Xaq Pitkow, Yashar Ahmadian, Ken D. Miller
Loopy belief propagation performs approximate inference on graphical models with loops. One might hope to compensate for the approximation by adjusting model parameters. Learning algorithms for this purpose have been explored previously, and the claim has been made that every set of locally consistent marginals can arise from belief propagation run on a graphical model. On the contrary, here we show that many probability distributions have marginals that cannot be reached by belief propagation using any set of model parameters or any learning algorithm. We call such marginals `unbelievable.' This problem occurs whenever the Hessian of the Bethe free energy is not positive-definite at the target marginals. All learning algorithms for belief propagation necessarily fail in these cases, producing beliefs or sets of beliefs that may even be worse than the pre-learning approximation. We then show that averaging inaccurate beliefs, each obtained from belief propagation using model parameters perturbed about some learned mean values, can achieve the unbelievable marginals.
Rajkumar Vasudeva Raju, Xaq Pitkow
May 20, 2016·q-bio.NC·PDF Behavioral experiments on humans and animals suggest that the brain performs probabilistic inference to interpret its environment. Here we present a new general-purpose, biologically-plausible neural implementation of approximate inference. The neural network represents uncertainty using Probabilistic Population Codes (PPCs), which are distributed neural representations that naturally encode probability distributions, and support marginalization and evidence integration in a biologically-plausible manner. By connecting multiple PPCs together as a probabilistic graphical model, we represent multivariate probability distributions. Approximate inference in graphical models can be accomplished by message-passing algorithms that disseminate local information throughout the graph. An attractive and often accurate example of such an algorithm is Loopy Belief Propagation (LBP), which uses local marginalization and evidence integration operations to perform approximate inference efficiently even for complex models. Unfortunately, a subtle feature of LBP renders it neurally implausible. However, LBP can be elegantly reformulated as a sequence of Tree-based Reparameterizations (TRP) of the graphical model. We re-express the TRP updates as a nonlinear dynamical system with both fast and slow timescales, and show that this produces a neurally plausible solution. By combining all of these ideas, we show that a network of PPCs can represent multivariate probability distributions and implement the TRP updates to perform probabilistic inference. Simulations with Gaussian graphical models demonstrate that the neural network inference quality is comparable to the direct evaluation of LBP and robust to noise, and thus provides a promising mechanism for general probabilistic inference in the population codes of the brain.
Shell Xu Hu, Md Rifat Arefin, Viet-Nhat Nguyen, Alish Dipani, Xaq Pitkow, Andreas Savas Tolias
To extract the voice of a target speaker when mixed with a variety of other sounds, such as white and ambient noises or the voices of interfering speakers, we extend the Transformer network to attend the most relevant information with respect to the target speaker given the characteristics of his or her voices as a form of contextual information. The idea has a natural interpretation in terms of the selective attention theory. Specifically, we propose two models to incorporate the voice characteristics in Transformer based on different insights of where the feature selection should take place. Both models yield excellent performance, on par or better than published state-of-the-art models on the speaker extraction task, including separating speech of novel speakers not seen during training.
Rajkumar Vasudeva Raju, Zhe Li, Scott Linderman, Xaq Pitkow
Patterns of microcircuitry suggest that the brain has an array of repeated canonical computational units. Yet neural representations are distributed, so the relevant computations may only be related indirectly to single-neuron transformations. It thus remains an open challenge how to define canonical distributed computations. We integrate normative and algorithmic theories of neural computation into a mathematical framework for inferring canonical distributed computations from large-scale neural activity patterns. At the normative level, we hypothesize that the brain creates a structured internal model of its environment, positing latent causes that explain its sensory inputs, and uses those sensory inputs to infer the latent causes. At the algorithmic level, we propose that this inference process is a nonlinear message-passing algorithm on a graph-structured model of the world. Given a time series of neural activity during a perceptual inference task, our framework finds (i) the neural representation of relevant latent variables, (ii) interactions between these variables that define the brain's internal model of the world, and (iii) message-functions specifying the inference algorithm. These targeted computational properties are then statistically distinguishable due to the symmetries inherent in any canonical computation, up to a global transformation. As a demonstration, we simulate recordings for a model brain that implicitly implements an approximate inference algorithm on a probabilistic graphical model. Given its external inputs and noisy neural activity, we recover the latent variables, their neural representation and dynamics, and canonical message-functions. We highlight features of experimental design needed to successfully extract canonical computations from neural data. Overall, this framework provides a new tool for discovering interpretable structure in neural recordings.
