Do unique node identifiers help in deciding whether a network G has a prescribed property P? We study this question in the context of distributed local decision, where the objective is to decide whether G has property P by having each node run a constant-time distributed decision algorithm. In a yes-instance all nodes should output yes, while in a no-instance at least one node should output no. Recently, (OPODIS 2012) gave several conditions under which identifiers are not needed, and they conjectured that identifiers are not needed in any decision problem. In the present work, we disprove the conjecture. More than that, we analyse two critical variations of the underlying model of distributed computing: (B): the size of the identifiers is bounded by a function of the size of the input network, (¬B): the identifiers are unbounded, (C): the nodes run a computable algorithm, (¬C): the nodes can compute any, possibly uncomputable function. While it is easy to see that under (¬B, ¬C) identifiers are not needed, we show that under all other combinations there are properties that can be decided locally if and only if identifiers are present.
Therefore to ask meaningful questions related to the role of unique identifiers in construction tasks, we usually compare the @math model with models that retain some symmetry-breaking information---two such models are @math , , and @math , . [noitemsep] In the @math model @cite_2 , the output of an algorithm is not allowed to change if we reassign the identifier while preserving their relative order. In the @math model @cite_5 , there is an ordering on the incident edges, and all edges carry an orientation. Note that model @math is stronger than the Id-oblivious model: in the Id-oblivious model, @math for two assignments @math , while in the @math model, we only require this for assignments @math that satisfy @math . This difference makes the @math model much stronger.
@cite_5 @cite_2
2771374750 2170546552 2564029303 2034230784
When training a deep neural network for supervised image classification, one can broadly distinguish between two types of latent features of images that will drive the classification of class Y. Following the notation of (2016), we can divide features broadly into the classes of (i) “core” or “conditionally invariant” features X^ci whose distribution P(X^ci | Y) does not change substantially across domains and (ii) “style” or “orthogonal” features X^orth whose distribution P(X^orth | Y) can change substantially across domains. These latter orthogonal features would generally include features such as position, rotation, image quality or brightness but also more complex ones like hair color or posture for images of persons. We try to guard against future adversarial domain shifts by ideally just using the “conditionally invariant” features for classification. In contrast to previous work, we assume that the domain itself is not observed and hence a latent variable. We can hence not directly see the distributional change of features across different domains. We do assume, however, that we can sometimes observe a so-called identifier or ID variable. We might know, for example, that two images show the same person, with ID referring to the identity of the person. In data augmentation, we generate several images from the same original image, with ID referring to the relevant original image. The method requires only a small fraction of images to have an ID variable. We provide a causal framework for the problem by adding the ID variable to the model of (2016). However, we are interested in settings where we cannot observe the domain directly and we treat domain as a latent variable. If two or more samples share the same class and identifier, (Y, ID)=(y,i), then we treat those samples as counterfactuals under different style interventions on the orthogonal or style features. Using this grouping-by-ID approach, we regularize the network to provide near constant output across samples that share the same ID by penalizing with an appropriate graph Laplacian. This is shown to substantially improve performance in settings where domains change in terms of image quality, brightness, color changes, and more complex changes such as changes in movement and posture. We show links to questions of interpretability, fairness and transfer learning. Mulmuley [Mul12a] recently gave an explicit version of Noether’s Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley’s reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors ([FS12]), who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors. As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation. We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model. Finally, we consider the depth-3 diagonal circuit model as defined by Saxena [Sax08], as PIT algorithms for this model also have implications in Mulmuley’s work. Previous work (such as [ASS12] and [FS12]) have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich [SV09]. This paper investigates the relation between three different notions of privacy: identifiability, differential privacy, and mutual-information privacy. Under a unified privacy-distortion framework, where the distortion is defined to be the expected Hamming distance between the input and output databases, we establish some fundamental connections between these three privacy notions. Given a maximum allowable distortion @math , we define the privacy-distortion functions @math , @math , and @math to be the smallest (most private best) identifiability level, differential privacy level, and mutual information between the input and the output, respectively. We characterize @math and @math , and prove that @math for @math within certain range, where @math is a constant determined by the prior distribution of the original database @math , and diminishes to zero when @math is uniformly distributed. Furthermore, we show that @math and @math can be achieved by the same mechanism for @math within certain range, i.e., there is a mechanism that simultaneously minimizes the identifiability level and achieves the best mutual-information privacy. Based on these two connections, we prove that this mutual-information optimal mechanism satisfies @math -differential privacy with @math . The results in this paper reveal some consistency between two worst case notions of privacy, namely, identifiability and differential privacy, and an average notion of privacy, mutual-information privacy. Summary form only. Inspired by Kolmogorov's structure function for finite sets as models of data in the algorithmic theory of information we adapt the construct to families of probability models to avoid the noncomputability problem. The picture of modeling looks then as follows: The models in the family have a double index, where the first specifies a structure, ranging over a finite or a countable set, and the second consists of parameter values, ranging over a continuum. An optimal structure index can be determined by the MDL (Minimum Description Length) principle in a two-part code, where the sum of the code lengths for the structure and the data is minimized. The latter is obtained from the universal NML (Normalized Maximum Likelihood) model for the subfamily of models having a specified structure. The determination of the optimal model in the optimized structure is more difficult. It requires a partition of the parameter space into equivalence classes, each associated with a model, in such a way that the Kullback-Leibler distance between any two adjacent models is equal and that the models are optimally distinguishable from the given amount of data. This notion of distinguishability is a modification of a related idea of Balasubramanian. The particular model, specified by the observed data, is the simplest one that incorporates all the properties in the data that can be extracted with the model class considered.