Procedural Memory Transferability

• Procedural memory transferability is defined as the ability to share dynamic, action-oriented processes between agents and systems, enhancing skill learning across tasks.
• Adaptive algorithms like BAT and BTT optimize sample transfer by balancing estimation and transfer errors, effectively mitigating negative transfer.
• Empirical studies show that adaptive procedural transfer accelerates convergence in reinforcement learning, especially in sparse target data regimes.

Procedural memory transferability refers to the capability of transferring not merely static knowledge but dynamic, action-oriented processes—such as skills, routines, and strategies—between agents, systems, or contexts. This concept is foundational in both biological and artificial cognitive domains, supporting tasks ranging from reinforcement learning across Markov decision processes (MDPs) to modular memory in multi-agent LLM systems. The following sections synthesize technical insights, mathematical formalisms, algorithms, and empirical results underpinning procedural memory transferability.

1. Theoretical Foundations and Error Decomposition

Procedural memory in agent learning systems is closely aligned with the representation and transfer of dynamic action policies. In reinforcement learning, this is formalized via the transfer of transition and reward samples from source MDPs to aid learning in a target MDP. The key method, “All-Sample Transfer” (AST), employs fitted Q-iteration (FQI) within a linear function space over an MDP $\mathcal{M} = \langle X, A, R, P, \gamma\rangle$.

When transferring samples, one defines an average MDP using weights $\lambda = (\lambda_1,\ldots,\lambda_M)$, where mixing source and target tasks induces an average Bellman operator: $$(T_{(\lambda)}Q)(x,a) = \left[\sum_m \lambda_m R_m(x,a)\right] + \gamma \int_x \max_{a'} Q(y, a')\left[\sum_m \lambda_m P_m(dy|x,a)\right]$$ A critical insight is the finite-sample performance bound for each FQI iteration: $$\|T Q^k - Q_1^{(k-1)}\|_\mu \leq 4 \|f_{\alpha_*^k} - Q_1^{(k-1)}\|_\mu + 5\sqrt{E_\lambda(Q^{(k-1)})} + \text{(estimation errors)}$$ where $E_\lambda(Q^{(k-1)})$ is the transfer error capturing deviations between the target and averaged Bellman operators. Error decomposition thus isolates: (i) the approximation error (function space expressiveness), (ii) estimation error (sample size), and (iii) transfer error (task mismatch). Transfer error is central—excessive dissimilarity among source tasks leads to negative transfer, undermining performance.

2. Adaptive Algorithms for Procedural Memory Transfer

Advances have focused on mitigating negative transfer through adaptive weighting of source task contributions. Two mechanisms exemplify this strategy:

Best Average Transfer (BAT):

• Computes the optimal $\lambda_*^k$ at each iteration to minimize empirical transfer error, using auxiliary samples $(x_s, a_s)$ from the generative models of sources.
• Quantifies transfer error via

$$\widehat{E}_\lambda(Q) = \frac{1}{S}\sum_{s=1}^S\left[R_s^{(1)} - \sum_{m=2}^M \lambda_m R_s^{(m)} + \frac{\gamma}{T}\sum_{t=1}^T \left( \max_{a'} Q(Y^t_{s,1},a') - \sum_{m=2}^M \lambda_m \max_{a'} Q(Y^t_{s,m},a') \right) \right]^2$$

• Theoretical guarantees ensure upper-bounded transfer error and estimation error, see Lemma 1.

Best Tradeoff Transfer (BTT):

• Explicitly models available sample constraints per source via fractions $\beta_m$, solving: $$\widehat{\beta}^k = \arg\min_{\beta \in [0,1]^M}\left(\widehat{E}_\beta(Q^{k-1}) + \tau \sqrt{\frac{d}{\sum_{m=1}^M \beta_m N_m}}\right)$$
• Heuristically trades off the transfer error with estimation error under finite sampling; shows robust results in scenarios where negative transfer is likely.

Together, these schemes model procedural memory transfer as a data-driven, iteration-wise adaptive process grounded in empirical similarity, effectively transferring not just outcome distributions but the procedures for solving tasks.

3. Empirical Analysis: Continuous Chain Problem

Empirical validation in continuous chain-walk domains demonstrates the nuanced dynamics of procedural transferability:

• Source tasks with similar structural dynamics enable adaptive methods (BAT, BTT) to closely match the target by optimally weighting source contributions, dynamically de-emphasizing divergent sources as learning progresses.
• When the source set cannot closely approximate the target, adaptive algorithms shift reliance toward target samples as they accumulate, preventing negative transfer and illustrating the inherent limits of procedural memory generalization.
• In situations of sparse target data, adaptive transfer can significantly accelerate learning by leveraging transferable procedural knowledge; however, as single-task estimation error shrinks, the residual transfer error sets a lower bound for performance improvement.

This experimental paradigm exemplifies the essential role of dynamic, procedure-aware transfer in RL and its connection to efficient exploitation of heterogeneous experience.

4. Mathematical Formalization of Transferability

Procedural memory transferability is rigorously conceptualized through operators and risk bounds:

• The effectivity of sample transfer is governed by the interplay between the empirical risk (estimation error decay as $\mathcal{O}(1/\sqrt{L})$ or $\mathcal{O}(1/L)$ under fast rates), approximation error (projection in the function space $F$), and transfer error $E_\lambda$, which acts as a bias term from model mismatch.
• Optimal mixture weights ($\lambda^*$ or $\beta^*$) are iteratively computed to minimize this bias at each policy improvement step, operationalizing the notion of “procedural” transfer as the alignment of Bellman operators over evolving Q-functions.

The general methodology applies beyond RL; any learning paradigm involving process-based transfers (e.g., imitation learning, multitask policy networks) inherits this error structure.

5. Conceptual and Practical Implications

The principal implications of procedural memory transferability include:

• Dynamic Adaptivity: Effective transfer is not a one-time static process; algorithms must adapt the contribution of each source according to the evolving target dynamics, measured through explicit similarity of Bellman images or policy effects.
• Sample-Efficient Learning: Procedural transfer accelerates convergence especially in small-sample regimes, but its utility diminishes as the contribution from static source tasks cannot further reduce the irreducible transfer error.
• Bias–Variance Tradeoff: Practitioners must balance increased sample size (reducing variance) against the risk of negative transfer (increased bias when source procedures are mismatched).
• Transfer Beyond Rewards: The core insight is transfer of the procedural dynamics—policy improvement mechanisms, not just outcome statistics—enabling the reuse of solution strategies, not merely data.

6. Broader Context and Ongoing Directions

Procedural memory transferability, classically developed in the RL literature (Lazaric et al., 2011), underpins much of modern practice in multitask and lifelong learning. Adaptive transfer algorithms as partial instantiations of “procedural knowledge transfer” catalyze advances across imitation learning, meta-RL, modular LLM systems, and beyond. Open challenges remain in characterizing optimal transfer bounds under more expressive (nonlinear, deep) function approximators, understanding transfer in partially observable and non-stationary domains, and scaling procedural memory structures to continual, lifelong settings. The field continues to explore robust mechanisms for quantifying, optimizing, and deploying procedural memory transfers across increasingly heterogeneous and dynamic task environments.