arXiv subjects

David H. Wolpert

Publications and source records attributed to David H. Wolpert.

At least 37 records · Page 2Linked to original sources

arXiv2023

Stochastic Mathematical Systems

We introduce a framework that can be used to model both mathematics and human reasoning about mathematics. This framework involves {stochastic mathematical systems} (SMSs), which are stochastic processes that generate pairs of questions and associated answers (with no explicit referents). We use the SMS framework to define normative conditions for mathematical reasoning, by defining a ``calibration'' relation between a pair of SMSs. The first SMS is the human reasoner, and the second is an ``oracle'' SMS that can be interpreted as deciding whether the question-answer pairs of the reasoner SMS are valid. To ground thinking, we understand the answers to questions given by this oracle to be the answers that would be given by an SMS representing the entire mathematical community in the infinite long run of the process of asking and answering questions. We then introduce a slight extension of SMSs to allow us to model both the physical universe and human reasoning about the physical universe. We then define a slightly different calibration relation appropriate for the case of scientific reasoning. In this case the first SMS represents a human scientist predicting the outcome of future experiments, while the second SMS represents the physical universe in which the scientist is embedded, with the question-answer pairs of that SMS being specifications of the experiments that will occur and the outcome of those experiments, respectively. Next we derive conditions justifying two important patterns of inference in both mathematical and scientific reasoning: i) the practice of increasing one's degree of belief in a claim as one observes increasingly many lines of evidence for that claim, and ii) abduction, the practice of inferring a claim's probability of being correct from its explanatory power with respect to some other claim that is already taken to hold for independent reasons.

math.LO↗

arXiv2022

Combining lower bounds on entropy production in complex systems with multiple interacting components

The past two decades have seen a revolution in statistical physics, generalizing it to apply to systems of arbitrary size, evolving while arbitrarily far from equilibrium. Many of these new results are based on analyzing the dynamics of the entropy of a system that is evolving according to a Markov process. These results comprise a sub-field called ``stochastic thermodynamics''. Some of the most powerful results in stochastic thermodynamics were traditionally concerned with single, monolithic systems, evolving by themselves, ignoring any internal structure of those systems. In this chapter I review how in complex systems, composed of many interacting constituent systems, it is possible to substantially strengthen many of these traditional results of stochastic thermodynamics. This is done by ``mixing and matching'' those traditional results, to each apply to only a subset of the interacting systems, thereby producing a more powerful result at the level of the aggregate, complex system.

cond-mat.stat-mech↗

arXiv2022

The Implications of the No-Free-Lunch Theorems for Meta-induction

The important recent book by G. Schurz appreciates that the no-free-lunch theorems (NFL) have major implications for the problem of (meta) induction. Here I review the NFL theorems, emphasizing that they do not only concern the case where there is a uniform prior -- they prove that there are "as many priors" (loosely speaking) for which any induction algorithm $A$ out-generalizes some induction algorithm $B$ as vice-versa. Importantly though, in addition to the NFL theorems, there are many {free lunch} theorems. In particular, the NFL theorems can only be used to compare the {marginal} expected performance of an induction algorithm $A$ with the marginal expected performance of an induction algorithm $B$. There is a rich set of free lunches which instead concern the statistical correlations among the generalization errors of induction algorithms. As I describe, the meta-induction algorithms that Schurz advocate as a "solution to Hume's problem" are just an example of such a free lunch based on correlations among the generalization errors of induction algorithms. I end by pointing out that the prior that Schurz advocates, which is uniform over bit frequencies rather than bit patterns, is contradicted by thousands of experiments in statistical physics and by the great success of the maximum entropy procedure in inductive inference.

cs.LG↗

arXiv2022

Dependence of integrated, instantaneous, and fluctuating entropy production on the initial state in quantum and classical processes

We consider the additional entropy production (EP) incurred by a fixed quantum or classical process on some initial state $ρ$, above the minimum EP incurred by the same process on any initial state. We show that this additional EP, which we term the "mismatch cost of $ρ$", has a universal information-theoretic form: it is given by the contraction of the relative entropy between $ρ$ and the least-dissipative initial state $φ$ over time. We derive versions of this result for integrated EP incurred over the course of a process, for trajectory-level fluctuating EP, and for instantaneous EP rate. We also show that mismatch cost for fluctuating EP obeys an integral fluctuation theorem. Our results demonstrate a fundamental relationship between "thermodynamic irreversibility" (generation of EP) and "logical irreversibility" (inability to know the initial state corresponding to a given final state). We use this relationship to derive quantitative bounds on the thermodynamics of quantum error correction and to propose a thermodynamically-operationalized measure of the logical irreversibility of a quantum channel. Our results hold for both finite and infinite dimensional systems, and generalize beyond EP to many other thermodynamic costs, including nonadiabatic EP, free energy loss, and entropy gain.

