Matthew Farrugia-Roberts

Joar Skalse, Matthew Farrugia-Roberts, Stuart Russell et al.

It is often very challenging to manually design reward functions for complex, real-world tasks. To solve this, one can instead use reward learning to infer a reward function from data. However, there are often multiple reward functions that fit the data equally well, even in the infinite-data limit. This means that the reward function is only partially identifiable. In this work, we formally characterise the partial identifiability of the reward function given several popular reward learning data sources, including expert demonstrations and trajectory comparisons. We also analyse the impact of this partial identifiability for several downstream tasks, such as policy optimisation. We unify our results in a framework for comparing data sources and downstream tasks by their invariances, with implications for the design and selection of data sources for reward learning.

Matthew Farrugia-Roberts

Modern large-scale deep learning exhibits two striking empirical phenomena: behavioural scaling laws (predictable performance gains with increasing scale) and emergent mechanisms (structured internal representations and circuits in deep neural networks). We hypothesise that these two phenomena are connected: that predictable changes in behaviour are the result of predictable changes in internal computational structure. In this paper, we report preliminary evidence of such a connection. We find a correlation between scaling patterns in performance and representations in small transformers trained to predict the outputs of a hidden Markov model, for which residual activations are known to linearly encode a belief distribution over latent states in a probability simplex.

Afiq Abdillah Effiezal Aswadi, Oliver Britton, Ross Baker et al.

Modern deep learning science often assumes that neural networks learn from a fixed data distribution. However, many practically important learning problems involve data distributions that change throughout training. How does such non-stationarity impact the inductive biases of deep learning towards models with different structural, generalisation, and safety properties? A fruitful testbed for studying inductive bias is in-context linear regression sequence modelling, where small transformers display strikingly different generalisation patterns depending on the diversity of the (fixed) training task distribution. In this paper, we explore the effect of diversifying the task distribution across training time, finding that such temporal diversity leads to an increased bias towards generalisation over memorisation.

Chris Elliott, Einar Urdshals, David Quarel et al.

Singular learning theory characterizes Bayesian learning as an evolving tradeoff between accuracy and complexity, with transitions between qualitatively different solutions as sample size increases. We extend this theory to deep reinforcement learning, proving that the concentration of the generalized posterior over policies is governed by the local learning coefficient (LLC), an invariant of the geometry of the regret function. This theory predicts that Bayesian phase transitions in reinforcement learning should proceed from simple policies with high regret to complex policies with low regret. We verify this prediction empirically in a gridworld environment exhibiting stagewise policy development: phase transitions over SGD training manifest as "opposing staircases" where regret decreases sharply while the LLC increases. Notably, the LLC detects phase transitions even when estimated on a subset of states where the policies appear identical in terms of regret, suggesting it captures changes in the underlying algorithm rather than just performance.

Jesse Hoogland, George Wang, Matthew Farrugia-Roberts et al.

Deep learning involves navigating a high-dimensional loss landscape over the neural network parameter space. Over the course of training, complex computational structures form and re-form inside the neural network, leading to shifts in input/output behavior. It is a priority for the science of deep learning to uncover principles governing the development of neural network structure and behavior. Drawing on the framework of singular learning theory, we propose that model development is deeply linked to degeneracy in the local geometry of the loss landscape. We investigate this link by monitoring loss landscape degeneracy throughout training, as quantified by the local learning coefficient, for a transformer language model and an in-context linear regression transformer. We show that training can be divided into distinct periods of change in loss landscape degeneracy, and that these changes in degeneracy coincide with significant changes in the internal computational structure and the input/output behavior of the transformers. This finding provides suggestive evidence that degeneracy and development are linked in transformers, underscoring the potential of a degeneracy-based perspective for understanding modern deep learning.

Simon Pepin Lehalleur, Jesse Hoogland, Matthew Farrugia-Roberts et al.

