Label optimal regret bounds for online local learning

We resolve an open question from (Christiano, 2014b) posed in COLT'14 regarding the optimal dependency of the regret achievable for online local learning on the size of the label set. In this framework the algorithm is shown a pair of items at each step, chosen from a set of $n$ items. The learner then predicts a label for each item, from a label set of size $L$ and receives a real valued payoff. This is a natural framework which captures many interesting scenarios such as collaborative filtering, online gambling, and online max cut among others. (Christiano, 2014a) designed an efficient online learning algorithm for this problem achieving a regret of $O(\sqrt{nL^3T})$, where $T$ is the number of rounds. Information theoretically, one can achieve a regret of $O(\sqrt{n \log L T})$. One of the main open questions left in this framework concerns closing the above gap. In this work, we provide a complete answer to the question above via two main results. We show, via a tighter analysis, that the semi-definite programming based algorithm of (Christiano, 2014a), in fact achieves a regret of $O(\sqrt{nLT})$. Second, we show a matching computational lower bound. Namely, we show that a polynomial time algorithm for online local learning with lower regret would imply a polynomial time algorithm for the planted clique problem which is widely believed to be hard. We prove a similar hardness result under a related conjecture concerning planted dense subgraphs that we put forth. Unlike planted clique, the planted dense subgraph problem does not have any known quasi-polynomial time algorithms. Computational lower bounds for online learning are relatively rare, and we hope that the ideas developed in this work will lead to lower bounds for other online learning scenarios as well.Comment: 13 pages; Changes from previous version: small changes to proofs of Theorems 1 & 2, a small rewrite of introduction as well (this version is the same as camera-ready copy in COLT '15

Evaluating Asset Pricing Implications of DSGE Models

This paper conducts an econometric evaluation of structural macroeconomic asset pricing models. A one-sector dynamic stochastic general equilibrium model (DSGE) with habit formation and capital adjustment costs is considered. Based on the log-linearized DSGE model, a Gaussian probability model for the joint distribution of aggregate consumption, investment, and a vector of asset returns R(t) is specified. We facilitate the stochastic discount factor M(t) representation obtained from the DSGE model and impose the no-arbitrage condition E[M(t)R(t)|t-1]=1. In addition to the full general equilibrium model, we also consider consumption and production based partial equilibrium specifications, and a more general reference model. To evaluate the various asset pricing models we compute posterior model probabilities and loss function based measures of model adequacy.

Shadow Tomography of Quantum States

We introduce the problem of *shadow tomography*: given an unknown $D$-dimensional quantum mixed state $\rho$, as well as known two-outcome measurements $E_{1},\ldots,E_{M}$, estimate the probability that $E_{i}$ accepts $\rho$, to within additive error $\varepsilon$, for each of the $M$ measurements. How many copies of $\rho$ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only $\widetilde{O}\left( \varepsilon^{-4}\cdot\log^{4} M\cdot\log D\right)$ copies. This means, for example, that we can learn the behavior of an arbitrary $n$-qubit state, on all accepting/rejecting circuits of some fixed polynomial size, by measuring only $n^{O\left( 1\right)}$ copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, Brand\~ao et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.Comment: 29 pages, extended abstract appeared in Proceedings of STOC'2018, revised to give slightly better upper bound (1/eps^4 rather than 1/eps^5) and lower bounds with explicit dependence on the dimension

Navigating Central Path with Electrical Flows: from Flows to Matchings, and Back

We present an $\tilde{O}(m^{10/7})=\tilde{O}(m^{1.43})$-time algorithm for the maximum s-t flow and the minimum s-t cut problems in directed graphs with unit capacities. This is the first improvement over the sparse-graph case of the long-standing $O(m \min(\sqrt{m},n^{2/3}))$ time bound due to Even and Tarjan [EvenT75]. By well-known reductions, this also establishes an $\tilde{O}(m^{10/7})$-time algorithm for the maximum-cardinality bipartite matching problem. That, in turn, gives an improvement over the celebrated celebrated $O(m \sqrt{n})$ time bound of Hopcroft and Karp [HK73] whenever the input graph is sufficiently sparse

(How) Do the ECB and the Fed React to Financial Market Uncertainty?: The Taylor Rule in Times of Crisis

We assess differences that emerge in Taylor rule estimations for the Fed and the ECB before and after the start of the subprime crisis. For this purpose, we apply an explicit estimate of the equilibrium real interest rate and of potential output in order to account for variations within these variables over time. We argue that measures of money and credit growth, interest rate spreads and asset price inflation should be added to the classical Taylor rule because these variables are proxies of a change in the equilibrium interest rate and are, thus, also ikely to have played a major role in setting policy rates during the crisis. Our empirical results gained from a state-space model and GMM estimations reveal that, as far as the Fed is concerned, the impact of consumer price inflation, and money and credit growth turns negative during the crisis while the sign of the asset price inflation coefficient turns positive. Thus we are able to establish significant differences in the parameters of the reaction functions of the Fed before and after the start of the subprime crisis. In case of the ECB, there is no evidence of a change in signs. Instead, the positive reaction to credit growth, consumer and house price inflation becomes even stronger than before. Moreover we find evidence of a less inertial policy of both the Fed and the ECB during the crisis.Subprime crisis, Federal Reserve, European Central Bank, equilibrium real interest rate, Taylor rule

