This cruise is going to be choppy, so grab a life preserver and join the adventure! Play the latest in the Delicious series of time management games Manage different places with Francois, Angela, and your other favorite Delicious characters Keep Patrick and Emily happy to earn golden hearts Enjoy bonus features like an extra episode, concept art, and a Delicious wallpaper.
System Requirements. Download free games at FreeRide Games. Iwaiya is a popular hotel with a pool. Many guests care about the quality of the rooms they stay in and want to ensure that their rooms have fresh, clean air.
When staying at a hotel, internet access is important for both vacationers and business travelers. Guesthouse Famusuton is a popular hotel in Iwami with free Wi-Fi.
App Download the App for Free. Search Bookings. Indigo Geohouse DS1. Guesthouse Famusuton. Minshuku New Otani. Hamasaka Onsen Totoya. Kasuikyo Izutsuya. Yukai Resort Yumuraonsen Miyoshiya.
Tottori City Hotel. Tottori Guest House Miraie Base. We will also explicitly define and track a flow u t over the tangent model; what we care about is w t , but we will show that indeed u t and w t stay close in this setting.
Note that u t is not needed for the analysis of w t. Remark 8. Smoothness constant. Singular values. This gives a concrete sense under which these eigenvalue assumptions are representation assumptions. Combining all parameters. An issue occurs once we perform time discretization. Next, since the gradient of the least squares risk is the residual, then decreasing risk implies decreasing gradient norms, and in particular we can not travel far. As a consequence we also immediately get that we never escape this ball: the gradient norms decay sufficiently rapidly.
Do we still have a nice convergence theory? Smoothness and differentiability do not in general hold for us ReLU, max-pooling, hinge loss, etc. Much of convex analysis and convex opt can use subgradients in place of gradients; cf. As an example from these notes, Lemma 7. Typically, we lack convexity, and the subdifferential set is empty.
Our main formalism is the Clarke differential Clarke et al. If R satisfies some technical structural conditions, then the following nice properties hold; these properties are mostly taken from Lemma 5. Chain rule. For a. This is the key strong property; since it holds for every element v of the Clarke differential simultaneously, it implies the next property.
Minimum norm path. Kakade and Lee gives some bad examples, e. Kakade and Lee also give a randomized algorithm for finding good subdifferentials. Does it matter? In the NTK regime, few activations change. As a function of x , this mapping is 1-homogeneous and piecewise affine. The boundary regions form a set of Lebesgue measure zero wrt to either weights or parameters. Fixing x and considering w , interior to each piece, the mapping is differentiable. Due to the definition of Clarke differential, it therefore suffices to compute the gradients in all adjacent pieces, and then take their convex hull.
If predictions are positive homogeneous with respect to each layer, then gradient flow preserves norms of layers. During , various works pointed out that deep networks generalize well, even though parameter norms are large, and there is no explicit generalization Neyshabur, Tomioka, and Srebro ; Zhang et al.
This prompted authors to study implicit bias of gradient descent , the first such result being an analysis of linear predictors with linearly separable data , showing that gradient descent on the cross-entropy loss is implicitly biased towards a maximum margin direction Soudry, Hoffer, and Srebro This in turn inspired many other works, handling other types of data, networks, and losses Ji and Telgarsky b, , ; Gunasekar et al. Margin maximization of first-order methods applied to exponentially-tailed losses was first proved for coordinate descent Telgarsky The results go through for similar losses.
Moreover, we do not even have unique directions, nor a way to tell different ones apart! We can use margins, now appropriately generalized to the L -homogeneous case, to build towards a better-behaved objective function. Later, when we study the L -homogeneous case, we are only able to show for every unit norm to the power L , the unnormalized margin increases by at least the current margin , which implies nondecreasing, but not margin maximization.
By Theorem 7. This nicely shows that we decrease the risk to 0 , but not that we maximize margins. For this, we need a more specialized analysis. Combining these inequalities gives the bound. In fact, the two gradient flows are the same!
Bounding these terms is now much simpler than for the regular gradient flow. In the nonlinear case, we do not have a general result, and instead only prove that smoothed margins are nondecreasing.
Lemma Proof of Theorem Making use of Lemma The purpose of this generalization part is to bound the gap between testing and training error for standard multilayer ReLU deep networks via the classical uniform convergence tools, and also to present and develop these classical tools based on Rademacher complexity.
These bounds are very loose, and there is extensive criticism now both of them and of the general approach, as will be discussed shortly Neyshabur, Tomioka, and Srebro ; Zhang et al. Zhou, Sutherland, and Srebro ; P.
