# Progressive Neural Networks

Andrei A. Rusu*, Neil C. Rabinowitz*, Guillaume Desjardins, Hubert Soyer,
James Kirkpatrick, Koray Kavukcuoglu, Razvan Pascanu, Raia Hadsell
* These authors contributed equally to this work
Google DeepMind
London, UK
{andreirusu, ncr, gdesjardins, soyer, kirkpatrick, korayk, razp,
<EMAIL_ADDRESS>

###### Abstract

Learning to solve complex sequences of tasks—while both leveraging transfer
and avoiding catastrophic forgetting—remains a key obstacle to achieving
human-level intelligence. The _progressive networks_ approach represents a
step forward in this direction: they are immune to forgetting and can leverage
prior knowledge via lateral connections to previously learned features. We
evaluate this architecture extensively on a wide variety of reinforcement
learning tasks (Atari and 3D maze games), and show that it outperforms common
baselines based on pretraining and finetuning. Using a novel sensitivity
measure, we demonstrate that transfer occurs at both low-level sensory and
high-level control layers of the learned policy.

##  1 Introduction

Finetuning remains the method of choice for transfer learning with neural
networks: a model is pretrained on a source domain (where data is often
abundant), the output layers of the model are adapted to the target domain,
and the network is finetuned via backpropagation. This approach was pioneered
in [7] by transferring knowledge from a generative to a discriminative model,
and has since been generalized with great success [11]. Unfortunately, the
approach has drawbacks which make it unsuitable for transferring across
multiple tasks: if we wish to leverage knowledge acquired over a sequence of
experiences, which model should we use to initialize subsequent models? This
seems to require not only a learning method that can support transfer learning
without catastrophic forgetting, but also foreknowledge of task similarity.
Furthermore, while finetuning may allow us to recover expert performance in
the target domain, it is a destructive process which discards the previously
learned function. One could copy each model before finetuning to explicitly
remember all previous tasks, but the issue of selecting a proper
initialization remains. While distillation [8] offers one potential solution
to multitask learning [17], it requires a reservoir of persistent training
data for all tasks, an assumption which may not always hold.

This paper introduces progressive networks, a novel model architecture with
explicit support for transfer across sequences of tasks. While finetuning
incorporates prior knowledge only at initialization, progressive networks
retain a pool of pretrained models throughout training, and learn lateral
connections from these to extract useful features for the new task. By
combining previously learned features in this manner, progressive networks
achieve a richer compositionality, in which prior knowledge is no longer
transient and can be integrated at each layer of the feature hierarchy.
Moreover, the addition of new capacity alongside pretrained networks gives
these models the flexibility to both reuse old computations and learn new
ones. As we will show, progressive networks naturally accumulate experiences
and are immune to catastrophic forgetting by design, making them an ideal
springboard for tackling long-standing problems of continual or lifelong
learning.

The contributions of this paper are threefold. While many of the individual
ingredients used in progressive nets can be found in the literature, their
combination and use in solving complex sequences of tasks is novel. Second, we
extensively evaluate the model in complex reinforcement learning domains. In
the process, we also evaluate alternative approaches to transfer (such as
finetuning) within the RL domain. In particular, we show that progressive
networks provide comparable (if not slightly better) transfer performance to
traditional finetuning, but without the destructive consequences. Finally, we
develop a novel analysis based on Fisher Information and perturbation which
allows us to analyse in detail how and where transfer occurs across tasks.

##  2 Progressive Networks

Continual learning is a long-standing goal of machine learning, where agents
not only learn (and remember) a series of tasks experienced in sequence, but
also have the ability to transfer knowledge from previous tasks to improve
convergence speed [20]. Progressive networks integrate these desiderata
directly into the model architecture: catastrophic forgetting is prevented by
instantiating a new neural network (a column) for each task being solved,
while transfer is enabled via lateral connections to features of previously
learned columns. The scalability of this approach is addressed at the end of
this section.

A progressive network starts with a single column: a deep neural network
having $L$ layers with hidden activations $h_{i}^{(1)}\in\mathbb{R}^{n_{i}}$,
with $n_{i}$ the number of units at layer $i\leq L$, and parameters
$\Theta^{(1)}$ trained to convergence. When switching to a second task, the
parameters $\Theta^{(1)}$ are “frozen” and a new column with parameters
$\Theta^{(2)}$ is instantiated (with random initialization), where layer
$h_{i}^{(2)}$ receives input from both $h_{i-1}^{(2)}$ and $h_{i-1}^{(1)}$ via
lateral connections. This generalizes to $K$ tasks as follows: 111Progressive
networks can also be generalized in a straightforward manner to have arbitrary
network width per column/layer, to accommodate varying degrees of task
difficulty, or to compile lateral connections from multiple, independent
networks in an ensemble setting. Biases are omitted for clarity.:

$\displaystyle
h_{i}^{(k)}=f\left(W_{i}^{(k)}h_{i-1}^{(k)}+\sum_{j<k}U_{i}^{(k:j)}h_{i-1}^{(j)}\right),$
(1)

where $W_{i}^{(k)}\in\mathbb{R}^{n_{i}\times n_{i-1}}$ is the weight matrix of
layer $i$ of column $k$, $U_{i}^{(k:j)}\in\mathbb{R}^{n_{i}\times n_{j}}$ are
the lateral connections from layer $i-1$ of column $j$, to layer $i$ of column
$k$ and $h_{0}$ is the network input. $f$ is an element-wise non-linearity: we
use $f(x)=\max(0,x)$ for all intermediate layers. A progressive network with
$K=3$ is shown in Figure 1.

Figure 1: Depiction of a three column progressive network. The first two
columns on the left (dashed arrows) were trained on task 1 and 2 respectively.
The grey box labelled $a$ represent the adapter layers (see text). A third
column is added for the final task having access to all previously learned
features.

These modelling decisions are informed by our desire to: (1) solve $K$
independent tasks at the end of training; (2) accelerate learning via transfer
when possible; and (3) avoid catastrophic forgetting.

In the standard pretrain-and-finetune paradigm, there is often an implicit
assumption of “overlap” between the tasks. Finetuning is efficient in this
setting, as parameters need only be adjusted slightly to the target domain,
and often only the top layer is retrained [23]. In contrast, we make no
assumptions about the relationship between tasks, which may in practice be
orthogonal or even adversarial. While the finetuning stage could potentially
unlearn these features, this may prove difficult. Progressive networks side-
step this issue by allocating a new column for each new task, whose weights
are initialized randomly. Compared to the task-relevant initialization of
pretraining, columns in progressive networks are free to reuse, modify or
ignore previously learned features via the lateral connections. As the lateral
connections $U_{i}^{(k:j)}$ are only from column $k$ to columns $j<k$,
previous columns are not affected by the newly learned features in the forward
pass. Because also the parameters $\\{\Theta^{(j)};j<k\\}$ are kept frozen
(i.e. are constants for the optimizer) when training $\Theta^{(k)}$, there is
no interference between tasks and hence no catastrophic forgetting.

#### Application to Reinforcement Learning.

Although progressive networks are widely applicable, this paper focuses on
their application to deep reinforcement learning. In this case, each column is
trained to solve a particular Markov Decision Process (MDP): the $k$-th column
thus defines a policy $\pi^{(k)}(a\mid s)$ taking as input a state $s$ given
by the environment, and generating probabilities over actions $\pi^{(k)}(a\mid
s):=h_{L}^{(k)}(s)$. At each time-step, an action is sampled from this
distribution and taken in the environment, yielding the subsequent state. This
policy implicitly defines a stationary distribution $\rho_{\pi^{(k)}}(s,a)$
over states and actions.

#### Adapters.

In practice, we augment the progressive network layer of Equation 2 with non-
linear lateral connections which we call adapters. They serve both to improve
initial conditioning and perform dimensionality reduction. Defining the vector
of anterior features $h_{i-1}^{(<k)}=[h_{i-1}^{(1)}\cdots h_{i-1}^{(j)}\cdots
h_{i-1}^{(k-1)}]$ of dimensionality $n_{i-1}^{(<k)}$, in the case of dense
layers, we replace the linear lateral connection with a single hidden layer
MLP. Before feeding the lateral activations into the MLP, we multiply them by
a learned scalar, initialized by a random small value. Its role is to adjust
for the different scales of the different inputs. The hidden layer of the non-
linear adapter is a projection onto an $n_{i}$ dimensional subspace. As the
index $k$ grows, this ensures that the number of parameters stemming from the
lateral connections is in the same order as $\left|\Theta^{(1)}\right|$.
Omitting bias terms, we get:

$\displaystyle
h_{i}^{(k)}=\sigma\left(W_{i}^{(k)}h_{i-1}^{(k)}+U_{i}^{(k:j)}\sigma(V_{i}^{(k:j)}\alpha_{i-1}^{(<k)}h_{i-1}^{(<k)})\right),$
(2)

where $V_{i}^{(k:j)}\in\mathbb{R}^{n_{i-1}\times n_{i-1}^{(<k)}}$ is the
projection matrix. For convolutional layers, dimensionality reduction is
performed via $1\times 1$ convolutions [10].

