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Finite-Time Performance of Distributed Temporal Difference Learning with Linear Function Approximation

Thinh T. Doan, Siva Theja Maguluri, Justin Romberg

arXiv:1907.12530v2math.OCcs.LG

TL;DR

The paper addresses finite-time analysis of distributed TD(λ) for cooperative policy evaluation, where agents estimate global discounted reward from local rewards. The method combines consensus among agents with local TD(λ) updates to estimate the value-function weights. λ=1 gives the best approximation, while λ=0 yields faster convergence under large variance.

  • Problem

    The paper addresses finite-time analysis of distributed TD(λ) for cooperative policy evaluation, where agents estimate global discounted reward from local rewards.

    Source: §1 · p. 2, §2 · p. 3

  • Method

    The method combines consensus among agents with local TD(λ) updates to estimate the value-function weights.

    Source: §2.1 · p. 4

  • Results

    λ=1 gives the best approximation, while λ=0 yields faster convergence under large variance.

    Source: §3.2 · p. 9, §5 · p. 15

  • Takeaways & Limitations

    The convergence analysis makes explicit a trade-off between value-function approximation quality and convergence speed as λ changes.

    Source: §3.2 · p. 9, §5 · p. 15

    The stated convergence result requires σmin > 0.

    Source: §3.2 · p. 9

Abstract

from arXiv · show

We study the policy evaluation problem in multi-agent reinforcement learning, modeled by a Markov decision process. In this problem, the agents operate in a common environment under a fixed control policy, working together to discover the value (global discounted accumulative reward) associated with each environmental state. Over a series of time steps, the agents act, get rewarded, update their local estimate of the value function, then communicate with their neighbors. The local update at each agent can be interpreted as a distributed variant of the popular temporal difference learning methods {\sf TD}$ (\lambda)$. Our main contribution is to provide a finite-analysis on the performance of this distributed {\sf TD}$(\lambda)$ algorithm for both constant and time-varying step sizes. The key idea in our analysis is to use the geometric mixing time $\tau$ of the underlying Markov chain, that is, although the "noise" in our algorithm is Markovian, its dependence is very weak at samples spaced out at every $\tau$. We provide an explicit upper bound on the convergence rate of the proposed method as a function of the network topology, the discount factor, the constant $\lambda$, and the mixing time $\tau$. Our results also provide a mathematical explanation for observations that have appeared previously in the literature about the choice of $\lambda$. Our upper bound illustrates the trade-off between approximation accuracy and convergence speed implicit in the choice of $\lambda$. When $\lambda=1$, the solution will correspond to the best possible approximation of the value function, while choosing $\lambda = 0$ leads to faster convergence when the noise in the algorithm has large variance.

1 Introduction

The paper studies policy evaluation in multi-agent reinforcement learning, using linear function approximation to handle large state spaces. It analyzes a consensus-based distributed TD(λ) method under Markovian noise, addressing a gap in finite-time convergence results and the trade-off between approximation accuracy and convergence speed.

1 Introduction

  • Linear function approximation makes value estimation tractable for large state spaces, while policy evaluation estimates expected long-term reward under a fixed policy.The paper focuses on agents cooperatively estimating global accumulative reward from local rewards and limited neighbor communication.Source: §1 · pp. 1–2

1.1 Existing literature

  • Prior finite-time analyses covered single-agent TD or distributed TD(0) under independent noise, leaving distributed TD(λ) with Markovian noise without a convergence-rate analysis.Earlier distributed TD(0) work also required a projection step and a priori knowledge of the solution.Source: §1.1 · pp. 2–3

1.2 Main contributions

  • The paper analyzes a consensus-based distributed TD(λ) algorithm for constant and time-varying step sizes, using the Markov chain’s geometric mixing time τ to handle dependent noise.Dependence becomes quantifiably weak for observations spaced τ steps apart.Source: §1.2 · p. 3
  • The convergence-rate bound depends on network topology, discount factor, λ, and mixing time; λ=1 gives the best approximation, while λ=0 converges faster under large variance.These results provide a theoretical explanation for previously observed TD(λ) behavior.Source: §1.2 · p. 3

2 Multi-agent reinforcement learning

The section formulates cooperative policy evaluation for agents sharing an environment and approximates the global value function with a low-dimensional linear model. It then presents a consensus-based TD(λ) approach and characterizes how λ affects the target solution and convergence behavior.

