Expectiles are tail-sensitive distributional summaries with important applications in financial risk measurement and forecasting, yet statistical theory for their online estimation from dependent data remains limited. We study a one-pass, stochastic-gradient algorithm with constant learning rate γ for estimating the marginal expectile of a stationary causal process. The recursion admits a unique stationary causal solution and geometrically forgets its initialization. Under short-range dependence, its last iterate is asymptotically Gaussian at the γ scale, with variance governed by the population curvature and the long-run variance of the expectile score. For long-memory linear processes the scale changes to γβ−1/2ℓ(1/γ), with a Gaussian limit arising from a fractional Brownian motion integral. We further derive the joint long-memory limit for multiple expectile levels and characterize its limiting covariance structure. Various numerical experiments corroborate the predicted Gaussian approximations and regime-specific fluctuation scales.
Figures & tables
Figure 1: (a)–(d) Kolmogorov distance to the corresponding Gaussian limit along the expectile SGD trajectory for AR(1), GARCH (1,1) , nonlinear AR, and ARFIMA (0,0.25,0) processes at τ=0.9 . Each curve corresponds to a different learning rate γ . The shaded region shows the 95% Monte Carlo reference band for the estimated Kolmogorov distance under the N(0,1) reference, excluding calibration uncertainty. (e) Joint normalized SGD errors at τ=0.5,0.9 and γ=10−4 under ARFIMA (0,0.25,0) with 50% and 95% contours of the joint Gaussian limit. (f) Central 90% interval widths of the normalized centered contrast between τ=0.5 and 0.9 under ARFIMA (0,0.25,0) . Batch uses the last m=⌈1/γ⌉ observations, with open squares for m=105,106 . Error bars are pointwise 95% bootstrap intervals.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Model
τ
θτ∗
s.e.
A
Ωg
V
AR(1)
0.5
0
—
0.50000
1.00000
1.00000
AR(1)
0.9
0.99488
—
0.25556
0.32527
0.63638
GARCH (1,1)
0.5
0
—
0.50000
0.25000
0.25000
GARCH (1,1)
0.9
0.85814
4.5×10−5
0.25153
0.17481
0.34750
Nonlinear AR
0.5
1.64726
2.7×10−4
0.50000
5.13856
5.13856
Nonlinear AR
0.9
3.11902
4.3×10−4
0.23916
2.93354
6.13301
Appendix
Table 1: Targets and normalization constants. The s.e. column gives the Monte Carlo standard error of the calibrated target, with a dash for analytically determined targets. For short memory, V=Ωg/(2A) . For ARFIMA, V is the coefficient of γ1−2d in s(γ)2 , and the score long-run variance is infinite.
MSEγ/s(γ)2
Slope
Model
τ
0.1
0.05
0.01
10−3
10−4
Fitted
Theory
AR(1)
0.5
0.951
0.965
0.981
0.993
0.996
0.994
1.0
AR(1)
0.9
0.997
1.012
1.011
1.017
1.022
0.997
1.0
GARCH (1,1)
0.5
1.020
1.014
1.013
1.008
1.000
1.003
1.0
GARCH (1,1)
0.9
0.899
0.945
0.994
0.998
1.013
0.985
1.0
Nonlinear AR
0.5
0.837
0.907
0.983
0.998
1.003
0.977
1.0
Appendix
Table 2: Empirical MSE divided by s(γ)2 at the five learning rates, and fitted log–log slopes. Each MSE uses R=30,000 replicates at n=1,842,069 . The final column gives the exponent of the theoretical variance scale.
We consider the policy evaluation and control in a finite horizon reinforcement learning (RL) setting under an expectile-based objective. First, we derive the mean-squared error (MSE) and concentration bounds for the classic estimator of expectiles based on independent and identically distributed (i.i.d.) samples. To the best of our knowledge, expectiles have not been analyzed in the non-asymptotic regime and the bounds we derive may be of independent interest. Next, we analyze a Monte Carlo type estimator of expectile of the Markov chain underlying a given policy. We derive upper bounds that hold in expectation as well as with high probability for this estimator. Further, we show the order-optimality of our estimator by deriving a lower bound on expectile-based policy evaluation. For the problem of control, we adopt a policy gradient approach and derive a policy gradient theorem for expectiles. Using this result, we propose a gradient estimator with a O(1/m) mean-squared error bounds, where m is the number of trajectories. Further, under standard assumptions for policy gradient-type algorithms, we establish smoothness of the expectile-sensitive objective, in turn leading to stationary convergence rate bounds for the overall risk-sensitive policy gradient algorithm that we propose. Finally, we conduct numerical experiments to show the utility of expectiles on popular RL benchmarks.
This paper considers the estimation of quantiles via a smoothed version of the stochastic gradient descent (SGD) algorithm. By smoothing the score function with a bandwidth tied to the learning rate, we obtain estimates that are monotone in the quantile level at every iteration, while retaining the memory and computational efficiency required for streaming data. We establish non-asymptotic tail probability bounds for the smoothed estimate with and without Polyak-Ruppert averaging, which are sub-exponential with a multi-regime structure. For the averaged estimate we further derive a Bahadur representation that is uniform in the quantile level and across coordinates, and a resulting Gaussian approximation by the maximum of Brownian bridges, with the dimension p allowed to grow exponentially in the sample size. This yields simultaneous inference across coordinates and quantile levels. As an alternative that avoids estimating the sparsity function, we propose an online multiplier bootstrap that preserves monotonicity, runs in a single pass and is asymptotically valid. Extending the theory to a localized recursion, we obtain online nonparametric conditional quantile estimates with uniform bands over design points and quantile levels. Simulations confirm accurate finite-sample coverage, and we illustrate the method on conditional value-at-risk curves.
Likai Chen, Georg Keilbar, Wei Biao Wu
Department of Mathematics and Statistics · Washington University in St.Louis · St.Louis, MO, USA +6
We study the asymptotic shape of the trajectory of the stochastic gradient descent algorithm applied to a convex objective function. Under mild regularity assumptions, we prove a functional central limit theorem for the properly rescaled trajectory. Our result characterizes the long-term fluctuations of the algorithm around the minimizer by providing a diffusion limit for the trajectory. In contrast with classical central limit theorems for the last iterate or Polyak-Ruppert averages, this functional result captures the temporal structure of the fluctuations and applies to non-smooth settings such as robust location estimation, including the geometric median.