Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction

Virtually all questions that one can ask about the behavioral and structural complexity of a stochastic process reduce to a linear algebraic framing of a time evolution governed by an appropriate hidden-Markov process generator. Each type of question---correlation, predictability, predictive cost, observer synchronization, and the like---induces a distinct generator class. Answers are then functions of the class-appropriate transition dynamic. Unfortunately, these dynamics are generically nonnormal, nondiagonalizable, singular, and so on. Tractably analyzing these dynamics relies on adapting the recently introduced meromorphic functional calculus, which specifies the spectral decomposition of functions of nondiagonalizable linear operators, even when the function poles and zeros coincide with the operator's spectrum. Along the way, we establish special properties of the projection operators that demonstrate how they capture the organization of subprocesses within a complex system. Circumventing the spurious infinities of alternative calculi, this leads in the sequel, Part II, to the first closed-form expressions for complexity measures, couched either in terms of the Drazin inverse (negative-one power of a singular operator) or the eigenvalues and projection operators of the appropriate transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht

Structural Drift: The Population Dynamics of Sequential Learning

We introduce a theory of sequential causal inference in which learners in a chain estimate a structural model from their upstream teacher and then pass samples from the model to their downstream student. It extends the population dynamics of genetic drift, recasting Kimura's selectively neutral theory as a special case of a generalized drift process using structured populations with memory. We examine the diffusion and fixation properties of several drift processes and propose applications to learning, inference, and evolution. We also demonstrate how the organization of drift process space controls fidelity, facilitates innovations, and leads to information loss in sequential learning with and without memory.Comment: 15 pages, 9 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/sdrift.ht

Explainable and Advisable Learning for Self-driving Vehicles

Deep neural perception and control networks are likely to be a key component of self-driving vehicles. These models need to be explainable - they should provide easy-to-interpret rationales for their behavior - so that passengers, insurance companies, law enforcement, developers, etc., can understand what triggered a particular behavior. Explanations may be triggered by the neural controller, namely introspective explanations, or informed by the neural controller's output, namely rationalizations. Our work has focused on the challenge of generating introspective explanations of deep models for self-driving vehicles. In Chapter 3, we begin by exploring the use of visual explanations. These explanations take the form of real-time highlighted regions of an image that causally influence the network's output (steering control). In the first stage, we use a visual attention model to train a convolution network end-to-end from images to steering angle. The attention model highlights image regions that potentially influence the network's output. Some of these are true influences, but some are spurious. We then apply a causal filtering step to determine which input regions actually influence the output. This produces more succinct visual explanations and more accurately exposes the network's behavior. In Chapter 4, we add an attention-based video-to-text model to produce textual explanations of model actions, e.g. "the car slows down because the road is wet". The attention maps of controller and explanation model are aligned so that explanations are grounded in the parts of the scene that mattered to the controller. We explore two approaches to attention alignment, strong- and weak-alignment. These explainable systems represent an externalization of tacit knowledge. The network's opaque reasoning is simplified to a situation-specific dependence on a visible object in the image. This makes them brittle and potentially unsafe in situations that do not match training data. In Chapter 5, we propose to address this issue by augmenting training data with natural language advice from a human. Advice includes guidance about what to do and where to attend. We present the first step toward advice-giving, where we train an end-to-end vehicle controller that accepts advice. The controller adapts the way it attends to the scene (visual attention) and the control (steering and speed). Further, in Chapter 6, we propose a new approach that learns vehicle control with the help of long-term (global) human advice. Specifically, our system learns to summarize its visual observations in natural language, predict an appropriate action response (e.g. "I see a pedestrian crossing, so I stop"), and predict the controls, accordingly

Informational and Causal Architecture of Discrete-Time Renewal Processes

Renewal processes are broadly used to model stochastic behavior consisting of isolated events separated by periods of quiescence, whose durations are specified by a given probability law. Here, we identify the minimal sufficient statistic for their prediction (the set of causal states), calculate the historical memory capacity required to store those states (statistical complexity), delineate what information is predictable (excess entropy), and decompose the entropy of a single measurement into that shared with the past, future, or both. The causal state equivalence relation defines a new subclass of renewal processes with a finite number of causal states despite having an unbounded interevent count distribution. We use these formulae to analyze the output of the parametrized Simple Nonunifilar Source, generated by a simple two-state hidden Markov model, but with an infinite-state epsilon-machine presentation. All in all, the results lay the groundwork for analyzing processes with infinite statistical complexity and infinite excess entropy.Comment: 18 pages, 9 figures, 1 table; http://csc.ucdavis.edu/~cmg/compmech/pubs/dtrp.ht

Above and Beyond the Landauer Bound: Thermodynamics of Modularity

Information processing typically occurs via the composition of modular units, such as universal logic gates. The benefit of modular information processing, in contrast to globally integrated information processing, is that complex global computations are more easily and flexibly implemented via a series of simpler, localized information processing operations which only control and change local degrees of freedom. We show that, despite these benefits, there are unavoidable thermodynamic costs to modularity---costs that arise directly from the operation of localized processing and that go beyond Landauer's dissipation bound for erasing information. Integrated computations can achieve Landauer's bound, however, when they globally coordinate the control of all of an information reservoir's degrees of freedom. Unfortunately, global correlations among the information-bearing degrees of freedom are easily lost by modular implementations. This is costly since such correlations are a thermodynamic fuel. We quantify the minimum irretrievable dissipation of modular computations in terms of the difference between the change in global nonequilibrium free energy, which captures these global correlations, and the local (marginal) change in nonequilibrium free energy, which bounds modular work production. This modularity dissipation is proportional to the amount of additional work required to perform the computational task modularly. It has immediate consequences for physically embedded transducers, known as information ratchets. We show how to circumvent modularity dissipation by designing internal ratchet states that capture the global correlations and patterns in the ratchet's information reservoir. Designed in this way, information ratchets match the optimum thermodynamic efficiency of globally integrated computations.Comment: 17 pages, 9 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/idolip.ht