Weight Space Learning in Neural Networks

• Weight space learning is a paradigm that treats neural network parameters as the primary source for inferring properties like dataset characteristics, generalization gaps, and model metadata.
• It employs both static mechanistic methods and dynamic probing strategies to extract invariant features and query black-box responses from network weights.
• Innovative frameworks such as ProbeGen demonstrate significant efficiency gains and improved prediction accuracy, reducing computational cost by up to 1,000× over graph-based baselines.

Weight space learning is a research paradigm in which the parameters of trained neural networks—weights and biases—are treated as the primary modality for extracting information, analysis, prediction, or synthesis. Rather than inferring properties by evaluating a network on held-out data, weight space learning algorithms directly analyze the weight vector of each network within a model zoo. Its objectives include predicting the original training dataset, estimating generalization error, enabling model property inference, and synthesizing performant new networks, all by operating solely on network parameters. This approach inherently presents challenges due to high dimensionality and permutation symmetries, and has motivated the development of both static and dynamic methodologies, efficient representations, augmentation strategies, and probing protocols (Kahana et al., 2024).

1. Core Principles and Objectives

The fundamental objective in weight space learning is to construct a mapping from network parameters to meaningful properties:

$f_W \mapsto y$

where $f_W$ is a neural network fully specified by weight vector $W$ (with fixed architecture), and $y$ is the target property such as dataset class, generalization gap, or other model metadata. The weight-space learner $C: W \to \hat{y}$ should accurately predict $y$ given only $W$, with no access to training data or additional extensive evaluation (Kahana et al., 2024).

The primary methodological distinction is between static and dynamic approaches:

• Static (mechanistic): Directly learn from $W$ via invariant (statistical) features or permutation-equivariant architectures.
• Dynamic (probing): Query $f_W$ as a black-box on crafted or learned inputs (probes), and aggregate model responses to infer $y$.

Key challenges common to both paradigms are:

• High dimensionality: $W$ comprises millions of parameters, making naive learning prone to overfitting and computational inefficiency.
• Permutation symmetries: Hidden units within layers are symmetric under permutation, meaning that $W$ must be represented or processed in an equivariant fashion.

2. Methodological Families in Weight Space Learning

Mechanistic Approaches

Mechanistic approaches fall into two principal categories (Kahana et al., 2024):

Method	Features	Strengths	Limitations
Weight statistics	Layerwise norms/moments	Perm.-invariant, efficient	Loses relational info
Graph-based equivariant	Neurons as nodes in computational graph	Captures rich interactions	Computationally expensive, symmetry handling complexity

StatNN computes layer-wise statistics (norms, moments) and concatenates them for classification. These statistics respect permutation invariance but discard structural relationships between neurons.

Neural Graphs and transformer-based GNNs model the network structure as a graph, where nodes represent neurons and edges represent connections, processed in a permutation-equivariant way. Although more expressive, these approaches scale poorly with model size.

Probing-Based Approaches

Probing treats the model as a black box, collecting outputs for a fixed set of learned probe inputs. For $k$ probes $P = \{p_1, ..., p_k\}$, the process is:

$$\hat{y} = h_\theta\left( f_W(p_1), f_W(p_2), ..., f_W(p_k) \right)$$

with trainable probes $\{p_i\}$ and a predictor $h_\theta$, typically an MLP (Kahana et al., 2024). Probes are optimized by backpropagation through $f_W$. Probing bypasses explicit symmetry handling and is highly efficient, but naive probe learning leads to probes that overfit model idiosyncrasies.

3. Deep Linear Probe Generators: ProbeGen

To combat overfitting and introduce structure to probe learning, Deep Linear Probe Generators (ProbeGen) factorize the probe creation process:

$p_i = G_\phi(z_i)$

where $G_\phi$ is a shared deep linear generator (stacked linear/transposed convolutional layers) and $z_i$ is a per-probe latent code (Kahana et al., 2024).