quant-ph↗

arXiv2021

The Past as a Stochastic Process

Historical processes manifest remarkable diversity. Nevertheless, scholars have long attempted to identify patterns and categorize historical actors and influences with some success. A stochastic process framework provides a structured approach for the analysis of large historical datasets that allows for detection of sometimes surprising patterns, identification of relevant causal actors both endogenous and exogenous to the process, and comparison between different historical cases. The combination of data, analytical tools and the organizing theoretical framework of stochastic processes complements traditional narrative approaches in history and archaeology.

stat.AP↗

arXiv2021

Thermodynamic speed limits for co-evolving systems

Previously derived "global" thermodynamic speed limit theorems state that increasing the maximum speed with which a system can evolve between two given probability distributions over its states requires the system to produce more entropy in its evolution. However, these theorems ignore that many systems are not monolithic, but instead comprise multiple subsystems that interact according to an (often sparse) network. Indeed, most naturally-occurring and human-engineered systems of increasing complexity can be decomposed into sets of co-evolving subsystems, where there exist a priori constraints on the dynamics of each subsystem, restricting which other subsystems can affect its dynamics. Here we derive three new SLTs that account for the thermodynamic effects of such constraints. Our first new speed limit strengthens the global speed limit. While our other two SLTs do not have this guarantee, in some situations they are even stronger than our first speed limit. Our results establish that a stochastically evolving system will, on average, produce more entropy in evolving between two distributions within a given time simply due to its comprising multiple, co-evolving subsystems. We illustrate our results with numerical calculations involving a model of two cells sensing and storing information about their environment.

cond-mat.stat-mech↗

arXiv2021

Work, entropy production, and thermodynamics of information under protocol constraints

In many real-world situations, there are constraints on the ways in which a physical system can be manipulated. We investigate the entropy production (EP) and extractable work involved in bringing a system from some initial distribution $p$ to some final distribution $p'$, given that the set of master equations available to the driving protocol obeys some constraints. We first derive general bounds on EP and extractable work, as well as a decomposition of the nonequilibrium free energy into an "accessible free energy" (which can be extracted as work, given a set of constraints) and an "inaccessible free energy" (which must be dissipated as EP). In a similar vein, we consider the thermodynamics of information in the presence of constraints, and decompose the information acquired in a measurement into "accessible" and "inaccessible" components. This decomposition allows us to consider the thermodynamic efficiency of different measurements of the same system, given a set of constraints. We use our framework to analyze protocols subject to symmetry, modularity, and coarse-grained constraints, and consider various examples including the Szilard box, the 2D Ising model, and a multi-particle flashing ratchet.

cond-mat.stat-mech↗

arXiv2021

Entropy production given constraints on the energy functions

We consider the problem of driving a finite-state classical system from some initial distribution $p$ to some final distribution $p'$ with vanishing entropy production (EP), under the constraint that the driving protocols can only use some limited set of energy functions $\mathcal{E}$. Assuming no other constraints on the driving protocol, we derive a simple condition that guarantees that such a transformation can be carried out, which is stated in terms of the smallest probabilities in $\{p,p'\}$ and a graph-theoretic property defined in terms of $\mathcal{E}$. Our results imply that a surprisingly small amount of control over the energy function is sufficient (in particular, any transformation $p\to p'$ can be carried out as soon as one can control some one-dimensional parameter of the energy function, e.g., the energy of a single state). We also derive a lower bound on the EP under more general constraints on the transition rates, which is formulated in terms of a convex optimization problem.

cond-mat.stat-mech↗

arXiv2021

Fluctuation theorems for multiple co-evolving systems

All previously derived thermodynamic fluctuation theorems (FTs) that concern multiple co-evolving systems have required that each system can only change its state during an associated pre-fixed, limited set of time intervals. However, in many real-world cases the times when systems change their states are randomly determined, e.g., in almost all biological examples of co-evolving systems. Such randomness in the timing drastically modifies the thermodynamics. Here I derive FTs that apply whether or not the timing is random. These FTs provide new versions of the second law, and of all conventional thermodynamic uncertainty relations (TURs). These new results are often stronger than the conventional versions, which ignore how an overall system may decompose into a set of co-evolving systems. In addition, the new TURs often bound entropy production (EP) of the overall system even if none of the criteria for a conventional TUR (e.g., being a non-equilibrium steady state) hold for that overall system.

