submissions, providing an additional training signal and allowing use of data which could otherwise
mislead the model. Similar approaches were used in, e.g., Vinyals et al. (2019). In value conditioning,
we inserted whether or not a submission was correct into the problem description so that the model
can condition on this information, as shown in Figure 5. At sampling time, the model was always
conditioned on the sample being correct. In value prediction, we added an auxiliary value prediction
task during training such that the last layer token representations before projecting to logits are also
used in a small Transformer to classify whether the submission is correct. Value prediction was not
alphacode