471 research outputs found
Distributionally Time-Varying Online Stochastic Optimization under Polyak-{\L}ojasiewicz Condition with Application in Conditional Value-at-Risk Statistical Learning
In this work, we consider a sequence of stochastic optimization problems
following a time-varying distribution via the lens of online optimization.
Assuming that the loss function satisfies the Polyak-{\L}ojasiewicz condition,
we apply online stochastic gradient descent and establish its dynamic regret
bound that is composed of cumulative distribution drifts and cumulative
gradient biases caused by stochasticity. The distribution metric we adopt here
is Wasserstein distance, which is well-defined without the absolute continuity
assumption or with a time-varying support set. We also establish a regret bound
of online stochastic proximal gradient descent when the objective function is
regularized. Moreover, we show that the above framework can be applied to the
Conditional Value-at-Risk (CVaR) learning problem. Particularly, we improve an
existing proof on the discovery of the PL condition of the CVaR problem,
resulting in a regret bound of online stochastic gradient descent
Social welfare and profit maximization from revealed preferences
Consider the seller's problem of finding optimal prices for her
(divisible) goods when faced with a set of consumers, given that she can
only observe their purchased bundles at posted prices, i.e., revealed
preferences. We study both social welfare and profit maximization with revealed
preferences. Although social welfare maximization is a seemingly non-convex
optimization problem in prices, we show that (i) it can be reduced to a dual
convex optimization problem in prices, and (ii) the revealed preferences can be
interpreted as supergradients of the concave conjugate of valuation, with which
subgradients of the dual function can be computed. We thereby obtain a simple
subgradient-based algorithm for strongly concave valuations and convex cost,
with query complexity , where is the additive
difference between the social welfare induced by our algorithm and the optimum
social welfare. We also study social welfare maximization under the online
setting, specifically the random permutation model, where consumers arrive
one-by-one in a random order. For the case where consumer valuations can be
arbitrary continuous functions, we propose a price posting mechanism that
achieves an expected social welfare up to an additive factor of
from the maximum social welfare. Finally, for profit maximization (which may be
non-convex in simple cases), we give nearly matching upper and lower bounds on
the query complexity for separable valuations and cost (i.e., each good can be
treated independently)
Second-order Online Nonconvex Optimization
We present the online Newton's method, a single-step second-order method for
online nonconvex optimization. We analyze its performance and obtain a dynamic
regret bound that is linear in the cumulative variation between round optima.
We show that if the variation between round optima is limited, the method leads
to a constant regret bound. In the general case, the online Newton's method
outperforms online convex optimization algorithms for convex functions and
performs similarly to a specialized algorithm for strongly convex functions. We
simulate the performance of the online Newton's method on a nonlinear,
nonconvex moving target localization example and find that it outperforms a
first-order approach
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