#### Limitations.

Progressive networks are a stepping stone towards a full continual learning
agent: they contain the necessary ingredients to learn multiple tasks, in
sequence, while enabling transfer and being immune to catastrophic forgetting.
A downside of the approach is the growth in number of parameters with the
number of tasks. The analysis of Appendix 2 reveals that only a fraction of
the new capacity is actually utilized, and that this trend increases with more
columns. This suggests that growth can be addressed, e.g. by adding fewer
layers or less capacity, by pruning [9], or by online compression [17] during
learning. Furthermore, while progressive networks retain the ability to solve
all $K$ tasks at test time, choosing which column to use for inference
requires knowledge of the task label. These issues are left as future work.

##  3 Transfer Analysis

Unlike finetuning, progressive nets do not destroy the features learned on
prior tasks. This enables us to study in detail which features and at which
depth transfer actually occurs. We explored two related methods: an intuitive,
but slow method based on a perturbation analysis, and a faster analytical
method derived from the Fisher Information [2].

#### Average Perturbation Sensitivity (APS).

To evaluate the degree to which source columns contribute to the target task,
we can inject Gaussian noise at isolated points in the architecture (e.g. a
given layer of a single column) and measure the impact of this perturbation on
performance. A significant drop in performance indicates that the final
prediction is heavily reliant on the feature map or layer. We find that this
method yields similar results to the faster Fisher-based method presented
below. We thus relegate details and results of the perturbation analysis to
the appendix.

#### Average Fisher Sensitivity (AFS).

We can get a local approximation to the perturbation sensitivity by using the
Fisher Information matrix [2]. While the Fisher matrix is typically computed
with respect to the model parameters, we compute a modified diagonal Fisher
$\hat{F}$ of the network policy $\pi$ with respect to the normalized
activations 222The Fisher of individual neurons (fully connected) and feature
maps (convolutional layers) are computed over $\rho_{\pi^{(k)}}(s,a)$. The use
of a normalized representation $\hat{h}$ is non-standard, but makes the scale
of $\hat{F}$ comparable across layers and columns. at each layer
$\hat{h}_{i}^{(k)}$. For convolutional layers, we define $\hat{F}$ to
implicitly perform a summation over pixel locations. $\hat{F}$ can be
interpreted as the sensitivity of the policy to small changes in the
representation. We define the diagonal matrix $\hat{F}$, having elements
$\hat{F}(m,m)$, and the derived Average Fisher Sensitivity (AFS) of feature
$m$ in layer $i$ of column $k$ as:

$\displaystyle\hat{F}_{i}^{(k)}$
$\displaystyle=\mathbb{E}_{\rho(s,a)}\left[\frac{\partial\log\pi}{\partial\hat{h}_{i}^{(k)}}\,\frac{\partial\log\pi}{\partial\hat{h}_{i}^{(k)}}^{T}\right]$
$\displaystyle\text{AFS}(i,k,m)$
$\displaystyle=\frac{\hat{F}_{i}^{(k)}(m,m)}{\sum_{k}\hat{F}_{i}^{(k)}(m,m)}$

where the expectation is over the joint state-action distribution $\rho(s,a)$
induced by the progressive network trained on the target task. In practice, it
is often useful to consider the AFS score per-layer
$\text{AFS}(i,k)=\sum_{m}\text{AFS}(i,k,m)$, i.e. summing over all features of
layer $i$. The AFS and APS thus estimate how much the network relies on each
feature or column in a layer to compute its output.

##  4 Related Literature

There exist many different paradigms for transfer and multi-task reinforcement
learning, as these have long been recognized as critical challenges in AI
research [15, 19, 20]. Many methods for transfer learning rely on linear and
other simple models (e.g. [18]), which is a limiting factor to their
applicability. Recently, there have been new methods proposed for multi-task
or transfer learning with deep RL: [22, 17, 14]. In this work we present an
architecture for deep reinforcement learning that in sequential task regimes
that enables learning without forgetting while supporting individual feature
transfer from previous learned tasks.

Pretraining and finetuning was proposed in [7] and applied to transfer
learning in [4, 11], generally in unsupervised-to-supervised or supervised-to-
supervised settings. The actor-mimic approach [14] applied these principles to
reinforcement learning, by fine-tuning a DQN multi-task network on new Atari
games and showing that some responded with faster learning, while others did
not. Progressive networks differ from the finetuning direction substantially,
since capacity is added as new tasks are learned.

Progressive nets are related to the incremental and constructive architectures
proposed in neural network literature. The cascade-correlation architecture
was designed to eliminate forgetting while incrementally adding and refining
feature extractors [6]. Auto-encoders such as [24] use incremental feature
augmentation to track concept drift, and deep architectures such as [16] have
been designed that specifically support feature transfer. More recently, in
[1], columns are separately trained on individual noise types, then linearly
combined, and [5] use columns for image classification. The block-modular
architecture of [21] has many similarities to our approach but focuses on a
visual discrimination task. The progressive net approach, in contrast, uses
lateral connections to access previously learned features for deep
compositionality. It can be used in any sequential learning setting but is
especially valuable in RL.

##  5 Experiments

We evaluate progressive networks across three different RL domains. First, we
consider synthetic versions of Pong, altered to have visual or control-level
similarities. Next, we experiment broadly with random sequences of Atari games
and perform a feature-level transfer analysis. Lastly, we demonstrate
performance on a set of 3D maze games. Fig. 2 shows examples from selected
tasks.

|
---|---
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(a) Pong variants

|
---|---
|
(b) Labyrinth games

|
---|---
|
(c) Atari games

Figure 2: Samples from different task domains: (a) Pong variants include
flipped, noisy, scaled, and recoloured transforms; (b) Labyrinth is a set of
3D maze games with diverse level maps and diverse positive and negative reward
items; (c) Atari games offer a more challenging setting for transfer.

###  5.1 Setup

We rely on the Async Advantage Actor-Critic (A3C) framework introduced in
[13]. Compared to DQN [12], the model simultaneously learns a policy and a
value function for predicting expected future rewards. A3C is trained on CPU
using multiple threads and has been shown to converge faster than DQN on GPU.
This made it a more natural fit for the large amount of sequential experiments
required for this work.

We report results by averaging the top 3 out of 25 jobs, each having different
seeds and random hyper-parameter sampling. Performance is evaluated by
measuring the area under the learning curve (average score per episode during
training), rather than final score. The _transfer score_ is then defined as
the relative performance of an architecture compared with a single column
baseline, trained only on the target task (baseline 1). We present transfer
score curves for selected source-target games, and summarize all such pairs in
_transfer matrices_. Models and baselines we consider are illustrated in
Figure 3. Details of the experimental setup are provided in section 3 of the
Appendix.

Figure 3: Illustration of different baselines and architectures. _Baseline 1_
is a single column trained on the target task; _baseline 2_ is a single
column, pretrained on a source task and finetuned on the target task (output
layer only); _baseline 3_ is the same as baseline 2 but the whole model is
finetuned; and _baseline 4_ is a 2 column progressive architecture, with
previous column(s) initialized randomly and frozen.

###  5.2 Pong Soup

The first evaluation domain is a set of synthetic variants of the Atari game
of Pong ("Pong Soup") where the visuals and gameplay have been altered, thus
providing a setting where we can be confident that there are transferable
aspects of the tasks. The variants are _Noisy_ (frozen Gaussian noise is added
to the inputs); _Black_ (black background); _White_ (white background); _Zoom_
(input is scaled by 75% and translated); _V-flip, H-flip, and VH-flip_ (input
is horizontally and/or vertically flipped). Example frames are shown in Fig.
2. The results of training two columns on the Pong variants, including all
relevant baselines are shown in Figure 4. Transfer scores are summarized over
all target tasks in Table 1.

Figure 4: (a) Transfer matrix. Colours indicate transfer scores (clipped at
2). For progressive nets, the first column is trained on Pong, Noisy, or
H-flip (table rows); the second column is trained on each of the other pong
variants (table columns). (b) Example learning curves.

We can make several observations from these results. Baseline 2 (single
column, only output layer is finetuned; see Fig. 3) fails to learn the target
task in most experiments and thus has negative transfer. This approach is
quite standard in supervised learning settings, where features from ImageNet-
trained nets are routinely repurposed for new domains. As expected, we observe
high positive transfer with baseline 3 (single column, full finetuning), a
well established paradigm for transfer. Progressive networks outperform this
baseline however in terms of both median and mean score, with the difference
being more pronounced for the latter. As the mean is more sensitive to
outliers, this suggests that progressive networks are better able to exploit
transfer when transfer is possible (i.e. when source and target domains are
compatible). Fig. 4 (b) lends weight to this hypothesis, where progressive
networks are shown to significantly outperform the baselines for particular
game pairs. Progressive nets also compare favourably to baseline 4, confirming
that progressive nets are indeed taking advantage of the features learned in
previous columns.

