126 research outputs found
Effect of spin-orbit interaction on the critical temperature of an ideal Bose gas
We consider Bose-Einstein condensation of an ideal bose gas with an equal
mixture of `Rashba' and `Dresselhaus' spin-orbit interactions and study its
effect on the critical temperature.
In uniform bose gas a `cusp' and a sharp drop in the critical temperature
occurs due to the change in the density of states at a critical Raman coupling
where the degeneracy of the ground states is lifted. Relative drop in the
critical temperature depends on the diluteness of the gas as well as on the
spin-orbit coupling strength. In the presence of a harmonic trap, the cusp in
the critical temperature smoothened out and a minimum appears. Both the drop in
the critical temperature and lifting of `quasi-degeneracy' of the ground states
exhibit crossover phenomena which is controlled by the trap frequency. By
considering a 'Dicke' like model we extend our calculation to bosons with large
spin and observe a similar minimum in the critical temperature near the
critical Raman frequency, which becomes deeper for larger spin. Finally in the
limit of infinite spin, the critical temperature vanishes at the critical
frequency, which is a manifestation of Dicke type quantum phase transition.Comment: 9 pages, 6 figure
Misspecified Linear Bandits
We consider the problem of online learning in misspecified linear stochastic
multi-armed bandit problems. Regret guarantees for state-of-the-art linear
bandit algorithms such as Optimism in the Face of Uncertainty Linear bandit
(OFUL) hold under the assumption that the arms expected rewards are perfectly
linear in their features. It is, however, of interest to investigate the impact
of potential misspecification in linear bandit models, where the expected
rewards are perturbed away from the linear subspace determined by the arms
features. Although OFUL has recently been shown to be robust to relatively
small deviations from linearity, we show that any linear bandit algorithm that
enjoys optimal regret performance in the perfectly linear setting (e.g., OFUL)
must suffer linear regret under a sparse additive perturbation of the linear
model. In an attempt to overcome this negative result, we define a natural
class of bandit models characterized by a non-sparse deviation from linearity.
We argue that the OFUL algorithm can fail to achieve sublinear regret even
under models that have non-sparse deviation.We finally develop a novel bandit
algorithm, comprising a hypothesis test for linearity followed by a decision to
use either the OFUL or Upper Confidence Bound (UCB) algorithm. For perfectly
linear bandit models, the algorithm provably exhibits OFULs favorable regret
performance, while for misspecified models satisfying the non-sparse deviation
property, the algorithm avoids the linear regret phenomenon and falls back on
UCBs sublinear regret scaling. Numerical experiments on synthetic data, and on
recommendation data from the public Yahoo! Learning to Rank Challenge dataset,
empirically support our findings.Comment: Thirty-First AAAI Conference on Artificial Intelligence, 201
No-Regret Reinforcement Learning with Value Function Approximation: a Kernel Embedding Approach
We consider the regret minimization problem in reinforcement learning (RL) in
the episodic setting. In many real-world RL environments, the state and action
spaces are continuous or very large. Existing approaches establish regret
guarantees by either a low-dimensional representation of the stochastic
transition model or an approximation of the -functions. However, the
understanding of function approximation schemes for state-value functions
largely remains missing. In this paper, we propose an online model-based RL
algorithm, namely the CME-RL, that learns representations of transition
distributions as embeddings in a reproducing kernel Hilbert space while
carefully balancing the exploitation-exploration tradeoff. We demonstrate the
efficiency of our algorithm by proving a frequentist (worst-case) regret bound
that is of order , where is the
episode length, is the total number of time steps and is an
information theoretic quantity relating the effective dimension of the
state-action feature space. Our method bypasses the need for estimating
transition probabilities and applies to any domain on which kernels can be
defined. It also brings new insights into the general theory of kernel methods
for approximate inference and RL regret minimization
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