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Knowledge distillation leverages a teacher model to improve the training of a student model. A persistent challenge is that a better teacher does not always yield a better student, to which a common mitigation is to use additional supervision from several “intermediate” teachers. One empirically validated variant of this principle is progressive distillation, where the student learns from successive intermediate checkpoints of the teacher. Using sparse parity as a sandbox, we identify an implicit curriculum as one mechanism through which progressive distillation accelerates the student’s learning. This curriculum is available only through the intermediate checkpoints but not the final converged one, and imparts both empirical acceleration and a provable sample complexity benefit to the student. We then extend our investigation to Transformers trained on probabilistic context-free grammars (PCFGs) and real-world pre-training datasets (Wikipedia and Books). Through probing the teacher model, we identify an analogous implicit curriculum where the model progressively learns features that capture longer context. Our theoretical and empirical findings on sparse parity, complemented by empirical observations on more complex tasks, highlight the benefit of progressive distillation via implicit curriculum across setups. 

Transformer interpretability aims to understand the algorithm implemented by a learned Transformer by examining various aspects of the model, such as the weight matrices or the attention patterns. In this work, through a combination of theoretical results and carefully controlled experiments on synthetic data, we take a critical view of methods that exclusively focus on individual parts of the model, rather than consider the network as a whole. We consider a simple synthetic setup of learning a (bounded) Dyck language. Theoretically, we show that the set of models that (exactly or approximately) solve this task satisfy a structural characterization derived from ideas in formal languages (the pumping lemma). We use this characterization to show that the set of optima is qualitatively rich; in particular, the attention pattern of a single layer can be “nearly randomized”, while preserving the functionality of the network. We also show via extensive experiments that these constructions are not merely a theoretical artifact: even with severe constraints to the architecture of the model, vastly different solutions can be reached via standard training. Thus, interpretability claims based on inspecting individual heads or weight matrices in the Transformer can be misleading. 

Abstract In this work, we analyze data collected by an HF transmitter/receiver radio link, operating as an oblique ionosonde between the McMurdo Station (transmitter) and South Pole Station (receiver) at 4.1, 5.1, 6.0, 6.4, and 7.2 MHz between 28 February and 14 March 2019. To help contextualize the link's data we have performed numerical raytrace simulations to help understand the observations. By considering both the data and simulations, we have identified both single‐ and two‐hop E‐ and F‐region propagation modes in the data, where the multi‐hop modes were observed in the hours around sunrise and sunset in the 4.1 and 5.1 MHz channels. This is an unexpected result given the accepted wisdom that multi‐hop modes, which require a ground scatter component, cannot be supported in Antarctica because of the highly absorptive ice covering much of the continent. Our results show that multi‐hop propagation modes can be supported in the region under specific ionospheric conditions—around sunrise and sunset—if the mode's ground scatter component is collocated with the Transantarctic Mountains. The mountains are located along the great‐circle path between the link's transmitter and receiver. However, the combination of favorable ionospheric and ground scattering conditions makes the detection of the multi‐hop mode a rare occurrence in the data set analyzed here. These findings are critical to data analysis efforts of any current or future oblique ionosonde systems operating in Antarctica and other regions such as the Arctic.