Detailed analysis

Figure 5: (a) Transfer analysis for 2-column nets on Pong variants. The
relative sensitivity of the network’s outputs on the columns within each layer
(the AFS) is indicated by the darkness of shading. (b) AFS values for the 8
feature maps of conv. 1 of a 1-column Pong net. Only one feature map is
effectively used by the net; the same map is also used by the 2-column
versions. Below: spatial filter components (red = positive, blue = negative).
(c) Activation maps of the filter in (b) from example states of the four
games.

We use the metric derived in Sec. 3 to analyse what features are being
transferred between Pong variants. We see that when switching from Pong to
H-Flip, the network reuses the same components of low and mid-level vision
(the outputs of the two convolutional layers; Figure 5a). However, the fully
connected layer must be largely re-learned, as the policy relevant features of
the task (the relative locations/velocities of the paddle and ball) are now in
a new location. When switching from Pong to Zoom, on the other hand, low-level
vision is reused for the new task, but new mid-level vision features are
learned. Interestingly, only one low-level feature appears to be reused: (see
Fig. 5b): this is a spatio-temporal filter with a considerable temporal DC
component. This appears sufficient for detecting both ball motion and paddle
position in the original, flipped, and zoomed Pongs.

Finally, when switching from Pong to Noisy, some new low-level vision is
relearned. This is likely because the first layer filter learned on the clean
task is not sufficiently tolerant to the added noise. In contrast, this
problem does not apply when moving from Noisy to Pong (Figure 5a, rightmost
column), where all of vision transfers to the new task.

###  5.3 Atari Games

We next investigate feature transfer between randomly selected Atari games
[3]. This is an interesting question, because the visuals of Atari games are
quite different from each other, as are the controls and required strategy.
Though games like Pong and Breakout are conceptually similar (both involve
hitting a ball with a paddle), Pong is vertically aligned while Breakout is
horizontal: a potentially insurmountable feature-level difference. Other Atari
game pairs have _no_ discernible overlap, even at a conceptual level.

Figure 6: Transfer scores and example learning curves for Atari target games,
as per Figure 4.

To this end we start by training single columns on three _source_ games (Pong,
River Raid, and Seaquest) 333Progressive columns having more than one “source”
column are trained sequentially on these source games, i.e. Seaquest-River
Raid-Pong means column 1 is first trained on Seaquest, column 2 is added
afterwards and trained on River Raid, and then column 3 added and trained on
Pong. and assess if the learned features transfer to a different subset of
randomly selected _target_ games (Alien, Asterix, Boxing, Centipede, Gopher,
Hero, James Bond, Krull, Robotank, Road Runner, Star Gunner, and Wizard of
Wor). We evaluate progressive networks with 2, 3 and 4 columns, comparing to
the baselines of Figure 3). The transfer matrix and selected transfer curves
are shown in Figure 6, and the results summarized in Table 1.

Across all games, we observe from Fig. 6, that progressive nets result in
positive transfer in 8 out of 12 target tasks, with only two cases of negative
transfer. This compares favourably to baseline 3, which yields positive
transfer in only 5 of 12 games. This trend is reflected in Table 1, where
progressive networks convincingly outperform baseline 3 when using additional
columns. This is especially promising as we show in the Appendix that
progressive network use a diminishing amount of capacity with each added
column, pointing a clear path to online compression or pruning as a means to
mitigate the growth in model size.

Now consider the specific sequence Seaquest-to-Gopher, an example of two
dissimilar games. Here, the pretrain/finetune paradigm (baseline 3) exhibits
negative transfer, unlike progressive networks (see Fig.6b, bottom), perhaps
because they are more able to ignore the irrelevant features. For the sequence
Seaquest[+River Raid][+Pong]-to-Boxing, using additional columns in the
progressive networks can yield a significant increase in transfer (see Fig.
6b, top).

| Pong Soup | Atari | Labyrinth
---|---|---|---
| Mean (%) | Median (%) | Mean (%) | Median (%) | Mean (%) | Median (%)
Baseline 1 | 100 | 100 | 100 | 100 | 100 | 100
Baseline 2 | 35 | 7 | 41 | 21 | 88 | 85
Baseline 3 | 181 | 160 | 133 | 110 | 235 | 112
Baseline 4 | 134 | 131 | 96 | 95 | 185 | 108
Progressive 2 col | 209 | 169 | 132 | 112 | 491 | 115
Progressive 3 col | 222 | 183 | 140 | 111 | — | —
Progressive 4 col | — | — | 141 | 116 | — | —
Table 1: Transfer percentages in three domains. Baselines are defined in Fig.
3.

Detailed Analysis

Figure 6 demonstrates that both positive and negative transfer is possible
with progressive nets. To differentiate these cases, we consider the Average
Fisher Sensitivity for the 3 column case (e.g., see Fig. 7a). A clear pattern
emerges amongst these and other examples: the most negative transfer coincides
with complete dependence on the convolutional layers of the previous columns,
and no learning of new visual features in the new column. In contrast, the
most positive transfer occurs when the features of the first two columns are
augmented by new features. The statistics across all 3-column nets (Figure 7b)
show that positive transfer in Atari occurs at a "sweet spot" between heavy
reliance on features from the source task, and heavy reliance on all new
features for the target task.

Figure 7: (a) AFS scores for 3-column nets with lowest (left) and highest
(right) transfer scores on the 12 target Atari games. (b) Transfer statistics
across 72 three-column nets, as a function of the mean AFS across the three
convolutional layers of the new column (i.e. how much new vision is learned).

At first glance, this result appears unintuitive: if a progressive net finds a
valuable feature set from a source task, shouldn’t we expect a high degree of
transfer? We offer two hypotheses. First, this may simply reflect an
optimization difficulty, where the source features offer fast convergence to a
poor local minimum. This is a known challenge in transfer learning [20]:
learned source tasks confer an inductive bias that can either help or hinder
in different cases. Second, this may reflect a problem of exploration, where
the transfered representation is "good enough" for a functional, but sub-
optimal policy.

###  5.4 Labyrinth

The final experimental setting for progressive networks is Labyrinth, a 3D
maze environment where the inputs are rendered images granting partial
observability and the agent outputs discrete actions, including looking up,
down, left, or right and moving forward, backwards, left, or right. The tasks
as well as the level maps are diverse and involve getting positive scores for
‘eating’ good items (apples, strawberries) and negative scores for eating bad
items (mushrooms, lemons). Details can be found in the appendix. While there
is conceptual and visual overlap between the different tasks, the tasks
present a challenging set of diverse game elements (Figure 2).

Figure 8: Transfer scores and example learning curves for Labyrinth tasks.
Colours indicate transfer (clipped at 2). The learning curves show two
examples of two-column progressive performance vs. baselines 1 and 3.

As in the other domains, the progressive approach yields more positive
transfer than any of the baselines (see Fig. 8a and Table 1). We observe less
transfer on the Seek Track levels, which have dense reward items throughout
the maze and are easily learned. Note that even for these easy cases, baseline
2 shows negative transfer because it cannot learn new low-level visual
features, which are important because the reward items change from task to
task. The learning curves in Fig. 8b exemplify the typical results seen in
this domain: on simpler games, such as Track 1 and 2, learning is rapid and
stable by all agents. On more difficult games, with more complex game
structure, the baselines struggle and progressive nets have an advantage.

##  6 Conclusion

Continual learning, the ability to accumulate and transfer knowledge to new
domains, is a core characteristic of intelligent beings. Progressive neural
networks are a stepping stone towards continual learning, and this work has
demonstrated their potential through experiments and analysis across three RL
domains, including Atari, which contains orthogonal or even adversarial tasks.
We believe that we are the first to show positive transfer in deep RL agents
within a continual learning framework. Moreover, we have shown that the
progressive approach is able to effectively exploit transfer for compatible
source and task domains; that the approach is robust to harmful features
learned in incompatible tasks; and that positive transfer increases with the
number of columns, thus corroborating the constructive, rather than
destructive, nature of the progressive architecture.

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Supplementary Material

##  Appendix A Perturbation Analysis

We explored two related methods for analysing transfer in progressive
networks. One based on Fisher information yields the Average Fisher
Sensitivity (AFS) and is described in Section 3 of the paper. We describe the
second method based on perturbation analysis in this appendix, as it proved
too slow to use at scale. Given its intuitive appeal however, we provide
details of the method along with results on Pong Variants (see Section 5.2),
as a means to corroborate the AFS score.

