Memory-Amortized Inference

• Memory-Amortized Inference (MAI) is a framework that reconceptualizes inference as a process using structured, reusable memory cycles to reduce computational demands.
• It employs bootstrapping and retrieval operators to adapt previously optimized latent codes for new observations, ensuring energy efficiency and context sensitivity.
• By integrating topological data analysis, reinforcement learning, and neuroscience, MAI supports robust generalization and a unified inference-planning cycle.

Memory-Amortized Inference (MAI) is a formal framework for inference in artificial and biological systems that emphasizes the structured reuse of past inference trajectories, or cycles in memory, as opposed to recomputation by optimization from scratch. MAI models cognition as inference over latent memory cycles, enabling context-aware, structure-preserving, and energy-efficient inference by leveraging persistent topological memories. This contrasts with traditional approaches such as ergodic sampling or iterative gradient descent, providing a biologically grounded and computationally efficient theory for intelligent systems.

1. Core Concept: Inference via Structured Memory Cycles

MAI reconceptualizes inference as a process in which, for each new observation or context Ψ, the system does not solve for the latent variables Φ by fresh optimization. Instead, an operator retrieves a latent code Φ from a memory ℳ of past optimized context–content pairs and adapts it to fit the current context. The core operators are:

• Bootstrapping Update: Φₜ₊₁ = 𝓕(Φₜ, Ψₜ)
• Retrieval/Adaptation: Φₜ ≈ ℛ(Φₜ₊₁, Ψₜ)

Inference thus progresses as a cycle:

$$Φ_{t+1} = \mathcal{F}(Φ_t, Ψ_t),\quad Φ_t ≈ \mathcal{R}(Φ_{t+1}, Ψ_t)$$

The system "amortizes" the computational cost by drawing on previously stored latent trajectories (cycles), with local adaptation, so the amortization gap relative to full re-optimization remains small. This framework makes inference both memory-driven and path-dependent, rather than memoryless and uniform.

2. Structural Reuse and Inductive Biases

A critical aspect of MAI is structural reuse: the past trajectories through the latent space ℤ are not discarded after inference, but persist as cycles γ ⊂ ℤ. These cycles encode strong inductive biases by constraining inference for new contexts Ψ to prefer portions of latent space previously validated by experience. The likelihood of inferring a latent state along an existing cycle is much higher than random sampling:

$$\Pr(Φ_{t+1} \in γ \mid Φ_t) \gg \text{Uniform}(\mathcal{Z})$$

This ensures that entropy is minimized along these recurrent paths, facilitating generalization and context sensitivity. The persistency of cycles is topologically characterized, supporting the view that robust memory and efficient inference emerge from non-ergodic, cyclic traversals within the latent space.

3. Comparison with Traditional Inference Paradigms

In contrast to MAI, traditional inference methods include:

• Gradient Descent Optimization: Each new inference request requires re-optimization of Φ by minimizing a loss, often incurring substantial computational cost and missing opportunities for reuse of structural knowledge extracted from past experience.
• Ergodic Sampling: Ergodic inference assumes all states are equally likely to be visited over time, treating each observation independently and failing to capture temporal or structural dependencies inherent in intelligent processing.

MAI is fundamentally non-ergodic, emphasizing that cognition is not sampling over the full latent space, but recurrently traversing low-entropy, structured cycles in memory. Rather than treating each problem as new, MAI systems recall and reuse compatible prior cycles, making inference both more adaptive and less resource-intensive.

4. Delta-Homology and Biological Substrate

The MAI framework draws on delta-homology from algebraic topology to formalize the persistence and gluing of latent cycles. When symmetry in inference is broken—such as the selection of a unique latent cause among many possibilities—the residue manifests as a nontrivial topological cycle γ ∈ H₁(ℤ), encoding invariant structure through repeated experiences.

This topological formalism maps onto Mountcastle’s Universal Cortical Algorithm: each neocortical column is modeled as a local inference loop (a MAI operator), implementing feedforward bootstrapping (𝓕) and feedback retrieval (ℛ). The brain-wide network of such columns realizes efficient, context-aware inference through cycle-consistent memory, making cortical computation a biological instantiation of MAI principles.

5. Time-Reversal Duality with Reinforcement Learning

A central insight in MAI is its time-reversal duality with reinforcement learning (RL):

• Reinforcement Learning: Propagates value estimates forward from present state to future reward (e.g., value iteration).

$$V(s_t) \gets V(s_t) + \alpha [r_t + \gamma V(s_{t+1}) - V(s_t)]$$

• MAI: Retraces latent causes or past states from predictions of future states, i.e., bootstrapping in reverse:

$Φ_t ≈ \mathcal{R}(Φ_{t+1}, Ψ_t)$

While RL reduces uncertainty by anticipating future outcomes, MAI minimizes inference uncertainty by reconstructing the structured causes of current observations from memory. This time-reversed perspective bridges planning and inference within a unified algorithmic cycle.

6. Implications for AGI

MAI's foundational aspects have direct implications for the development of AGI:

• Energy Efficiency: By recycling structured memory rather than recomputing inferences wholesale, MAI systems achieve lower energy and computational costs.
• Robust Generalization: Inductive biases encoded in persistent cycles support strong generalization from few examples, as inference is guided by the structure of past successes.
• Biological Plausibility: The MAI paradigm maps onto observed neural architectures (e.g., hippocampal replay, cortical microcircuits) and cognitive processes such as contextual retrieval and consolidated memory.
• Unified Inference-Planning Loop: The duality with RL suggests that a genuinely general intelligent agent must integrate planning and memory-driven inference, enabling forward prediction and backward explanation within a coherent cycle.
• Modality Agnosticism: As structural cycles are topological, MAI frameworks can unify multi-modal reasoning, sensory perception, motor planning, and abstraction without specialized modules for each.

In aggregate, MAI provides a rigorous, structure-centered alternative to optimization-from-scratch, positioning memory-guided inference as a fundamental computational primitive for both biological and artificial intelligent systems (Li, 19 Aug 2025).

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Memory-Amortized Inference thus reframes the foundations of inference and cognition, prioritizing the structured reuse of memory cycles, topological constraints, and energy efficiency as the bedrock of general intelligence. This theoretical synthesis links topological data analysis, probabilistic inference, reinforcement learning, and neuroscience into a unified, non-ergodic perspective on intelligence.