cond-mat.stat-mech↗

arXiv2021

Thermodynamic Uncertainty Relations for Multipartite Processes

The thermodynamic uncertainty relations (TURs) provide lower bounds on the entropy production (EP) of a system in terms of the statistical precision of an arbitrary current in that system. All conventional TURs derived so far have concerned a single physical system, differing from one another in what properties they require the system to have. However, many physical scenarios of interest involve multiple interacting systems, e.g. organelles within a biological cell. Here we show how to extend the conventional TURs to those scenarios. A common feature of these extended versions of the TURs is that they bound the global EP, jointly generated by the set of interacting systems, in terms of a weighted sum of the precisions of the local currents generated within those systems -- plus an information-theoretic correction term. Importantly, these extended TURs can bound the global EP even when the global system does not meet any of the requirements of the conventional TURs. After deriving these extended TURs we use them to obtain bounds that do not involve the global EP, but instead relate the local EPs of the individual systems and the statistical coupling among the currents generated within those systems. We derive such bounds for both scalar-valued and vector-valued currents within each system. We illustrate our results with numerical experiments.

cond-mat.stat-mech↗

arXiv2021

Thermodynamics of nonequilibrium systems with uncertain parameters

In the real world, one almost never knows the parameters of a thermodynamic process to infinite precision. Reflecting this, here we investigate how to extend stochastic thermodynamics to systems with uncertain parameters, including uncertain number of heat baths / particle reservoirs, uncertainty in the precise values of temperatures / chemical potentials of those reservoirs, uncertainty in the energy spectrum, uncertainty in the control protocol, etc. We formalize such uncertainty with an (arbitrary) probability measure over all transition rate matrices satisfying local detailed balance. This lets us define the effective thermodynamic quantities by averaging over all LDB-obeying rate matrices. We show that the resultant effective entropy violates the second law of thermodynamics. In contrast to the effective entropy though, the expected stochastic entropy, defined as the ensemble average of the effective trajectory-level entropy, satisfies the second law. We then and explicitly calculate the second-order correction to the second law for the case of one heat bath with uncertain temperature. We also derive the detailed fluctuation theorem for expected effective trajectory entropy production for this case, and derive a lower bound for the associated expected work. Next, to ground these formal considerations with experimentally testable bounds on allowed energetics, we derive a bound on the maximal work that can be extracted from systems with arbitrarily uncertain temperature. We end by extending previous work on "thermodynamic value of information", to allow for uncertainty in the time-evolution of the rate matrix.

cond-mat.stat-mech↗

arXiv2020

Stochastic thermodynamics and fluctuation theorems for non-linear systems

We extend stochastic thermodynamics by relaxing the two assumptions that the Markovian dynamics must be linear and that the equilibrium distribution must be a Boltzmann distribution. We show that if we require the second law to hold when those assumptions are relaxed, then it cannot be formulated in terms of Shannon entropy. However, thermodynamic consistency is salvaged if we reformulate the second law in terms of generalized entropy; our first result is an equation relating the precise form of the non-linear master equation to the precise associated generalized entropy which results in thermodynamic consistency. We then build on this result to extend the usual trajectory-level definitions of thermodynamic quantities that are appropriate even when the two assumptions are relaxed. We end by using these trajectory-level definitions to derive extended versions of the Crooks fluctuation theorem and Jarzynski equality which apply when the two assumptions are relaxed.

cond-mat.stat-mech↗

arXiv2020

Noisy Deductive Reasoning: How Humans Construct Math, and How Math Constructs Universes

We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, on our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a matter of certainty, but is instead governed by a probability distribution. We then show that this framework gives a compelling account of several aspects of mathematical practice. These include: 1) the way in which mathematicians generate research programs, 2) the applicability of Bayesian models of mathematical heuristics, 3) the role of abductive reasoning in mathematics, 4) the way in which multiple proofs of a proposition can strengthen our degree of belief in that proposition, and 5) the nature of the hypothesis that there are multiple formal systems that are isomorphic to physically possible universes. Thus, by embracing a model of mathematics as not perfectly predictable, we generate a new and fruitful perspective on the epistemology and practice of mathematics.

math.LO↗

arXiv2020

Thermodynamic costs of Turing Machines

Turing Machines (TMs) are the canonical model of computation in computer science and physics. We combine techniques from algorithmic information theory and stochastic thermodynamics to analyze the thermodynamic costs of TMs. We consider two different ways of realizing a given TM with a physical process. The first realization is designed to be thermodynamically reversible when fed with random input bits. The second realization is designed to generate less heat, up to an additive constant, than any realization that is computable (i.e., consistent with the physical Church-Turing thesis). We consider three different thermodynamic costs: the heat generated when the TM is run on each input (which we refer to as the "heat function"), the minimum heat generated when a TM is run with an input that results in some desired output (which we refer to as the "thermodynamic complexity" of the output, in analogy to the Kolmogorov complexity), and the expected heat on the input distribution that minimizes entropy production. For universal TMs, we show for both realizations that the thermodynamic complexity of any desired output is bounded by a constant (unlike the conventional Kolmogorov complexity), while the expected amount of generated heat is infinite. We also show that any computable realization faces a fundamental tradeoff between heat generation, the Kolmogorov complexity of its heat function, and the Kolmogorov complexity of its input-output map. We demonstrate this tradeoff by analyzing the thermodynamics of erasing a long string.

cond-mat.stat-mech↗

arXiv2020

What is important about the No Free Lunch theorems?

