We propose to scale RL to unverifiable data with a novel algorithm JEPO (Jensen's Evidence lower bound for Policy Optimization). While most prior effort on scaling RL for LLMs focuses on verifiable data where ground truth answers are typically short-form and can be matched easily, we investigate the case where such assumptions are less valid (e.g., when answers are long-form such as mathematical proofs). To scale RL training to unverifiable data with contemporary training constraints, we propose JEPO. JEPO applies Jensen's evidence lower bound, a pragmatic simplification of the evidence lower bound which views chain-of-thought as a latent variable in the generative process. We show that on verifiable datasets (math), JEPO is as effective as RL with verifiable reward; on semi-verifiable and unverifiable datasets (numina and numina-proof), JEPO improves on soft-match based evaluations compared to RL with verifiable reward which can only leverage a subset of the data source as well as test set likelihood evaluations.

Predicted sequences of a moving pendulum conditioned on z1 q(z1|x1:5,u1:4) or, in case of the EKVAE, on z1 p(z1|a1:5,u1:4), where the auxiliary variables are obtained through a1:5 q(a1:5|x1:5). The average prediction accuracy, measured by the MSE, can be found in Tab.

In our approach, the intrinsic reward can be separated into two parts. One is related to action-aware diversity, while the other is related to observation-aware diversity. We revisit the formulation of our information-theoretic objective (Eq. A.1 Intrinsic Rewards for Action-Aware Diversity First we analyze term 2, which is related to action-aware diversity. T 1 T 1 X p(at| t,id) Xp(at| t,id) 2 = Eid, log q(at| t) DKL (p(at| t)kq(at| t)) Eid, log q(at| t) .

The average prediction accuracy, measured by the MSE, can be found in Tab.

The approximate posterior distributions of the latent variables are derived with variational inference, and the evidence lower bound is evaluated and optimized by the proposed recursive sampling scheme.

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Furthermore, we generate visual analogies using IIAE which are presented in table 11.