Saurabh Daptardar, Paul Schrater, Xaq Pitkow
Continuous control and planning remains a major challenge in robotics and machine learning. Neuroscience offers the possibility of learning from animal brains that implement highly successful controllers, but it is unclear how to relate an animal's behavior to control principles. Animals may not always act optimally from the perspective of an external observer, but may still act rationally: we hypothesize that animals choose actions with highest expected future subjective value according to their own internal model of the world. Their actions thus result from solving a different optimal control problem from those on which they are evaluated in neuroscience experiments. With this assumption, we propose a novel framework of model-based inverse rational control that learns the agent's internal model that best explains their actions in a task described as a partially observable Markov decision process (POMDP). In this approach we first learn optimal policies generalized over the entire model space of dynamics and subjective rewards, using an extended Kalman filter to represent the belief space, a neural network in the actor-critic framework to optimize the policy, and a simplified basis for the parameter space. We then compute the model that maximizes the likelihood of the experimentally observable data comprising the agent's sensory observations and chosen actions. Our proposed method is able to recover the true model of simulated agents within theoretical error bounds given by limited data. We illustrate this method by applying it to a complex naturalistic task currently used in neuroscience experiments. This approach provides a foundation for interpreting the behavioral and neural dynamics of highly adapted controllers in animal brains.
Lokesh Boominathan, Xaq Pitkow
Oct 15, 2021·q-bio.NC·PDF Sensory observations about the world are invariably ambiguous. Inference about the world's latent variables is thus an important computation for the brain. However, computational constraints limit the performance of these computations. These constraints include energetic costs for neural activity and noise on every channel. Efficient coding is one prominent theory that describes how such limited resources can best be used. In one incarnation, this leads to a theory of predictive coding, where predictions are subtracted from signals, reducing the cost of sending something that is already known. This theory does not, however, account for the costs or noise associated with those predictions. Here we offer a theory that accounts for both feedforward and feedback costs, and noise in all computations. We formulate this inference problem as message-passing on a graph whereby feedback serves as an internal control signal aiming to maximize how well an inference tracks a target state while minimizing the costs of computation. We apply this novel formulation of inference as control to the canonical problem of inferring the hidden scalar state of a linear dynamical system with Gaussian variability. The best solution depends on architectural constraints, such as Dale's law, the ubiquitous law that each neuron makes solely excitatory or inhibitory postsynaptic connections. This biological structure can create asymmetric costs for feedforward and feedback channels. Under such conditions, our theory predicts the gain of optimal predictive feedback and how it is incorporated into the inference computation. We show that there is a non-monotonic dependence of optimal feedback gain as a function of both the computational parameters and the world dynamics, leading to phase transitions in whether feedback provides any utility in optimal inference under computational constraints.
Xaq Pitkow
This paper shows how sparse, high-dimensional probability distributions could be represented by neurons with exponential compression. The representation is a novel application of compressive sensing to sparse probability distributions rather than to the usual sparse signals. The compressive measurements correspond to expected values of nonlinear functions of the probabilistically distributed variables. When these expected values are estimated by sampling, the quality of the compressed representation is limited only by the quality of sampling. Since the compression preserves the geometric structure of the space of sparse probability distributions, probabilistic computation can be performed in the compressed domain. Interestingly, functions satisfying the requirements of compressive sensing can be implemented as simple perceptrons. If we use perceptrons as a simple model of feedforward computation by neurons, these results show that the mean activity of a relatively small number of neurons can accurately represent a high-dimensional joint distribution implicitly, even without accounting for any noise correlations. This comprises a novel hypothesis for how neurons could encode probabilities in the brain.
Xaq Pitkow
To understand the computations of our visual system, it is important to understand also the natural environment it evolved to interpret. Unfortunately, existing models of the visual environment are either unrealistic or too complex for mathematical description. Here we describe a naturalistic image model and present a mathematical solution for the statistical relationships between the image features and model variables. The world described by this model is composed of independent, opaque, textured objects which occlude each other. This simple structure allows us to calculate the joint probability distribution of image values sampled at multiple arbitrarily located points, without approximation. This result can be converted into probabilistic relationships between observable image features as well as between the unobservable properties that caused these features, including object boundaries and relative depth. Using these results we explain the causes of a wide range of natural scene properties, including highly non-gaussian distributions of image features and causal relations between pairs of edges. We discuss the implications of this description of natural scenes for the study of vision.