The No Free Lunch theorems prove that under a uniform distribution over induction problems (search problems or learning problems), all induction algorithms perform equally. As I discuss in this chapter, the importance of the theorems arises by using them to analyze scenarios involving {non-uniform} distributions, and to compare different algorithms, without any assumption about the distribution over problems at all. In particular, the theorems prove that {anti}-cross-validation (choosing among a set of candidate algorithms based on which has {worst} out-of-sample behavior) performs as well as cross-validation, unless one makes an assumption -- which has never been formalized -- about how the distribution over induction problems, on the one hand, is related to the set of algorithms one is choosing among using (anti-)cross validation, on the other. In addition, they establish strong caveats concerning the significance of the many results in the literature which establish the strength of a particular algorithm without assuming a particular distribution. They also motivate a ``dictionary'' between supervised learning and improve blackbox optimization, which allows one to ``translate'' techniques from supervised learning into the domain of blackbox optimization, thereby strengthening blackbox optimization algorithms. In addition to these topics, I also briefly discuss their implications for philosophy of science.

cs.LG↗

arXiv2019

Nonlinear Information Bottleneck

Information bottleneck (IB) is a technique for extracting information in one random variable $X$ that is relevant for predicting another random variable $Y$. IB works by encoding $X$ in a compressed "bottleneck" random variable $M$ from which $Y$ can be accurately decoded. However, finding the optimal bottleneck variable involves a difficult optimization problem, which until recently has been considered for only two limited cases: discrete $X$ and $Y$ with small state spaces, and continuous $X$ and $Y$ with a Gaussian joint distribution (in which case optimal encoding and decoding maps are linear). We propose a method for performing IB on arbitrarily-distributed discrete and/or continuous $X$ and $Y$, while allowing for nonlinear encoding and decoding maps. Our approach relies on a novel non-parametric upper bound for mutual information. We describe how to implement our method using neural networks. We then show that it achieves better performance than the recently-proposed "variational IB" method on several real-world datasets.

cs.IT↗

arXiv2019

Number of hidden states needed to physically implement a given conditional distribution

We consider the problem of how to construct a physical process over a finite state space $X$ that applies some desired conditional distribution $P$ to initial states to produce final states. This problem arises often in the thermodynamics of computation and nonequilibrium statistical physics more generally (e.g., when designing processes to implement some desired computation, feedback controller, or Maxwell demon). It was previously known that some conditional distributions cannot be implemented using any master equation that involves just the states in $X$. However, here we show that any conditional distribution $P$ can in fact be implemented---if additional "hidden" states not in $X$ are available. Moreover, we show that it is always possible to implement $P$ in a thermodynamically reversible manner. We then investigate a novel cost of the physical resources needed to implement a given distribution $P$: the minimal number of hidden states needed to do so. We calculate this cost exactly for the special case where $P$ represents a single-valued function, and provide an upper bound for the general case, in terms of the nonnegative rank of $P$. These results show that having access to one extra binary degree of freedom, thus doubling the total number of states, is sufficient to implement any $P$ with a master equation in a thermodynamically reversible way, if there are no constraints on the allowed form of the master equation. (Such constraints can greatly increase the minimal needed number of hidden states.) Our results also imply that for certain $P$ that can be implemented without hidden states, having hidden states permits an implementation that generates less heat.

cond-mat.stat-mech↗

arXiv2019

Overview of Information Theory, Computer Science Theory, and Stochastic Thermodynamics for Thermodynamics of Computation

I give a quick overview of some of the theoretical background necessary for using modern non-equilibrium statistical physics to investigate the thermodynamics of computation. I first present some of the necessary concepts from information theory, and then introduce some of the most important types of computational machine considered in computer science theory. After this I present a central result from modern non-equilibrium statistical physics: an exact expression for the entropy flow out of a system undergoing a given dynamics with a given initial distribution over states. This central expression is crucial for analyzing how the total entropy flow out of a computer depends on its global structure, since that global structure determines the initial distributions into all of the computer's subsystems, and therefore (via the central expression) the entropy flows generated by all of those subsystems. I illustrate these results by analyzing some of the subtleties concerning the benefits that are sometimes claimed for implementing an irreversible computation with a reversible circuit constructed out of Fredkin gates.

cond-mat.stat-mech↗