Binomial Gradient-Based Meta-Learning for Enhanced Meta-Gradient Estimation

Gradient descent scales well to large models, but becomes unstable over long time horizons. Gradient-free optimizers can scale to arbitrary timespans, but are hobbled by high dimensions. Since learning occurs in large models over long timescales, neither of these approaches is likely to produce traits which can accelerate the learning process. Instead, we propose a meta-learning algorithm in which the agent learns to modify its own weights and biases. Our algorithm consists of an inner loop, wherein the agent performs some high-dimensional optimization upon itself, and an outer loop, wherein we perform some low-dimensional optimization upon the inner loop. Since the outer loop handles very few parameters, standard zeroth-order methods may be used.

Model-Agnostic Meta-Learning (MAML) is a widely used framework for reinforcement learning (RL) that enables efficient transfer by learning global policy parameters that can be rapidly adapted to new tasks. MAML training proceeds in two loops: an inner loop where the global parameters are adapted to task-specific parameters, and an outer loop where these task-specific parameters are evaluated and losses are back-propagated to improve the global parameters. Traditionally, the inner loop adaptation is performed by collecting trajectories from the task environment and applying gradient updates on the empirical expected return, which can be a costly operation. We note that it is the outer loop that drives the actual learning of global parameters, and therefore the inner loop adaptation mechanism need not be restricted to be gradient-based. This observation leads us to ask: Can we replace the inner loop trajectory collection and gradient update with a simpler, task-specific signal? In many practical settings, tasks are naturally accompanied by language instructions. Leveraging these instructions as a direct task-specific signal, we propose LA-MAML (Language Adapted MAML), which modifies the inner loop by adapting the global policy parameters in a single step through a learned embedding of the task instruction, replacing the inner loop trajectory collection and gradient-based updates. Experiments on the BabyAI benchmark demonstrate that LA-MAML achieves competitive or improved performance compared to baselines at a significantly lower per-iteration wall-clock training time. These results demonstrate that language instructions are an effective and efficient substitute for trajectory-based inner loop adaptation in meta RL.

We study learning to learn through the lens of hyperparameter tuning. We propose the Langevin Gradient Descent Algorithm (LGD), which approximates the mean of the posterior distribution defined by the loss function and regularizer of a regression task with convex objective. For classification tasks, the LGD algorithm estimates the posterior probabilities of each class on the test set. We prove the existence of an optimal hyperparameter configuration for which the LGD algorithm achieves the Bayes' optimal solution for squared loss on regression tasks, and for which LGD closely approximates the posterior probabilities for well-specified classification tasks. Subsequently, we study generalization guarantees on meta learning optimal hyperparameters for the LGD algorithm from a given set of tasks in the data-driven setting. For a number of parameters $d$ and hyperparameter dimension $h$, we show a pseudo-dimension bound of $O(dh)$, up to logarithmic terms under mild assumptions on LGD. This matches the dependence of the bounds on number of parameters obtained in prior work for linear regression using the elastic net, which only allows for $h=2$ hyperparameters, and extends their bounds to regression on convex loss. Compared to bounds on regularized logistic regression that allow for only $h=1$ hyperparameter, our bounds improve greatly on the dependence on samples per task at the cost of worse dependence on the number of parameters by accounting for hardware-aware procedures. Finally, we show empirical evidence of the success of LGD and the meta learning procedure for few-shot learning on linear and logistic regression using synthetically created datasets.