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Inverse Optimization with Noisy Data
Anil Aswani, Zuo-Jun Max Shen, Auyon Siddiq
TL;DR
The paper studies how to estimate parameters of convex optimization problems when observed optimal solutions are noisy. It combines a duality-based bilevel reformulation with regularization, proves statistical consistency despite NP-hardness, and develops algorithms that preserve this consistency while performing competitively with existing heuristics.
Problem
Inverse optimization must estimate unknown model parameters from noisy measurements of optimal solutions, but existing heuristics are statistically inconsistent.
Method
The paper combines a duality-based bilevel reformulation with regularization and proposes enumeration and semiparametric algorithms for solving the resulting formulation.
Results
The formulation produces statistically consistent parameter estimates, although inverse optimization with noisy data is NP-hard, and the approach performs competitively with existing heuristics.
Takeaways & Limitations
The proposed formulation provides a statistically consistent basis for inverse optimization with noisy data, and its numerical algorithms retain that consistency.
Takeaways & Limitations
The consistency analysis relies on stated continuity assumptions, and the general risk-consistency result is unavailable because Q(θ) is typically discontinuous.
Abstract
from arXiv · showhide
Inverse optimization refers to the inference of unknown parameters of an optimization problem based on knowledge of its optimal solutions. This paper considers inverse optimization in the setting where measurements of the optimal solutions of a convex optimization problem are corrupted by noise. We first provide a formulation for inverse optimization and prove it to be NP-hard. In contrast to existing methods, we show that the parameter estimates produced by our formulation are statistically consistent. Our approach involves combining a new duality-based reformulation for bilevel programs with a regularization scheme that smooths discontinuities in the formulation. Using epi-convergence theory, we show the regularization parameter can be adjusted to approximate the original inverse optimization problem to arbitrary accuracy, which we use to prove our consistency results. Next, we propose two solution algorithms based on our duality-based formulation. The first is an enumeration algorithm that is applicable to settings where the dimensionality of the parameter space is modest, and the second is a semiparametric approach that combines nonparametric statistics with a modified version of our formulation. These numerical algorithms are shown to maintain the statistical consistency of the underlying formulation. Lastly, using both synthetic and real data, we demonstrate that our approach performs competitively when compared with existing heuristics.
1. Introduction
The paper frames inverse optimization as estimating parameters of latent convex optimization problems from noisy observed decisions. It develops a statistically consistent formulation, establishes computational and statistical properties, and proposes numerical algorithms for solving it.
- Inverse optimization estimates unknown parameters of an optimization problem from observed, potentially noisy optimal solutions.The setting is motivated by decisions generated by latent optimization problems and by measurement noise that can prevent all observations from being exactly optimal.
- The paper formulates inverse optimization to obtain statistically consistent parameter estimates, addressing discontinuity and continuity difficulties in the associated risk measures.The formulation is designed for consistency, while the analysis handles risk measures that are generally lower semicontinuous rather than continuous.
- A new duality-based bilevel reformulation and an upper-bounding optimality condition provide favorable convexity and continuity properties for analysis and algorithm design.The reformulation underlies the statistical results and the proposed numerical methods.
- The formulation jointly estimates model parameters and denoised underlying optimal solutions, using a loss function to fit the parametric model to noisy data.This differs from ordinary regression because inverse optimization may involve estimating a possibly multi-valued solution set rather than a single-valued closed-form function.
- Two numerical algorithms are proposed: an enumeration method for modest parameter dimension and a semiparametric method combining nonparametric statistics with the formulation.The algorithms use the bilevel formulation, and their statistical consistency follows from the paper’s theoretical results.
- The paper proves that inverse optimization with noisy data is NP-hard and that existing convex-optimization heuristics are statistically inconsistent.The heuristics can converge to incorrect parameter values as the amount of data increases.
2. Challenges with Noisy Inverse Optimization
The paper formalizes inverse optimization from noisy measurements of convex forward-problem solutions and establishes that both population and sample-average formulations are computationally hard. It then contrasts this hardness with statistical guarantees, showing that the proposed formulation is consistent while existing heuristics are not.
- Model for Inverse Optimization Problem: Noisy inverse optimization infers a fixed unknown θ from observed pairs (u,y), where y is a noisy observation of an optimal decision.The sample-average formulation uses iid measurements when the joint distribution P(u,y) is unknown.
- Model for Forward Problem: The forward problem minimizes f(x,u,θ) over constraints g(x,u,θ) ≤ 0, with x as the decision, u as the input, and θ as the unknown parameter.The solution set is S(u,θ), and the value function is V(u,θ).
- NP-Hardness of Inverse Optimization Problem: Under continuity and convexity assumptions, solving the inverse optimization problem remains NP-hard.The paper states this as Theorem 1, despite convexity of the functions and sets involved.
- NP-Hardness of Inverse Optimization Problem: The hardness reduction connects inverse optimization to computing a best rank-1 approximation of an order-3 tensor.With a discrete parameter set, related inapproximability results apply up to factors of 2(log d)^(1−ϵ).
- NP-Hardness of Inverse Optimization Problem: The sample-average inverse problem is also NP-hard because it is a bilevel program, including quadratic bilevel structure when the forward problem is linear.Specific quadratic-program instances remain polynomial-time solvable, but these are exceptional settings.