Bartlett and Long Generalization properties of more architectures. One key omission is of convolution layers; for one generalization analysis, see Long and Sedghi In the present notes, we only focus on uniform convergence bounds , which give high probability bounds between training and test error which hold simultaneously for every element of some class. By contrast, PAC-Bayes consider a distribution over predictors, and bound the expected gap between testing and training error for these predictors in terms of how close this distribution is to some prior distribution over the predictors.
The looseness of the uniform-convergence bounds presented in these notes leads many authors to instead use them as explanatory tools, e. A correlation was claimed and presented in P. Bartlett, Foster, and Telgarsky , however it was on a single dataset and architecture. More extensive investigations have appeared recently Jiang et al. Compression-based approaches Arora, Ge, et al.
Zhou et al. For further connections between PAC-Bayes methodology and compression, see Blum and Langford , and for more on the concept of compression schemes , see for instance Moran and Yehudayoff Double descent Belkin et al. Wei and Ma give a bound which requires smooth activations; if we convert it to ReLU, it introduces a large factor which does not seem to improve over those presented here. That said, it is an interesting bound and approach.
Azuma-Hoeffding gave us control on the errors in SGD; note that we averaged together many errors before studying concentration! Our main concentration tool will be the Chernoff bounding method , which works nicely with sub-Gaussian random variables.
Example The Chernoff bounding technique is as follows. What if we apply this to an average of sub-Gaussian r. The point is: this starts to look like an empirical risk! Gaussian sanity check. Note also the bound is neat for the Gaussian since it says the tail mass and density are of the same order algebraically this makes sense, as with geometric series. There are more sophisticated bounds e. I should say something about necessary and sufficient, like convex lipschitz bounded vs lipschitz gaussian.
Suppose marginal on X has finite support. Suppose marginal on X is continuous. Restrict access to data within training algorithm: SGD does this, and has a specialized martingale-based deviation analysis. Uniform deviations: define a new r. On the other hand, we can adapt the approach to the output of the algorithm in various ways, as we will discuss after presenting the main Rademacher bound. Sanity checks. Here are a few basic checks:. Absolute value version. The original definition of Rademacher complexity P.
Most texts now drop the absolute value. Here are my reasons:. The proof of this bound has many interesting points and is spread out over the next few subsections. It has these basic steps:.
The expected uniform deviations are upper bounded by the expected Rademacher complexity. This itself is done in two steps:. The expected deviations are upper bounded by expected deviations between two finite samples.
This is interesting since we could have reasonably defined generalization in terms of this latter quantity. These two-sample deviations are upper bounded by expected Rademacher complexity by introducing random signs. As above, in this section we are working only in expectation for now.
As mentioned before, the preceding lemma says we can instead work with two samples. Working with two samples could have been our starting point and definition of generalization : by itself it is a meaningful and interpretable quantity!
The second step swaps points between the two samples; a magic trick with random signs boils this down into Rademacher complexity. A standard way is via a Martingale variant of the Chernoff bounding method. The Martingale adds one point at a time, and sees how things grow. The third bullet item follows from the first two by union bounding.
Average case vs worst case. This bound scales as the SGD logistic regression bound proved via Azuma, despite following a somewhat different route Azuma and McDiarmid are both proved with Chernoff bounding method; the former approach involves no symmetrization, whereas the latter holds for more than the output of an algorithm.
Relatedly: regularizing the gradient is sometimes used in practice? In the logistic regression example, we peeled off the loss and bounded the Rademacher complexity of the predictors. If most training labels are predicted not only accurately, but with a large margin, as in section 10 , then we can further reduce the generalization bound. As a generalization notion, this was first introduced for 2-layer networks in P. Bartlett , and then carried to many other settings SVM, boosting, …. There are many different proof schemes; another one uses sparsification Schapire et al.
This approach is again being extensively used for deep networks, since it seems that while weight matrix norms grow indefinitely, the margins grow along with them P. Here we will recover that, via Rademacher complexity. Moreover, the bound has a special form which will be useful in the later VC dimension and especially covering sections. This matters later when we encounter the Dudley Entropy integral. See the entire latin catalogue. See entire Soundtrack catalogue.
See entire library of World music. Cart 0 Your cart is empty. Available in Bit Digital Download Purchase and download this album in a wide variety of formats depending on your needs.
Your browser does not support the audio element. Copy the following link to share it Copy. You are currently listening to samples. Keep It Up. Coming Home.
Deepest Blue. Fall in Love Relax Mix. Sweet Spot. DISC 2. Back to Innocence.
0コメント