The complexity of gradient descent: CLS = PPAD ∩ PLS

Fearnley, John; Goldberg, Paul W.; Hollender, Alexandros; Savani, Rahul

doi:10.1145/3406325.3451052

Cited by 33 publications

(43 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Nevertheless, we show below that finding a CF using Equation 6is CLScomplete while finding a globally optimal CF is CLS-hard. These complexity results complement those of Tsirtsis and Gomez Rodriguez [51] and also suggest that any instance of CFX-POTENTIAL-LOCALOPT can be transformed in polynomial time into problems such as GD-FINITE-DIFF [18]. This affords us access to a powerful toolbox of methods from numerical computing and optimisation.…”

Section: Potential-based Searchsupporting

confidence: 75%

“…Proof (Sketch). Show inclusion by reducing to GENERAL-CONTINUOUS-LOCALOPT, then hardness follows by reducing from GD-LOCAL-SEARCH [18]; see Appendix A.…”

Section: Cfx-potential-(localopt/globalopt) (Informal)mentioning

confidence: 99%

“…2. Prove that, under this formalism, finding an optimal CF is CLS-complete [18], and that deciding CF existence more generally is NP-complete.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Counterfactual Explanations for Arbitrary Regression Models

Spooner,

Dervovic,

Long

et al. 2021

Preprint

View full text Add to dashboard Cite

We present a new method for counterfactual explanations (CFEs) based on Bayesian optimisation that applies to both classification and regression models. Our method is a globally convergent search algorithm with support for arbitrary regression models and constraints like feature sparsity and actionable recourse, and furthermore can answer multiple counterfactual questions in parallel while learning from previous queries. We formulate CFE search for regression models in a rigorous mathematical framework using differentiable potentials, which resolves robustness issues in threshold-based objectives. We prove that in this framework, (a) verifying the existence of counterfactuals is NP-complete; and (b) that finding instances using such potentials is CLS-complete. We describe a unified algorithm for CFEs using a specialised acquisition function that composes both expected improvement and an exponential-polynomial (EP) family with desirable properties. Our evaluation on real-world benchmark domains demonstrate high sample-efficiency and precision.Preprint. Under review.

show abstract

Section: Potential-based Searchsupporting

confidence: 75%

“…Proof (Sketch). Show inclusion by reducing to GENERAL-CONTINUOUS-LOCALOPT, then hardness follows by reducing from GD-LOCAL-SEARCH [18]; see Appendix A.…”

Section: Cfx-potential-(localopt/globalopt) (Informal)mentioning

confidence: 99%

See 1 more Smart Citation

Counterfactual Explanations for Arbitrary Regression Models

Spooner,

Dervovic,

Long

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…A potential candidate for such a smaller class could be the class PPAD ∩ PLS, which was recently shown by Fearnley et al [24] and Babichenko and Rubinstein [2] to capture the complexity of interesting problems related to optimization via gradient descent, and computing mixed Nash equilibria in congestion games [63] respectively. The class PLS was introduced by Johnson et al [38] and captures the computation of local minima of some objective function, and notably characterizes the complexity of finding pure Nash equilibria in congestion games [23].…”

Section: Discussionmentioning

confidence: 99%

On the Complexity of Equilibrium Computation in First-Price Auctions

Filos-Ratsikas

Giannakopoulos

Hollender

et al. 2021

Proceedings of the 22nd ACM Conference on Economics and Computation

Self Cite

View full text Add to dashboard Cite

We consider the problem of computing a (pure) Bayes-Nash equilibrium in the first-price auction with continuous value distributions and discrete bidding space. We prove that when bidders have independent subjective prior beliefs about the value distributions of the other bidders, computing an -equilibrium of the auction is PPAD-complete, and computing an exact equilibrium is FIXP-complete.CCS Concepts: • Theory of computation → Problems, reductions and completeness; Exact and approximate computation of equilibria; Algorithmic mechanism design; Computational pricing and auctions.

show abstract

“…Formally speaking these circuits also need to be well-behaved in a certain sense (see[26]). It is easy to check that the circuits we construct in this paper all have this property.Technical Program Presentation • EC '21, July 18-23, 2021, Budapest, Hungary…”

mentioning

confidence: 99%

Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents

Deligkas

Filos-Ratsikas

Hollender

2021

Proceedings of the 22nd ACM Conference on Economics and Computation

Self Cite

View full text Add to dashboard Cite

We consider the -Consensus-Halving problem, in which a set of heterogeneous agents aim at dividing a continuous resource into two (not necessarily contiguous) portions that all of them simultaneously consider to be of approximately the same value (up to ). This problem was recently shown to be PPA-complete, for agents and cuts, even for very simple valuation functions. In a quest to understand the root of the complexity of the problem, we consider the setting where there is only a constant number of agents, and we consider both the computational complexity and the query complexity of the problem.For agents with monotone valuation functions, we show a dichotomy: for two agents the problem is polynomial-time solvable, whereas for three or more agents it becomes PPA-complete. Similarly, we show that for two monotone agents the problem can be solved with polynomially-many queries, whereas for three or more agents, we provide exponential query complexity lower bounds. These results are enabled via an interesting connection to a monotone Borsuk-Ulam problem, which may be of independent interest. For agents with general valuations, we show that the problem is PPA-complete and admits exponential query complexity lower bounds, even for two agents.

show abstract

The complexity of gradient descent: CLS = PPAD ∩ PLS

Cited by 33 publications

References 30 publications

Counterfactual Explanations for Arbitrary Regression Models

Counterfactual Explanations for Arbitrary Regression Models

On the Complexity of Equilibrium Computation in First-Price Auctions

Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents

Contact Info

Product

Resources

About