Key properties of ProbeGen:

• Shared generator imposes structure and regularization across all probes.
• Linear architecture insulates against overfitting, leveraging the implicit regularization characteristics of SGD-trained deep linear nets.
• In image-based contexts, transposed convolutions induce locality and multi-scale coherence in probes, favoring image-like patterns.
• The full pipeline passes generated probes through $f_W$ and aggregates outputs:

$\hat{y} = h_\theta( o_1 \| o_2 \| ... \| o_k )$

where $o_i = f_W(p_i)$.

ProbeGen is trained end-to-end to minimize expected loss $\mathcal{L}$ over model zoo instances, without explicit regularizers:

$$\min_{\phi, \{z_i\}, \theta} \mathbb{E}_{(f_W, y)}\left[ \mathcal{L} \left( h_\theta( f_W(G_\phi(z_1)), ..., f_W(G_\phi(z_k)) ), y \right) \right]$$

Empirical Outcomes:

• ProbeGen outperforms vanilla probing and mechanistic approaches, with significant gains in both classification and regression benchmarks.
• Computational cost is drastically reduced, requiring $30$–$1{,}000\times$ fewer FLOPs than leading graph-based baselines (Kahana et al., 2024).

4. Empirical Benchmarks, Efficiency, and Generalization

Benchmarks:

• FMNIST-INR classification: vanilla probing at $95.5\%$, Neural Graphs at $74.5\%$, ProbeGen at $87.7\%$.
• CIFAR10 Wild Park regression ($\tau$): vanilla probing $0.885$, ProbeGen $0.932$ (Kahana et al., 2024).

Efficiency:

• On CIFAR10-GS, $k=128$ probes, batch size $64$: Neural Graphs $\sim 94$ GFLOPs, ProbeGen $\sim 3.4$ GFLOPs.

Generalization:

• ProbeGen’s deep linear shared generator reduces overfitting, avoiding nearly-adversarial probe patterns seen in vanilla probing.
• Structured probes capture robust and task-relevant variation rather than model-specific artifacts.

5. Weight Space Structure, Symmetries, and Loss Geometry

Modern weight space learning is tightly constrained by the underlying geometry and symmetries of neural network parameter space:

• Permutation symmetry: Equivalence under hidden neuron permutation compels all direct $W$-based models to be permutation-invariant or equivariant (Kahana et al., 2024).
• Overparameterization: Large networks exhibit vast continuous flat manifolds of equal loss—SGD typically navigates within these high-dimensional regions rather than seeking isolated minima (Lipton, 2016).
• Mode connectivity and functional invariance: Networks can undergo substantial weight transformation along continuous paths (e.g. scaling, linear geodesics) without altering functional output (Raghavan et al., 2021).

Weight space learning algorithms must either respect or exploit these properties to achieve generalization and avoid overfitting.

6. Overfitting and Regularization Strategies

Vanilla weight space learners (probing or direct $W$-based models) are highly susceptible to overfitting due to the low diversity and huge dimensionality of input parameter vectors. Major regularization and augmentation strategies include:

• Shared generator architectures (ProbeGen): Impose inductive biases toward structured, coherent probe representations (Kahana et al., 2024).
• Permutation-equivariant networks: Graph-based or DeepSets designs guarantee invariance (Kahana et al., 2024).
• Data augmentation at the level of weights: MixUp, synthetic probe creation, permutation alignment, and activation-based transformations expand the apparent dataset and inject diversity (Shamsian et al., 2024, Shamsian et al., 2023).
• Contrastive learning: Structured latent embeddings via contrastive loss further prevent memorization and encourage semantic separation (Schürholt et al., 2024).

For discriminative tasks, cross-entropy or regression losses are standard; generative weight synthesis (as in SANE) leverages latent sampling and bootstrapped refinement.

7. Outlook and Advancements

ProbeGen’s deep linear shared structure marks a significant stride in weight space learning, offering a balance between efficiency, regularization, and expressivity. Mechanistic approaches continue to evolve with more sophisticated graph structures and permutation handling. The general trend is toward hybrid models that incorporate dynamic probing with structured representation learning, responding to the unique challenges posed by high-dimensional, permutation-symmetric parameter spaces.

The paradigm supports applications in model zoo analysis, property inference, efficient knowledge transfer, and meta-learning, with potential for continued impact in model understanding and scalable neural model design (Kahana et al., 2024).