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On Stronger Computational Separations Between Multimodal and Unimodal Machine Learning
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Shows an average-case computational separation between bimodal and unimodal learning under low-noise LPN, and proves any such separation implies a cryptographic key agreement protocol.
Problem
Prior work (Lu) showed only worst-case computational separations between multimodal and unimodal learning; it is unclear whether typical (average-case) instances of unimodal tasks are hard and how natural such separations are.
Approach
Construct an average-case bimodal task where the second modality is a parity (LPN secret) hidden from the first modality via low-noise LPN, making unimodal learning hard on typical instances while bimodal learning is easy; then show any average-case super-polynomial separation yields a bit agreement and hence key agreement protocol.
Key results
- Under low-noise LPN (theta = n^-0.5), an average-case bimodal learning task is learnable in poly time (population risk n^-0.5 with prob 1-exp(-Omega(n))) while the corresponding unimodal tasks are hard for any polynomial t.
- Any average-case super-polynomial computational separation between bimodal and unimodal learning implies a bit agreement protocol and hence a cryptographic key agreement protocol (win-win result).
- The KA reduction extends to polynomial separations, yielding polynomial-security key agreement.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use both modalities instead of either modality alone | Average-case learning task μ, under the low-noise LPN assumption · Average-case bimodal and unimodal tasks | learnability in polynomial time | Helped |
| Use both modalities during training | A_μ · Average-case bimodal learning task μ | population risk (ℓ₀): ≤ n^{-0.5} | Helped |
| Restrict learning to either modality alone | Average-case task μ, under the low-noise LPN assumption · Average-case unimodal tasks ρ_X,Z and ρ_Y,Z | population risk (ℓ₀): > 1/2-1/t(n) | Hurt |
| Derive key agreement from an average-case super-polynomial computational separation | Average-case bimodal learning task · Cryptographic key agreement | existence of a key agreement protocol | Helped |
Surprising findings
- Average-case super-polynomial computational advantages of multimodal learning would imply cryptographic key agreement, suggesting such advantages are rare in natural data, while statistical advantages are not ruled out.
Limitations
- Separation relies on a pathologically constructed task tailored to low-noise LPN, not a natural task.
- Standard LPN does not known to imply key agreement, so the approach cannot yield separations under standard LPN.
- Interpretation of rarity of computational advantages is heuristic and does not directly address practical multimodal models.
Experimental setup
Baselines: Lu worst-case computational separation
Builds on: bimodal PAC learning (Lu) LPN assumption covert learning LUPI
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