Xaq Pitkow, Dora Angelaki
Feb 12, 2017·q-bio.NC·PDF It is widely believed that the brain performs approximate probabilistic inference to estimate causal variables in the world from ambiguous sensory data. To understand these computations, we need to analyze how information is represented and transformed by the actions of nonlinear recurrent neural networks. We propose that these probabilistic computations function by a message-passing algorithm operating at the level of redundant neural populations. To explain this framework, we review its underlying concepts, including graphical models, sufficient statistics, and message-passing, and then describe how these concepts could be implemented by recurrently connected probabilistic population codes. The relevant information flow in these networks will be most interpretable at the population level, particularly for redundant neural codes. We therefore outline a general approach to identify the essential features of a neural message-passing algorithm. Finally, we argue that to reveal the most important aspects of these neural computations, we must study large-scale activity patterns during moderately complex, naturalistic behaviors.
Richard D. Lange, Ari Benjamin, Ralf M. Haefner, Xaq Pitkow
Sampling and Variational Inference (VI) are two large families of methods for approximate inference that have complementary strengths. Sampling methods excel at approximating arbitrary probability distributions, but can be inefficient. VI methods are efficient, but may misrepresent the true distribution. Here, we develop a general framework where approximations are stochastic mixtures of simple component distributions. Both sampling and VI can be seen as special cases: in sampling, each mixture component is a delta-function and is chosen stochastically, while in standard VI a single component is chosen to minimize divergence. We derive a practical method that interpolates between sampling and VI by solving an optimization problem over a mixing distribution. Intermediate inference methods then arise by varying a single parameter. Our method provably improves on sampling (reducing variance) and on VI (reducing bias+variance despite increasing variance). We demonstrate our method's bias/variance trade-off in practice on reference problems, and we compare outcomes to commonly used sampling and VI methods. This work takes a step towards a highly flexible yet simple family of inference methods that combines the complementary strengths of sampling and VI.
Mengye Ren, Eleni Triantafillou, Kuan-Chieh Wang, James Lucas, Jake Snell, Xaq Pitkow, Andreas S. Tolias, Richard Zemel
Despite impressive progress in deep learning, generalizing far beyond the training distribution is an important open challenge. In this work, we consider few-shot classification, and aim to shed light on what makes some novel classes easier to learn than others, and what types of learned representations generalize better. To this end, we define a new paradigm in terms of attributes -- simple building blocks of which concepts are formed -- as a means of quantifying the degree of relatedness of different concepts. Our empirical analysis reveals that supervised learning generalizes poorly to new attributes, but a combination of self-supervised pretraining with supervised finetuning leads to stronger generalization. The benefit of self-supervised pretraining and supervised finetuning is further investigated through controlled experiments using random splits of the attribute space, and we find that predictability of test attributes provides an informative estimate of a model's generalization ability.
Zhe Li, Wieland Brendel, Edgar Y. Walker, Erick Cobos, Taliah Muhammad, Jacob Reimer, Matthias Bethge, Fabian H. Sinz, Xaq Pitkow, Andreas S. Tolias
Despite impressive performance on numerous visual tasks, Convolutional Neural Networks (CNNs) --- unlike brains --- are often highly sensitive to small perturbations of their input, e.g. adversarial noise leading to erroneous decisions. We propose to regularize CNNs using large-scale neuroscience data to learn more robust neural features in terms of representational similarity. We presented natural images to mice and measured the responses of thousands of neurons from cortical visual areas. Next, we denoised the notoriously variable neural activity using strong predictive models trained on this large corpus of responses from the mouse visual system, and calculated the representational similarity for millions of pairs of images from the model's predictions. We then used the neural representation similarity to regularize CNNs trained on image classification by penalizing intermediate representations that deviated from neural ones. This preserved performance of baseline models when classifying images under standard benchmarks, while maintaining substantially higher performance compared to baseline or control models when classifying noisy images. Moreover, the models regularized with cortical representations also improved model robustness in terms of adversarial attacks. This demonstrates that regularizing with neural data can be an effective tool to create an inductive bias towards more robust inference.