Our perturbation analysis aims to estimate which components of the source
columns materially contribute to the performance of the final column on the
target tasks. To this end, we injected Gaussian noise into each of the (post-
ReLU) hidden representations, with a new sample on every forward pass, and
calculated the average effect of these perturbations on the game score over 10
episodes. We did this at a coarse scale, by adding noise across all features
of a given layer, though a fine scale analysis is also possible per feature
(map). In order to be invariant to any arbitrary scale factors in the network
weights, we scale the noise variance proportional to the variance of the
activations in each feature map and fully-connected neuron. Scaling the
variance in this manner is analogous to computing the Fisher w.r.t. normalized
activations for the AFS score.

Figure 9: (a) Perturbation analysis for the two second-layer convolutional
representations in the two columns of the Pong/Pong-noise net. Blue: adding
noise to second convolutional layer from column 1; green: from column 2. Grey
line determines critical noise magnitude for each representation,
$\sigma_{i}^{2}$. (b-c) Comparison of per-layer sensitivities obtained using
the APS method (b) and the AFS method (c; as per main text). These are highly
similar.

Define ${\Lambda_{i}^{(k)}}=1/\sigma_{i}^{2(k)}$ as the precision of the noise
injected at layer $i$ of column $k$, which results in a $50\%$ drop in
performance. The Average Perturbation Sensitivity (APS) for this layer is
simply:

$\displaystyle\text{APS}(i,k)=\frac{\Lambda_{i}^{(k)}}{\sum_{k}\Lambda_{i}^{(k)}}$
(3)

Note that this value is normalized across columns for a given layer. The APS
score can thus be interpreted as the responsibility of each column in a given
layer to final performance. The APS score of 2-column progressive networks
trained on Pong Variants is shown in Fig9 (b). These clearly corroborate the
AFS shown in (c).

##  Appendix B Compressibility of Progressive Networks

As described in the main text, one of the limitations of progressive networks
is the growth in the size of the network with added tasks. In the basic
approach we pursue in the main text, the number of hidden units and feature
maps grows linearly with the number of columns, and the number of parameters
grows quadratically.

Here, we sought to determine the degree to which this full capacity is
actually used by the network. We leveraged the Average Fisher Sensitivity
measure to study how increasing the number of columns in the Atari task set
changes the need for additional resources. In Figure 10a, we measure the
average fractional use of existing feature maps in a given layer (here, layer
2). We do this for each network by concatenating the per-feature-map AFS
values from all source columns in this layer, sorting the values to produce a
spectrum, and then averaging across networks. We find that as the number of
columns increases, the average spectrum becomes sparser: the network relies on
a smaller proportion of features from the source columns. Similar results were
found for all layers.

Similarly, in Figure 10b, we measure the capacity required in the final added
column as a function of the total number of columns. Again, we measure the
spectrum of AFS values in an example layer, but here from only the final
column. As the progressive network grows, the new column’s features are both
less important overall (indicated by the declining area under the graph), and
have a sparser AFS spectrum. Combined, these results suggest that significant
pruning of lateral connections is possible, and the quadratic growth of
parameters might be contained.

Figure 10: (a) Spectra of AFS values (for layer 2) across all feature maps
from source columns, for the Atari dataset. The spectra show the range of AFS
values, and are averaged across networks. While the 2 column / 3 column / 4
column nets all have different values of $N_{maps}$ (here, 12, 24, and 36
respectively), these have been dilated to fit the same axis to show the
proportional use of these maps. (b) Spectra of AFS values (for layer 2) for
the feature maps from only the final column.

##  Appendix C Setup Details

In our grid we sample hyper-parameters from categorical distributions:

  * •

Learning rate was sampled from $\\{10^{-3},5\cdot 10^{-4},10^{-4}\\}$.

  * •

Strength of the entropy regularization from $\\{10^{-2},10^{-3},10^{-4}\\}$

  * •

Gradient clipping cut-off from $\\{20,40\\}$

  * •

scalar multiplier on the lateral feature is initialized randomly to one from
$\\{1,10^{-1},10^{-2}\\}$

For the Atari experiments we used a model with 3 convolutional layers followed
by a fully connected layer and from which we predict the policy and value
function. The convolutional layers are as follows. All have 12 feature maps.
The first convolutional layer has a kernel of size 8x8 and a stride of 4x4.
The second layer has a kernel of size 4 and a stride of 2. The last
convolutional layer has size 3x4 with a stride of 1. The fully connected layer
has 256 hidden units.

Learning follows closely the paradigm described in [13]. We use 16 workers and
the same RMSProp algorithm without momentum or centring of the variance. The
score for each point of a training curve is the average over all the episodes
the model gets to finish in $25e4$ environment steps.

The whole experiments are run for a maximum of $1.6e8$ environment step. The
agent has an action repeat of 4 as in [13], which means that for 4 consecutive
steps the agent will use the same action picked at the beginning of the
series. For this reason through out the paper we actually report results in
terms of agent perceived steps rather than environment steps. That is, the
maximal number of agent perceived step that we do for any particular run is
$4e7$.

##  Appendix D Learning curves

Figure 11 shows training curves for all the target games in the Atari domain.
We plot learning curves for two column, three column and four column
progressive networks alongside Baseline 3 (gray dashed line), a model
pretrained on Seaquest and then finetuned on the particular target game and
Baseline 1 (gray dotted line), where a single column is trained on the source
game Seaquest.

|  |
---|---|---
|  |
|  |
|  |
Figure 11: Training curves for transferring to the target games after seeing
first Seaquest followed by River Raid and lastly Pong. For the baselines, the
source game used for pretraining is Seaquest.

We can see that overall baseline 3 performs well. However there are situations
when having features learned from more previous task actually helps with
transfer (e.g. when target game is Boxing).

Figure 12 shows how two-column progressive networks perform as compared to
Baseline 3 (gray dashed line), a model pretrained on the source game, here
standard Pong, and then finetuned on a particular target game, and Baseline 1
(black dotted line), where a single column is trained on standard Pong. Figure
13 shows two-column progressive networks and baselines on Labyrinth tasks; the
source game was Maze Y.

Target: Pong | Target: Black
---|---
|
Target: H-flip | Target: HV-flip
|
Target: Noisy | Target: V-flip
|
Target: White | Target: Zoom
|
Figure 12: Training curves for transferring to 8 target games after learning standard Pong first.  Target: Track 1 | Target: Track 2
---|---
|
Target: Track 3 | Target: Track 4
|
Target: Avoid 1 | Target: Avoid 2
|
Target: Maze Y | Target: Maze M
|
Figure 13: Training curves for transferring to 8 target games after learning
Maze Y first.

##  Appendix E Labyrinth

Section 5.4 evaluates progressive networks on foraging tasks in complex 3D
maze environments. Positive rewards are given to the agent for collecting
apples and strawberries, and negative rewards for mushrooms and lemons.
Episodes terminate when either all (positive) rewards are collected, or after
a fixed time interval.

Levels differ in their maze layout, the type of items present and the sparsity
of the reward structure. The levels we employed can be characterized as
follows:

  * •

Seek Track 1: simple corridor with many apples

  * •

Seek Track 2: U-shaped corridor with many strawberries

  * •

Seek Track 3: $\Omega$-shaped, with $90^{o}$ turns, with few apples

  * •

Seek Track 4: $\Omega$-shaped, with $45^{o}$ turns, with few apples

  * •

Seek Avoid 1: large square room with apples and lemons

  * •

Seek Avoid 2: large square room with apples and mushrooms

  * •

Seek Maze M : M-shaped maze, with apples at dead-ends

  * •

Seek Maze Y : Y-shaped maze, with apples at dead-ends


Model-Based Deep Learning of Joint Probabilistic and Geometric Shaping for
Optical Communication

Vladislav Neskorniuk1,3, Andrea Carnio2, Domenico Marsella2,

Sergei K. Turitsyn3, Jaroslaw E. Prilepsky3, Vahid Aref1

1Nokia, 70469 Stuttgart, Germany 2Nokia, Vimercate 20871, Italy

3Aston Institute of Photonic Technologies, Aston University, B4 7ET
Birmingham, UK

<EMAIL_ADDRESS>
###### Abstract

Autoencoder-based deep learning is applied to jointly optimize geometric and
probabilistic constellation shaping for optical coherent communication. The
optimized constellation shaping outperforms the 256 QAM Maxwell-Boltzmann
probabilistic distribution with extra 0.05 bits/4D-symbol mutual information
for 64 GBd transmission over 170 km SMF link.

##  1 Introduction

Geometric (GS) and probabilistic (PS) constellation shaping, i.e. the
optimization of, respectively, locations and occurrence probabilities of the
constellation points, can improve considerably the transmission rate of
coherent fiber-optic communication systems. While for linear channels, the
family of Maxwell-Boltzmann (MB) distributions leads to the optimal PS [1],
finding the optimal shaping for nonlinear-dispersive optical fiber channels is
a subtle problem [2]. Usually, a time-consuming numerical statistical modeling
is used for the optimization of either GS [3], PS [2], or constrained joint
geometric and probabilistic shaping (JS) [4].

