Complexity-calibrated benchmarks for machine learning reveal when prediction algorithms succeed and mislead.

Recurrent neural networks are used to forecast time series in finance, climate, language, and from many other domains. Reservoir computers are a particularly easily trainable form of recurrent neural network. Recently, a "next-generation" reservoir computer was introduced in which the memory trace involves only a finite number of previous symbols. We explore the inherent limitations of finite-past memory traces in this intriguing proposal. A lower bound from Fano's inequality shows that, on highly non-Markovian processes generated by large probabilistic state machines, next-generation reservoir computers with reasonably long memory traces have an error probability that is at least ∼ 60 % higher than the minimal attainable error probability in predicting the next observation. More generally, it appears that popular recurrent neural networks fall far short of optimally predicting such complex processes. These results highlight the need for a new generation of optimized recurrent neural network architectures. Alongside this finding, we present concentration-of-measure results for randomly-generated but complex processes. One conclusion is that large probabilistic state machines-specifically, large ϵ -machines-are key to generating challenging and structurally-unbiased stimuli for ground-truthing recurrent neural network architectures.

From reinforcement learning to agency: Frameworks for understanding basal cognition.

Organisms play, explore, and mimic those around them. Is there a purpose to this behavior? Are organisms just behaving, or are they trying to achieve goals? We believe this is a false dichotomy. To that end, to understand organisms, we attempt to unify two approaches for understanding complex agents, whether evolved or engineered. We argue that formalisms describing multiscale competencies and goal-directedness in biology (e.g., TAME), and reinforcement learning (RL), can be combined in a symbiotic framework. While RL has been largely focused on higher-level organisms and robots of high complexity, TAME is naturally capable of describing lower-level organisms and minimal agents as well. We propose several novel questions that come from using RL/TAME to understand biology as well as ones that come from using biology to formulate new theory in AI. We hope that the research programs proposed in this piece shape future efforts to understand biological organisms and also future efforts to build artificial agents.

Prediction in cultured cortical neural networks.

Theory suggest that networks of neurons may predict their input. Prediction may underlie most aspects of information processing and is believed to be involved in motor and cognitive control and decision-making. Retinal cells have been shown to be capable of predicting visual stimuli, and there is some evidence for prediction of input in the visual cortex and hippocampus. However, there is no proof that the ability to predict is a generic feature of neural networks. We investigated whether random in vitro neuronal networks can predict stimulation, and how prediction is related to short- and long-term memory. To answer these questions, we applied two different stimulation modalities. Focal electrical stimulation has been shown to induce long-term memory traces, whereas global optogenetic stimulation did not. We used mutual information to quantify how much activity recorded from these networks reduces the uncertainty of upcoming stimuli (prediction) or recent past stimuli (short-term memory). Cortical neural networks did predict future stimuli, with the majority of all predictive information provided by the immediate network response to the stimulus. Interestingly, prediction strongly depended on short-term memory of recent sensory inputs during focal as well as global stimulation. However, prediction required less short-term memory during focal stimulation. Furthermore, the dependency on short-term memory decreased during 20 h of focal stimulation, when long-term connectivity changes were induced. These changes are fundamental for long-term memory formation, suggesting that besides short-term memory the formation of long-term memory traces may play a role in efficient prediction.

How Well Can We Infer Selection Benefits and Mutation Rates from Allele Frequencies?

Experimentalists observe allele frequency distributions and try to infer mutation rates and selection coefficients. How easy is this? We calculate limits to their ability in the context of the Wright-Fisher model by first finding the maximal amount of information that can be acquired using allele frequencies about the mutation rate and selection coefficient- at least 2 bits per allele- and then by finding how the organisms would have shaped their mutation rates and selection coefficients so as to maximize the information transfer.

Not so optimal: The evolution of mutual information in potassium voltage-gated channels.

