19 research outputs found
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Stochastic Methods in Optimization and Machine Learning
Stochastic methods are indispensable to the modeling, analysis and design of complex systems involving randomness. In this thesis, we show how simulation techniques and simulation-based computational methods can be applied to a wide spectrum of applied domains including engineering, optimization and machine learning. Moreover, we show how analytical tools in statistics and computer science including empirical processes, probably approximately correct learning, and hypothesis testing can be used in these contexts to provide new theoretical results. In particular, we apply these techniques and present how our results can create new methodologies or improve upon existing state-of-the-art in three areas: decision making under uncertainty (chance-constrained programming, stochastic programming), machine learning (covariate shift, reinforcement learning) and estimation problems arising from optimization (gradient estimate of composite functions) or stochastic systems (solution of stochastic PDE).
The work in the above three areas will be organized into six chapters, where each area contains two chapters. In Chapter 2, we study how to obtain feasible solutions for chance-constrained programming using data-driven, sampling-based scenario optimization (SO) approach. When the data size is insufficient to statistically support a desired level of feasibility guarantee, we explore how to leverage parametric information, distributionally robust optimization and Monte Carlo simulation to obtain a feasible solution of chance-constrained programming in small-sample situations.
In Chapter 3, We investigate the feasibility of sample average approximation (SAA) for general stochastic optimization problems, including two-stage stochastic programming without the relatively complete recourse. We utilize results from the Vapnik-Chervonenkis (VC) dimension and Probably Approximately Correct learning to provide a general framework.
In Chapter 4, we design a robust importance re-weighting method for estimation/learning problem in the covariate shift setting that improves the best-know rate. In Chapter 5, we develop a model-free reinforcement learning approach to solve constrained Markov decision processes (MDP). We propose a two-stage procedure that generates policies with simultaneous guarantees on near-optimality and feasibility.
In Chapter 6, we use multilevel Monte Carlo to construct unbiased estimators for expectations of random parabolic PDE. We obtain estimators with finite variance and finite expected computational cost, but bypassing the curse of dimensionality. In Chapter 7, we introduce unbiased gradient simulation algorithms for solving stochastic composition optimization (SCO) problems. We show that the unbiased gradients generated by our algorithms have finite variance and finite expected computational cost
Do price trajectory data increase the efficiency of market impact estimation?
Market impact is an important problem faced by large institutional investor
and active market participant. In this paper, we rigorously investigate whether
price trajectory data from the metaorder increases the efficiency of
estimation, from an asymptotic view of statistical estimation. We show that,
for popular market impact models, estimation methods based on partial price
trajectory data, especially those containing early trade prices, can outperform
established estimation methods (e.g., VWAP-based) asymptotically. We discuss
theoretical and empirical implications of such phenomenon, and how they could
be readily incorporated into practice
Short-term Temporal Dependency Detection under Heterogeneous Event Dynamic with Hawkes Processes
Many event sequence data exhibit mutually exciting or inhibiting patterns.
Reliable detection of such temporal dependency is crucial for scientific
investigation. The de facto model is the Multivariate Hawkes Process (MHP),
whose impact function naturally encodes a causal structure in Granger
causality. However, the vast majority of existing methods use direct or
nonlinear transform of standard MHP intensity with constant baseline,
inconsistent with real-world data. Under irregular and unknown heterogeneous
intensity, capturing temporal dependency is hard as one struggles to
distinguish the effect of mutual interaction from that of intensity
fluctuation. In this paper, we address the short-term temporal dependency
detection issue. We show the maximum likelihood estimation (MLE) for
cross-impact from MHP has an error that can not be eliminated but may be
reduced by order of magnitude, using heterogeneous intensity not of the target
HP but of the interacting HP. Then we proposed a robust and
computationally-efficient method modified from MLE that does not rely on the
prior estimation of the heterogeneous intensity and is thus applicable in a
data-limited regime (e.g., few-shot, no repeated observations). Extensive
experiments on various datasets show that our method outperforms existing ones
by notable margins, with highlighted novel applications in neuroscience.Comment: Conference on Uncertainty in Artificial Intelligence 202
Accelerated Policy Evaluation: Learning Adversarial Environments with Adaptive Importance Sampling
The evaluation of rare but high-stakes events remains one of the main
difficulties in obtaining reliable policies from intelligent agents, especially
in large or continuous state/action spaces where limited scalability enforces
the use of a prohibitively large number of testing iterations. On the other
hand, a biased or inaccurate policy evaluation in a safety-critical system
could potentially cause unexpected catastrophic failures during deployment. In
this paper, we propose the Accelerated Policy Evaluation (APE) method, which
simultaneously uncovers rare events and estimates the rare event probability in
Markov decision processes. The APE method treats the environment nature as an
adversarial agent and learns towards, through adaptive importance sampling, the
zero-variance sampling distribution for the policy evaluation. Moreover, APE is
scalable to large discrete or continuous spaces by incorporating function
approximators. We investigate the convergence properties of proposed algorithms
under suitable regularity conditions. Our empirical studies show that APE
estimates rare event probability with a smaller variance while only using
orders of magnitude fewer samples compared to baseline methods in both
multi-agent and single-agent environments.Comment: 10 pages, 5 figure
Provably Convergent Schr\"odinger Bridge with Applications to Probabilistic Time Series Imputation
The Schr\"odinger bridge problem (SBP) is gaining increasing attention in
generative modeling and showing promising potential even in comparison with the
score-based generative models (SGMs). SBP can be interpreted as an
entropy-regularized optimal transport problem, which conducts projections onto
every other marginal alternatingly. However, in practice, only approximated
projections are accessible and their convergence is not well understood. To
fill this gap, we present a first convergence analysis of the Schr\"odinger
bridge algorithm based on approximated projections. As for its practical
applications, we apply SBP to probabilistic time series imputation by
generating missing values conditioned on observed data. We show that optimizing
the transport cost improves the performance and the proposed algorithm achieves
the state-of-the-art result in healthcare and environmental data while
exhibiting the advantage of exploring both temporal and feature patterns in
probabilistic time series imputation.Comment: Accepted by ICML 202