Kernel-based Methods for Bandit Convex Optimization

We consider the adversarial convex bandit problem and we build the first poly( T )-time algorithm with poly( n ) √ T -regret for this problem. To do so, we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves Õ( n 9.5 √ T )-regret, and we show that a simple variant of this algorithm can be run in poly( n log ( T ))-time per step (for polytopes with polynomially many constraints) at the cost of an additional poly( n ) T o(1) factor in the regret. These results improve upon the Õ( n 11 √ T -regret and exp (poly( T ))-time result of the first two authors and the log ( T ) poly( n ) √ T -regret and log( T ) poly( n ) -time result of Hazan and Li. Furthermore, we conjecture that another variant of the algorithm could achieve Õ( n 1.5 √ T )-regret, and moreover that this regret is unimprovable (the current best lower bound being Ω ( n √ T ) and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order n 3 / ɛ 2 .

Conjugates of Pisot numbers

In this paper, we investigate the Galois conjugates of a Pisot number [Formula: see text], [Formula: see text]. In particular, we conjecture that for [Formula: see text] we have [Formula: see text] for all conjugates [Formula: see text] of [Formula: see text]. Further, for [Formula: see text], we conjecture that for all Pisot numbers [Formula: see text] we have [Formula: see text]. A similar conjecture if made for [Formula: see text]. We conjecture that all of these bounds are tight. We provide partial supporting evidence for this conjecture. This evidence is both of a theoretical and computational nature. Lastly, we connect this conjecture to a result on the dimension of Bernoulli convolutions parameterized by [Formula: see text], whose conjugate is the reciprocal of a Pisot number.

Lebesgue structure of asymmetric Bernoulli convolution based on Jacobsthal–Lucas sequence

We study the problem of deepening the Jessen–Wintner theorem for asymmetric Bernoulli convolutions. In particular, we investigate the Lebesgue structure of a random incomplete sum of series, whose terms are reciprocal to Jacobsthal–Lucas numbers.

LEBESGUE STRUCTURE OF DISTRIBUTION OF GENERALIZED BERNOULLI CONVOLUTIONS OF SPECIAL TYPE

The paper deals with the problem of deepening the Jessen-Wintner theorem for generalized Bernoulli convolutions of a special kind. The main attention is paid to the case when the terms of a random series acquire three values: 0, 1, 2. In the case when the probability that the term of a random series becomes 2 is 0, the corresponding generalized Bernoulli convolutions coincide with classic Bernoulli convolutions, which were actively studied domestic scientists (Pratsovyty M., Turbin G., Torbin G., Honcharenko Ya., Baranovsky O., Savchenko I. and others) as well as foreign researchers (Erdos P., Peres Y., Schlag W, Solomyak B., Albeverio S. and others). The problem of deepening the Jessen-Wintner theorem concerning the necessary and sufficient conditions for the distribution of a probably convergent random series with discrete additions to each of the three pure types, is extremely difficult to formulate and is not completely solved even for classical Bernoulli convolutions. The results of the study are a deepening in relation to the analysis of the Lebesgue structure of random series formed by s-expansions of real numbers. In the case when the corresponding Bernoulli convolution is generated by the sequence 3-n, we have a random variable with independent triple digits, which was studied by scientists in different directions: Lebesgue structure (Chaterji S., Marsaglia G.), topological-metric structure of the distribution spectrum (Pratsovityi M., Turbin G.), fractal analysis of the distribution carrier (Pratsovyty M., Torbin G.), asymptotic properties of the characteristic function at infinity (Honcharenko Ya., Pratsovyty M., Torbin G.). The paper presents certain sufficient conditions for the absolute continuity and singularity of the distribution, with certain restrictions on the stochastic distribution matrix and the asymptotics of the values of the random terms of the series. In the case when the Lebesgue measure of the set of realizations of the generalized Bernoulli convolution is different from zero, it is possible together with Levy's theorem to formulate criteria for belonging of the Bernoulli convolution distribution to each of the three pure Lebesgue types, namely: purely discrete, purely continuous or purely singular.

Lower Assouad Dimension of Measures and Regularity

In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the infimum of the measure’s lower local dimensions. Although strict inequality is possible in general, equality holds for the class of self-similar measures of finite type. This class includes all self-similar, equicontractive measures satisfying the open set condition, as well as certain “overlapping” self-similar measures, such as Bernoulli convolutions with contraction factors that are inverses of Pisot numbers. We give lower bounds for the lower Assouad dimension for measures arising from a Moran construction, prove that self-affine measures are uniformly perfect and have positive lower Assouad dimension, prove that the Assouad spectrum of a measure converges to its quasi-Assouad dimension and show that coincidence of the upper and lower Assouad dimension of a measure does not imply that the measure is s-regular.