two-timescale-markovian-sa-convergence

name	two-timescale-markovian-sa-convergence
description	Convergence theory for two-timescale stochastic approximations under Markovian noise, applicable to TDC and actor-critic methods. First almost sure convergence proof for TDC with eligibility traces under off-policy learning with linear function approximation. ICML 2026.

Two-Timescale Markovian SA Convergence

Paper: arXiv:2605.31172 | Submitted: 29 May 2026 | ICML 2026

Authors: Vagul Mahadevan, Claire Chen, Shuze Daniel Liu, Shangtong Zhang

Core Concept

This work establishes stability and convergence of two-timescale stochastic approximation (SA) algorithms under Markovian noise, a more realistic setup for RL compared to prior i.i.d. noise assumptions.

Key Innovation

Technical Novelty: Control fast timescale parameter with running max of slow timescale parameter instead of current slow parameter (as prior works do).

This enables:

• Convergence without projection operators
• Noise not required to live in compact space
• First convergence proof for TDC with eligibility traces (off-policy)

Two-Timescale Stochastic Approximation

Framework

θ_{k+1} = θ_k + β_k h(θ_k, w_k, X_k)  // Slow timescale
w_{k+1} = w_k + α_k g(θ_k, w_k, X_k)  // Fast timescale

where α_k >> β_k (fast >> slow step sizes)

Key RL Applications

1. TDC (Temporal Difference with Gradient Correction):

   • θ: policy parameters
   • w: value function parameters

2. Actor-Critic Methods:

   • θ: actor (policy)
   • w: critic (value)

3. Off-policy learning with eligibility traces

Theoretical Results

Main Theorem

Under Markovian noise {X_k}:

1. Stability (Boundedness): Both θ_k and w_k remain bounded
2. Convergence: θ_k → θ* (optimal solution)
3. No projection needed: Natural convergence without artificial constraints

Technical Key

Running max control:

bound_w = max_{i≤k} ||θ_i||

Fast parameter w_k bounded by running max of slow parameter, not current value.

This handles Markovian dependencies that i.i.d. analysis cannot.

TDC with Eligibility Traces

First Convergence Result

TDC with eligibility traces under:

• Off-policy learning
• Linear function approximation
• Markovian environment

Achieves almost sure convergence - first such proof.

TDC Algorithm

δ_k = R_k + γ V_w(X_{k+1}) - V_w(X_k)
e_k = γ λ e_{k-1} + φ(X_k)  // Eligibility trace

θ update: θ_{k+1} = θ_k + β_k δ_k e_k
w update: w_{k+1} = w_k + α_k (δ_k - w_k^T φ(X_k)) φ(X_k)

λ: eligibility trace decay parameter

Implementation Guidelines

1. Step Size Selection

α_k = α_0 / (k + 1)^α
β_k = β_0 / (k + 1)^β

where α > β (fast >> slow)

Common: α ∈ [0.5, 1], β ∈ [0.5, 0.8] with α > β

2. Stability Without Projection

No need for projection operator Ω:

// Prior works required:
θ_{k+1} = Π_Ω(θ_k + β_k h(...))

This work: natural convergence without projection.

3. Markovian Noise Handling

• Environment state transitions Markovian
• Observations have temporal dependencies
• Prior i.i.d. assumption unrealistic for RL

Applications

Primary Use Cases

1. Off-policy TD learning - value function estimation
2. Actor-critic convergence - policy gradient methods
3. Eligibility traces - temporal credit assignment
4. Theoretical RL foundations - convergence guarantees

When This Theory Applies

• Two-timescale algorithms (actor-critic, TDC)
• Markovian environments (most RL settings)
• Need convergence guarantees
• Off-policy learning scenarios

Comparison with Prior Works

Aspect	Prior Works	This Work
Noise	i.i.d. required	Markovian ✓
Projection	Required	Not needed ✓
Compact noise	Required	Not needed ✓
TDC+traces	No convergence	First proof ✓

Practical Implications

For Algorithm Design

1. Actor-critic: Convergence guaranteed under realistic assumptions
2. TDC: Eligibility traces work off-policy with convergence
3. Step sizes: Theory guides fast/slow separation

For Implementation

• No projection operator needed
• Handles real environment noise
• Validates existing RL algorithm stability

Activation Keywords

• two-timescale convergence
• Markovian stochastic approximation
• TDC convergence
• actor-critic stability
• eligibility traces convergence
• off-policy TD learning
• running max control

Related Skills

• [[rat-randomized-advantage-transformation]] - Policy optimization
• [[agpo-adaptive-group-policy-optimization]] - GRPO variants
• [[conditional-equivalence-dpo-rlhf]] - RLHF theory
• [[pg-dpo-non-exponential-discounting]] - Policy gradient theory

References

• Paper: arXiv:2605.31172 - "Convergence of Two-Timescale Markovian Stochastic Approximations with Applications in Reinforcement Learning"
• TDC algorithm: Sutton et al. (2009)
• Two-timescale SA: prior theoretical works (i.i.d. case)

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Category: reinforcement-learning, convergence-theory, stochastic-approximation, theoretical-RL