Beyond Feedforward: Verifying Diverse Architectures

Beyond Feedforward: Verifying Diverse Architectures#

Most verification research focuses on feedforward convolutional networks. But modern neural networks span far beyond simple CNNs. Recurrent networks process sequences. Transformers use attention mechanisms. Graph neural networks operate on structured data. Each architecture introduces new verification challenges and requires different approaches. This guide explores verification across diverse modern architectures and the unique challenges each presents.

Why Architecture Matters for Verification#

Verification methods rely on architectural properties. Feedforward networks allow layer-by-layer bound propagation without loops. This won’t work for recurrent networks with feedback. Softmax in attention mechanisms requires special handling—it’s not a standard ReLU or sigmoid. Graph networks operate on discrete structures that don’t fit standard tensor frameworks.

Different architectures mean different verification problems. You can’t just plug a transformer into a CNN verifier. The architecture’s structure shapes which verification techniques are applicable.

Architectural Assumptions in Verification

Most verification tools make implicit assumptions about architecture: feedforward structure, standard activation functions, standard layer types. Deviating from these assumptions often means tools won’t work. Understanding these assumptions helps you recognize which tools apply to your network.

Recurrent Neural Networks and LSTMs#

Recurrent networks process sequences, with hidden state that carries information across timesteps. Verification of recurrent networks raises a fundamental question: over how many timesteps do you verify?

Unbounded time horizons: A feedforward network has fixed depth. A recurrent network can process sequences of arbitrary length. Verifying a property for all possible sequence lengths is intractable. So verification typically focuses on bounded horizons: “for sequences up to length T.”

This creates a specification challenge. “The network is robust” is unclear for an RNN. “The network makes correct predictions for sequences up to 100 timesteps, given bounded perturbations at each step” is more concrete but may not capture your full intent.

Hidden state tracking: Verification must track bounds on hidden state across timesteps. A single hidden state vector contains information accumulated from all previous timesteps. Bounding this vector tightly is difficult—approximation error accumulates with sequence length.

\[h_t = \text{RNN}(h_{t-1}, x_t)\]

Verification must compute bounds on \(h_t\) that account for uncertainty in all previous \(h_{t-1}, h_{t-2}, \ldots, h_0\) and all input perturbations.

Truncation strategies: In practice, verification uses truncation. Either truncate to a fixed horizon, or verify that properties hold asymptotically (the network converges to a safe region). Neither is fully satisfactory, but they’re practical compromises.

Hint

RNN verification is harder than feedforward verification. Expect longer verification times and looser bounds. Use complete RNNs in feedforward architectures when possible (e.g., encoding a sequence to a fixed representation, then processing feedforward).

Transformers and Attention Mechanisms#

Transformers have become dominant in NLP and increasingly used in vision. Their architecture—self-attention, layer normalization, feed-forward sublayers—is very different from CNNs.

Self-attention complexity: Self-attention computes relationships between all pairs of positions. The operation involves softmax, which is inherently non-linear and difficult to verify. Softmax doesn’t have simple bounds—computing tight bounds requires understanding how input changes affect the softmax’s behavior.

\[\text{Attention}(Q, K, V) = \text{softmax}\left(\frac{QK^T}{\sqrt{d}}\right)V\]

Verification must track uncertainty through this entire computation, including the softmax normalization.

Quadratic complexity: Attention has quadratic complexity in sequence length—for a sequence of length \(N\), computing attention is \(O(N^2)\). For verification, this becomes \(O(N^4)\) or worse in some bounds-computation schemes. Long sequences quickly become intractable.

Layer normalization: Transformers use layer normalization instead of batch normalization. Layer norm computes statistics per sample and normalizes accordingly. This couples all positions in a sequence, making isolation and analysis difficult.

Position embeddings: Positional information is added to embeddings. These embeddings interact with other embedding dimensions in complex ways. Verifying how position changes affect outputs is non-obvious.

Transformer Verification is Very Hard

Transformers combine multiple architectural features—attention, layer norm, residual connections—that individually complicate verification. No general-purpose transformer verifier exists. Some specialized methods are emerging, but they handle only specific properties or small models.

Current approaches: Researchers are developing transformer-specific verification: - Simplified attention approximations (ignoring some softmax properties) - Layerwise verification of attention components - Combining verification with statistical analysis for uncertain regions

But these are early-stage. Transformers remain a frontier for verification.

Graph Neural Networks#

Graph neural networks (GNNs) operate on irregular structures—nodes and edges that form graphs. Verification requires reasoning about graph structure and message-passing operations.

Message passing verification: GNNs aggregate information from neighbors. Verification must account for how neighbor outputs affect each node. But neighbor count and identity can vary—the computational graph isn’t fixed like in feedforward networks.

\[h_v^{(t+1)} = \text{AGG}(\{h_u^{(t)} : u \in N(v)\})\]

where \(N(v)\) is the neighborhood of node \(v\). Verification must bound this aggregation across all possible neighborhoods.