Lokesh Boominathan, Yizhou Chen, Matthew McGinley, Xaq Pitkow
Jan 13, 2025·q-bio.NC·PDF Being attentive to task-relevant features can improve task performance, but paying attention comes with its own metabolic cost. Therefore, strategic allocation of attention is crucial in performing the task efficiently. This work aims to understand this strategy. Recently, de Gee et al. conducted experiments involving mice performing an auditory sustained attention-value task. This task required the mice to exert attention to identify whether a high-order acoustic feature was present amid the noise. By varying the trial duration and reward magnitude, the task allows us to investigate how an agent should strategically deploy their attention to maximize their benefits and minimize their costs. In our work, we develop a reinforcement learning-based normative model of the mice to understand how it balances attention cost against its benefits. The model is such that at each moment the mice can choose between two levels of attention and decide when to take costly actions that could obtain rewards. Our model suggests that efficient use of attentional resources involves alternating blocks of high attention with blocks of low attention. In the extreme case where the agent disregards sensory input during low attention states, we see that high attention is used rhythmically. Our model provides evidence about how one should deploy attention as a function of task utility, signal statistics, and how attention affects sensory evidence.
Minhae Kwon, Saurabh Daptardar, Paul Schrater, Xaq Pitkow
A fundamental question in neuroscience is how the brain creates an internal model of the world to guide actions using sequences of ambiguous sensory information. This is naturally formulated as a reinforcement learning problem under partial observations, where an agent must estimate relevant latent variables in the world from its evidence, anticipate possible future states, and choose actions that optimize total expected reward. This problem can be solved by control theory, which allows us to find the optimal actions for a given system dynamics and objective function. However, animals often appear to behave suboptimally. Why? We hypothesize that animals have their own flawed internal model of the world, and choose actions with the highest expected subjective reward according to that flawed model. We describe this behavior as rational but not optimal. The problem of Inverse Rational Control (IRC) aims to identify which internal model would best explain an agent's actions. Our contribution here generalizes past work on Inverse Rational Control which solved this problem for discrete control in partially observable Markov decision processes. Here we accommodate continuous nonlinear dynamics and continuous actions, and impute sensory observations corrupted by unknown noise that is private to the animal. We first build an optimal Bayesian agent that learns an optimal policy generalized over the entire model space of dynamics and subjective rewards using deep reinforcement learning. Crucially, this allows us to compute a likelihood over models for experimentally observable action trajectories acquired from a suboptimal agent. We then find the model parameters that maximize the likelihood using gradient ascent.
Yicheng Fei, Xaq Pitkow
Probabilistic graphical models provide a powerful tool to describe complex statistical structure, with many real-world applications in science and engineering from controlling robotic arms to understanding neuronal computations. A major challenge for these graphical models is that inferences such as marginalization are intractable for general graphs. These inferences are often approximated by a distributed message-passing algorithm such as Belief Propagation, which does not always perform well on graphs with cycles, nor can it always be easily specified for complex continuous probability distributions. Such difficulties arise frequently in expressive graphical models that include intractable higher-order interactions. In this paper we define the Recurrent Factor Graph Neural Network (RF-GNN) to achieve fast approximate inference on graphical models that involve many-variable interactions. Experimental results on several families of graphical models demonstrate the out-of-distribution generalization capability of our method to different sized graphs, and indicate the domain in which our method outperforms Belief Propagation (BP). Moreover, we test the RF-GNN on a real-world Low-Density Parity-Check dataset as a benchmark along with other baseline models including BP variants and other GNN methods. Overall we find that RF-GNNs outperform other methods under high noise levels.
KiJung Yoon, Renjie Liao, Yuwen Xiong, Lisa Zhang, Ethan Fetaya, Raquel Urtasun, Richard Zemel, Xaq Pitkow
A fundamental computation for statistical inference and accurate decision-making is to compute the marginal probabilities or most probable states of task-relevant variables. Probabilistic graphical models can efficiently represent the structure of such complex data, but performing these inferences is generally difficult. Message-passing algorithms, such as belief propagation, are a natural way to disseminate evidence amongst correlated variables while exploiting the graph structure, but these algorithms can struggle when the conditional dependency graphs contain loops. Here we use Graph Neural Networks (GNNs) to learn a message-passing algorithm that solves these inference tasks. We first show that the architecture of GNNs is well-matched to inference tasks. We then demonstrate the efficacy of this inference approach by training GNNs on a collection of graphical models and showing that they substantially outperform belief propagation on loopy graphs. Our message-passing algorithms generalize out of the training set to larger graphs and graphs with different structure.