- Statistical Consistency: Existing VIA and KKA heuristics are not estimation consistent, whereas the proposed IOP-SAA approach produces an estimated set with statistical consistency.The paper attributes the inconsistency of VIA and KKA to minimizing an incorrect measure.
3. Consistent Estimation for Inverse Optimization Problem
The paper develops a statistically consistent noisy inverse-optimization formulation by combining a duality-based bilevel reformulation with regularization. Continuity and consistency strengthen under strict convexity, while the general case yields weak risk consistency.
- Existing heuristics are statistically inconsistent, motivating a noisy inverse-optimization formulation with consistency guarantees.
- Under strict convexity, Qn(θ) is continuous, enabling standard uniform-law-of-large-numbers arguments for risk and estimation consistency.
- Without strict convexity, Qn(θ) may be discontinuous, so uniform convergence fails and the analysis instead establishes weak risk consistency.
- The duality-based reformulation characterizes forward optimality through the Lagrangian dual and yields a convex single-level formulation under the stated assumptions.
- Regularization produces a jointly continuous objective and supports risk consistency for fixed positive ϵ, with predictions asymptotically attaining the best possible R-RISK performance.
- With strict convexity, solving IOP–SAA additionally yields asymptotically optimal prediction risk and estimation consistency under the stated conditions.
4. Numerical Approaches to Solving IOP–SAA
The paper proposes enumeration and semiparametric algorithms for solving IOP–SAA, matching different dimensionality and noise settings. Both algorithms preserve the statistical consistency guarantees of the underlying formulation, with computational and statistical trade-offs.
- The enumeration algorithm targets parameter spaces with modest dimensionality, while the semiparametric algorithm addresses higher-dimensional parameters under a specified noise distribution.
- Both algorithms produce estimates satisfying the conditions required for statistical consistency.
- Enumeration requires computation exponential in p, whereas the semiparametric method uses a nonparametric step expected to converge more slowly.
- Enumeration Algorithm: The enumeration algorithm discretizes Θ with a δ-net, evaluates the regularized objective over that finite set, and returns its minimizing parameter.
- Enumeration Algorithm: For sufficiently small ϵ and δ, enumeration returns an estimate within distance d of the IOP–SAA optimal-solution set.
- Semiparametric Approach: The semiparametric algorithm is statistically consistent under additional assumptions, including θ-independent constraints and an objective affine in θ.
5. Numerical Experiments
Numerical experiments evaluate enumeration and semiparametric algorithms on synthetic and real data, finding statistical consistency and competitive performance against KKA and VIA.
- Experimental design: The experiments test estimation and risk consistency as sample size grows, and assess feasibility as the parameter dimension increases.Synthetic experiments vary either the number of data points or the number of parameters; a real dataset estimates an energy-comfort utility function.
- Enumeration algorithm: Enumeration estimates become increasingly accurate as n grows, demonstrating estimation consistency on synthetic instances.Scatter plots use alignment of estimated and true parameters along the diagonal as the consistency criterion.
- Enumeration algorithm: KKA and VIA perform more poorly on FOP-A than FOP-B because FOP-A is a linear program whose optima mostly lie on the feasible-set boundary.The paper attributes the difference to the location of optimal solutions, with FOP-B having more optima in the strict interior.
- Prediction performance: The enumeration algorithm has lower prediction error than KKA and VIA because it is risk consistent, whereas those benchmarks are not.Prediction-error results are normalized by subtracting measurement-noise variance or E(w′w), depending on the experiment.
- Semiparametric algorithm: The semiparametric algorithm shows decreasing estimation error with more data and lower estimation and prediction errors than KKA and VIA in tested examples.The paper also reports competitiveness in a high-dimensional nonlinear forward problem and competitive performance on real data.
6. Conclusion
The paper develops a statistically consistent formulation for inverse optimization with noisy measurements, despite computational hardness, and provides algorithms and experiments supporting its practical use.
- The formulation targets inverse optimization when noisy measurements of optimal points from a convex problem are available.
- A new duality-based reformulation converts the required bilevel program into a single-level program.
- The bilevel formulation is statistically consistent, unlike existing heuristics.
- Although the formulation is NP-hard, two numerical algorithms preserve its statistical consistency.
- Synthetic and empirical numerical experiments demonstrate improvements over existing methods.
A. Lemmas and Omitted Proofs
The omitted proofs analyze counterexamples, consistency behavior, and technical regularity conditions underlying the paper’s statistical claims.
- The proofs use kernel support, smooth density assumptions, bounded solutions, and sub-exponential noise to establish finite terms and concentration bounds.
- A counterexample shows that existing heuristics VIA and KKA are not estimation consistent.
- For VIA, the empirical objective is piecewise-defined in θ, enabling convergence analysis but yielding inconsistency in the counterexample.
- The proofs connect risk consistency and estimation consistency through continuity of the risk function.
B. Identifiability in Inverse Optimization
Identifiability is required for consistent parameter inference, but constraints can destroy it; external inputs can restore it in some settings.
- Identifiability requires model outputs to differ for distinct parameter values and depends on the model and error metric.
- Identifiability alone does not guarantee estimator consistency.
- Constraints can make inverse optimization unidentifiable even when the forward problem is strictly convex with unique minimizers.
- In FOP-II, observing yi = 1 is compatible with every θ ∈ [1,2], so the constraint x ≤ 1 prevents the identifiability condition from holding.
- External inputs can restore identifiability: for FOP-III, P(ui ≤ −1) > 0 is sufficient.