In this position paper, we argue that understanding the relation between structure in the data distribution and structure in trained models is central to AI alignment. First, we discuss how two neural networks can have equivalent performance on the training set but compute their outputs in essentially different ways and thus generalise differently. For this reason, standard testing and evaluation are insufficient for obtaining assurances of safety for widely deployed generally intelligent systems. We argue that to progress beyond evaluation to a robust mathematical science of AI alignment, we need to develop statistical foundations for an understanding of the relation between structure in the data distribution, internal structure in models, and how these structures underlie generalisation.

Liam Carroll, Jesse Hoogland, Matthew Farrugia-Roberts et al.

Modern deep neural networks display striking examples of rich internal computational structure. Uncovering principles governing the development of such structure is a priority for the science of deep learning. In this paper, we explore the transient ridge phenomenon: when transformers are trained on in-context linear regression tasks with intermediate task diversity, they initially behave like ridge regression before specializing to the tasks in their training distribution. This transition from a general solution to a specialized solution is revealed by joint trajectory principal component analysis. Further, we draw on the theory of Bayesian internal model selection to suggest a general explanation for the phenomena of transient structure in transformers, based on an evolving tradeoff between loss and complexity. We empirically validate this explanation by measuring the model complexity of our transformers as defined by the local learning coefficient.

Matthew Farrugia-Roberts

To better understand complexity in neural networks, we theoretically investigate the idealised phenomenon of lossless network compressibility, whereby an identical function can be implemented with fewer hidden units. In the setting of single-hidden-layer hyperbolic tangent networks, we define the rank of a parameter as the minimum number of hidden units required to implement the same function. We give efficient formal algorithms for optimal lossless compression and computing the rank of a parameter. Losslessly compressible parameters are atypical, but their existence has implications for nearby parameters. We define the proximate rank of a parameter as the rank of the most compressible parameter within a small L-infinity neighbourhood. We give an efficient greedy algorithm for bounding the proximate rank of a parameter, and show that the problem of tightly bounding the proximate rank is NP-complete. These results lay a foundation for future theoretical and empirical work on losslessly compressible parameters and their neighbours.

Karim Abdel Sadek, Matthew Farrugia-Roberts, Usman Anwar et al. · cambridge, gatech

Safe generalization in reinforcement learning requires not only that a learned policy acts capably in new situations, but also that it uses its capabilities towards the pursuit of the designer's intended goal. The latter requirement may fail when a proxy goal incentivizes similar behavior to the intended goal within the training environment, but not in novel deployment environments. This creates the risk that policies will behave as if in pursuit of the proxy goal, rather than the intended goal, in deployment -- a phenomenon known as goal misgeneralization. In this paper, we formalize this problem setting in order to theoretically study the possibility of goal misgeneralization under different training objectives. We show that goal misgeneralization is possible under approximate optimization of the maximum expected value (MEV) objective, but not the minimax expected regret (MMER) objective. We then empirically show that the standard MEV-based training method of domain randomization exhibits goal misgeneralization in procedurally-generated grid-world environments, whereas current regret-based unsupervised environment design (UED) methods are more robust to goal misgeneralization (though they don't find MMER policies in all cases). Our findings suggest that minimax expected regret is a promising approach to mitigating goal misgeneralization.

Matthew Farrugia-Roberts

Understanding the learning process of artificial neural networks requires clarifying the structure of the parameter space within which learning takes place. A neural network parameter's functional equivalence class is the set of parameters implementing the same input--output function. For many architectures, almost all parameters have a simple and well-documented functional equivalence class. However, there is also a vanishing minority of reducible parameters, with richer functional equivalence classes caused by redundancies among the network's units. In this paper, we give an algorithmic characterisation of unit redundancies and reducible functional equivalence classes for a single-hidden-layer hyperbolic tangent architecture. We show that such functional equivalence classes are piecewise-linear path-connected sets, and that for parameters with a majority of redundant units, the sets have a diameter of at most 7 linear segments.