Ralf M. Haefner, Jeff Beck, Cristina Savin, Mehrdad Salmasi, Xaq Pitkow
This perspective piece is the result of a Generative Adversarial Collaboration (GAC) tackling the question `How does neural activity represent probability distributions?'. We have addressed three major obstacles to progress on answering this question: first, we provide a unified language for defining competing hypotheses. Second, we explain the fundamentals of three prominent proposals for probabilistic computations -- Probabilistic Population Codes (PPCs), Distributed Distributional Codes (DDCs), and Neural Sampling Codes (NSCs) -- and describe similarities and differences in that common language. Third, we review key empirical data previously taken as evidence for at least one of these proposal, and describe how it may or may not be explainable by alternative proposals. Finally, we describe some key challenges in resolving the debate, and propose potential directions to address them through a combination of theory and experiments.
Ryan Liu, Dilip Arumugam, Cedegao E. Zhang, Sean Escola, Xaq Pitkow, Thomas L. Griffiths
While contemporary large language models (LLMs) are increasingly capable in isolation, there are still many difficult problems that lie beyond the abilities of a single LLM. For such tasks, there is still uncertainty about how best to take many LLMs as parts and combine them into a greater whole. This position paper argues that potential blueprints for designing such modular language agents can be found in the existing literature on cognitive models and artificial intelligence (AI) algorithms. To make this point clear, we formalize the idea of an agent template that specifies roles for individual LLMs and how their functionalities should be composed. We then survey a variety of existing language agents in the literature and highlight their underlying templates derived directly from cognitive models or AI algorithms. By highlighting these designs, we aim to call attention to agent templates inspired by cognitive science and AI as a powerful tool for developing effective, interpretable language agents.
A. Emin Orhan, Xaq Pitkow
Training recurrent neural networks (RNNs) is a hard problem due to degeneracies in the optimization landscape, a problem also known as vanishing/exploding gradients. Short of designing new RNN architectures, previous methods for dealing with this problem usually boil down to orthogonalization of the recurrent dynamics, either at initialization or during the entire training period. The basic motivation behind these methods is that orthogonal transformations are isometries of the Euclidean space, hence they preserve (Euclidean) norms and effectively deal with vanishing/exploding gradients. However, this ignores the crucial effects of non-linearity and noise. In the presence of a non-linearity, orthogonal transformations no longer preserve norms, suggesting that alternative transformations might be better suited to non-linear networks. Moreover, in the presence of noise, norm preservation itself ceases to be the ideal objective. A more sensible objective is maximizing the signal-to-noise ratio (SNR) of the propagated signal instead. Previous work has shown that in the linear case, recurrent networks that maximize the SNR display strongly non-normal, sequential dynamics and orthogonal networks are highly suboptimal by this measure. Motivated by this finding, here we investigate the potential of non-normal RNNs, i.e. RNNs with a non-normal recurrent connectivity matrix, in sequential processing tasks. Our experimental results show that non-normal RNNs outperform their orthogonal counterparts in a diverse range of benchmarks. We also find evidence for increased non-normality and hidden chain-like feedforward motifs in trained RNNs initialized with orthogonal recurrent connectivity matrices.
Kijung Yoon, Emin Orhan, Juhyun Kim, Xaq Pitkow
Neurons in the brain are complex machines with distinct functional compartments that interact nonlinearly. In contrast, neurons in artificial neural networks abstract away this complexity, typically down to a scalar activation function of a weighted sum of inputs. Here we emulate more biologically realistic neurons by learning canonical activation functions with two input arguments, analogous to basal and apical dendrites. We use a network-in-network architecture where each neuron is modeled as a multilayer perceptron with two inputs and a single output. This inner perceptron is shared by all units in the outer network. Remarkably, the resultant nonlinearities often produce soft XOR functions, consistent with recent experimental observations about interactions between inputs in human cortical neurons. When hyperparameters are optimized, networks with these nonlinearities learn faster and perform better than conventional ReLU nonlinearities with matched parameter counts, and they are more robust to natural and adversarial perturbations.