Potassium voltage-gated (Kv) channels need to detect and respond to rapidly changing ionic concentrations in their environment. With an essential role in regulating electric signaling, they would be expected to be optimal sensors that evolved to predict the ionic concentrations. To explore these assumptions, we use statistical mechanics in conjunction with information theory to model how animal Kv channels respond to changes in potassium concentrations in their environment. By measuring mutual information in representative Kv channel types across a variety of environments, we find two things. First, under weak conditions, there is a gating charge that maximizes mutual information with the environment. Second, as Kv channels evolved, they have moved towards decreasing mutual information with the environment. This either suggests that Kv channels do not need to act as sensors of their environment or that Kv channels have other functionalities that interfere with their role as sensors of their environment.

On the Role of Theory and Modeling in Neuroscience.

In recent years, the field of neuroscience has gone through rapid experimental advances and a significant increase in the use of quantitative and computational methods. This growth has created a need for clearer analyses of the theory and modeling approaches used in the field. This issue is particularly complex in neuroscience because the field studies phenomena that cross a wide range of scales and often require consideration at varying degrees of abstraction, from precise biophysical interactions to the computations they implement. We argue that a pragmatic perspective of science, in which descriptive, mechanistic, and normative models and theories each play a distinct role in defining and bridging levels of abstraction, will facilitate neuroscientific practice. This analysis leads to methodological suggestions, including selecting a level of abstraction that is appropriate for a given problem, identifying transfer functions to connect models and data, and the use of models themselves as a form of experiment.

Inference, Prediction, & Entropy-Rate Estimation of Continuous-Time, Discrete-Event Processes.

Inferring models, predicting the future, and estimating the entropy rate of discrete-time, discrete-event processes is well-worn ground. However, a much broader class of discrete-event processes operates in continuous-time. Here, we provide new methods for inferring, predicting, and estimating them. The methods rely on an extension of Bayesian structural inference that takes advantage of neural network's universal approximation power. Based on experiments with complex synthetic data, the methods are competitive with the state-of-the-art for prediction and entropy-rate estimation.

Maximum entropy models provide functional connectivity estimates in neural networks.

Tools to estimate brain connectivity offer the potential to enhance our understanding of brain functioning. The behavior of neuronal networks, including functional connectivity and induced connectivity changes by external stimuli, can be studied using models of cultured neurons. Cultured neurons tend to be active in groups, and pairs of neurons are said to be functionally connected when their firing patterns show significant synchronicity. Methods to infer functional connections are often based on pair-wise cross-correlation between activity patterns of (small groups of) neurons. However, these methods are not very sensitive to detect inhibitory connections, and they were not designed for use during stimulation. Maximum Entropy (MaxEnt) models may provide a conceptually different method to infer functional connectivity. They have the potential benefit to estimate functional connectivity during stimulation, and to infer excitatory as well as inhibitory connections. MaxEnt models do not involve pairwise comparison, but aim to capture probability distributions of sets of neurons that are synchronously active in discrete time bins. We used electrophysiological recordings from in vitro neuronal cultures on micro electrode arrays to investigate the ability of MaxEnt models to infer functional connectivity. Connectivity estimates provided by MaxEnt models correlated well with those obtained by conditional firing probabilities (CFP), an established cross-correlation based method. In addition, stimulus-induced connectivity changes were detected by MaxEnt models, and were of the same magnitude as those detected by CFP. Thus, MaxEnt models provide a potentially powerful new tool to study functional connectivity in neuronal networks.

Probabilistic Deterministic Finite Automata and Recurrent Networks, Revisited.

Reservoir computers (RCs) and recurrent neural networks (RNNs) can mimic any finite-state automaton in theory, and some workers demonstrated that this can hold in practice. We test the capability of generalized linear models, RCs, and Long Short-Term Memory (LSTM) RNN architectures to predict the stochastic processes generated by a large suite of probabilistic deterministic finite-state automata (PDFA) in the small-data limit according to two metrics: predictive accuracy and distance to a predictive rate-distortion curve. The latter provides a sense of whether or not the RNN is a lossy predictive feature extractor in the information-theoretic sense. PDFAs provide an excellent performance benchmark in that they can be systematically enumerated, the randomness and correlation structure of their generated processes are exactly known, and their optimal memory-limited predictors are easily computed. With less data than is needed to make a good prediction, LSTMs surprisingly lose at predictive accuracy, but win at lossy predictive feature extraction. These results highlight the utility of causal states in understanding the capabilities of RNNs to predict.