Permutation invariance: GNNs are permutation invariant—reordering nodes shouldn’t change the output. This property helps with verification (you can focus on properties that respect the invariance) but also constrains what you can verify.

Node/edge/graph level properties: You might verify properties at the node level (each node’s representation), edge level (relationships), or graph level (global properties). Each requires different verification approaches.

GNN Property Type	What’s Verified	Verification Approach	Difficulty
Node-level	Individual node representations	Local verification per node	Moderate
Edge-level	Pairwise relationships	Edge-wise constraints	Moderate to hard
Graph-level	Global graph properties	Aggregate constraints	Hard (many dependencies)

Current state: GNN verification is emerging but limited. Some work verifies simple properties on small graphs. Verification of large graphs with many nodes and complex message passing remains largely open.

Convolutional Networks and Spatial Structure#

CNNs have been well-studied for verification. They’re actually easier to verify than many alternatives because convolution creates structured, reusable computations. Verification can exploit this structure.

Spatial structure: Convolution operates on local patches. Verification can reason about how perturbations to one region affect outputs. This localization helps tighten bounds compared to fully-connected networks.

Pooling operations: Max-pooling requires careful handling—it selects the maximum, which creates non-smoothness. Average pooling is simpler. But both require careful bound computation.

Why CNNs are relatively easy: No attention mechanisms, no complex normalization tricks, no unbounded-horizon problems. Standard architecture design choices make verification tractable.

This is partly why CIFAR-10 and other CNN benchmarks are standard. Not because CNNs are interesting—they’re actually among the easier networks to verify. They’re standard because tools handle them well.

Hybrid and Novel Architectures#

Recent architectures blend multiple paradigms.

Vision Transformers (ViTs): Treat images as sequences of patches and apply transformer attention. Combine CNN’s spatial structure reasoning with transformer’s attention mechanisms. Verification must handle both.

Encoder-decoder architectures: Combine different components—an encoder processing input, a decoder generating output. Verification must handle the two-stage structure and how information flows between stages.

Multi-modal networks: Process images, text, audio, or other modalities. Might use different architectures for each modality, then combine. Verification must handle each modality’s specifics.

Generative models: VAEs, GANs, diffusion models. Verification of generative models is a frontier—what does it even mean to verify a generative model? Properties are different from classifiers.

These hybrids inherit challenges from their components. A ViT faces both CNN verification and transformer verification challenges.

Practical Recommendations#

Choose architectures with verification in mind. If verification matters for your application, prefer architectures that are more verifiable: feedforward networks, simple RNNs, CNNs. Avoid transformers and complex GNNs unless necessary for accuracy.

Decompose when possible. Break complex architectures into components. Verify components separately, then reason about how they compose. This is easier than verifying monolithic architectures.

Use simplifications for verification. You might use a complex model for accuracy, but verify a simplified surrogate model that captures the essential behavior. If the surrogate is verified, that provides some confidence in the original.

Accept incomplete solutions. For architectures like transformers and GNNs, complete general-purpose verification isn’t available. Use incomplete methods, accept conservative bounds, or combine verification with testing.

Tip

When deploying unverifiable architectures, compensate with: (1) robust training and testing, (2) runtime monitoring to detect out-of-distribution inputs, (3) fallback mechanisms when verification assumptions are violated, (4) interpretability methods to understand network behavior.

Research Frontiers#

Verification for diverse architectures is an active research frontier. Recent trends include:

Attention verification: Specialized approaches for handling softmax and attention in transformers. Still early but improving.

Graph neural network verification: Extending verification to message-passing and graph structure. Emerging methods handle small graphs, scaling is ongoing.

Temporal property verification: For RNNs and sequence models, verifying temporal properties beyond fixed horizons. Important for safety-critical sequential systems.

Compositional verification: Breaking architectures into components, verifying components, then composing guarantees. Enables verification of larger, more complex systems.

Learning-based verification: Using neural networks or learning to assist verification—predicting verification difficulty, learning refinement strategies. A research frontier with potential.

Final Thoughts#

Feedforward CNNs are well-trodden ground for verification. But modern neural networks are far more diverse. Recurrent networks, transformers, graph networks—each brings new challenges and requires different verification approaches.

For practitioners, this means: understand your architecture’s verification challenges. Simple feedforward networks are verifiable with existing tools. Complex architectures might not be. Plan accordingly. When new architectures are essential, combine verification (where possible) with robust training and comprehensive testing.

For researchers, diverse architectures represent frontier challenges. Each architecture type has unique properties to exploit and unique obstacles to overcome. The field is expanding beyond feedforward networks, slowly but steadily.

The future of verification likely involves architecture-specific tools and hybrid verification strategies that combine multiple techniques. No single approach will handle all architectures. Specialization and flexibility will be key.

Note