End-to-end (E2E) learning [5] offers a different solution to this problem with
a gradient-based optimization of constellation shaping which does not require
any constraints on the spatial or probabilistic distributions of constellation
points. Nonetheless, the previous works on E2E learning of coherent systems
were focused on the optimization of GS [6, 7, 8]. Here, we apply the E2E
learning technique proposed in [9] to jointly optimize geometric and
probabilistic shaping in simulation for high symbol-rate coherent transmission
over single-span fiber link. Using training through the nonlinear interference
noise (NLIN) model [10] in the numerical simulation, we demonstrate that the
optimized JS constellation leads to a mutual information (MI) gain over the MB
PS of a quadarature-amplitude modulation (QAM) constellation.

$X$
---
$Y$
---
$X$
---
symbols $S$
---
probabilities $P_{S}$
---
sampler
---
$p(s_{k}|y_{i})$
---
loss $\mathcal{L}$
---
receiver
---
channel $p(y_{i}|x_{i})$
---
Fig. 1: The principal scheme of an end-to-end learning algorithm.

##  2 Joint Learning of Geometric and Probabilistic Shaping

In an autoencoder-based E2E learning, the trainable transmitter (TX), receiver
(RX) and the channel are implemented by cascades of neural networks (NN) which
are jointly trained to best reproduce the TX inputs from the RX outputs (Fig.
1) [5]. Following [9], the considered E2E architecture starts with the
trainable TX sampling a symbol sequence $X$. We implemented the TX utilizing a
trainable sampler based on the Gumbel-Softmax trick [9]. The TX has two sets
of trainable parameters: the set of the constellation symbols location,
denoted by $S=\\{s_{1},s_{2},...,s_{N}\\}$, and their occurrence
probabilities, denoted by $P_{S}=\\{p_{1},p_{2},...,p_{N}\\}$. Each symbol
location $s_{k}$ and probability $p_{k}$ is trained separately with the only
constraint on the average power: $\sum_{k}p_{k}|s_{k}|^{2}=1$.

Next, a pre-defined stochastic channel model, $p(y_{i}|x_{i})$, maps the
transmitted symbols ${X=\\{x_{i}\\},\,x_{i}\in S}$ to the received ones,
$Y=\\{y_{i}\\}$. We modeled the channel by a nonlinear interference noise
(NLIN) model [10]. In details, we approximated the symbol distortion as a
Gaussian noise with the variance
$\sigma^{2}=\sigma^{2}_{ASE}+P^{3}\left(\chi_{0}+\chi_{1}(\mu_{4}-2)+\chi_{2}(\mu_{6}-9\mu_{4}+12)+\chi_{3}(\mu_{4}-2)^{2}\right)$,
where $\sigma^{2}_{ASE}$ is the variance of amplified spontaneous emission
(ASE) noise injected by optical amplifiers (OAs), $P$ is the average launch
power, $\mu_{4},\,\mu_{6}$ are the 4th-order and 6th-order standardized
moments of the input sequence $X$, and
$\chi_{0},\,\chi_{1},\,\chi_{2},\,\chi_{3}$, are the modulation-format-
independent NLIN model coefficients. Then, RX maps each received symbol
$y_{i}$ to the vector of posterior probabilities $p(s_{k}|y_{i})$ for all
$s_{k}\in S$, estimating how likely $s_{k}$ was sent when $y_{i}$ is received.
Our RX estimates the posteriors distribution in closed form via Bayesian rule
from the conditional density defining the channel $p(y_{i}|x_{i})$ and the
symbol probabilities $P_{S}$. As shown in [9], the E2E MI, $I(X;Y)$, can be
characterized in terms of $p(s_{k}|y_{i})$ and $P_{S}$. To numerically
maximize MI, the batch gradient descent algorithm with Adam optimizer is used
to optimize the training parameters, i.e. $S$ and $P_{S}$. The gradients are
computed using back-propagation through the autoenconder.

To assess the learned constellations, we generated two independent random
sequences of constellation symbols sampled according to the optimized $P_{S}$.
These sequences are the transmission data over two polarizations. The
transmission is simulated using a precise Manakov equations split-step (MSS)
solver. Finally, the E2E mutual information of received symbols are calculated
using Gaussian kernel density estimator which numerically estimates the MI
with high accuracy.

Fig. 2: Numerical estimations of (a) mutual information (MI) over 4D symbols,
(d) effective signal-to-noise ratios (SNRs), (e) $\mu_{4}$ 4th-order, and (f)
$\mu_{6}$ 6th-order standardized moments for unshaped 256QAM, MB-shaped
256QAM, and E2E-learnt constellations over single-channel 64 GBd 170 km SMF
link; (b) E2E-learnt and (c) MB constellations at optimal power level, marker
size proportional to symbol probability $p(s_{k})$. SMF parameters are taken
as $D=16.8$ ps/(nm*km), $\gamma=$ 1.14 1/(W*km), $\alpha=$ 0.21 dB/km, OA
noise figure = 4.5 dB. The pulses are shaped by a root-raised-cosine filter
with roll-off 0.1.

##  3 Results and Conclusions

In order to obtain a strong dependence of the nonlinear distortion on the
modulation format a single-span link was studied [2]. Particularly, we
considered a dual-polarized single-channel 64 GBd transmission over a 170 km
standard single-mode fiber (SSMF) span, followed by an ideal OA.

The effectiveness of the E2E-learned 256-symbol JS constellation compared with
MB-shaped 256QAM and unshaped 256QAM was evaluated in terms of MI and reported
in Figure 2. The optimization process of E2E-learned and MB constellations was
done separately for each power level by considering the NLIN channel model,
and the MSS solver, respectively. The E2E-learned constellation provided a
peak-to-peak MI gain of 0.05 bits/4D over MB-shaped 256QAM and 0.45 bits/4D
over unshaped 256QAM. The learned JS offers an effective trade-off between
linear regime performance and tolerance to nonlinear distortion: in linear
regime, similarly to the MB PS, it leads to a better MI performance than the
unshaped constellation, while in the nonlinear regime it produces higher
effective SNR (Fig. 2d) than the MB PS, due to lower moments $\mu_{4},\mu_{6}$
(Figs. 2e, 2f). Note that a small mismatch between the optimal launch powers
in terms of SNR and MI curves is because the MI is computed by density
estimator in complex domain but SNR computation neglects correlations in
complex domain.

Acknowledgements: This project has received funding from EU Horizon 2020
program under the Marie Skłodowska-Curie grant agreement No. 766115 (FONTE).
SKT acknowledges the support of EPSRC project TRANSNET.

## References

  * [1] F. R. Kschischang et al. ”Optimal nonuniform signaling for Gaussian channels”. IEEE Trans. Inf. Theory 39.3 (1993), pp. 913-929.
  * [2] T. Fehenberger et al. ”On probabilistic shaping of quadrature amplitude modulation for the nonlinear fiber channel”. IEEE JLT 2020.
  * [3] B. Chen et al. “Increasing achievable information rates via geometric shaping”. ECOC 2020, pap. We1F.4.
  * [4] J. Cai et al. “Performance comparison of probabilistically shaped QAM formats and hybrid shaped APSK formats with coded modulation”. IEEE JLT 38.12 (2020), pp. 3280–3288.
  * [5] T. O’shea et al. “An introduction to deep learning for the physical layer”. IEEE Trans. Cogn. Commun. Netw. 3.4 (2017), pp. 563–575.
  * [6] R. T. Jones et al. “End-to-end learning for GMI optimized geometric constellation shape”. ECOC 2019, pap. W.1.B.3.
  * [7] T. Uhlemann et al. “Deep-learning Autoencoder for Coherent and Nonlinear Optical Communication”. arXiv:2006.15027v2
  * [8] J. Song et al. “End-to-end Autoencoder for Superchannel Transceivers with Hardware Impairment”.
  * [9] M. Stark et al. “Joint learning of geometric and probabilistic constellation shaping”. arXiv:1906.07748v3 (2019).
  * [10] R. Dar et al. “Accumulation of nonlinear interference noise in fiber-optic systems”. In: Opt. Express 22.12 (2014), pp. 14199–14211.
  * [11] D. P. Kingma et al. “Adam: A method for stochastic optimization”. arXiv:1412.6980 (2014).


# The extremal problem for weighted combined energy and the generalization of
Nitsche inequality

Xiaogao Feng∗ College of Mathematics and Information, China West Normal
University, Nanchong 637009,P.R.CHINA<EMAIL_ADDRESS>,  Ruyue Tang
College of Mathematics and Information, China West Normal University, Nanchong
637009,P.R.CHINA<EMAIL_ADDRESS>and  Ting Peng  College of Mathematics and
Information, China West Normal University, Nanchong 637009,P.R.CHINA
<EMAIL_ADDRESS>
###### Abstract.

