Algorithmic Analysis of Dense Associative Memory: Finite-Size Guarantees and Adversarial Robustness

Abstract: Dense Associative Memory (DAM) generalizes Hopfield networks through higher-order interactions and achieves storage capacity that scales as $O(N^{n-1})$ under suitable pattern separation conditions. Existing dynamical analyses primarily study the thermodynamic limit $N\to\infty$ with randomly sampled patterns and therefore do not provide finite-size guarantees or explicit convergence rates. We develop an algorithmic analysis of DAM retrieval dynamics that yields finite-$N$ guarantees under explicit, verifiable pattern conditions. Under a separation assumption and a bounded-interference condition at high loading, we prove geometric convergence of asynchronous retrieval dynamics, which implies $O(\log N)$ convergence time once the trajectory enters the basin of attraction. We further establish adversarial robustness bounds expressed through an explicit margin condition that quantifies the number of corrupted bits tolerable per sweep, and derive capacity guarantees that scale as $Θ(N^{n-1})$ up to polylogarithmic factors in the worst case, while recovering the classical $Θ(N^{n-1})$ scaling for random pattern ensembles. Finally, we show that DAM retrieval dynamics admit a potential-game interpretation that ensures convergence to pure Nash equilibria under asynchronous updates. Complete proofs are provided in the appendices, together with preliminary experiments that illustrate the predicted convergence, robustness, and capacity scaling behavior.

• The paper establishes finite-size algorithmic guarantees for DAM by leveraging explicit separation and interference conditions.
• It proves geometric convergence rates under asynchronous updates and provides adversarial robustness bounds tolerating up to a fixed fraction of bit corruptions.
• The study derives capacity bounds scaling as Θ(N^(n-1)) and interprets retrieval dynamics as best-response updates in potential games, linking theory to practical memory design.

Finite-Size Algorithmic Guarantees and Adversarial Robustness in Dense Associative Memory

Introduction

This work systematically characterizes the retrieval dynamics, robustness, and storage capacity of Dense Associative Memory (DAM) models in the finite-$N$ regime. Building on the core insight that higher-order interaction associative memories can far surpass the capacity of Hopfield networks, the analysis departs from classical statistical-physics approaches—which predominantly focus on the thermodynamic limit and ensemble-averaged random patterns—to provide nonasymptotic, explicit performance guarantees. By introducing verifiable separation and interference assumptions on the set of stored patterns, the paper establishes algorithmic guarantees for asynchronous retrieval dynamics, including convergence rates, robustness thresholds against adversarial corruption, and capacity bounds that scale near-optimally with system size. The results are accompanied by a potential-game interpretation of the update dynamics and an empirical validation across synthetic and real-world datasets.

Model and Analytical Framework

DAM models store $p$ binary patterns in an $N$-neuron network using $n$-way higher-order interactions. The energy function is generalized as $$E(\bm{x}) = -N^{-(n-1)}\sum_{\mu=1}^p (\sum_i \xi_i^\mu x_i)^n$$. Retrieval proceeds via asynchronous updates, where a randomly selected neuron is updated according to the sign of its local field, ensuring monotonicity of the Lyapunov (potential) function and convergence to fixed points.

A salient technical step is the explicit separation assumption: inside the basin of a target pattern, the overlap with all distractor patterns remains bounded by $\beta<\gamma$, where $\gamma$ denotes the minimum target overlap. This, together with a strong componentwise interference bound, allows one to prove retrieval even in the adversarial or structured (non-random) regime, moving decisively beyond thermodynamic-limit ensemble claims.

Main Theoretical Results

Finite-Time Convergence

Under the separation and interference conditions, the asynchronous retrieval dynamics are proven to contract geometrically towards the target pattern. Specifically, given initialization within the basin, convergence to the fixed point occurs in

$T = O\left(\frac{1}{\alpha}\log N\right)$

full asynchronous sweeps, where $\alpha = 1/n - 2(n-1)p/N^{n-1}$ quantifies the effective contraction rate. This explicit quantitative guarantee is unattainable via standard statistical mechanics arguments, which typically ignore finite-size and worst-case pattern effects.