2 Multi-agent reinforcement learning

  • Agents follow local policies in a common MDP and cooperatively estimate the network’s global discounted accumulative reward from their local rewards.The environment has a global state, and agents communicate over a connected, undirected graph.Source: §2 · p. 3
  • A low-dimensional linear function approximation represents the value function, and agents cooperate to choose weights minimizing its distance from the true value function.The approximation uses L basis vectors with L much smaller than the number of states.Source: §2 · p. 4

2.1 Distributed TD(λ) methods

  • The distributed TD(λ) method combines neighboring agents’ weight estimates through consensus and then applies local temporal-difference updates, sharing estimates rather than local rewards.Each agent maintains its own weight estimate and uses its observed state transition and local reward in the update.Source: §2.1 · pp. 4–6, §2 · p. 4

2.2 Characterization of θ∗

  • At λ=1, the limiting approximation is the best projection of the value function onto the feature subspace; λ<1 can converge faster when sampling noise has large variance.The characterization identifies a trade-off between approximation quality and convergence speed.Source: §2.2 · p. 6

3 Finite-time performance of distributed TD(λ)

The analysis uses the Markov chain’s geometric mixing time and iterate coupling to derive explicit finite-time convergence bounds for distributed TD(λ). It states results for constant and time-varying step sizes, under assumptions on the network, rewards, features, and chain.

3 Finite-time performance of distributed TD(λ)

  • The bounds use geometric mixing time τ and coupling between iterates to handle Markovian noise whose dependence is weak across samples at least τ steps apart.The analysis provides explicit convergence-rate formulas for both step-size regimes.Source: §3 · p. 6

3.1 Notation and Assumptions

  • The analysis assumes doubly stochastic network weights, bounded rewards, linearly independent bounded features, and an irreducible, aperiodic Markov chain.These conditions support the network, projection, and ergodicity properties used in the analysis.Source: §3.1 · p. 7

3.2 Main Results

  • With a suitable constant α, average mean-squared error converges linearly to a ball around θ∗ whose size depends on α.The result requires δ∈(0,1), with network mixing represented by the second-largest singular value σ2 of W.Source: §3.2 · p. 8
  • λ expresses a trade-off: λ=1 gives the best approximation within the feature span, while λ=0 yields faster convergence when algorithmic variance is large.The paper says the variance quantity increases as λ rises toward 1.Source: §3.2 · p. 9
  • With αk=α0/(k+1), the agents’ average mean-square error converges to zero at a sublinear rate.This is the stated result for time-varying step sizes.Source: §3.2 · p. 9

4 Finite-time analysis of distributed TD(λ)

The proof analysis bounds consensus error and bias for distributed TD(λ), treating constant and time-varying step sizes separately.

4 Finite-time analysis of distributed TD(λ)

  • Consensus error, the difference between each agent’s estimate and the agents’ average estimate, is bounded as a common component of both theorem analyses.The section introduces this bound for use in Theorems 1 and 2.Source: §4 · p. 10

4.1 Constant Step Sizes

  • For constant α satisfying the theorem’s condition, the analysis uses a bias-term bound to establish a convergence bound after mixing time τ(α).The bias-term lemma is introduced as a step toward Theorem 1.Source: §4.1 · pp. 10–11

4.2 Time-Varying Step Sizes

  • For αt=α0/(t+1), a bias-term lemma supports the time-varying-step-size analysis, whose stated bound applies from K∗ onward.Lemma 3 specifies a nonincreasing step-size sequence tending to zero and gives its bound for k≥K∗.Source: §4.2 · p. 13

5 Conclusion and Discussion

The paper gives a finite-time analysis of distributed TD(λ) for stationary-policy evaluation and relates its conclusions to the choice of λ.

5 Conclusion and Discussion

  • Finite-time performance analysis of distributed TD(λ) is the paper’s main contribution.The algorithm is a consensus-based variant for estimating a stationary policy’s value function.Source: §5 · p. 15
  • λ = 1 gives the best function-value approximation, while λ = 0 performs better when algorithmic noise has large variance.Source: §5 · p. 15
  • Finite-time analysis for nonstationary policies, including distributed actor-critic methods, is left for future work.Source: §5 · p. 15

A Proof of Lemma 1

The proof uses the doubly stochastic property of W and considers separate constant and time-varying step-size cases.

A Proof of Lemma 1

  • The proof invokes the doubly stochastic property of W to derive a relation used in the argument.Source: A Proof of Lemma 1 · p. 18
  • The proof combines intermediate relations and applies the 2-norm and triangle inequality in its derivation.Source: A Proof of Lemma 1 · p. 19
  • The step-size analysis treats a constant choice and a sequence αk = α0/(k + 1) as separate cases.Source: A Proof of Lemma 1 · p. 19

B Constant step sizes

For constant step sizes, the analysis controls the change in the averaged iterate over a mixing-time interval and uses that bound in subsequent results.

B Constant step sizes

  • With an appropriate constant step size, the change in averaged iterate over any τ(α) mixing-time interval is bounded.The passage identifies this bound as a key ingredient for later results.Source: B Constant step sizes · p. 20
  • The proof considers k ≥ τ(α) and uses the Markov-chain mixing time to analyze the iterates over the preceding interval.Source: B Constant step sizes · p. 20
  • The argument uses the step-size condition (1 + γ)ατ(α) ≤ (1 − γλ) log(2).Source: B Constant step sizes · p. 21

C Time-varying step sizes

The time-varying-step-size proof uses a mixing time associated with αk and develops bounds under stated step-size conditions.

C Time-varying step sizes

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