decision Processes [62.90204655228324]
  We study regret minimization for reinforcement learning (RL) in Latent Markov Decision Processes (LMDPs) with context in hindsight. We design a novel model-based algorithmic framework which can be instantiated with both a model-optimistic and a value-optimistic solver.
  arXiv  Detail & Related papers  (2022-10-20T21:32:01Z)
• Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret [144.6358229217845]
  We study the problem of learning in the shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus. We prove that EB-SSP achieves the minimax regret rate $widetildeO(B_star sqrtS A K)$, where $K$ is the number of episodes, $S$ is the number of states, $A$
  arXiv  Detail & Related papers  (2021-04-22T17:20:48Z)
• Variance-Aware Confidence Set: Variance-Dependent Bound for Linear Bandits and Horizon-Free Bound for Linear Mixture MDP [76.94328400919836]
  We show how to construct variance-aware confidence sets for linear bandits and linear mixture Decision Process (MDP) For linear bandits, we obtain an $widetildeO(mathrmpoly(d)sqrt1 + sum_i=1Ksigma_i2) regret bound, where $d is the feature dimension. For linear mixture MDP, we obtain an $widetildeO(mathrmpoly(d)sqrtK)$ regret bound, where
  arXiv  Detail & Related papers  (2021-01-29T18:57:52Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)

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.