Rolling Memory: Principles & Applications

• Rolling memory is a strategy that retains a fixed window of recent data while discarding older information to balance context retention with computational efficiency.
• It is applied in reinforcement learning using frame stacking and adaptive stacking to selectively preserve critical observations, reducing memory footprint and processing cost.
• Rolling memory underpins methods in stochastic optimization and LLM inference, using sliding windows, exponential decay, and reversible compression to improve stability and long-context modeling.

Rolling memory refers to a class of memory management strategies where information is implicitly or explicitly maintained over a moving window or set of states, discarding or compressing earlier content as new information arrives. This paradigm, widely used in sequence modeling and optimization, encompasses techniques ranging from frame stacking in reinforcement learning agents to finite-window and adaptive memory schemes in stochastic optimization and LLMs. Rolling memory mechanisms address the dual objectives of maintaining relevant historical context while controlling computational and storage costs—an essential balance in domains with long-range dependencies or resource constraints.

1. Fundamental Concepts of Rolling Memory

The canonical form of rolling memory is the sliding-window (frame stacking) mechanism, wherein at each time step, the system retains the most recent $k$ items and discards anything older. In formal terms, for an agent or model consuming a sequence $\{x_1,\ldots,x_t\}$, the state at time $t$ is

$s_t = [x_{t-k+1}, \ldots, x_t].$

This first-in/first-out (FIFO) buffer strategy implies that both inference and learning costs depend linearly (MLP, LSTM) or quadratically (Transformers) on the window length $k$ (Tasse et al., 22 Dec 2025). Finite-window averaging in optimization algorithms is also a direct application of rolling memory, where a moving average of the last $N$ gradients is used to update parameters (Orvieto et al., 2019).

Rolling memory is not limited to naïve windowing: modern extensions adapt the management policy, hierarchically compress information, or dynamically allocate memory based on predicted relevance, enabling efficient modeling of long-term dependencies.

2. Rolling Memory in Sequence Agents and Reinforcement Learning

Frame stacking is the standard approach to symbolically extend agent state in partially observable Markov decision processes (POMDPs). The agent augments its observation with the most recent $k$ frames, making the policy depend on $[x_{t-k+1}, \ldots, x_t]$ ("frame stacking" or "rolling memory" in RL literature) (Tasse et al., 22 Dec 2025).

However, this approach is intractable for environments exhibiting long or sparse dependencies—if the true dependency length $k^*$ is much larger than the buffer size ($k^* \gg k$), frame stacking cannot capture the necessary context, while raising $k$ increases computational and memory cost to $\Omega(k)$ or $\Omega(k^2)$ per step.

Adaptive Stacking (AS) generalizes rolling memory by making the choice of which memory slot to evict a learnable, reward-driven sub-policy. Instead of dropping the oldest element by default, the agent selects a slot to pop:

$$s_{t+1} = \operatorname{push}(\operatorname{pop}(s_t,i_t), x_{t+1}),$$

with the slot index $i_t$ chosen by a joint action policy $\pi_k(a, i|s)$ (Tasse et al., 22 Dec 2025). This adaptive policy learns to retain "high-value" observations (e.g., cues in memory tasks) while evicting irrelevant frames, dramatically reducing the required memory footprint (to $\kappa \ll k^*$) and computational cost.

Theoretical results guarantee that, for any $k \geq \kappa$ (minimal sufficient memory length), an RL algorithm using unbiased-return estimates will converge to an optimal policy, and standard TD methods also converge under suitable consistency assumptions. Empirically, AS matches frame stacking with $k^*$ frames in cumulative reward, using only $\kappa$ slots, with similar benefits for MLP, LSTM, and Transformer agents.

3. Rolling Memory in Stochastic Optimization

Rolling memory provides a unifying abstraction for several gradient-based optimization algorithms. Classical approaches include:

• Finite-window memory: The update rule uses the mean of the most recent $N$ gradients:

$$x_{k+1} = x_k - \eta \left(\frac{1}{N} \sum_{i=k-N+1}^k \nabla f_i(x_i)\right).$$

This leads to the "RollSGD–N" algorithm, directly analogous to sliding-window averaging (Orvieto et al., 2019).

• Exponential decay (momentum): The update is a weighted average with exponentially decaying weights:

$$m_{k+1} = \beta m_k + (1-\beta) \nabla f(x_k), \qquad x_{k+1} = x_k - \eta m_{k+1}.$$

Here, all past gradients influence the current direction, but more recent ones count more.

The continuous-time analysis via SDEs shows that a general memory kernel $K(\cdot)$ encodes how information from the past is forgotten or retained, with finite window and exponential decay as special cases:

$$dV(t) = -\int_0^t K(t-s) \nabla f(X(s))ds\,dt - \alpha(t)V(t)dt.$$

Empirical findings indicate that finite-window or polynomially decaying memory can deliver superior stability in convex (and noisy) optimization, avoiding the variance explosion seen with pure momentum. In deep, nonconvex scenarios, shorter windows or moderate-order polynomial forgetting suffice, as distant historical gradients often become stale.

The same principles extend to adaptive optimizers' second moments (e.g., Adam, RMSprop). Finite-window variance estimation

$$v_{k+1} = \frac{1}{N} \sum_{i=k-N+1}^k [\nabla f_i(x_i)]^2$$

tunes the adaptivity and stability trade-off in preconditioning (Orvieto et al., 2019).

