Adam has been shown to outperform gradient descent in optimizing large language transformers empirically, and by a larger margin than on other tasks, but it is unclear why this happens. We show that the heavy-tailed class imbalance found in language modeling tasks leads to difficulties in the optimization dynamics. When training with gradient descent, the loss associated with infrequent words decreases slower than the loss associated with frequent ones. As most samples come from relatively infrequent words, the average loss decreases slowly with gradient descent. On the other hand, Adam and sign-based methods do not suffer from this problem and improve predictions on all classes. To establish that this behavior is indeed caused by class imbalance, we show empirically that it persist through different architectures and data types, on language transformers, vision CNNs, and linear models. We further study this phenomenon on a linear classification with cross-entropy loss, showing that heavy-tailed class imbalance leads to ill-conditioning, and that the normalization used by Adam can counteract it.

We introduce Monte Carlo Forest Search (MCFS), a class of reinforcement learning (RL) algorithms for learning policies in {tree MDPs}, for which policy execution involves traversing an exponential-sized tree. Examples of such problems include proving unsatisfiability of a SAT formula; counting the number of solutions of a satisfiable SAT formula; and finding the optimal solution to a mixed-integer program. MCFS algorithms can be seen as extensions of Monte Carlo Tree Search (MCTS) to cases where, rather than finding a good path (solution) within a tree, the problem is to find a small tree within a forest of candidate trees. We instantiate and evaluate our ideas in an algorithm that we dub Knuth Synthesis, an MCFS algorithm that learns DPLL branching policies for solving the Boolean satisfiability (SAT) problem, with the objective of achieving good average-case performance on a given distribution of unsatisfiable problem instances. Knuth Synthesis leverages two key ideas to avoid the prohibitive costs of policy evaluations in an exponentially-sized tree. First, we estimate tree size by randomly sampling paths and measuring their lengths, drawing on an unbiased approximation due to Knuth (1975). Second, we query a strong solver at a user-defined depth rather than learning a policy across the whole tree, to focus our policy search on early decisions that offer the greatest potential for reducing tree size. We matched or improved performance over a strong baseline on three well-known SAT distributions (R3SAT, sgen, satfc).