Deep Linear Probe Generators (ProbeGen)

• Deep Linear Probe Generators (ProbeGen) is a framework that unifies structured probing with deep linear generators to yield highly predictive and interpretable representations.
• It employs deep linear architectures combined with parameter sharing and sparsity, reducing FLOPs and parameter count while outperforming traditional probing methods.
• ProbeGen's modular design enables applications in weight-space learning, deep supervision for world models, and sparse local linear predictions.

Deep Linear Probe Generators (ProbeGen) are a class of models that unify efficient, structured probing with deep-learning-based feature generation in order to yield highly predictive yet interpretable representations from neural networks. ProbeGen has been developed and instantiated in several contexts: for learning about neural network weights via black-box probing (Kahana et al., 2024), for regularizing and supervising world models in recurrent architectures (Zahorodnii, 4 Apr 2025), and as generators of sparse, locally linear models to enhance both prediction accuracy and interpretability (Yoshikawa et al., 2020). The central motif is the use of a parameter-efficient, (often deep) linear or near-linear generator to construct informative probes whose responses are used for target downstream tasks such as classification, regression, or feature supervision.

1. Foundations of ProbeGen: Probing and Weight-Space Learning

ProbeGen was introduced to address key limitations in weight-space learning, where the challenge is to extract predictive information (e.g., training data identity, generalization error) from the parameters of an unknown neural network. Direct approaches that feed the flat parameter vector into an MLP are stymied by high dimensionality and permutation symmetries among neurons. Probing, the foundational alternative, queries the black-box model with inputs (“probes”) and collects the output responses. By processing the outputs $\phi(f) = (f(p_1), \dots, f(p_k))$, permutation symmetry across neuron weights is avoided, enabling a classifier $C$ to map $\phi(f)$ to the prediction target $y$.

ProbeGen augments this framework by replacing the arbitrary or independently learned probe vectors with a parameter-efficient, structured probe generator $G$, increasing both generalization and computational efficiency (Kahana et al., 2024).

2. ProbeGen Architectures

2.1 Deep Linear Probe Generator for Weight-Space Learning

The central architecture of ProbeGen consists of two modules:

• Deep Linear Generator $G$: Maps a low-dimensional latent vector $z_i$ to a high-dimensional input probe $p_i$, via a stack of $L$ linear layers:

$$p_i = G(z_i) = W_G^{(L)}(\cdots W_G^{(1)} z_i + b^{(1)} \cdots ) + b^{(L)}$$

No non-linearities are present between layers, to enforce strict linearity and its associated inductive biases.

• Probe Classifier $C$: Consumes the concatenation or flattening of outputs $\phi(f)$ and produces prediction $\hat y$ using either a softmax (classification) or linear head (regression).

The generator’s parameters $\{W_G^{(\ell)}, b^{(\ell)}\}$ and the probe latent codes $\{z_i\}$, as well as the classifier parameters $\theta$, are all jointly optimized. In vision domains, $G$ can incorporate transposed convolutions (while remaining linear) to encourage spatially local, multi-scale structure in probes (Kahana et al., 2024).

2.2 Linear Probing for Deep Supervision

In world model architectures, a linear probe generator is attached to the recurrent hidden state $r_t$:

$\hat f_t = W r_t + b$

where $W \in \mathbb{R}^{K \times H}$ and $b \in \mathbb{R}^K$. The probe loss $L_{probe}$ is a mean-squared error between $\hat f_t$ and the true environment features $f_t$, added to the standard next-observation predictive loss. The linear probe is effective in regularizing the representation and enhancing its correspondence to known, interpretable world variables (Zahorodnii, 4 Apr 2025).

2.3 Neural Generators of Sparse Local Linear Models

ProbeGen can also be realized as a sample-wise generator of sparse linear models (“local linear probes”) (Yoshikawa et al., 2020). For each sample, a weight generator network $f_\theta$ maps the original input $x_{orig}$ to a dense weight vector $w$. A $K$-hot gate module, implemented via Gumbel-softmax, selects $K$ nonzero entries, producing a sparse weight vector $w' = g \odot w$. The prediction is then $\hat y = z_{simp}^\top w'$, directly interpretable due to the enforced sparsity.

3. Optimization Objectives and Training

The ProbeGen framework jointly optimizes all parameters via standard cross-entropy (for classification) or mean-squared error (for regression). In the canonical probing-for-weight-space-learning setting, the objective is

$$\min_{\{W_G^{(\ell)}, b^{(\ell)}\}, Z, \theta} \frac{1}{N}\sum_{j=1}^N \mathcal{L}(y_j, C(\phi(f_j); \theta))$$

where $\mathcal{L}$ is selected per task (Kahana et al., 2024).

In deep supervision scenarios for world models, the total loss is

$L_{total} = L_{pred} + \lambda L_{probe}$

with $\lambda$ as a regularization hyperparameter (Zahorodnii, 4 Apr 2025).

