Attention-based representations for multi-task computation

Multi-head attention layers produce vector representations that support multiple downstream tasks. We establish bounds on the number of heads required in two simple and concrete multi-task scenarios. In the first scenario, a vector representation is sought so that linear predictors can compute both the smallest and largest numbers in a given list. In this case, it is known two attention heads with small embedding dimension and bit precision level suffice. We prove that a single attention head requires exponentially higher embedding dimension or precision level. In the second scenario, a vector representation is sought so that a polynomial threshold function can compute the XOR of a given string of $n$ bits. This scenario is analogous to the first one for $n=2$, since XOR is readily computed by a linear function using a vector representation that encodes both the AND and the OR of the two bits. We observe that $n$-bit XOR requires the product of the number of heads and the polynomial degree to be at least $n$, and we construct multi-head attention layers that match this lower bound. These results generalize to arbitrary (symmetric) Boolean functions, where the bound is given in terms of the threshold degree.

Behavior Cloning is Not All You Need: The Optimality of On-Policy Distillation for Noisy Expert Feedback

Imitation Learning is a natural framework for learning in sequential decision-making systems and has emerged as the dominant paradigm through which we understand language model training. A central puzzle is that, while in theory offline IL can be horizon-free and optimal, in practice online methods such as on-policy distillation often outperform offline methods such as supervised fine-tuning. We propose a noisy expert model to explain this gap, in which the learner only has access to a noisy version of the expert's policy, but wishes to compete against the reward achieved by a clean expert, motivated by the fact that in many applications, e.g. training language models to perform long chains of thought, the expert is often imperfect. In this setting, we show a sharp separation between offline and online IL. Offline learning from noisy trajectories is fundamentally hard: to compete with the clean expert, the sample complexity must grow exponentially, in contradistinction to the clean expert setting where no explicit horizon dependence exists. In contrast, we prove that online interaction with the noisy expert via a novel variant of OPD enables polynomial dependence on the horizon in general. We further show that, under a natural condition on the expert noise distribution, which we show to be necessary for any horizon-free sample complexity, one can obtain such a guarantee, although our proposed algorithm sacrifices statistical efficiency in its dependence on the size of the policy class. Our analysis leads to an alternative loss function that is commonly considered empirically for LM training. We further provide algorithms and lower bounds, and extend our results to the more realistic setting of unknown corruption when the clean expert is deterministic, thereby providing a theoretical foundation for why OPD can outperform SFT when training language models from imperfect teachers.

The price of multi-group transductive learning

We show every multi-group learner in the transductive setting may incur a multiplicative penalty in its error rate on some group relative to the error rate achievable in the single-group setting, and the penalty can increasing linearly with the number of groups, up to roughly the square-root of the sample size. This stands in stark contrast to optimal multi-group learners in an analogous (group-realizable) statistical setting, where the penalty is always at most logarithmic in the sample size and independent of the number of groups.

Fixed Universal Transformers

We introduce \emph{universal transformers}: fixed transformers that can simulate any transformer in a given class via a suitable input embedding. Analogous to a universal Turing machine, the input embedding encodes a description of the target model while all internal parameters remain fixed. We provide explicit sparse constructions achieving universality when the embedding dimension is sufficiently large, and further show that universality is generic: randomly initialized transformers are universal almost surely, which aligns with recent empirical results of Zhong and Andreas (2024). We empirically validate our theory on the algorithmic tasks of parenthesis balancing and multi-hop reasoning. Our results suggest that much of a transformer's expressive power may reside in its input representation rather than its learned weights.

Lower bounds for one-layer transformers that compute parity

This note shows that no self-attention layer post-processed by a rational function can sign-represent the parity function unless the product of the number of heads and the degree of the post-processing function grows linearly with the input length. Combining this lower bound with rational approximation of ReLU networks yields a margin-dependent extension for self-attention layers post-processed by ReLU networks.

A One-Inclusion Graph Approach to Multi-Group Learning

We prove the tightest-known upper bounds on the sample complexity of multi-group learning. Our algorithm extends the one-inclusion graph prediction strategy using a generalization of bipartite $b$-matching. In the group-realizable setting, we provide a lower bound confirming that our algorithm's $\log n / n$ convergence rate is optimal in general. If one relaxes the learning objective such that the group on which we are evaluated is chosen obliviously of the sample, then our algorithm achieves the optimal $1/n$ convergence rate under group-realizability.

Group-realizable multi-group learning by minimizing empirical risk

The sample complexity of multi-group learning is shown to improve in the group-realizable setting over the agnostic setting, even when the family of groups is infinite so long as it has finite VC dimension. The improved sample complexity is obtained by empirical risk minimization over the class of group-realizable concepts, which itself could have infinite VC dimension. Implementing this approach is also shown to be computationally intractable, and an alternative approach is suggested based on improper learning.

Time-Aware Synthetic Control

The synthetic control (SC) framework is widely used for observational causal inference with time-series panel data. SC has been successful in diverse applications, but existing methods typically treat the ordering of pre-intervention time indices interchangeable. This invariance means they may not fully take advantage of temporal structure when strong trends are present. We propose Time-Aware Synthetic Control (TASC), which employs a state-space model with a constant trend while preserving a low-rank structure of the signal. TASC uses the Kalman filter and Rauch-Tung-Striebel smoother: it first fits a generative time-series model with expectation-maximization and then performs counterfactual inference. We evaluate TASC on both simulated and real-world datasets, including policy evaluation and sports prediction. Our results suggest that TASC offers advantages in settings with strong temporal trends and high levels of observation noise.

Panprediction: Optimal Predictions for Any Downstream Task and Loss

Supervised learning is classically formulated as training a model to minimize a fixed loss function over a fixed distribution, or task. However, an emerging paradigm instead views model training as extracting enough information from data so that the model can be used to minimize many losses on many downstream tasks. We formalize a mathematical framework for this paradigm, which we call panprediction, and study its statistical complexity. Formally, panprediction generalizes omniprediction and sits upstream from multi-group learning, which respectively focus on predictions that generalize to many downstream losses or many downstream tasks, but not both. Concretely, we design algorithms that learn deterministic and randomized panpredictors with $\tilde{O}(1/\varepsilon^3)$ and $\tilde{O}(1/\varepsilon^2)$ samples, respectively. Our results demonstrate that under mild assumptions, simultaneously minimizing infinitely many losses on infinitely many tasks can be as statistically easy as minimizing one loss on one task. Along the way, we improve the best known sample complexity guarantee of deterministic omniprediction by a factor of $1/\varepsilon$, and match all other known sample complexity guarantees of omniprediction and multi-group learning. Our key technical ingredient is a nearly lossless reduction from panprediction to a statistically efficient notion of calibration, called step calibration.

Fast attention mechanisms: a tale of parallelism

Transformers have the representational capacity to simulate Massively Parallel Computation (MPC) algorithms, but they suffer from quadratic time complexity, which severely limits their scalability. We introduce an efficient attention mechanism called Approximate Nearest Neighbor Attention (ANNA) with sub-quadratic time complexity. We prove that ANNA-transformers (1) retain the expressive power previously established for standard attention in terms of matching the capabilities of MPC algorithms, and (2) can solve key reasoning tasks such as Match2 and $k$-hop with near-optimal depth. Using the MPC framework, we further prove that constant-depth ANNA-transformers can simulate constant-depth low-rank transformers, thereby providing a unified way to reason about a broad class of efficient attention approximations.