4. Hierarchical and Reversible Compression in LLMs

Recent advances in LLMs employ rolling memory to enable long-context modeling without unbounded memory growth. The R$^3$Mem architecture implements rolling compression by segmenting input sequences and using virtual memory tokens to "zip" information in each segment (Wang et al., 21 Feb 2025):

• Each segment $c^s$ is flanked by read ($\theta^r$) and write ($\theta^w$) virtual tokens.
• After processing a segment, the write tokens summarize its content and later become the read tokens for the next segment, propagating compressed context forward ("rolling" memory).
• The reversible Transformer backbone allows exact decompression (reconstruction) via a backward pass, providing a bijective mapping between raw context and compressed memory. This enables retrieval of any segment by rolling memory backward.
• Hierarchical compression further refines information from document to entity level, using a structured training curriculum and powerful LLMs (e.g., GPT-4o) for context decomposition.

Empirical results show that R$^3$Mem achieves state-of-the-art (SOTA) perplexity on long-context language modeling tasks (e.g., 5.21 on PG19, 2.39 on arXiv), retrieval-augmented generation, and long-horizon conversational agents—substantially outperforming previous memory modules (Wang et al., 21 Feb 2025). Ablations confirm that both the reversible design and hierarchical curriculum are essential for robust retention and retrieval.

5. Efficient Rolling Memory for Long-Context LLM Inference

The Adaptive Soft Rolling KV Freeze with Entropy-Guided Recovery (ASR-KF-EGR) framework provides a training-free, inference-time rolling memory mechanism for LLMs (Metinov et al., 12 Dec 2025). It operates as follows:

• The active key-value (KV) cache (in GPU) is supplemented by a frozen KV store (on CPU).
• Token relevance is dynamically scored (e.g., via attention strength):

$$s_j = \frac{1}{H} \sum_{h=1}^H |Q_i^{(h)} \cdot K_j^{(h)T}|,$$

and tokens below a threshold $\tau$ are "soft-frozen"—evicted from the active cache but preserved on CPU.

• Freeze duration grows sublinearly as $d_j = \lfloor \sqrt{c_j} / k \rfloor$ with the number of times $c_j$ a token is flagged as low-importance, ensuring that no token is permanently excluded.
• Upon need (detected, for example, by entropy of the next-token distribution), frozen tokens are restored for attention.

This mechanism achieves substantial active KV cache reduction—up to 67%—without quality loss, passes "needle-in-haystack" retrieval, and allows full recovery of all historical context, outperforming irreversible hard eviction methods (Metinov et al., 12 Dec 2025). The resulting active cache grows sublinearly ($O(N^\gamma)$ with $\gamma<1$), with practical overhead limited by engineering choices, and is compatible with any model architecture without retraining.

6. Comparative Summary of Rolling Memory Implementations

The following table summarizes major rolling memory variants and their distinctive features as reported in the cited works:

Approach	Retention Mechanism	Key Advantages / Limitations
Sliding Window / Frame Stacking (Tasse et al., 22 Dec 2025)	FIFO buffer of last $k$ elements	Simple, but poor for long/sparse dependencies
Adaptive Stacking (Tasse et al., 22 Dec 2025)	Learned eviction of memory slots	Learns to keep only predictive frames; provably optimal with small memory
RollSGD–N, finite-window SGD (Orvieto et al., 2019)	Average over recent $N$ gradients	High stability in convex settings, direct control of bias/variance
Exponential decay (momentum) (Orvieto et al., 2019)	Weighted by decaying exponential	All history influences update, but may be less stable; variance can explode
R$^3$Mem rolling virtual tokens (Wang et al., 21 Feb 2025)	Infinitely rolling, compress-decompress	SOTA long-context retention/retrieval, reversible, hierarchical
ASR-KF-EGR rolling KV freeze (Metinov et al., 12 Dec 2025)	Soft, reversible freeze of KVs	Sublinear growth, no lost context, full reversibility at inference

7. Practical Guidelines and Limitations

Practical application of rolling memory requires adapting the window size, retention policy, and eviction criterion to the task and architecture:

• In RL, Adaptive Stacking dramatically reduces memory requirements and improves sample efficiency when only a subset of past observations are causally relevant (Tasse et al., 22 Dec 2025).
• For stochastic optimization, moderate-length rolling windows or polynomial forgetting deliver faster, more stable convergence in convex regimes, while too-long memory degrades adaptation in deep nonconvex problems (Orvieto et al., 2019).
• In LLMs, rolling and reversible memory strategies enable unbounded context retention, with full-retrievability and competitive empirical gains (Wang et al., 21 Feb 2025, Metinov et al., 12 Dec 2025).
• Rolling memory mechanisms that admit reversible eviction (e.g., soft freeze, virtual tokens) outperform permanent eviction when context dependence is sparse and unpredictable.
• Complexity control: Window size or equivalent parameters should match the minimal sufficient context $\kappa$ rather than the maximal theoretical dependency $k^*$.
• Monitoring strategies such as memory regret and entropy-guided recovery support the diagnosis and tuning of rolling memory systems.

Limitations persist in the handling of rare, impactful long-term dependencies, and the precise tuning of rolling memory parameters remains domain-sensitive. Empirical ablations reveal trade-offs between retention, retrieval fidelity, and generalization across in-domain and out-of-domain tasks (Wang et al., 21 Feb 2025, Orvieto et al., 2019).