Research map / Video / Video segmentation and tracking
An Analysis of Quantile Temporal-Difference Learning
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proves almost-sure convergence of synchronous quantile temporal-difference learning (QTD) to the fixed points of a family of quantile dynamic programming operators, via differential inclusions and a Lyapunov function, and bounds the fixed-point approximation error.
Problem
Theoretical convergence of quantile temporal-difference learning (QTD), a distributional RL algorithm using asymmetric L1 (quantile regression) losses, whose updates are non-linear, non-contractive and may have multiple fixed points.
Approach
Relates QTD to a family of quantile dynamic programming (QDP) operators parametrised by interpolation parameters lambda; shows the projected distributional Bellman operator is a gamma-contraction in Wasserstein-infinity, models QTD as a noisy discretisation of a differential inclusion, and constructs a Lyapunov function to prove almost-sure convergence under weaker assumptions than classical TD (n
Key results
- Proves QTD converges almost surely to the set of fixed points of projected distributional Bellman operators under step sizes with sum alpha_k = infinity and alpha_k = o(1/log k), without finite-variance assumptions.
- Shows the projected operator Pi^lambda T^pi is a gamma-contraction in Wasserstein-infinity, with a unique fixed point for m quantiles.
- Bounds the fixed-point error: w1-bar(eta_hat, eta) <= (Vmax - Vmin)/(2m(1-gamma)), improving with more quantiles m.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use QTD with the stated step-size conditions | QTD · Finite tabular MDPs | convergence probability: with probability 1 | Helped |
| Iterate the projected distributional Bellman operator Π^λ T^π | QDP · Finite tabular MDPs | convergence rate: γ^k | Helped |
| Increase the number of quantiles m | QTD/QDP fixed point · Bounded-reward MDPs | Wasserstein-1 fixed-point error bound: (V_max - V_min)/(2m(1-γ)) | Helped |
| Use equally spaced quantile levels τ_i = (2i-1)/(2m) | QTD · Bound on distribution approximation | Wasserstein-1 fixed-point error bound | Helped |
| Have transition and reward distributions become Dirac deltas after k steps | QTD/QDP fixed point · MDPs deterministic after k steps | Wasserstein-1 fixed-point error bound: (V_max - V_min)(1-γ^k)/(2m(1-γ)) | Helped |
| Increase m from 2 to 100 | QDP · Two-state MDP return distribution at x₁ | distribution approximation quality: m=2; m=100 | Helped |
| Use m=2 quantiles | QDP · Two-state MDP return distribution at x₁ | return overestimation: m=2; 20 | Hurt |
Surprising findings
- QTD can converge to a set rather than a point, or have multiple fixed points, depending on reward distribution structure (e.g. deterministic rewards with non-trivial transition probabilities).
- With few quantiles (m=2), fixed-point return estimates can be heavily over-estimated due to self-loop backups in the quantile backup diagram.
Limitations
- Main convergence proof is for synchronous updates; asynchronous case only sketched.
- Analysis is tabular with finite state and action spaces; no function approximation.
Experimental setup
Metrics: Wasserstein-1 distance (bound) Wasserstein-infinity contraction
Baselines: classical TD learning categorical temporal-difference learning
Builds on: quantile regression quantile temporal-difference learning distributional Bellman operator stochastic approximation with differential inclusions (Benaim-Hofbauer-Sorin)
Closest papers by meaning
- The Statistical Benefits of Quantile Temporal-Difference Learning for Value Estimation (ICML 2023 · similarity 0.87)
- Temporal Difference Learning as Gradient Splitting (ICML 2021 · similarity 0.85)
- An Improved Finite-time Analysis of Temporal Difference Learning with Deep Neural Networks (ICML 2024 · similarity 0.82)
- The surprising efficiency of temporal difference learning for rare event prediction (NeurIPS 2024 · similarity 0.81)
- Discerning Temporal Difference Learning (AAAI 2024 · similarity 0.81)
- Statistical Efficiency of Distributional Temporal Difference Learning (NeurIPS 2024 · similarity 0.81)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.