We consider the existence and uniqueness of a minimizer of the extremal
problem for weighted combined energy between two concentric annuli and obtain
that the extremal mapping is a certain radial mapping. Meanwhile, this in turn
implies a Nitsche type phenomenon and we get a
$\frac{1}{|w|^{\lambda}}-$Nitsche type inequality ($\lambda\neq 1$). As an
application, on the basis of the relationship between weighted combined energy
and weighted combined distortion, we also investigate the extremal problem for
weighted combined distortion on annuli. This extends the result obtained by
Kalaj in [10].

###### Key words and phrases:

weighted combined energy, Nitsche type inequality, ODE

###### 2010 Mathematics Subject Classification:

30C62

Research supported by the National Natural Science Foundation of China (Grant
Nos.11701459 and 12271218).

∗ Corresponding author.

##  1\. Introduction

For two positive and distinct constants $r$ and $R$, let

$\mathbb{A}_{1}=\\{z:1\leq|z|\leq
r\\}\quad\text{and}\quad\mathbb{A}_{2}=\\{w:1\leq|w|\leq R\\}$

be rounded annuli in the complex plane $\mathbb{C}$. we consider the class
$\mathfrak{H}(\mathbb{A}_{1},\mathbb{A}_{2})$ of all orientation preserving
homeomorphisms $h$ from $\mathbb{A}_{1}$ onto $\mathbb{A}_{2}$ keeping orders
of the boundaries, namely,

$|h(z)|=1\quad\text{for}\quad|z|=1\quad\text{and}\quad|h(z)|=R\quad\text{for}\quad|z|=r.$
$None$

For $z=te^{i\theta}$, the normal and tangential derivatives of $h$ are

$h_{N}=h_{t}\quad\text{and}\quad h_{T}=\frac{1}{t}h_{\theta}.$ $None$

The derivatives $h_{z}$ and $h_{\overline{z}}$ are expressed as

$h_{z}=\frac{e^{-i\theta}}{2}(h_{N}-ih_{T})\quad\text{and}\quad
h_{\overline{z}}=\frac{e^{i\theta}}{2}(h_{N}+ih_{T}).$ $None$

We find the Hilbert-Schmidet norm of the differential matrix

$|Dh|^{2}=2(|h_{z}|^{2}+|h_{\overline{z}}|^{2})=|h_{N}|^{2}+|h_{T}|^{2},$
$None$

and the Jocobian determinant of $h$

$J(z,h)=|h_{z}|^{2}-|h_{\overline{z}}|^{2}=\Im(\overline{h_{N}}h_{T}).$ $None$

In [6] Iwaniec and Oninnen have considered the $n-$harmonic energy between two
annuli. When $n=2$, in [5] they gave a new proof. For a similar problem but
for non-circular annuli we refer to the paper [8] and its generalization in
[11]. In 2020, Kalaj [10] has generalized the harmonic energy to the combined
energy. He studied the following extremal problem

$\inf_{h\in\mathfrak{H}(\mathbb{A}_{1},\mathbb{A}_{2})}\iint_{\mathbb{A}_{1}}\left(a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}\right)dz$
$None$

and proved that the minimizer is certain radial mapping (see Theorem 3.1 in
[10]). Very recently, Yang, Tang and Feng [19] furthered (1.6) to the weighted
case and discussed the following the extremal problem

$\inf_{h\in\mathfrak{H}(\mathbb{A}_{1},\mathbb{A}_{2})}\iint_{\mathbb{A}_{1}}\left(a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}\right)\frac{1}{|h(z)|^{4}}dz.$
$None$

In this note, we mainly investigate the mapping in
$\mathfrak{H}(\mathbb{A}_{1},\mathbb{A}_{2})$ which minimizes the following
extremal problem for weighted combined energy

$\mathbb{E}_{\lambda}[h]=\iint_{\mathbb{A}_{1}}\left(a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}\right)\frac{1}{|h(z)|^{2\lambda}}dz,$
$None$

where $a>0$, $b>0$ and $\lambda\in\mathbb{R}$.

The harmonic energy has close relationship to Nitsche phenomena. In 1962,
Nitsche [16] conjectured that a necessary and sufficient condtion for
existence of a harmonic homeomorphism between two annuli $\mathbb{A}_{1}$ and
$\mathbb{A}_{2}$ is the following inequality

$r\leq R+\sqrt{R^{2}-1}\quad\text{or}\quad R\geq\frac{1}{2}(r+\frac{1}{r}).$
$None$

After various lower bounds for $R$ ([14], [18], [17]), the Nitsche conjecture
was finally solved by Iwaniec, Kovalev and Onninen [7] in 2011.

Recall that, given a Riemannian metric $\rho$ on $\mathbb{A}_{2}$, a $C^{2}$
homeomorphism $h$ is said to be harmonic with respect to $\rho$ (or
$\rho-$harmonic) if

$h_{z\overline{z}}(z)+\left(\log\rho^{2}\right)_{w}\circ
hh_{z}h_{\overline{z}}=0.$

On the basis of Nitsche conjecture, Kalaj [13] proposed the so-called
$\rho-$Nitsche conjecture as follows

$\rho-$Nitsche conjecture _If there exists a $\rho-$harmonic homeomorphism of
the annuli $\mathbb{A}_{1}$ onto $\mathbb{A}_{2}$, then_

$r\leq\exp\left(\int_{1}^{R}\frac{\rho(s)}{\sqrt{s^{2}\rho^{2}(s)-\alpha_{0}}}ds\right),$
$None$

_where_ $\alpha_{0}=\inf_{1\leq s\leq R}(\rho^{2}(s))s^{2}$.

It should be pointed out that the $\rho-$Nitsche conjecture corresponds to the
classical Nitsche conjecture when $\rho=1$. In [12] a partial result is proved
on the $\rho-$Nitsche conjecture. In particular, when $\rho(w)=|w|^{-2}$, then
inequality (1.10) becomes

$r\leq R+\sqrt{R^{2}-1}.$ $None$

According to this property, Feng and Tang [4] gave a positive answer when
$\rho(w)=|w|^{-2}$. Recently, Kalaj [10] has considered the extremal problem
(1.6) and obtained a Nitsche type inequality

$R\geq\cosh(\frac{b}{a}\log r)=\frac{1+r^{\frac{2b}{a}}}{2r^{\frac{b}{a}}}.$
$None$

Notice that when $a=b$, (1.12) turns out to be the classic Nitsche inequality
(1.9).

Very recently, in [19] we have considered the extremal problem (1.7) and got
$\frac{1}{|w|^{2}}-$Nitsche type inequality

$r\leq(R+\sqrt{R^{2}-1})^{\frac{a}{b}}.$ $None$

Moreover, we obtain the following result.

Theorem A (see Theorem 3.1 in [19]) Under condition (1.13) and among all
mappings in $\mathfrak{H}(\mathbb{A}_{1},\mathbb{A}_{2})$ , the infimum of

$\iint_{\mathbb{A}_{1}}\left(a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}\right)\frac{1}{|h(z)|^{4}}dz$

is attained at the radial mapping

$h_{\ast}(z)=\frac{2(1+\sqrt{1+\frac{\alpha}{b^{2}}})|z|^{\frac{b}{a}}}{(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{2}-|z|^{\frac{2b}{a}}\frac{\alpha}{b^{2}}}e^{i\theta},$
$None$

where $\alpha$ satisfies

$R=\frac{2r^{\frac{b}{a}}(1+\sqrt{1+\frac{\alpha}{b^{2}}})}{(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{2}-r^{\frac{2b}{a}}\frac{\alpha}{b^{2}}}.$
$None$

The minimizer is unique up to a rotation of annuli.

In this paper, we shall generalize the extremal problems (1.6) and (1.7) to
(1.8) and get the following main theorem.

######  Theorem 1.1.

(I) When $\lambda\neq 1$ in (1.8), under condition

$r\leq\left(R^{|\lambda-1|}+\sqrt{R^{2|\lambda-1|}-1}\right)^{\frac{1}{|\lambda-1|}\frac{a}{b}}$
$None$

and among all mapping $\mathfrak{H}(\mathbb{A}_{1},\mathbb{A}_{2})$, the
infimum of

$\iint_{\mathbb{A}_{1}}\left(a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}\right)\frac{dz}{|h(z)|^{2\lambda}}$

is attained at the radial mapping

$h^{\ast}_{\lambda}(z)=\frac{2^{\frac{1}{\lambda-1}}(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{\frac{1}{\lambda-1}}|z|^{\frac{b}{a}}}{\left[(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{2}-|z|^{\frac{2(\lambda-1)b}{a}}\frac{\alpha}{b^{2}}\right]^{\frac{1}{\lambda-1}}}e^{i\theta},$
$None$

where $\alpha$ satisfies

$R=\frac{2^{\frac{1}{\lambda-1}}(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{\frac{1}{\lambda-1}}r^{\frac{b}{a}}}{\left[(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{2}-r^{\frac{2(\lambda-1)b}{a}}\frac{\alpha}{b^{2}}\right]^{\frac{1}{\lambda-1}}}e^{i\theta}.$
$None$

That is to say,

$\iint_{\mathbb{A}_{1}}\left(a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}\right)\frac{dz}{|h(z)|^{2\lambda}}\geq\iint_{\mathbb{A}_{1}}\left(a^{2}|h^{\ast}_{\lambda
N}|^{2}+b^{2}|h^{\ast}_{\lambda
T}|^{2}\right)\frac{dz}{|h^{\ast}_{\lambda}(z)|^{2\lambda}}.$ $None$

The minimizer is unique up to a rotation of annuli.

