Forward-Pass Amortization in Neural Models

• Forward-pass amortization is a computational paradigm that replaces iterative procedures with single feed-forward operations to reduce inference cost and complexity.
• It leverages techniques like quasi-Newton estimates, randomized contractions, and uncertainty parameterization across domains including VAEs, DEQs, particle filtering, and privacy-preserving models.
• Applications span efficient bi-level optimization, scalable neural inference, and biologically plausible learning in both scientific and multimodal machine learning tasks.

Forward-pass amortization is a computational paradigm whereby the cost or complexity of inference, optimization, or representation learning is "spread" over a single feed-forward computation through a model, as opposed to iterative, backward, or multi-pass procedures. This approach has gained prominence in diverse areas such as amortized variational inference, bi-level optimization, planner policy distillation, efficient kernel selection, differentiable particle filtering, privacy-preserving neural modeling, and high-order differential operator estimation. The unifying theme is leveraging properties of the forward propagation—whether by parameterizing uncertainty, sharing local computations, or compressing expensive operations—to achieve scalable, efficient, or biologically plausible learning frameworks.

1. Core Principles of Forward-Pass Amortization

Forward-pass amortization is defined by the replacement of computationally intensive, instance-specific or iterative inference procedures by single-pass, globally parameterized estimators or learning signals. In the context of variational autoencoders (VAEs), the canonical example is amortized inference: a learned neural network encodes input $x$ and outputs variational parameters $q(z|x)$ with one feed-forward pass, thereby avoiding per-instance optimization (Kim et al., 2021). In bi-level optimization and implicit models, expensive backward solves (e.g. implicit gradients requiring matrix inversions) are replaced by estimates extracted from forward computations, such as quasi-Newton matrices (Ramzi et al., 2021).

Forward-pass amortization is also employed to amortize discrete procedures (such as resampling in particle filters (Ścibior et al., 2021)) and the evaluation of high-order derivative tensors (e.g., for physics-informed neural networks and differential equations (Shi et al., 27 Nov 2024)), where randomized contraction in the forward calculation yields unbiased estimators at much lower cost. In privacy-preserving modeling, local differential privacy and robustness against inversion attacks can be achieved by perturbing embeddings preemptively in the forward pass (Du et al., 2023).

A representative summary table of domains and forward-pass amortization mechanisms:

Domain	Forward-Pass Amortization Technique	Main Benefit
Variational Inference	Feed-forward encoder for posterior parameters	Fast, scalable inference
Implicit/DEQ Models	Share inverse Jacobian from forward quasi-Newton	Efficient implicit gradient computation
Particle Filtering	Stop-gradient correction for discrete steps	AD-compatible, unbiased gradients
Privacy (LMs)	Matrix Gaussian embedding perturbation in forward	Local DP, efficient fine-tuning/inference
High-order Operators	Taylor-mode AD with randomized contraction	Speed/memory reduction, large-scale PDEs

2. Bayesian Random Function Models in Single-Pass Inference

The amortization gap—degraded posterior approximation due to fixed encoder architectures—is a persistent challenge in VAEs. Bayesian random function models address this by treating the encoder output itself as a stochastic process. Specifically, mean and variance functions $f(x)$ and $h(x)$ are modeled as Gaussian processes (GPs):

$$q(z|x, f, h) = \mathcal{N}(z; f(x), \text{diag}(h(x))^2)$$

with GP priors $f(x) \sim \text{GP}(b(x), k^m(x,x'))$ and $h(x) \sim \text{GP}(c(x), k^s(x,x'))$. This approach allows uncertainty in posterior approximation to be quantified and propagated in a single pass (Kim et al., 2021). Implementation uses deep kernel parameterization, resulting in inference networks with stochastic weights and the ELBO objective marginalized over random functions. Empirically, GPVAE attains systematically higher test likelihoods and quantifiable uncertainty without incurring the iterative costs of semi-amortized schemes.

3. Efficient Bi-Level and Implicit Model Optimization

Bi-level optimization problems typical in implicit deep learning models (such as Deep Equilibrium Models, DEQs) require the computation of hypergradients via implicit differentiation. Standard approaches involve solving $g_\theta(z^*)=0$ and inverting $J_{g_\theta}(z^*)$ via backward passes, often at cubic cost.