Virtual to Real Reinforcement Learning for Autonomous Driving

Reinforcement learning is considered as a promising direction for driving policy learning. However, training autonomous driving vehicle with reinforcement learning in real environment involves non-affordable trial-and-error. It is more desirable to first train in a virtual environment and then transfer to the real environment. In this paper, we propose a novel realistic translation network to make model trained in virtual environment be workable in real world. The proposed network can convert non-realistic virtual image input into a realistic one with similar scene structure. Given realistic frames as input, driving policy trained by reinforcement learning can nicely adapt to real world driving. Experiments show that our proposed virtual to real (VR) reinforcement learning (RL) works pretty well. To our knowledge, this is the first successful case of driving policy trained by reinforcement learning that can adapt to real world driving data

Price Stability and The ECB's Monetary Policy Strategy

This paper focuses on the price stability objective within the framework of the single monetary policy strategy. It starts by reviewing what this objective, which is common to all central banks, means. Secondly, this paper will focus exclusively on the anchoring of short- to medium-term inflation expectations (Part 2). Several measures show that this anchoring is effective. Modern New Keynesian theory is an appropriate framework for analysing the impact that this anchoring of expectations has on the determination of the short- to medium-term inflation rate. From this point of view, observed inflation in the euro area seems to be in line with the theory and the ECB's action seems to be very effective. Thirdly, we will focus on the other aspect of monetary stability: the degree of price-level uncertainty and the anchoring of inflation expectations in the medium to long term. Even though this assessment is more difficult than it is in the short to medium term, since we only have a track record covering five years, various indicators from the theoretical analysis paint a fairly reassuring picture of the effectiveness of the device used by the ECB.Monetary policy ; European Central Bank ; Inflation

Type VII Collagen Gene Mutations (c.8569G>T and c.4879G>A) Result in the Moderately Severe Phenotype of Recessive Dystrophic Epidermolysis Bullosa in a Korean Patient

Dystrophic epidermolysis bullosa (DEB) are caused by mutations in the COL7A1 gene, which encodes type VII collagen. Even though more than 500 different COL7A1 mutations have been identified in DEB, it still remains to be under-investigated. To investigate the mutation of COL7A1 in moderately severe phenotype of recessive DEB (RDEB) in a Korean patient, the mutation detection strategy was consisted of polymerase chain reaction (PCR) amplification of genomic DNA, followed by heteroduplex analysis, nucleotide sequencing of the PCR products demonstrating altered mobility. In this study, we found that one mutation (c.8569G>T) was detected within exon 116. The mutation of c.8569G>T in exon 116 changed the GAG (Glu) to TAG, eventually resulted in premature termination of type VII collagen polypeptide. Furthermore the mother did not have the mutation c.8569G>T in exon 116. The other novel mutation (c.4879G>A) was detected within exon 51 of both patient and mother, thereby resulting in changing valine (Val) to isoleucine (Ile) in type VII collagen polypeptide. Taken together, in this study we identified compound heterozygosity for COL7A1 mutations (c.8569G>T and c.4879G>A) in moderately severe RDEB in a Korean patient. We hope that this data contribute to the expanding database on COL7A1 mutations in DEB

Electrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs

We introduce a new approach to computing an approximately maximum s-t flow in a capacitated, undirected graph. This flow is computed by solving a sequence of electrical flow problems. Each electrical flow is given by the solution of a system of linear equations in a Laplacian matrix, and thus may be approximately computed in nearly-linear time. Using this approach, we develop the fastest known algorithm for computing approximately maximum s-t flows. For a graph having n vertices and m edges, our algorithm computes a (1-\epsilon)-approximately maximum s-t flow in time \tilde{O}(mn^{1/3} \epsilon^{-11/3}). A dual version of our approach computes a (1+\epsilon)-approximately minimum s-t cut in time \tilde{O}(m+n^{4/3}\eps^{-8/3}), which is the fastest known algorithm for this problem as well. Previously, the best dependence on m and n was achieved by the algorithm of Goldberg and Rao (J. ACM 1998), which can be used to compute approximately maximum s-t flows in time \tilde{O}(m\sqrt{n}\epsilon^{-1}), and approximately minimum s-t cuts in time \tilde{O}(m+n^{3/2}\epsilon^{-3})