We consider the problem of recovering an unknown latent code vector under a known generative model.

The paper shows that the optimal rule for such metrics has the form of sign(P(y|x)-\delta^*), where \delta^* is a threshold that is metric dependent. This unified framework recovers many results for metrics studied in the literature. Authors devise 2 simple algorithms for estimating p(y|x) and the threshold delta^*. Bayes and statistical consistency of the proposed algorithms is then analyzed.

Here we state some results on Gaussian Matrices, which will be used in the proofs later. The following theorem is the concentration of (Gaussian) measure inequality for Lipschitz functions. Here we only state a one-sided version, though it is more commonly stated with a two-sided one, i.e., The result follows since E k A k p m + p n (see, e.g., [31, Section 7.3]). In particular, the following Bernstein's Inequality [31, Section 2.8] holds: P First, we establish that Z ( u, v; w) has a mixed tail. Next, by induction on k (i.e., apply (12) with r = r ( i 1) The result then follows by induction.

We consider the problem of recovering an unknown latent code vector under a known generative model. We introduce a simple novel algorithm, Partially Linearized Update for Generative Inversion (PLUGIn), to estimate x (and thus \mathcal{G}(x)). We prove that, when weights are Gaussian and layer widths n_i \gtrsim 5 i n_0 (up to log factors), the algorithm converges geometrically to a neighbourhood of x with high probability. Note the inequality on layer widths allows n_i n_{i 1} when i\geq 1 . To our knowledge, this is the first such result for networks with some contractive layers.

These specifications can be seen as non-Markovian reward functions. Thus, this work is related to inverse reinforcement learning (IRL) which aims to infer the reward function of an agent by observing these successive states and actions. By defining the probability of a trajectory knowing a specification (using the maximum entropy principle) the development leads to a posterior distribution. Two algorithms result from this and allow to test the approach on the system presented in introduction (motivating the paper).

We provide several applications of Optimistic Mirror Descent, an online learning algorithm based on the idea of predictable sequences. First, we recover the Mirror Prox algorithm for offline optimization, prove an extension to Hölder-smooth functions, and apply the results to saddle-point type problems. Next, we prove that a version of Optimistic Mirror Descent (which has a close relation to the Exponential Weights algorithm) can be used by two strongly-uncoupled players in a finite zero-sum matrix game to converge to the minimax equilibrium at the rate of O((log T) T).

Similarity functions measure how comparable pairs of elements are, and play a key role in a wide variety of applications, e.g., notions of Individual Fairness abiding by the seminal paradigm of Dwork et al., as well as Clustering problems. However, access to an accurate similarity function should not always be considered guaranteed, and this point was even raised by Dwork et al. For instance, it is reasonable to assume that when the elements to be compared are produced by different distributions, or in other words belong to different ``demographic'' groups, knowledge of their true similarity might be very difficult to obtain. In this work, we present an efficient sampling framework that learns these across-groups similarity functions, using only a limited amount of experts' feedback. We show analytical results with rigorous theoretical bounds, and empirically validate our algorithms via a large suite of experiments.

We study the problem of best-item identification from choice-based feedback. In this problem, a company sequentially and adaptively shows display sets to a population of customers and collects their choices. The objective is to identify the most preferred item with the least number of samples and at a high confidence level. We propose an elimination-based algorithm, namely Nested Elimination (NE), which is inspired by the nested structure implied by the information-theoretic lower bound. NE is simple in structure, easy to implement, and has a strong theoretical guarantee for sample complexity. Specifically, NE utilizes an innovative elimination criterion and circumvents the need to solve any complex combinatorial optimization problem. We provide an instance-specific and non-asymptotic bound on the expected sample complexity of NE. We also show NE achieves high-order worst-case asymptotic optimality. Finally, numerical experiments from both synthetic and real data corroborate our theoretical findings.

Lipschitz Bound Estimation is an effective method of regularizing deep neural networks to make them robust against adversarial attacks. This is useful in a variety of applications ranging from reinforcement learning to autonomous systems. In this paper, we highlight the significant gap in obtaining a non-trivial Lipschitz bound certificate for Convolutional Neural Networks (CNNs) and empirically support it with extensive graphical analysis. We also show that unrolling Convolutional layers or Toeplitz matrices can be employed to convert Convolutional Neural Networks (CNNs) to a Fully Connected Network. Further, we propose a simple algorithm to show the existing 20x-50x gap in a particular data distribution between the actual lipschitz constant and the obtained tight bound. We also ran sets of thorough experiments on various network architectures and benchmark them on datasets like MNIST and CIFAR-10. All these proposals are supported by extensive testing, graphs, histograms and comparative analysis.

Lidar is an acronym for light detection and ranging. Lidar is like radar, except that it uses light instead of radio waves. The light source is a laser. A lidar sends out light pulses and measures the time it takes for a reflection bouncing off a remote object to return to the device. As the speed of light is a known constant, the distance to the object can be calculated from the travel time of the light pulse (Figure 1).