In sparse local linear ProbeGen (Yoshikawa et al., 2020), sparsity is imposed via architectural constraint ($K$ nonzero coefficients), and the overall loss jointly supervises both accuracy and feature selection via the end-to-end pipeline.

4. Inductive Biases of Deep Linear Generators

ProbeGen architectures exploit several forms of inductive bias:

• Implicit regularization: Deep linear mappings, when optimized by stochastic gradient descent, naturally converge to low-rank solutions, preventing overfitting via degenerate or “adversarial” probes (Kahana et al., 2024).
• Parameter sharing: All probes are generated by the same $G$; only their latent codes differ, reducing total parameter count from $k \cdot d_{in}$ (for $k$ independent probes) to $\sum_\ell h_\ell h_{\ell-1} + k d_z$.
• Data structure bias: Task-specific architectural adaptations (e.g., linear transposed convolutions) encourage probes to respect local or multi-scale regularities of the input domain.
• Hard sparsity (in sparse local linear ProbeGen): Interpretability is enforced by $K$-hot gating, ensuring that each probe is supported on at most $K$ features (Yoshikawa et al., 2020).

Ablation studies demonstrate that adding non-linear activations to the generator increases overfitting and test-train gap, and that removing structure (e.g., replacing convolutional expansions with fully-connected layers) degrades performance (Kahana et al., 2024).

5. Computational Complexity and Efficiency

Compared to graph-based methods (e.g., Neural Graphs with Transformers or GNNs), ProbeGen achieves significant reductions in floating-point operations (FLOPs). For example, in MNIST INR and CIFAR10-GS tasks with $k=128$ probes and batch size $B=64$, ProbeGen requires $0.02 \times 10^9$ FLOPs (MNIST INR) and $3.4 \times 10^9$ FLOPs (CIFAR10-GS), in contrast to $63.4 \times 10^9$ and $94.6 \times 10^9$ for equivariant graph methods—a $30\times$–$1\,000\times$ reduction. Parameter counts also favor ProbeGen, with a few million parameters in generator and classifier versus tens of millions in large GNN/Transformer baselines (Kahana et al., 2024).

In sparse local linear ProbeGen, runtime per sample is two to three orders of magnitude lower than model-agnostic explainers (e.g., SHAP, LIME) (Yoshikawa et al., 2020).

6. Empirical Performance and Interpretability

6.1 Weight-Space Learning and Probing

ProbeGen establishes new SOTA in several probing benchmarks.

Task (Dataset)	StatNN	Neural Graphs	Vanilla Probe	ProbeGen
FMNIST-INR accuracy	0.418	0.745	0.808	0.877
CIFAR10 Wild Park (Kendall's $\tau$)	0.719	0.885	0.889	0.933

ProbeGen outperforms vanilla probing even with fewer probes ($k=32$ outperforming vanilla at $k=128$) (Kahana et al., 2024).

6.2 Deep Supervision in World Models

In predictive world models, adding a linear probe generator improves next-state prediction loss by $\approx 15\%$ at $\lambda=64$ versus $\lambda=0$. Decodability (measured as $R^2$ of linear regression from hidden state to supervised world features) increases from $\sim0.3$ to $\sim0.8$, and unsupervised features become linearly decodable only with strong probe supervision. Distribution drift is reduced by $30$–$50\%$, and training stability (fraction of non-divergent runs) is significantly improved (Zahorodnii, 4 Apr 2025).

6.3 Sparse Local Linear Models

ProbeGen on MNIST (binary, $K=1$–$10$) achieves test accuracies $\sim 0.99$, matching or exceeding full-depth DNNs, while Ridge/Lasso baselines are near $0.53$–$0.58$. For text datasets, ProbeGen outperforms linear baselines by $5$–$15$ percentage points and achieves near-DNN accuracy with only $K=1$–$5$ selected features. Interpretability is direct: image probes with $K=5$ or $10$ highlight class-discriminative strokes; in text, single-word or few-word probes select contextually salient tokens (Yoshikawa et al., 2020).

7. Limitations, Open Problems, and Future Directions

Limitations of ProbeGen include the requirement for consistent input/output semantics across models, restriction to black-box probing of model input–output (no direct access to intermediate layers), and possible computational challenges for very large models requiring many forward/backward passes (Kahana et al., 2024).

Promising avenues for future research include adaptive and sequential probing (choosing $p_i$ based on prior responses), extension of the structured linear generator to cross-modal induction (e.g., audio, text), and application in pure black-box scenarios (no gradients, API-only access).

This suggests that the broad paradigm underlying ProbeGen—deep, structured, and parameter-efficient probe generation—can be a foundation for a variety of efficient, interpretable, and high-performing neural model assessment and supervision strategies across domains.