A note on no-free-lunch theorem

The No-Free-Lunch theorem is an interesting and important theoretical result in machine learning. Based on philosophy of No-Free-Lunch theorem, we discuss extensively on the limitation of a data-driven approach in solving NP-hard problems.

Still No Free Lunches: The Price to Pay for Tighter PAC-Bayes Bounds

“No free lunch” results state the impossibility of obtaining meaningful bounds on the error of a learning algorithm without prior assumptions and modelling, which is more or less realistic for a given problem. Some models are “expensive” (strong assumptions, such as sub-Gaussian tails), others are “cheap” (simply finite variance). As it is well known, the more you pay, the more you get: in other words, the most expensive models yield the more interesting bounds. Recent advances in robust statistics have investigated procedures to obtain tight bounds while keeping the cost of assumptions minimal. The present paper explores and exhibits what the limits are for obtaining tight probably approximately correct (PAC)-Bayes bounds in a robust setting for cheap models.

Can You Hear the Shape of a Market? Geometric Arbitrage and Spectral Theory

Utilizing gauge symmetries, the Geometric Arbitrage Theory reformulates any asset model, allowing for arbitrage by means of a stochastic principal fibre bundle with a connection whose curvature measures the “instantaneous arbitrage capability”. The cash flow bundle is the associated vector bundle. The zero eigenspace of its connection Laplacian parameterizes all risk-neutral measures equivalent to the statistical one. A market satisfies the No-Free-Lunch-with-Vanishing-Risk (NFLVR) condition if and only if 0 is in the discrete spectrum of the Laplacian. The Jarrow–Protter–Shimbo theory of asset bubbles and their classification and decomposition extend to markets not satisfying the NFLVR. Euler’s characteristic of the asset nominal space and non-vanishing of the homology group of the cash flow bundle are both topological obstructions to NFLVR.

Guilds and Competition: A Response to Objections of Laissez Faire

This paper offers a critique of laissez-faire objections to the guild system from the perspective of Catholic social teaching a presents the economic reasons for a restoration of the guild system as a functional economic model. The laissez-faire argument is that 1) the free competition on the supply side is in the best interest of the consumer, 2) each member of the society is a consumer, 3) ergo: free competition on the supply side is a common good, i. e. a goal which the state should follow. The author argues that 1) unlike majority of goods and services, two goods are becoming scarcer as a result of free competition on the supply side: time and land, 2) consumers which prefer consumption of leisure time and land are worse off as a result of free competition on the supply side, 3) ergo: the conclusion that free competition on the supply side is always a common good is invalid. According to the author, leisure time and land are essential for a good operation of a well-functioning family which is essential for a well-functioning state. In this connection, the author contends that the primary goal and raison d’etre of the guilds is control of the entrance of new producers to the industry, so that the incumbents who want to pay just (family) wages and/or prefer leisure time necessary for operation of a well-functioning family are not forced to change their behavior by the competitive pressure of the newcomers. Besides, a control of the entrance will limit the pressure to lowering wages below the level of the just (family) wage. To the objection of “no free lunch” the author responds: yes, the employers are facing a trade-off. In exchange for a possibly high but uncertain profit margin they will receive a lower but certain profit margin. To the objection of allocational inefficiency and economic stagnation resulting from suppressing external innovations, the author responds: 1) an external innovator can have his discovery patented and instead of the transitory entrepreneurial profit enjoy the incomes from the licenses; 2) the competitive pressure is only one possible drive of the economic growth, another drive is the human laziness which is constant across the economic systems.

Coping With Plenitude: A Computational Approach to Selecting the Right Algorithm

Sociologists increasingly face choices among competing algorithms that represent reasonable approaches to the same task, with little guidance in choosing among them. We develop a strategy that uses simulated data to identify the conditions under which different methods perform well and applies what is learned from the simulations to predict which method will perform best on never-before-seen empirical data sets. We apply this strategy to a class of methods that group respondents to attitude surveys according to whether they share construals of a given domain. This allows us to identify the relative strengths and weaknesses of the methods we consider, including relational class analysis, correlational class analysis, and eight other such variants. Results support the “no free lunch” view that researchers should abandon the quest for one best algorithm in favor of matching algorithms to kinds of data for which each is most appropriate and provide direction on how to do so.