21,141 research outputs found

    Categorical invariance and structural complexity in human concept learning

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    An alternative account of human concept learning based on an invariance measure of the categorical\ud stimulus is proposed. The categorical invariance model (CIM) characterizes the degree of structural\ud complexity of a Boolean category as a function of its inherent degree of invariance and its cardinality or\ud size. To do this we introduce a mathematical framework based on the notion of a Boolean differential\ud operator on Boolean categories that generates the degrees of invariance (i.e., logical manifold) of the\ud category in respect to its dimensions. Using this framework, we propose that the structural complexity\ud of a Boolean category is indirectly proportional to its degree of categorical invariance and directly\ud proportional to its cardinality or size. Consequently, complexity and invariance notions are formally\ud unified to account for concept learning difficulty. Beyond developing the above unifying mathematical\ud framework, the CIM is significant in that: (1) it precisely predicts the key learning difficulty ordering of\ud the SHJ [Shepard, R. N., Hovland, C. L.,&Jenkins, H. M. (1961). Learning and memorization of classifications.\ud Psychological Monographs: General and Applied, 75(13), 1-42] Boolean category types consisting of three\ud binary dimensions and four positive examples; (2) it is, in general, a good quantitative predictor of the\ud degree of learning difficulty of a large class of categories (in particular, the 41 category types studied\ud by Feldman [Feldman, J. (2000). Minimization of Boolean complexity in human concept learning. Nature,\ud 407, 630-633]); (3) it is, in general, a good quantitative predictor of parity effects for this large class of\ud categories; (4) it does all of the above without free parameters; and (5) it is cognitively plausible (e.g.,\ud cognitively tractable)

    Approximate resilience, monotonicity, and the complexity of agnostic learning

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    A function ff is dd-resilient if all its Fourier coefficients of degree at most dd are zero, i.e., ff is uncorrelated with all low-degree parities. We study the notion of approximate\mathit{approximate} resilience\mathit{resilience} of Boolean functions, where we say that ff is α\alpha-approximately dd-resilient if ff is α\alpha-close to a [−1,1][-1,1]-valued dd-resilient function in ℓ1\ell_1 distance. We show that approximate resilience essentially characterizes the complexity of agnostic learning of a concept class CC over the uniform distribution. Roughly speaking, if all functions in a class CC are far from being dd-resilient then CC can be learned agnostically in time nO(d)n^{O(d)} and conversely, if CC contains a function close to being dd-resilient then agnostic learning of CC in the statistical query (SQ) framework of Kearns has complexity of at least nΩ(d)n^{\Omega(d)}. This characterization is based on the duality between ℓ1\ell_1 approximation by degree-dd polynomials and approximate dd-resilience that we establish. In particular, it implies that ℓ1\ell_1 approximation by low-degree polynomials, known to be sufficient for agnostic learning over product distributions, is in fact necessary. Focusing on monotone Boolean functions, we exhibit the existence of near-optimal α\alpha-approximately Ω~(αn)\widetilde{\Omega}(\alpha\sqrt{n})-resilient monotone functions for all α>0\alpha>0. Prior to our work, it was conceivable even that every monotone function is Ω(1)\Omega(1)-far from any 11-resilient function. Furthermore, we construct simple, explicit monotone functions based on Tribes{\sf Tribes} and CycleRun{\sf CycleRun} that are close to highly resilient functions. Our constructions are based on a fairly general resilience analysis and amplification. These structural results, together with the characterization, imply nearly optimal lower bounds for agnostic learning of monotone juntas

    A Survey of Quantum Learning Theory

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    This paper surveys quantum learning theory: the theoretical aspects of machine learning using quantum computers. We describe the main results known for three models of learning: exact learning from membership queries, and Probably Approximately Correct (PAC) and agnostic learning from classical or quantum examples.Comment: 26 pages LaTeX. v2: many small changes to improve the presentation. This version will appear as Complexity Theory Column in SIGACT News in June 2017. v3: fixed a small ambiguity in the definition of gamma(C) and updated a referenc

    The GIST of Concepts

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    A unified general theory of human concept learning based on the idea that humans detect invariance patterns in categorical stimuli as a necessary precursor to concept formation is proposed and tested. In GIST (generalized invariance structure theory) invariants are detected via a perturbation mechanism of dimension suppression referred to as dimensional binding. Structural information acquired by this process is stored as a compound memory trace termed an ideotype. Ideotypes inform the subsystems that are responsible for learnability judgments, rule formation, and other types of concept representations. We show that GIST is more general (e.g., it works on continuous, semi-continuous, and binary stimuli) and makes much more accurate predictions than the leading models of concept learning difficulty,such as those based on a complexity reduction principle (e.g., number of mental models,structural invariance, algebraic complexity, and minimal description length) and those based on selective attention and similarity (GCM, ALCOVE, and SUSTAIN). GIST unifies these two key aspects of concept learning and categorization. Empirical evidence from three\ud experiments corroborates the predictions made by the theory and its core model which we propose as a candidate law of human conceptual behavior

    Towards a Law of Invariance in Human Concept Learning

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    Invariance principles underlie many key theories in modern science. They provide the explanatory and predictive framework necessary for the rigorous study of natural phenomena ranging from the structure of crystals, to magnetism, to relativistic mechanics. Vigo (2008, 2009)introduced a new general notion and principle of invariance from which two parameter-free (ratio and exponential) models were derived to account for human conceptual behavior. Here we introduce a new parameterized \ud exponential “law” based on the same invariance principle. The law accurately predicts the subjective degree of difficulty that humans experience when learning different types of concepts. In addition, it precisely fits the data from a large-scale experiment which examined a total of 84 category structures across 10 category families (R-Squared =.97, p < .0001; r= .98, p < .0001). Moreover, it overcomes seven key challenges that had, hitherto, been grave obstacles for theories of concept learning

    A novel Boolean kernels family for categorical data

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    Kernel based classifiers, such as SVM, are considered state-of-the-art algorithms and are widely used on many classification tasks. However, this kind of methods are hardly interpretable and for this reason they are often considered as black-box models. In this paper, we propose a new family of Boolean kernels for categorical data where features correspond to propositional formulas applied to the input variables. The idea is to create human-readable features to ease the extraction of interpretation rules directly from the embedding space. Experiments on artificial and benchmark datasets show the effectiveness of the proposed family of kernels with respect to established ones, such as RBF, in terms of classification accuracy
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