Adversarial Robustness

An explicit margin-based robustness bound is established: DAM retrieval can tolerate adversarial corruption of up to $\rho N$ bits per sweep, so long as

$p$0

holds. This is an adversarial (worst-case) guarantee, specifying that up to a constant fraction of state bits may be perturbed each sweep without preventing eventual convergence, provided the loading and interference constraints are satisfied. These statements stand in stark contrast to typical-case (random) noise analyses.

Storage Capacity

The paper derives information-theoretic capacity bounds in the adversarial setting. DAM models provably store up to $p$1 patterns with explicit finite-size constants, with the worst-case guarantee recovering ensemble-averaged (random pattern) capacity up to polylogarithmic factors. Notably, at capacity loading, further strengthened componentwise interference conditions are required, reflecting the essential role of sign cancellations for high pattern counts.

Potential Game Interpretation

A novel insight is the formal equivalence between DAM asynchronous dynamics and best-response updates in finite potential games. The potential function is the negative energy, player payoffs align with neuron-wise local fields, and all dynamics are proven to converge to pure Nash equilibria under asynchronous play. This opens connections to theory in game dynamics and best-response convergence, further reinforcing the monotonicity and stability of retrieval.

Empirical Validation

Extensive experiments validate the theoretical claims. For cubic ($p$2) DAMs, logarithmic convergence in $p$3 and geometric contraction under adversarial perturbation are empirically observed. The adversarial corruption bound matches the worst-case robust retrieval threshold. Capacity scaling saturates the $p$4 curve, and the empirical gap to ensemble-optimal capacity shrinks with system size. Comparisons between asynchronous and synchronous dynamics reveal that asynchrony is strictly more robust and efficient when approaching capacity.

Experiments on both synthetic random patterns and correlated/adversarial patterns quantify the dependence of retrieval on explicit separation parameters; performance degrades sharply once these are violated, confirming the necessity of the explicit theoretical assumptions. Retrieval tests on binarized MNIST and CIFAR-10 exemplify that, while DAMs can remain surprisingly performant in regimes of high pattern correlation, formal finite-size guarantees no longer apply absent separation.

Theoretical and Practical Implications

The explicit, checkable nature of the separation and interference conditions directly bridges the algorithmic theory of energy-based models with the operational needs of memory system designers. The finite-size, worst-case nature of the primary guarantees ensures that the results are meaningful for real-world, moderate-size DAMs (where classical mean-field approximations are invalid). The adversarial robustness bounds provide not only theoretical insight but also practical certification of retrieval stability in safety-critical or adversarial environments. Explicit bounds on convergence time and pattern capacity inform choice of model and update strategy in hardware realizations.

On the theoretical front, the work demonstrates that capacity and robustness in higher-order associative memories critically depend on pattern geometry (separation, interference) rather than simply statistical ensemble properties, opening new avenues for architecture and pattern design. The potential-game formulation invites cross-fertilization between associative memory theory and computational game theory.

Future Directions

Several open problems and potential extensions are identified. Sharpening the interference control at high capacity, thereby closing the polylogarithmic factor between worst-case and ensemble-optimal storage, is a key challenge. Characterizing basin geometry and convergence for synchronous dynamics (where oscillations may arise) remains open. The systematic design of pattern sets maximizing capacity under explicit separation and interference is an open combinatorial question. Finally, extending these results to continuous-valued modern Hopfield networks or architectures incorporating learning-based or structured patterns connects the present work to current practice in neural network models and attention mechanisms.

Conclusion

This paper provides a comprehensive finite-size, worst-case analysis of DAM retrieval dynamics, adversarial robustness, and storage capacity, bridging a significant gap between statistical-physics arguments and algorithmic guarantees. The explicit, algorithmically verifiable assumptions and the accompanying potential game characterization push the theoretical understanding of associative memories closer to practical engineering realities, and motivate further advances in robust, high-capacity memory systems.