Convergence of Unregularized Online Learning Algorithms

In this paper we study the convergence of online gradient descent algorithms in reproducing kernel Hilbert spaces (RKHSs) without regularization. We establish a sufficient condition and a necessary condition for the convergence of excess generalization errors in expectation. A sufficient condition for the almost sure convergence is also given. With high probability, we provide explicit convergence rates of the excess generalization errors for both averaged iterates and the last iterate, which in turn also imply convergence rates with probability one. To our best knowledge, this is the first high-probability convergence rate for the last iterate of online gradient descent algorithms without strong convexity. Without any boundedness assumptions on iterates, our results are derived by a novel use of two measures of the algorithm's one-step progress, respectively by generalization errors and by distances in RKHSs, where the variances of the involved martingales are cancelled out by the descent property of the algorithm

Heterogeneity, Bounded Rationality and Market Dysfunctionality

As the main building blocks of the modern finance theory, homogeneity and rational expectation have faced difficulty in explaining many market anomalies, stylized factors, and market inefficiency in empirical studies. As a result, heterogeneity and bounded rationality have been used as an alterative paradigm of asset price dynamics and this paradigm has been widely recognized recently in both academic and financial market practitioners. Within the framework of Chiarella, Dieci and He (2006a, 2006b) on mean-variance analysis under heterogeneous beliefs in terms of either the payoffs or returns of the risky assets, this paper examines the effect of the heterogeneity. We first demonstrate that, in market equilibrium, the standard one fund theorem under homogeneous belief does not held under heterogeneous belief in general, however, the optimal portfolios of investors are very close to the market efficient frontier. By imposing certain distribution assumption on the heterogeneous beliefs, we then use Monte Carlo simulations to show that certain heterogeneity among investors can improve the Sharpe and Treynor ratios of the portfolios and investors can benefit from the diversity in investors’ beliefs. We also show that non-normality of market equilibrium return distributions is an outcome of the market aggregation of individual investors who make rational decisions based on their beliefs. Our results explain the empirical funding that that managed funds under-perform the market index on average and show that heterogeneity can improve the market efficiency.heterogeneity; bounded rationality; heterogeneous CAPM; mean-variance efficiency; Sharpe and Treynor ratios

Differences in Opinion and Risk Premium

When people agree to disagree, this paper examines the impact of the disagreement among agents on market equilibrium and equity premium. Within the standard mean variance framework, we consider a market of two risky assets, a riskless asset and two (and then a continuum of) agents who have different preferences and heterogeneous beliefs in the means and variance/covariances of the asset returns. By constructing a consensus belief, we introduce a boundedly rational equilibrium (BRE) to characterize the market equilibrium and derive a CAPM under heterogeneous beliefs. When the differences in opinion are formed as mean-preserving spreads of a benchmark homogeneous belief, we analyz eexplicitly the impact on the market equilibrium and risk premium, showing that the risk tolerance, optimism/pessimism and con?dence/doubt can jointly generate high risk premium and low risk-free rate. JELClassi?cation:.Assetprices;heterogeneousbeliefs;boundedlyrationalequilibriuasset prices; heterogeneous beliefs; boundedly rational equilibrium; zero-beta CAPM; risk premium

Portfolio Analysis and Zero-Beta CAPM with Heterogeneous Beliefs

With the standard mean variance framework, by assuming heterogeneity and bounded rationality of investors, this paper examines their impact on the market equilibrium and implications to the portfolio analysis. By constructing a market consensus belief, we establish market equilibrium prices of risky assets and show that the standard Black’s zero-beta CAPM under homogeneous beliefs holds under the heterogeneous belief. We demonstrate that the biased belief (from the market consensus belief) of investors makes their optimal portfolio not necessarily locate on the market mean-variance frontier. We show that the traditional geometric relation of the mean variance frontiers with and without the riskless asset under the homogeneous beliefs does not hold under the heterogeneous beliefs. The results shed light on the risk premium puzzle, Miller’s hypothesis, the lower market performance when the access to the riskfree asset is impossible, and the empirical finding that managed funds under-perform comparing to the market indices on average.asset prices; heterogeneous beliefs; portfolio analysis; zero-beta CAPM