Towards Exact Computation of Inductive Bias

Much research in machine learning involves finding appropriate inductive biases (e.g. convolutional neural networks, momentum-based optimizers, transformers) to promote generalization on tasks. However, quantification of the amount of inductive bias associated with these architectures and hyperparameters has been limited. We propose a novel method for efficiently computing the inductive bias required for generalization on a task with a fixed training data budget; formally, this corresponds to the amount of information required to specify well-generalizing models within a specific hypothesis space of models. Our approach involves modeling the loss distribution of random hypotheses drawn from a hypothesis space to estimate the required inductive bias for a task relative to these hypotheses. Unlike prior work, our method provides a direct estimate of inductive bias without using bounds and is applicable to diverse hypothesis spaces. Moreover, we derive approximation error bounds for our estimation approach in terms of the number of sampled hypotheses. Consistent with prior results, our empirical results demonstrate that higher dimensional tasks require greater inductive bias. We show that relative to other expressive model classes, neural networks as a model class encode large amounts of inductive bias. Furthermore, our measure quantifies the relative difference in inductive bias between different neural network architectures. Our proposed inductive bias metric provides an information-theoretic interpretation of the benefits of specific model architectures for certain tasks and provides a quantitative guide to developing tasks requiring greater inductive bias, thereby encouraging the development of more powerful inductive biases.

Delay Embedding Theory of Neural Sequence Models

To generate coherent responses, language models infer unobserved meaning from their input text sequence. One potential explanation for this capability arises from theories of delay embeddings in dynamical systems, which prove that unobserved variables can be recovered from the history of only a handful of observed variables. To test whether language models are effectively constructing delay embeddings, we measure the capacities of sequence models to reconstruct unobserved dynamics. We trained 1-layer transformer decoders and state-space sequence models on next-step prediction from noisy, partially-observed time series data. We found that each sequence layer can learn a viable embedding of the underlying system. However, state-space models have a stronger inductive bias than transformers-in particular, they more effectively reconstruct unobserved information at initialization, leading to more parameter-efficient models and lower error on dynamics tasks. Our work thus forges a novel connection between dynamical systems and deep learning sequence models via delay embedding theory.

Resampling-free Particle Filters in High-dimensions

State estimation is crucial for the performance and safety of numerous robotic applications. Among the suite of estimation techniques, particle filters have been identified as a powerful solution due to their non-parametric nature. Yet, in high-dimensional state spaces, these filters face challenges such as 'particle deprivation' which hinders accurate representation of the true posterior distribution. This paper introduces a novel resampling-free particle filter designed to mitigate particle deprivation by forgoing the traditional resampling step. This ensures a broader and more diverse particle set, especially vital in high-dimensional scenarios. Theoretically, our proposed filter is shown to offer a near-accurate representation of the desired posterior distribution in high-dimensional contexts. Empirically, the effectiveness of our approach is underscored through a high-dimensional synthetic state estimation task and a 6D pose estimation derived from videos. We posit that as robotic systems evolve with greater degrees of freedom, particle filters tailored for high-dimensional state spaces will be indispensable.

Do Diffusion Models Learn Semantically Meaningful and Efficient Representations?

Diffusion models are capable of impressive feats of image generation with uncommon juxtapositions such as astronauts riding horses on the moon with properly placed shadows. These outputs indicate the ability to perform compositional generalization, but how do the models do so? We perform controlled experiments on conditional DDPMs learning to generate 2D spherical Gaussian bumps centered at specified $x$- and $y$-positions. Our results show that the emergence of semantically meaningful latent representations is key to achieving high performance. En route to successful performance over learning, the model traverses three distinct phases of latent representations: (phase A) no latent structure, (phase B) a 2D manifold of disordered states, and (phase C) a 2D ordered manifold. Corresponding to each of these phases, we identify qualitatively different generation behaviors: 1) multiple bumps are generated, 2) one bump is generated but at inaccurate $x$ and $y$ locations, 3) a bump is generated at the correct $x$ and y location. Furthermore, we show that even under imbalanced datasets where features ($x$- versus $y$-positions) are represented with skewed frequencies, the learning process for $x$ and $y$ is coupled rather than factorized, demonstrating that simple vanilla-flavored diffusion models cannot learn efficient representations in which localization in $x$ and $y$ are factorized into separate 1D tasks. These findings suggest the need for future work to find inductive biases that will push generative models to discover and exploit factorizable independent structures in their inputs, which will be required to vault these models into more data-efficient regimes.

Neuro-Inspired Fragmentation and Recall to Overcome Catastrophic Forgetting in Curiosity

Deep reinforcement learning methods exhibit impressive performance on a range of tasks but still struggle on hard exploration tasks in large environments with sparse rewards. To address this, intrinsic rewards can be generated using forward model prediction errors that decrease as the environment becomes known, and incentivize an agent to explore novel states. While prediction-based intrinsic rewards can help agents solve hard exploration tasks, they can suffer from catastrophic forgetting and actually increase at visited states. We first examine the conditions and causes of catastrophic forgetting in grid world environments. We then propose a new method FARCuriosity, inspired by how humans and animals learn. The method depends on fragmentation and recall: an agent fragments an environment based on surprisal, and uses different local curiosity modules (prediction-based intrinsic reward functions) for each fragment so that modules are not trained on the entire environment. At each fragmentation event, the agent stores the current module in long-term memory (LTM) and either initializes a new module or recalls a previously stored module based on its match with the current state. With fragmentation and recall, FARCuriosity achieves less forgetting and better overall performance in games with varied and heterogeneous environments in the Atari benchmark suite of tasks. Thus, this work highlights the problem of catastrophic forgetting in prediction-based curiosity methods and proposes a solution.

