Pranay Tankala

Cynthia Dwork, Daniel Lee, Huijia Lin et al. · harvard

We identify and explore connections between the recent literature on multi-group fairness for prediction algorithms and the pseudorandomness notions of leakage-resilience and graph regularity. We frame our investigation using new variants of multicalibration based on statistical distance and closely related to the concept of outcome indistinguishability. Adopting this perspective leads us not only to new, more efficient algorithms for multicalibration, but also to our graph theoretic results and a proof of a novel hardcore lemma for real-valued functions.

Elbert Du, Cynthia Dwork, Pranay Tankala et al. · harvard

A recent line of work initiated by Chiesa and Gur and further developed by Herman and Rothblum investigates the sample and communication complexity of verifying properties of distributions with the assistance of a powerful, knowledgeable, but untrusted prover. In this work, we initiate the study of differentially private (DP) distribution property testing. After all, if we do not trust the prover to help us with verification, why should we trust it with our sensitive sample? We map a landscape of DP prover-aided proofs of properties of distributions. In the non-private case it is known that one-round (two message) private-coin protocols can have substantially lower complexity than public-coin AM protocols, but in the private case, the possibility for improvement depends on the parameter regime and privacy model. Drawing on connections to replicability and techniques for amplification, we show: (1) There exists a reduction from any one-round $(\varepsilon,δ)$-DP private-coin interactive proof to a one-round public-coin DP interactive proof with the same privacy parameters, for the parameter regime $\varepsilon=O(1/\sqrt{n})$ and $δ=O(1/n^{5/2})$, and with the same sample and communication complexities. (2) If the verifier's message in the private-coin interactive proof is $O(1/\sqrt{\log n})$ locally DP -- a far more relaxed privacy parameter regime in a different model -- then applying one additional transformation again yields a one-round public-coin protocol with the same privacy bound and the same sample and computational complexities. (3) However, when the privacy guarantee is very relaxed ($\varepsilon\inΩ(\log n)$), private coins indeed reduce complexity. We also obtain a Merlin-Arthur (one-message) proof for privately testing whether samples are drawn from a product distribution, and prove that its sample complexity is optimal.

Cynthia Dwork, Chris Hays, Nicole Immorlica et al. · harvard

Professional networks provide invaluable entree to opportunity through referrals and introductions. A rich literature shows they also serve to entrench and even exacerbate a status quo of privilege and disadvantage. Hiring platforms, equipped with the ability to nudge link formation, provide a tantalizing opening for beneficial structural change. We anticipate that key to this prospect will be the ability to estimate the likelihood of edge formation in an evolving graph. Outcome-indistinguishable prediction algorithms ensure that the modeled world is indistinguishable from the real world by a family of statistical tests. Omnipredictors ensure that predictions can be post-processed to yield loss minimization competitive with respect to a benchmark class of predictors for many losses simultaneously, with appropriate post-processing. We begin by observing that, by combining a slightly modified form of the online K29 star algorithm of Vovk (2007) with basic facts from the theory of reproducing kernel Hilbert spaces, one can derive simple and efficient online algorithms satisfying outcome indistinguishability and omniprediction, with guarantees that improve upon, or are complementary to, those currently known. This is of independent interest. We apply these techniques to evolving graphs, obtaining online outcome-indistinguishable omnipredictors for rich -- possibly infinite -- sets of distinguishers that capture properties of pairs of nodes, and their neighborhoods. This yields, inter alia, multicalibrated predictions of edge formation with respect to pairs of demographic groups, and the ability to simultaneously optimize loss as measured by a variety of social welfare functions.