(II)When $\lambda=1$ in (1.8), the weighted combined energy
$\mathbb{E}_{\lambda}[h]$ attains its minimum for a radial mapping

$h_{1}^{\ast}(z)=|z|^{\frac{\ln R}{\ln r}}e^{i\theta}.$ $None$

The minimizer is unique up to a rotation of annuli.

Remark 1.2 When $\lambda=0$, (I) in Theorem 1.1 happens to be Theorem 3.1 in
[10].

Remark 1.3 When $\lambda=2$, (I) in Theorem 1.1 happens to be Theorem A (see
Theorem 3.1 in [19]).

Remark 1.4 When $\lambda=0$, the inequality (1.16) becomes (1.12).

Remark 1.5 When $\lambda=2$, the inequality (1.16) becomes (1.13).

We end this introduction section with the organization of the paper. In
section 2 we shall investigate the weighted combined energy of radial mapping
and construct radial minimizers of weighted combined energy. In Section 3, we
will give the proof of Theorem 1.1. The last section, By means of the
relationship between weighted combined energy and weighted combined
distortion, we also investigate the extremal problem for weighted combined
distortion.

##  2\. weighted combined energy of radial mapping

For mappings of annuli, it is natural to examine the radial mappings. Assume

$h(z)=H(t)e^{i\theta}\quad\text{for}\quad z=te^{i\theta},$

where $H:[1,r]\rightarrow[1,R]$ is continuous strictly increasing and
satisfies the boundary conditions (1.1). We obtain that

$\mathbb{E}_{\lambda}[h]=2\pi\int_{1}^{r}\frac{a^{2}t^{2}\dot{H}^{2}(t)+b^{2}H^{2}(t)}{tH^{2\lambda}(t)}dt=2\pi\int_{1}^{r}F(t,H(t),\dot{H}(t))dt.$
$None$

Then

$\frac{\partial F}{\partial H}=-\frac{2\lambda
a^{2}t\dot{H}^{2}(t)}{H^{2\lambda+1}(t)}-\frac{(2\lambda-2)b^{2}}{tH^{2\lambda-1}(t)},$
$\frac{\partial
F}{\partial\dot{H}}=\frac{2a^{2}t\dot{H}(t)}{H^{2\lambda}(t)},$

and

$\frac{d}{dt}\left(\frac{\partial
F}{\partial\dot{H}}\right)=\frac{2a^{2}\dot{H}(t)}{H^{2\lambda}(t)}+\frac{2a^{2}t\ddot{H}(t)}{H^{2\lambda}(t)}-\frac{4\lambda
a^{2}t\dot{H}^{2}(t)}{H^{2\lambda+1}(t)}.$

According to the Euler-Lagrange equation

$\frac{\partial F}{\partial H}=\frac{d}{dt}\left(\frac{\partial
F}{\partial\dot{H}}\right),$ $None$

we get

$a^{2}t^{2}H(t)\ddot{H}(t)+a^{2}tH(t)\dot{H}(t)+(\lambda-1)b^{2}H^{2}(t)=\lambda
a^{2}t^{2}\dot{H}^{2}(t).$ $None$

Set $t=e^{x}$, $y=H(t)=H(e^{x})$, then $y^{\prime}=\dot{H}(t)e^{x}$.
Furthermore,

$y^{\prime\prime}=\ddot{H}(t)e^{2x}+\dot{H}(t)e^{x}.$

Then

$\dot{H}(t)=\frac{y^{\prime}}{e^{x}}\quad\text{and}\quad\ddot{H}(t)=\frac{y^{\prime\prime}-y^{\prime}}{e^{2x}}.$
$None$

We put (2.4) into (2.3) and obtain

$a^{2}yy^{\prime\prime}+(\lambda-1)b^{2}y^{2}=\lambda a^{2}y^{\prime 2}.$
$None$

Moreover, set $y^{\prime}=\zeta$, then

$y^{\prime\prime}=\frac{d\zeta}{dy}\frac{dy}{dx}=\zeta^{\prime}\zeta.$ $None$

According to (2.5) and (2.6), we have

$a^{2}\zeta\zeta^{\prime}+(\lambda-1)b^{2}y=\frac{\lambda}{y}a^{2}\zeta^{2}.$
$None$

Thus

$a^{2}\zeta\zeta^{\prime}-b^{2}y=\frac{\lambda}{y}(a^{2}\zeta^{2}-b^{2}y^{2}).$
$None$

Let

$\omega=a^{2}\zeta^{2}-b^{2}y^{2},$ $None$

then $\omega^{\prime}=2a^{2}\zeta\zeta^{\prime}-2b^{2}y$. Hence, the equality
(2.8) reduces to

$\frac{\omega^{\prime}}{\omega}=\frac{2\lambda}{y}.$ $None$

We get $\omega=\alpha y^{2\lambda}$. Together with (2.9), we get

$\alpha y^{2\lambda}=a^{2}\zeta^{2}-b^{2}y^{2},$ $None$

and

$y^{\prime}=\frac{dy}{dx}=\zeta=\frac{\sqrt{\alpha
y^{2\lambda}+b^{2}y^{2}}}{a}$ $None$

which also can be written as

$dx=\frac{a}{\sqrt{\alpha y^{2\lambda}+b^{2}y^{2}}}dy.$ $None$

Thus, for $1\leq y\leq R$, we have

$x(y)=\int_{1}^{y}\frac{a}{\sqrt{\alpha s^{2\lambda}+b^{2}s^{2}}}ds.$ $None$

Furthermore,

$t=e^{x}=\exp\left(\int_{1}^{y}\frac{a}{\sqrt{\alpha
s^{2\lambda}+b^{2}s^{2}}}ds\right).$ $None$

In order to discuss (2.15), we divide $\lambda$ into three cases: $\lambda>1$,
$\lambda<1$ and $\lambda=1$.

When $\lambda>1$ in (2.15), we take $\alpha=-\inf_{1\leq s\leq
R}(\frac{b^{2}}{s^{2(\lambda-1)}})=-\frac{b^{2}}{R^{2\lambda-2}}$ and have the
following inequality

$\displaystyle r$ $\displaystyle\leq$
$\displaystyle\exp\left[\frac{1}{1-\lambda}\frac{a}{b}\int_{1}^{R}\frac{d(\frac{b}{s^{\lambda-1}})}{\sqrt{\left(\frac{b}{s^{\lambda-1}}\right)^{2}-\frac{b^{2}}{R^{2\lambda-2}}}}\right]$
$\displaystyle=$
$\displaystyle\left(R^{\lambda-1}+\sqrt{R^{2(\lambda-1)}-1}\right)^{\frac{1}{\lambda-1}\frac{a}{b}},\quad\quad(\frac{1}{|w|^{\lambda}}-\text{Nitsche
type inequality})$ $None$

which happens to be (1.16) when $\lambda>1$. Notice that when $\lambda=2$,
(2.16) becomes (1.13). Next we will construct the radial mapping
$h^{\ast}_{\lambda}(z)$ (see (1.17)). For
$\alpha\geq-\frac{b^{2}}{R^{2\lambda-2}}$, we obtain

$\exp\left(\frac{1}{1-\lambda}\frac{a}{b}\int_{1}^{R}\frac{d(\frac{b}{s^{\lambda-1}})}{\sqrt{(\frac{b}{s^{\lambda-1}})^{2}+\alpha}}\right)=\left[\frac{R^{\lambda-1}(1+\sqrt{1+\frac{\alpha}{b^{2}}})}{1+\sqrt{1+\frac{R^{2\lambda-2}\alpha}{b^{2}}}}\right]^{\frac{1}{\lambda-1}\frac{a}{b}}.$
$None$

Set

$\varphi(\alpha)=\left[\frac{R^{\lambda-1}(1+\sqrt{1+\frac{\alpha}{b^{2}}})}{1+\sqrt{1+\frac{R^{2\lambda-2}\alpha}{b^{2}}}}\right]^{\frac{1}{\lambda-1}\frac{a}{b}},$
$None$

then

$\displaystyle\varphi^{\prime}(\alpha)$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda-1}\frac{a}{b}\left[\frac{R^{{}^{\lambda-1}}(1+\sqrt{1+\frac{\alpha}{b^{2}}})}{1+\sqrt{1+\frac{R^{2\lambda-2}\alpha}{b^{2}}}}\right]^{\frac{1}{\lambda-1}\frac{a}{b}-1}$
$\displaystyle\times\frac{R}{2b^{2}}\frac{\sqrt{1+\frac{R^{2\lambda-2}\alpha}{b^{2}}}-R^{2\lambda-2}\sqrt{1+\frac{\alpha}{b^{2}}}+1-R^{2\lambda-2}}{\sqrt{1+\frac{\alpha}{b^{2}}}\sqrt{1+\frac{R^{2\lambda-2}\alpha}{b^{2}}}\left(1+\sqrt{1+\frac{R^{2\lambda-2}\alpha}{b^{2}}}\right)^{2}}.$
$None$