Neuro-Inspired Efficient Map Building via Fragmentation and Recall

Animals and robots navigate through environments by building and refining maps of the space. These maps enable functions including navigating back to home, planning, search, and foraging. In large environments, exploration of the space is a hard problem: agents can become stuck in local regions. Here, we use insights from neuroscience to propose and apply the concept of Fragmentation-and-Recall (FarMap), with agents solving the mapping problem by building local maps via a surprisal-based clustering of space, which they use to set subgoals for spatial exploration. Agents build and use a local map to predict their observations; high surprisal leads to a ``fragmentation event'' that truncates the local map. At these events, the recent local map is placed into long-term memory (LTM), and a different local map is initialized. If observations at a fracture point match observations in one of the stored local maps, that map is recalled (and thus reused) from LTM. The fragmentation points induce a natural online clustering of the larger space, forming a set of intrinsic potential subgoals that are stored in LTM as a topological graph. Agents choose their next subgoal from the set of near and far potential subgoals from within the current local map or LTM, respectively. Thus, local maps guide exploration locally, while LTM promotes global exploration. We evaluate FarMap on complex procedurally-generated spatial environments to demonstrate that this mapping strategy much more rapidly covers the environment (number of agent steps and wall clock time) and is more efficient in active memory usage, without loss of performance.

Optimizing protein fitness using Gibbs sampling with Graph-based Smoothing

The ability to design novel proteins with higher fitness on a given task would be revolutionary for many fields of medicine. However, brute-force search through the combinatorially large space of sequences is infeasible. Prior methods constrain search to a small mutational radius from a reference sequence, but such heuristics drastically limit the design space. Our work seeks to remove the restriction on mutational distance while enabling efficient exploration. We propose Gibbs sampling with Graph-based Smoothing (GGS) which iteratively applies Gibbs with gradients to propose advantageous mutations using graph-based smoothing to remove noisy gradients that lead to false positives. Our method is state-of-the-art in discovering high-fitness proteins with up to 8 mutations from the training set. We study the GFP and AAV design problems, ablations, and baselines to elucidate the results. Code: https://github.com/kirjner/GGS

Beyond Geometry: Comparing the Temporal Structure of Computation in Neural Circuits with Dynamical Similarity Analysis

How can we tell whether two neural networks are utilizing the same internal processes for a particular computation? This question is pertinent for multiple subfields of both neuroscience and machine learning, including neuroAI, mechanistic interpretability, and brain-machine interfaces. Standard approaches for comparing neural networks focus on the spatial geometry of latent states. Yet in recurrent networks, computations are implemented at the level of neural dynamics, which do not have a simple one-to-one mapping with geometry. To bridge this gap, we introduce a novel similarity metric that compares two systems at the level of their dynamics. Our method incorporates two components: Using recent advances in data-driven dynamical systems theory, we learn a high-dimensional linear system that accurately captures core features of the original nonlinear dynamics. Next, we compare these linear approximations via a novel extension of Procrustes Analysis that accounts for how vector fields change under orthogonal transformation. Via four case studies, we demonstrate that our method effectively identifies and distinguishes dynamic structure in recurrent neural networks (RNNs), whereas geometric methods fall short. We additionally show that our method can distinguish learning rules in an unsupervised manner. Our method therefore opens the door to novel data-driven analyses of the temporal structure of neural computation, and to more rigorous testing of RNNs as models of the brain.

Model-agnostic Measure of Generalization Difficulty

The measure of a machine learning algorithm is the difficulty of the tasks it can perform, and sufficiently difficult tasks are critical drivers of strong machine learning models. However, quantifying the generalization difficulty of machine learning benchmarks has remained challenging. We propose what is to our knowledge the first model-agnostic measure of the inherent generalization difficulty of tasks. Our inductive bias complexity measure quantifies the total information required to generalize well on a task minus the information provided by the data. It does so by measuring the fractional volume occupied by hypotheses that generalize on a task given that they fit the training data. It scales exponentially with the intrinsic dimensionality of the space over which the model must generalize but only polynomially in resolution per dimension, showing that tasks which require generalizing over many dimensions are drastically more difficult than tasks involving more detail in fewer dimensions. Our measure can be applied to compute and compare supervised learning, reinforcement learning and meta-learning generalization difficulties against each other. We show that applied empirically, it formally quantifies intuitively expected trends, e.g. that in terms of required inductive bias, MNIST < CIFAR10 < Imagenet and fully observable Markov decision processes (MDPs) < partially observable MDPs. Further, we show that classification of complex images $<$ few-shot meta-learning with simple images. Our measure provides a quantitative metric to guide the construction of more complex tasks requiring greater inductive bias, and thereby encourages the development of more sophisticated architectures and learning algorithms with more powerful generalization capabilities.

Map Induction: Compositional spatial submap learning for efficient exploration in novel environments

Humans are expert explorers. Understanding the computational cognitive mechanisms that support this efficiency can advance the study of the human mind and enable more efficient exploration algorithms. We hypothesize that humans explore new environments efficiently by inferring the structure of unobserved spaces using spatial information collected from previously explored spaces. This cognitive process can be modeled computationally using program induction in a Hierarchical Bayesian framework that explicitly reasons about uncertainty with strong spatial priors. Using a new behavioral Map Induction Task, we demonstrate that this computational framework explains human exploration behavior better than non-inductive models and outperforms state-of-the-art planning algorithms when applied to a realistic spatial navigation domain.