We study the estimation error of constrained M-estimators, and derive explicit upper bounds on the expected estimation error determined by the Gaussian width of the constraint set. Both of the cases where the true parameter is on the boundary of the constraint set (matched constraint), and where the true parameter is strictly in the constraint set (mismatched constraint) are considered. For both cases, we derive novel universal estimation error bounds for regression in a generalized linear model with the canonical link function. Our error bound for the mismatched constraint case is minimax optimal in terms of its dependence on the sample size, for Gaussian linear regression by the Lasso.
We say that the constraint is if @math lies on the boundary of @math in ) (or @math ), and if @math lies strictly in @math (or @math ). The analyses in @cite_20 @cite_7 require the constraint to be matched, while in practice the exact value of @math is seldom known. The constraint in @cite_2 is always matched due to the special structure of quantum density operators. The error bounds in @cite_24 @cite_0 can be overly pessimistic, because they hold for all @math . The results in @cite_6 @cite_23 @cite_10 do not require a matched constraint and depend on @math ; our result is of this kind. Recall that, however, @cite_6 is limited to specific statistical models, and @cite_23 @cite_10 are limited to specific @math -estimators.
@cite_7 @cite_6 @cite_24 @cite_0 @cite_23 @cite_2 @cite_10 @cite_20
2950890626 2649051464 2952621790 2088419249
Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with @math variables and @math clauses, there is a value of @math beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when @math ). Intuitively, strong refutation should become easier as the clause density @math grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as @math -SAT and @math -XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, @math , and the clause density at which instances become unsatisfiable with high probability, @math . In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random @math -XOR instances with clause density @math in time @math or in @math rounds of the sum-of-squares hierarchy, for any @math and any integer @math . Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at @math , and brute-force refutation at the satisfiability threshold when @math . We also leverage our @math -XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors. Additionally, we extend our techniques to give new results for certifying upper bounds on the injective tensor norm of random tensors. It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error @math in trace distance required @math copies for a @math -dimensional density matrix of rank @math . Here, we give a theoretical measurement scheme (POVM) that requires @math copies to estimate @math to error @math in infidelity, and a matching lower bound up to logarithmic factors. This implies @math copies suffice to achieve error @math in trace distance. We also prove that for independent (product) measurements, @math copies are necessary in order to achieve error @math in infidelity. For fixed @math , our measurement can be implemented on a quantum computer in time polynomial in @math . Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. Many NP-complete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2-colorability (Property B) and random Not-All-Equal @math -SAT is @math . As a corollary, we establish that the threshold for random @math -SAT is of order @math , resolving a long-standing open problem.