DoMo-AC: Doubly Multi-step Off-policy Actor-Critic Algorithm

Multi-step learning extends policy evaluation and control beyond single-step lookaheads, but its practical application in optimal control remains limited because multi-step policy improvements require operations that cannot be approximated by stochastic samples. This paper introduces DoMo-VI, a novel oracle algorithm combining multi-step policy evaluation and improvement, and its practical instantiation, DoMo-AC, which achieves improved policy gradient estimates through a bias-variance trade-off when combined with the IMPALA architecture.

The gist

DoMo-AC introduces a practical instantiation of the DoMo-VI algorithm that combines multi-step policy improvements and policy evaluations to achieve guaranteed convergence speedup to the optimal policy in general off-policy learning settings.

Background and Motivation

Off-policy learning involves two critical components: off-policy evaluation (approximating the value function of a target policy) and off-policy control (approximating the optimal value function). While multi-step learning has provided robust improvements to policy evaluation, combining it with policy improvement in the control case is fundamentally challenging because multi-step control requires solving an optimal control problem in an inner loop, which hinders its direct application with sample-based learning and incremental learning. The paper addresses this by aiming to make multi-step off-policy learning practical and theoretically sound for the control case.

Doubly Multi-step Off-policy VI (DoMo-VI)

DoMo-VI is a multi-step learning algorithm consisting of two steps: multi-step policy evaluation and multi-step improvement. The recursions are defined as:

1. Policy Improvement: πi+1(·x) = arg max π∈Π Rπ,µc¯ Vi(x)

2. Policy Evaluation: Vi+1 = Rπi,µc¯ Vi

Setting the threshold parameter to zero reduces DoMo-VI to standard Value Iteration (VI). When the threshold is positive, it allows the improvement objective to effectively look ahead multiple steps starting from x, resulting in a stronger improvement when the maximization problem can be solved exactly. The paper proves that DoMo-VI converges to the optimal policy with an accelerated convergence rate, where the contraction rate depends on a scalar η∗.

Doubly Multi-step Off-policy Actor-Critic (DoMo-AC)

DoMo-AC is a practical instantiation of DoMo-VI, designed to allow for a bias-variance trade-off in constructing policy gradient estimates from off-policy data. The policy update is performed via:

1. Policy Update: θi+1 = θi + βEx∼b [∇θiRπθi,µc¯ Vi(x)]

2. Value Update: Vi+1 = Rπθi,µc¯ Vi

The paper shows that the choice of the trace coefficient threshold c¯ mediates this trade-off. When c¯ increases from 0 to 10 in deep RL experiments on Atari games, the bias generally decreases, whereas the variance increases rapidly, leading to an optimal middle ground (in this case log ¯c ≈ 0 and c¯ ≈ 1) where the squared error is lowest.

Key Theoretical Results and Analysis

The paper establishes several key theoretical results:

Lemma 1: For any real-valued function V over X, a scalar cbar, and a behavior policy µ, there exists a Markov policy π such that π = arg maxp Rπ,µc¯ V.

This lemma implies that an optimally improved policy according to the improvement objective Rπ,µc¯ Vi is feasible to find.

"Theorem 2: Assume that expected rewards take values in [-R, ¯R], and V0 is bounded by 1/(1−γ). Then, there exist a scalar η∗ ∈ [0, γ] and a sequence of scalars (ηj)∞j=1 in [0, γ] such that DoMoVI (Eqn (2)) generates a sequence of Markov policies (πi)∞i=1 with value functions satisfying the following guarantee: Vπi+1 − V

∞ ≤ max(η∗) i, Yij=1 ηj4R¯(1 − γ)2. This shows that DoMo-VI generates policy sequence πi whose performance Vπi converges to the optimal performance V∗."

Experimental Validation

The paper validates its claims through experiments on tabular MDPs and deep RL environments (Atari-57 games) using the IMPALA architecture. The results confirm that DoMo-AC outperforms the one-step variant and the IMPALA baseline, showing statistically significant improvements when using V-trace with a threshold c¯ in between 0.3 and 0.5, which is found to be optimal for balancing bias and variance.

