Overview
The deployment of privacy-preserving machine learning, particularly for intelligent systems handling confidential data on cloud platforms, presents substantial challenges. Fully Homomorphic Encryption (FHE) offers a mechanism for secure computation by preserving data confidentiality during cloud-based operations. However, the integration of FHE into reinforcement learning (RL) frameworks necessitates the substitution of non-linear operations with polynomial approximations. This substitution can lead to a recursive error phenomenon termed the Bellman drift, causing catastrophic divergence.
This research introduces the Homomorphic Advantage Operator (HAO), a stabilization framework designed to counteract polynomial approximation divergence within FHE-based deep RL. HAO achieves this by adapting the zero-mean centering projection, typically derived from advantage-based value estimation, directly to temporal-difference (TD) targets. This linear projection eliminates the uniform state-value baseline that is identified as the driver of the Bellman drift. The HAO maintains per-state action rankings without requiring additional non-linear multiplicative depth or expensive ciphertext bootstrapping.
Research Context
Privacy-preserving machine learning is critical for intelligent systems processing confidential data in cloud environments. FHE provides a solution for secure computation, ensuring data confidentiality during these cloud operations. A key hurdle in applying FHE to reinforcement learning lies in approximating non-linear operations with polynomials. This approximation introduces errors that recursively amplify, leading to a phenomenon called Bellman drift, which results in catastrophic divergence of the RL system.
Approach
The Homomorphic Advantage Operator (HAO) was developed as a stabilization framework. Its primary mechanism involves adapting the zero-mean centering projection, which is typically employed in advantage-based value estimation, to operate directly on temporal-difference (TD) targets. This linear projection serves to annihilate the uniform state-value baseline. This baseline is identified as the underlying cause of the Bellman drift.
The HAO framework is designed to prevent divergence of polynomial approximations in FHE-based deep RL. A core aspect of its design is that it maintains per-state action rankings. Furthermore, its implementation requires zero additional non-linear multiplicative depth, and it avoids the need for expensive ciphertext bootstrapping.
Findings
The proposed HAO framework was subjected to evaluation through a three-tier experimental methodology. This methodology included testing on a tabular Markov Decision Process (MDP), an encrypted CartPole environment utilizing real CKKS cryptographic operations, and a 20-node logistics routing benchmark characterized by dense continuous features.
The evaluation demonstrated that the HAO strictly bounds network pre-activations, confining them within the safe polynomial approximation domain. Specifically, HAO RL agents achieved 0% boundary breaches across all random seeds used in the experiments. In contrast, regularization techniques alone (L2 weight decay and gradient clipping) resulted in boundary breaches on 3 out of 5 seeds. An unstabilized baseline breached the bound in 83.8% of episodes.
Furthermore, HAO agents demonstrated an improvement in optimal policy accuracy, increasing it by 18.0 percentage points in tabular domains. The agents also maintained stability when DP-SGD-style Gaussian noise was introduced to the clipped gradients.
Why This Matters
The integration of FHE into reinforcement learning requires addressing issues related to polynomial approximation errors causing catastrophic divergence, known as Bellman drift. The HAO framework provides a method to stabilize FHE-based deep RL by preventing this divergence without increasing computational complexity related to non-linear operations or requiring expensive cryptographic operations. This stabilization enables the use of FHE for secure computation in reinforcement learning applications handling confidential data.
Potential Applications
The stabilization framework could facilitate the deployment of privacy-preserving machine learning in intelligent systems that operate on confidential data in cloud environments, given FHE's utility in securing cloud computations.