Choosing dynamical systems that predict weak input.

Somehow, our brain and other organisms manage to predict their environment. Behind this must be an input-dependent dynamical system, or recurrent neural network, whose present state reflects the history of environmental input. The design principles for prediction-in particular, what kinds of attractors allow for greater predictive capability-are still unknown. We offer some clues to design principles using an attractor picture when the environment perturbs the system's state weakly, motivating and developing some theory for continuous-time time-varying linear reservoirs along the way. Reservoirs that inherently support only stable fixed points are generically good predictors, while reservoirs with limit cycles are good predictors for noisy periodic input.

A Proposed Probabilistic Method for Distinguishing Between Delusions and Other Environmental Judgements, With Applications to Psychotherapy.

How can individuals with schizophrenia best be equipped to distinguish delusions from accurate judgements about their environment? This study presents an approach based on the principles of Bayesian probability and presents the results of a series of tests in which a simulated observer classifies randomly generated data characteristic of a simulated environment. The complexity of the data ranges from scalars to vectors of variable lengths, and the simulated observer makes its decisions based on either perfect or imperfect models of its environment. We find that when a low-dimensional observation is considered characteristic of both real observations and delusions, the prior probabilities of any observation being real or fake are of greater importance to the final decision than the attributes of the observation. However, when an observation is high-dimensional (complex), classification accuracy tends to improve toward 100% with increasing complexity of observations, as long as the patient's model of the world isn't drastically inaccurate. On the contrary, when the observer's model is sufficiently inaccurate, the accuracy rate decreases with increasing observational complexity. Overall, the results suggest applicability of the Bayesian model to the use of interventional therapy for those who suffer from psychosis.

Machine learning outperforms thermodynamics in measuring how well a many-body system learns a drive.

Diverse many-body systems, from soap bubbles to suspensions to polymers, learn and remember patterns in the drives that push them far from equilibrium. This learning may be leveraged for computation, memory, and engineering. Until now, many-body learning has been detected with thermodynamic properties, such as work absorption and strain. We progress beyond these macroscopic properties first defined for equilibrium contexts: We quantify statistical mechanical learning using representation learning, a machine-learning model in which information squeezes through a bottleneck. By calculating properties of the bottleneck, we measure four facets of many-body systems' learning: classification ability, memory capacity, discrimination ability, and novelty detection. Numerical simulations of a classical spin glass illustrate our technique. This toolkit exposes self-organization that eludes detection by thermodynamic measures: Our toolkit more reliably and more precisely detects and quantifies learning by matter while providing a unifying framework for many-body learning.

Inferring an Observer's Prediction Strategy in Sequence Learning Experiments.

Cognitive systems exhibit astounding prediction capabilities that allow them to reap rewards from regularities in their environment. How do organisms predict environmental input and how well do they do it? As a prerequisite to answering that question, we first address the limits on prediction strategy inference, given a series of inputs and predictions from an observer. We study the special case of Bayesian observers, allowing for a probability that the observer randomly ignores data when building her model. We demonstrate that an observer's prediction model can be correctly inferred for binary stimuli generated from a finite-order Markov model. However, we can not necessarily infer the model's parameter values unless we have access to several "clones" of the observer. As stimuli become increasingly complicated, correct inference requires exponentially more data points, computational power, and computational time. These factors place a practical limit on how well we are able to infer an observer's prediction strategy in an experimental or observational setting.

First-principles prediction of the information processing capacity of a simple genetic circuit.

Given the stochastic nature of gene expression, genetically identical cells exposed to the same environmental inputs will produce different outputs. This heterogeneity has been hypothesized to have consequences for how cells are able to survive in changing environments. Recent work has explored the use of information theory as a framework to understand the accuracy with which cells can ascertain the state of their surroundings. Yet the predictive power of these approaches is limited and has not been rigorously tested using precision measurements. To that end, we generate a minimal model for a simple genetic circuit in which all parameter values for the model come from independently published data sets. We then predict the information processing capacity of the genetic circuit for a suite of biophysical parameters such as protein copy number and protein-DNA affinity. We compare these parameter-free predictions with an experimental determination of protein expression distributions and the resulting information processing capacity of E. coli cells. We find that our minimal model captures the scaling of the cell-to-cell variability in the data and the inferred information processing capacity of our simple genetic circuit up to a systematic deviation.