Cynthia Dwork, Pranay Tankala, Linjun Zhang · harvard

We initiate the study of differentially private learning in the proportional dimensionality regime, in which the number of data samples $n$ and problem dimension $d$ approach infinity at rates proportional to one another, meaning that $d/n\toδ$ as $n\to\infty$ for an arbitrary, given constant $δ\in(0,\infty)$. This setting is significantly more challenging than that of all prior theoretical work in high-dimensional differentially private learning, which, despite the name, has assumed that $δ= 0$ or is sufficiently small for problems of sample complexity $O(d)$, a regime typically considered "low-dimensional" or "classical" by modern standards in high-dimensional statistics. We provide sharp theoretical estimates of the error of several well-studied differentially private algorithms for robust linear regression and logistic regression, including output perturbation, objective perturbation, and noisy stochastic gradient descent, in the proportional dimensionality regime. The $1+o(1)$ factor precision of our error estimates enables a far more nuanced understanding of the price of privacy of these algorithms than that afforded by existing, coarser analyses, which are essentially vacuous in the regime we consider. Using our estimates, we discover a previously unobserved "double descent"-like phenomenon in the training error of objective perturbation for robust linear regression. We also identify settings in which output perturbation outperforms objective perturbation on average, and vice versa, demonstrating that the relative performance of these algorithms is less clear-cut than suggested by prior work. To prove our main theorems, we introduce several probabilistic tools that have not previously been used to analyze differentially private learning algorithms, such as a modern Gaussian comparison inequality and recent universality laws with origins in statistical physics.

Abiy Tasissa, Pranay Tankala, Demba Ba

Sparse manifold learning algorithms combine techniques in manifold learning and sparse optimization to learn features that could be utilized for downstream tasks. The standard setting of compressive sensing can not be immediately applied to this setup. Due to the intrinsic geometric structure of data, dictionary atoms might be redundant and do not satisfy the restricted isometry property or coherence condition. In addition, manifold learning emphasizes learning local geometry which is not reflected in a standard $\ell_1$ minimization problem. We propose weighted $\ell_0$ and weighted $\ell_1$ metrics that encourage representation via neighborhood atoms suited for dictionary based manifold learning. Assuming that the data is generated from Delaunay triangulation, we show the equivalence of weighted $\ell_0$ and weighted $\ell_1$. We discuss an optimization program that learns the dictionaries and sparse coefficients and demonstrate the utility of our regularization on synthetic and real datasets.

Emmanouil Theodosis, Bahareh Tolooshams, Pranay Tankala et al.

Recent approaches in the theoretical analysis of model-based deep learning architectures have studied the convergence of gradient descent in shallow ReLU networks that arise from generative models whose hidden layers are sparse. Motivated by the success of architectures that impose structured forms of sparsity, we introduce and study a group-sparse autoencoder that accounts for a variety of generative models, and utilizes a group-sparse ReLU activation function to force the non-zero units at a given layer to occur in blocks. For clustering models, inputs that result in the same group of active units belong to the same cluster. We proceed to analyze the gradient dynamics of a shallow instance of the proposed autoencoder, trained with data adhering to a group-sparse generative model. In this setting, we theoretically prove the convergence of the network parameters to a neighborhood of the generating matrix. We validate our model through numerical analysis and highlight the superior performance of networks with a group-sparse ReLU compared to networks that utilize traditional ReLUs, both in sparse coding and in parameter recovery tasks. We also provide real data experiments to corroborate the simulated results, and emphasize the clustering capabilities of structured sparsity models.

Pranay Tankala, Abiy Tasissa, James M. Murphy et al.

We propose K-Deep Simplex(KDS) which, given a set of data points, learns a dictionary comprising synthetic landmarks, along with representation coefficients supported on a simplex. KDS employs a local weighted $\ell_1$ penalty that encourages each data point to represent itself as a convex combination of nearby landmarks. We solve the proposed optimization program using alternating minimization and design an efficient, interpretable autoencoder using algorithm unrolling. We theoretically analyze the proposed program by relating the weighted $\ell_1$ penalty in KDS to a weighted $\ell_0$ program. Assuming that the data are generated from a Delaunay triangulation, we prove the equivalence of the weighted $\ell_1$ and weighted $\ell_0$ programs. We further show the stability of the representation coefficients under mild geometrical assumptions. If the representation coefficients are fixed, we prove that the sub-problem of minimizing over the dictionary yields a unique solution. Further, we show that low-dimensional representations can be efficiently obtained from the covariance of the coefficient matrix. Experiments show that the algorithm is highly efficient and performs competitively on synthetic and real data sets.