It is well known that rotation invariant algorithms are sub-optimal for learning sparse linear problems, when the number of examples is below the input dimension. This includes any gradient descent trained neural net with a fully-connected input layer initialized with a rotationally symmetric distribution. The simplest sparse problem is learning a single feature out of d features. In that case the classification error or regression loss of rotation invariant algorithms grows with 1−n/d, where n is the number of examples seen. These lower bounds become vacuous when the number of examples n reaches the dimension d. After d examples, the gradient space has full rank and any weight vector can be expressed, including the unit vector that determines the target feature. In this work, we show that when noise is added to this sparse linear problem, rotation invariant algorithms are still sub-optimal after seeing d or more examples. We prove this via a lower bound for the Bayes optimal algorithm on a rotationally symmetrized problem. We then prove much better upper bounds on the same problem for a large variety of algorithms that are non-invariant by rotations. Finally, we analyze the gradient flow trajectories of many standard optimization algorithms (such as AdaGrad) on the same noisy feature learning problem, and show how they veer away from the noisy sparse targets. We then contrast them with a group of non-rotation invariant algorithms that veer towards the sparse targets. We believe that the lower bounds method and trajectory categorization will be crucial for analyzing other families of algorithms with different classes of invariances.

The evolution of biological neural systems has led to both modularity and sparse coding, which enables energy efficiency and robustness across the diversity of tasks in the lifespan. In contrast, standard neural networks rely on dense, non-specialized architectures, where all model parameters are simultaneously updated to learn multiple tasks, leading to interference. Current sparse neural network approaches aim to alleviate this issue but are hindered by limitations such as 1) trainable gating functions that cause representation collapse, 2) disjoint experts that result in redundant computation and slow learning, and 3) reliance on explicit input or task IDs that limit flexibility and scalability. In this paper we propose Conditionally Overlapping Mixture of ExperTs (COMET), a general deep learning method that addresses these challenges by inducing a modular, sparse architecture with an exponential number of overlapping experts. COMET replaces the trainable gating function used in Sparse Mixture of Experts with a fixed, biologically inspired random projection applied to individual input representations. This design causes the degree of expert overlap to depend on input similarity, so that similar inputs tend to share more parameters. This results in faster learning per update step and improved out-of-sample generalization. We demonstrate the effectiveness of COMET on a range of tasks, including image classification, language modeling, and regression, using several popular deep learning architectures.

We derive a novel, provably robust, efficient, and closed-form Bayesian update rule for online filtering in state-space models in the presence of outliers and misspecified measurement models. Our method combines generalised Bayesian inference with filtering methods such as the extended and ensemble Kalman filter. We use the former to show robustness and the latter to ensure computational efficiency in the case of nonlinear models. Our method matches or outperforms other robust filtering methods (such as those based on variational Bayes) at a much lower computational cost. We show this empirically on a range of filtering problems with outlier measurements, such as object tracking, state estimation in high-dimensional chaotic systems, and online learning of neural networks.

We propose a novel approach to sequential Bayesian inference based on variational Bayes (VB). The key insight is that, in the online setting, we do not need to add the KL term to regularize to the prior (which comes from the posterior at the previous timestep); instead we can optimize just the expected log-likelihood, performing a single step of natural gradient descent starting at the prior predictive. We prove this method recovers exact Bayesian inference if the model is conjugate. We also show how to compute an efficient deterministic approximation to the VB objective, as well as our simplified objective, when the variational distribution is Gaussian or a sub-family, including the case of a diagonal plus low-rank precision matrix. We show empirically that our method outperforms other online VB methods in the non-conjugate setting, such as online learning for neural networks, especially when controlling for computational costs.

We discuss the problem of assessing and aiding user performance in dynamic tasks that require rapid selection among multiple information sources. Motivated by research in human sequential learning, we develop a system that learns by observation to predict the information a user desires in different contexts. The model decides when the display should be updated, which is akin to the problem of scene segmentation, and then selects the situationally relevant information display. The model reduces the cognitive burden of selecting situation-relevant displays. We evaluate the system in a tank video game environment and find that the system boosts user performance. The fit of the model to user data provides a quantitative assessment of user behavior, which is useful in assessing individual differences and the progression from novice- to expert-level proficiency. We discuss the relative benefits of adopting a learning approach to predicting information preferences and possible avenues to reduce the negative consequences of automation.