The SHINE algorithm (Ramzi et al., 2021) implements forward-pass amortization by reusing quasi-Newton (qN) matrices—produced during forward root-finding—as inverse Jacobian approximations in the backward pass for implicit gradients. The theoretical guarantee arises from the convergence properties of the qN updates, which asymptotically estimate the true implicit gradient directions. Refinements such as Outer-Problem Awareness (extra secant updates) and Adjoint Broyden (left-multiplicative update) further align inversion quality to practical gradient computations, leading to empirical speedups (10× reduction in backward pass cost) and improved hyperparameter optimization convergence.

4. Forward-Pass Amortization in Sequential and Kernelized Models

Particle filtering and high-order differential operator estimation have historically faced AD-incompatibilities and exponential cost blow-up. In differentiable particle filtering (Ścibior et al., 2021), introducing stop-gradient operators in the weight correction step enables unbiased gradient estimators for likelihood and posterior expectations, with the forward pass unaltered and minimal computation overhead.

For arbitrary high-order operators in scientific ML, the Stochastic Taylor Derivative Estimator (Shi et al., 27 Nov 2024) leverages Taylor-mode AD and randomized contractions of derivative tensors. By pushing forward “jets” (primal input plus random tangents) and taking contractions with operator-specific coefficient tensors, the required derivatives are estimated efficiently in a single pass, yielding up to $1000\times$ speedup and $30\times$ memory reduction in PINN training for million-dimensional PDEs.

5. Amortization in Policy Distillation, Planning, and Contrastive Learning

Model-based planners (e.g., MPC) are computationally costly for continuous control. Amortization here refers to distilling planner action selection into a learned policy: the planning computation guides a behavioral cloning objective combined with off-policy RL, embedding the planner’s benefits into a compact policy deployable at test time (Byravan et al., 2021). This has proven effective for multi-goal, high-DoF robotics and enables rapid execution without online planning.

Similarly, in contrastive learning frameworks for large language-image models (e.g., CLIP), the amortization of expensive contrastive objectives—particularly partition function computations—in the forward pass via auxiliary lightweight neural networks (AmorLIP) (Sun et al., 25 May 2025) reduces gradient bias and inter-sample dependency, enabling smaller batch sizes without loss of representation quality.

Unsupervised sentence representation models (CSE-SFP) (Zhang et al., 1 May 2025) further exemplify forward-pass amortization, using prompt design and causal masking in generative PLMs to produce anchor and positive embeddings simultaneously, halving compute/memory demands relative to dropout-based pairwise strategies.

6. Biological and Hardware Perspectives

Biological plausibility and hardware compatibility are driving forces behind forward-pass amortization innovations. Signal propagation frameworks (sigprop) (Kohan et al., 2022) eliminate backward error transport, enabling local layer-wise updates during the forward pass. In spiking, recurrent, and neuromorphic architectures, sigprop supports global learning signals using only feedforward connectivity, with empirical evidence for reduced training time, memory usage, and preserved performance. Forward-Forward (FF) algorithms (Adamson, 15 Apr 2025, Gong et al., 29 Aug 2025), which use only forward passes with layer-local objectives, show delayed accuracy progress in deeper layers but significant correlations between shallow layer accuracy and global model accuracy. Their extension with similarity-based objectives (FAUST) (Gong et al., 29 Aug 2025) approaches backpropagation performance while maintaining single-pass inference, advancing accuracy and biological plausibility in neural optimization.

7. Implications, Limitations, and Future Directions

Forward-pass amortization, via uncertainty quantification, shared inverse estimates, randomized contraction, and policy/contrastive distillation, systematically enhances scalability, efficiency, and robustness in neural modeling. It mitigates amortization gaps in VAEs, enables robust gradient-based bi-level optimization, addresses AD incompatibility in discrete/sequential models, and reaches unprecedented scales in scientific ML and multimodal pretraining.

Limitations include sensitivity to kernel/feature selection in GP-based amortization, potential variance in randomized derivative estimators, and the risk of losing expressive capacity when decoupling global inter-sample dependencies. Extensions to structured data, advanced kernel approximations, improved variance reduction techniques, and deeper biological modeling are active directions. The broad adoption of forward-pass amortization mechanisms signals a paradigm shift toward scalable, efficient neural inference and learning, with implications across generative modeling, scientific computing, privacy-preserving ML, and biologically inspired systems.