Reinforcement Learning with Verifiable Rewards

Reinforcement learning systems fundamentally rely on reward signals to guide policy optimization. However, the integrity of these signals represents a critical vulnerability in deployed RL systems, particularly in adversarial or high-stakes environments.

Problem Formulation

In standard RL frameworks, an agent interacts with an environment according to the following protocol:

• Agent executes action $a_t$ in state $s_t$
• Environment returns reward $r_t$ and next state $s_{t+1}$
• Agent updates policy $\pi$ based on $(s_t, a_t, r_t, s_{t+1})$

The critical assumption here is that $r_t$ accurately reflects the true reward function $R(s,a)$. This assumption breaks down in several scenarios:

1. Adversarial manipulation: External actors inject false reward signals
2. Environment corruption: System faults produce incorrect rewards
3. Reward hacking: Agents exploit implementation bugs to generate unearned rewards

Verifiable Reward Architecture

A verifiable reward system augments the standard RL protocol with cryptographic proofs. Formally, the environment now returns a tuple $(r_t, \pi_t)$ where:

• $r_t$ is the claimed reward
• $\pi_t$ is a proof that $r_t = R(s_t, a_t)$ for predetermined $R$

The agent only accepts rewards that satisfy a verification predicate $V(r_t, \pi_t, s_t, a_t) = 1$.

Implementation Components

The construction typically employs three cryptographic primitives:

1. Commitment Schemes

The environment commits to reward function $R$ at initialization using a binding commitment scheme:

$C = \text{Commit}(R, r) \text{ where } r \text{ is randomness}$

This commitment is shared with the agent before training begins.

2. Zero-Knowledge Proof Systems

For each reward calculation, the environment generates a proof $\pi$ demonstrating:

• Knowledge of $R$ such that $\text{Open}(C, R, r) = 1$
• $r_t = R(s_t, a_t)$
• Auxiliary constraints (e.g., valid state transitions)

Modern implementations leverage succinct proof systems (SNARKs/STARKs) to achieve:

• Proof size: $O(\log n)$ or $O(1)$
• Verification time: $O(\log n)$ or $O(1)$
• Prover time: $O(n \log n)$ where $n$ is circuit size

3. State Authentication

To prevent replay attacks, states must be authenticated:

$h_t = H(s_t \parallel t \parallel \text{nonce})$

Where $H$ is a collision-resistant hash function. The proof $\pi$ must demonstrate consistency with $h_t$.

Computational Complexity Analysis

The overhead introduced by verification depends on several factors:

Where $f$ represents the complexity of the reward function. For polynomial-time reward functions, proof generation remains tractable.

Application Domains

Several domains exhibit characteristics that make verifiable rewards particularly relevant:

Autonomous Systems

In safety-critical applications, reward integrity directly impacts system reliability:

• Sensor verification: Cryptographic attestation of sensor readings
• Action verification: Proofs that executed actions match commanded actions
• Safety constraints: Verifiable compliance with safety specifications

Financial Trading

Trading environments require strong guarantees about profit calculations:

$R(s,a) = P(\text{portfolio}_\text{after}) - P(\text{portfolio}_\text{before}) - \text{costs}(a)$

The proof must demonstrate correct price data and transaction execution.

Federated Learning

When multiple parties contribute to training:

• Each participant generates proofs for local rewards
• Aggregator verifies all proofs before global update
• Prevents poisoning attacks through reward manipulation

Security Properties

A properly implemented verifiable reward system provides:

1. Soundness: Invalid rewards are rejected with high probability
2. Completeness: Valid rewards are always accepted
3. Zero-knowledge: Proofs reveal nothing beyond reward validity
4. Non-malleability: Proofs cannot be modified to verify different rewards

The security reduces to the underlying cryptographic assumptions (e.g., discrete log, knowledge-of-exponent).

Performance Optimizations

Recent work has identified several optimization strategies:

Batch Verification

Aggregate multiple proofs into a single verification:

$V(r_1...r_n, \pi_{\text{batch}}) \text{ instead of } V(r_i, \pi_i) \text{ for each } i$

This amortizes verification costs across episodes.

Proof Composition

Complex reward functions can be decomposed:

$R(s,a) = R_1(s,a) + R_2(s,a)$ $\pi = (\pi_1, \pi_2) \text{ where } \pi_i \text{ proves } R_i$

Enables parallel proof generation and modular verification.

Hardware Acceleration

Specialized hardware (FPGAs, ASICs) can accelerate:

• Field arithmetic for elliptic curve operations
• Multi-scalar multiplication
• FFT-based polynomial operations

Open Research Questions

The intersection of cryptography and RL presents numerous open problems:

1. Adaptive proof systems: Can proof complexity adapt to reward function complexity?
2. Privacy-preserving rewards: How to verify rewards without revealing sensitive state information?
3. Continuous verification: Efficient verification for continuous action/state spaces
4. Proof-of-learning: Extending beyond rewards to verify entire learning trajectories

Conclusion

Verifiable rewards represent a natural evolution in RL system design. As deployment contexts become more adversarial and stakes increase, cryptographic verification transitions from theoretical interest to practical necessity. The computational overhead, while non-trivial, is increasingly manageable given advances in proof systems and hardware acceleration.

The key insight remains: in environments where reward integrity matters, the cost of verification is often negligible compared to the cost of manipulation.