As a fastidious and diligent AI researcher, I have thoroughly analyzed the proposed DoMo-AC: Doubly Multi-step Off-policy Actor-Critic Algorithm. This algorithm introduces a novel combination of multi-step policy evaluation and improvement within an off-policy actor-critic framework, specifically designed to address the limitations of traditional one-step methods in optimal control.

Here are the specific improvements this paper enables for AI systems and what these improved systems can achieve:

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) Specific Improvements Enabled by DoMo-AC:

1. Policy Gradient Accuracy Enhancement (Bias Reduction):

DoMo-AC utilizes a policy gradient estimator, where the target used in the update is derived from an off-policy value function estimate, specifically the doubly robust back-up target estimate:

Vtarget(Xt) = v(Xt) + ˜ρtδt + γct (Vtarget(Xt+1) − v(Xt)) (Eqn. 9, where ct = min(¯c, ρt)).

This mechanism effectively incorporates multi-step lookahead into the policy gradient calculation.

The paper proves that for the stochastic gradient estimate, as the trace coefficient threshold increases (from 0 to 10 in experiments), the bias generally decreases.

2. Variance-Bias Trade-off Optimization:

DoMo-AC explicitly introduces a tunable hyperparameter, the trace coefficient threshold, denoted as 'c' (or '¯c' in some contexts), which mediates a critical trade-off between variance and bias in the gradient estimate:

The optimal value of c ∈ [0.3, 0.5] is noticeably lower than the typical value of the trace threshold applied in value-based learning (e.g., Retrace and V-trace all adopt c¯ = 1 in their implementations by default).

This allows researchers to select an estimate that minimizes the squared error between the stochastic gradient estimate and the true policy gradient, achieving a lowest squared error among this class of stochastic gradient estimates.

3. Accelerated Convergence Speed:

By combining multi-step evaluation with policy improvement (DoMo-VI), the algorithm demonstrates a guaranteed convergence speed-up to the optimal policy compared to one-step baseline VI and even multi-step policy evaluation alone.

The theoretical analysis proves that DoMoVI generates a sequence of Markov policies whose performance converges to the optimal performance with an accelerated convergence rate, particularly when intermediate values of c are used.

4. Robustness in Large-Scale Deep RL:

The algorithm is implemented using the IMPALA architecture (distributed actor-critic), which is inherently suited for large-scale, distributed training environments common in deep reinforcement learning:

DoMo-AC achieves stable performance improvements over baseline methods on Atari-57 game benchmarks.

) Capabilities of the Improved AI System:

The improved AI systems powered by DoMo-AC will be capable of performing tasks with significantly higher efficiency and precision than current state-of-the-art policy gradient methods:

1. Enhanced Control in Complex Environments (Atari & Beyond):

The system will exhibit superior performance on complex, high-dimensional control tasks (like the 57 Atari games benchmark) by achieving better overall human-normalized scores compared to the baseline IMPALA and one-step variants. It can learn optimal motor skills or strategies with fewer training steps due to the accelerated convergence rate.

2. Optimized Exploration and Policy Search:

Because DoMo-AC allows for the optimization of the policy improvement objective via multiple gradient ascent steps (N), it can more effectively search for policies that maximize long-term rewards, leading to more robust and less sub-optimal final policies in environments where precise optimization is computationally expensive.

3. Efficient Off-Policy Learning:

The system will leverage off-policy data (e.g., expert demonstrations or previous experiences) more effectively than methods that rely solely on immediate bootstrapped estimates, enabling faster learning from diverse and potentially delayed data streams without suffering from the high variance associated with very long lookahead horizons.

4. Adaptable Gradient Estimation:

The system can dynamically adjust its gradient estimation strategy (by tuning 'c') to balance the trade-off between getting a low bias estimate (high 'c') and maintaining low variance estimates (lower 'c'), ensuring the most accurate policy updates for the specific learning stage.

5. Scalable Distributed Training:

The system is designed to be instantiated on distributed actor-critic architectures like IMPALA, meaning it can scale to massive GPU clusters, allowing it to tackle environments with high state and action spaces that are intractable for single-agent algorithms.