Prediction and Dissipation in Nonequilibrium Molecular Sensors: Conditionally Markovian Channels Driven by Memoryful Environments.

Biological sensors must often predict their input while operating under metabolic constraints. However, determining whether or not a particular sensor is evolved or designed to be accurate and efficient is challenging. This arises partly from the functional constraints being at cross purposes and partly since quantifying the prediction performance of even in silico sensors can require prohibitively long simulations, especially when highly complex environments drive sensors out of equilibrium. To circumvent these difficulties, we develop new expressions for the prediction accuracy and thermodynamic costs of the broad class of conditionally Markovian sensors subject to complex, correlated (unifilar hidden semi-Markov) environmental inputs in nonequilibrium steady state. Predictive metrics include the instantaneous memory and the total predictable information (the mutual information between present sensor state and input future), while dissipation metrics include power extracted from the environment and the nonpredictive information rate. Success in deriving these formulae relies on identifying the environment's causal states, the input's minimal sufficient statistics for prediction. Using these formulae, we study large random channels and the simplest nontrivial biological sensor model-that of a Hill molecule, characterized by the number of ligands that bind simultaneously-the sensor's cooperativity. We find that the seemingly impoverished Hill molecule can capture an order of magnitude more predictable information than large random channels.

Time cells might be optimized for predictive capacity, not redundancy reduction or memory capacity.

Recently, researchers have found time cells in the hippocampus that appear to contain information about the timing of past events. Some researchers have argued that time cells are taking a Laplace transform of their input in order to reconstruct the past stimulus. We argue that stimulus prediction, not stimulus reconstruction or redundancy reduction, is in better agreement with observed responses of time cells. In the process, we introduce new analyses of nonlinear, continuous-time reservoirs that model these time cells.

Novelty detection improves performance of reinforcement learners in fluctuating, partially observable environments.

Evolved and engineered organisms must adapt to fluctuating environments that are often only partially observed. We show that adaptation to a second environment can be significantly harder after adapting to a first, completely unrelated environment, even when using second-order learning algorithms and a constant learning rate. In effect, there is a lack of fading memory in the organism's performance. However, organisms can adapt well to the second environment by incorporating a simple novelty detection algorithm that signals when the environment has changed and reinitializing the parameters that define their behavior if so. We propose that it may be fruitful to look for signs of this novelty detection in biological organisms, and to engineer novelty detection algorithms into artificial organisms.

Optimized bacteria are environmental prediction engines.

Experimentalists observe phenotypic variability even in isogenic bacteria populations. We explore the hypothesis that in fluctuating environments this variability is tuned to maximize a bacterium's expected log-growth rate, potentially aided by epigenetic (all inheritable nongenetic) markers that store information about past environments. Crucially, we assume a time delay between sensing and action, so that a past epigenetic marker is used to generate the present phenotypic variability. We show that, in a complex, memoryful environment, the maximal expected log-growth rate is linear in the instantaneous predictive information-the mutual information between a bacterium's epigenetic markers and future environmental states. Hence, under resource constraints, optimal epigenetic markers are causal states-the minimal sufficient statistics for prediction-or lossy approximations thereof. We propose new theoretical investigations into and new experiments on bacteria phenotypic bet-hedging in fluctuating complex environments.

Infinitely large, randomly wired sensors cannot predict their input unless they are close to deterministic.

Building predictive sensors is of paramount importance in science. Can we make a randomly wired sensor "good enough" at predicting its input simply by making it larger? We show that infinitely large, randomly wired sensors are nonspecific for their input, and therefore nonpredictive of future input, unless they are close to deterministic. Nearly deterministic, randomly wired sensors can capture ∼ 10% of the predictive information of their inputs for "typical" environments.