Since $\alpha\geq-\frac{b^{2}}{R^{2\lambda-2}}$ and $\lambda>1$, we get

$\displaystyle\sqrt{1+\frac{R^{2(\lambda-1)}\alpha}{b^{2}}}-R^{2(\lambda-1)}\sqrt{1+\frac{\alpha}{b^{2}}}$
$\displaystyle=$
$\displaystyle\frac{\left[1-R^{2(\lambda-1)}\right]\left[1+R^{2(\lambda-1)}+\frac{R^{2(\lambda-1)}\alpha}{b^{2}}\right]}{\sqrt{1+\frac{R^{2(\lambda-1)}\alpha}{b^{2}}}+R^{2(\lambda-1)}\sqrt{1+\frac{\alpha}{b^{2}}}}$
$\displaystyle<$ $\displaystyle 0.$ $None$

Combining (2.19) and (2.20), we have $\varphi^{\prime}(\alpha)<0$. Thus, for
$\alpha\geq-\frac{b^{2}}{R^{2\lambda-2}}$, $\varphi(\alpha)$ is strictly
decreasing. By (2.16) and

$\displaystyle\varphi(-\frac{b^{2}}{R^{2\lambda-2}})$ $\displaystyle=$
$\displaystyle\left[\frac{R^{\lambda-1}(1+\sqrt{1+\frac{-\frac{b^{2}}{R^{2\lambda-2}}}{b^{2}}})}{1+\sqrt{1+\frac{R^{2\lambda-2}}{b^{2}}(-\frac{b^{2}}{R^{2\lambda-2}})}}\right]^{\frac{1}{\lambda-1}\frac{a}{b}}$
$\displaystyle=$
$\displaystyle\left(R^{\lambda-1}+\sqrt{R^{2\lambda-2}-1}\right)^{\frac{1}{\lambda-1}\frac{a}{b}}.$
$None$

We obtain that there exists a unique $\alpha\geq-\frac{b^{2}}{R^{2\lambda-2}}$
such that $\varphi(\alpha)=r$, that is,

$\left[\frac{R^{\lambda-1}(1+\sqrt{1+\frac{\alpha}{b^{2}}})}{1+\sqrt{1+\frac{R^{2\lambda-2}\alpha}{b^{2}}}}\right]^{\frac{1}{\lambda-1}\frac{a}{b}}=r.$
$None$

We conclude that

$R=\frac{2^{\frac{1}{\lambda-1}}(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{\frac{1}{\lambda-1}}r^{\frac{b}{a}}}{\left[(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{2}-r^{\frac{2(\lambda-1)b}{a}}\frac{\alpha}{b^{2}}\right]^{\frac{1}{\lambda-1}}}.$
$None$

So we get the radial mapping

$h^{\ast}_{\lambda}(z)=\frac{2^{\frac{1}{\lambda-1}}(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{\frac{1}{\lambda-1}}|z|^{\frac{b}{a}}}{\left[(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{2}-|z|^{\frac{2(\lambda-1)b}{a}}\frac{\alpha}{b^{2}}\right]^{\frac{1}{\lambda-1}}}e^{i\theta},$
$None$

where $\alpha$ satisfies (2.23). By (2.23) and direct computation we obtain

$\mathbb{E}[h^{\ast}_{\lambda}]=\frac{2\pi
ab}{\lambda-1}\left(\sqrt{1+\frac{\alpha}{b^{2}}}-\frac{1}{R^{\lambda-1}}\sqrt{\frac{1}{R^{2(\lambda-1)}}+\frac{\alpha}{b^{2}}}\right).$
$None$

When $\lambda<1$ in (2.15), we take $\alpha=-\inf_{1\leq s\leq
R}\left(\frac{b^{2}}{s^{2(\lambda-1)}}\right)=-b^{2}$ and have the following
inequality

$\displaystyle r$ $\displaystyle\leq$
$\displaystyle\exp\left[\frac{1}{1-\lambda}\frac{a}{b}\int_{1}^{R}\frac{d(\frac{b}{s^{\lambda-1}})}{\sqrt{\left(\frac{b}{s^{\lambda-1}}\right)^{2}-\frac{b^{2}}{R^{2\lambda-2}}}}\right]$
$\displaystyle=$
$\displaystyle\left(\frac{1}{R^{\lambda-1}}+\sqrt{\frac{1}{R^{2(\lambda-1)}}-1}\right)^{\frac{1}{1-\lambda}\frac{a}{b}}$
$\displaystyle=$
$\displaystyle\left(R^{1-\lambda}+\sqrt{R^{2-2\lambda}-1}\right)^{\frac{1}{1-\lambda}\frac{a}{b}},\quad\quad(\frac{1}{|w|^{\lambda}}-\text{Nitsche
type inequality})$ $None$

which happens to be (1.16). Notice that when $\lambda=0$, (2.26) becomes
(1.12). For $\alpha\geq-b^{2}$ and $\lambda<1$, we get

$\displaystyle\sqrt{1+\frac{R^{2(\lambda-1)}\alpha}{b^{2}}}-R^{2(\lambda-1)}\sqrt{1+\frac{\alpha}{b^{2}}}$
$\displaystyle=$
$\displaystyle\frac{\left[1-R^{2(\lambda-1)}\right]\left[1+R^{2(\lambda-1)}+\frac{R^{2(\lambda-1)}\alpha}{b^{2}}\right]}{\sqrt{1+\frac{R^{2(\lambda-1)}\alpha}{b^{2}}}+R^{2(\lambda-1)}\sqrt{1+\frac{\alpha}{b^{2}}}}$
$\displaystyle>$ $\displaystyle 0.$

Since $\frac{1}{\lambda-1}<0$, then $\varphi^{\prime}(\alpha)<0$ (see (2.19)).
Thus, $\varphi(\alpha)$ is strictly decreasing. By (2.26) and

$\displaystyle\varphi(-b^{2})$ $\displaystyle=$
$\displaystyle\left[\frac{R^{\lambda-1}(1+\sqrt{1+\frac{-b^{2}}{b^{2}}})}{1+\sqrt{1+\frac{R^{2\lambda-2}}{b^{2}}(-b^{2})}}\right]^{\frac{1}{\lambda-1}\frac{a}{b}}$
$\displaystyle=$
$\displaystyle\left(R^{1-\lambda}+\sqrt{R^{2-2\lambda}-1}\right)^{\frac{1}{1-\lambda}\frac{a}{b}}.$
$None$

We obtain there exists a unique $\alpha\geq-b^{2}$, such that
$\varphi(\alpha)=r$. Thus we also conclude that (2.23-2.24) from discussion
like $\lambda>1$. The inequality (2.16) and (2.26) are unified as (1.16).

When $\lambda=1$, (2.15) becomes

$t=\exp\left(\int_{1}^{y}\frac{a}{\sqrt{\alpha+b^{2}}}\frac{1}{s}ds\right)=y^{\frac{a}{\sqrt{\alpha+b^{2}}}}.$
$None$

So we get $H(t)=t^{\frac{\sqrt{\alpha+b^{2}}}{a}}$. By $H(r)=R$, we have
$\frac{\sqrt{\alpha+b^{2}}}{a}=\frac{\ln R}{\ln r}$. Therefore we obtain the
radial mapping

$h_{1}^{\ast}(te^{i\theta})=t^{\frac{\ln R}{\ln r}}e^{i\theta}.$ $None$

And a direct computation yields

$\mathbb{E}[h_{1}^{\ast}]=2\pi\ln R\left(a^{2}\frac{\ln R}{\ln
r}+b^{2}\frac{\ln r}{\ln R}\right).$ $None$

##  3\. Proof of Theorem 1.1

###  3.1. When $\lambda\neq 1$

In view of $J(z,h)=\Im(\overline{h_{{}_{N}}}h_{T})$ and $|h|_{N}\leq|h_{N}|$,
for $p$, $q$, $A$, $B\in\mathbb{R}$, we obtain the following general
inequality

$\displaystyle\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2\lambda}}$
$\displaystyle\geq$
$\displaystyle\frac{(a^{2}-b^{2}p^{2})|h_{N}|^{2}+(b^{2}-a^{2}q^{2})|h_{T}|^{2}+2abpq|h_{N}||h_{T}|}{|h(z)|^{2\lambda}}$
$\displaystyle\geq$
$\displaystyle\frac{2(a^{2}-b^{2}p^{2})At\frac{|h|_{N}}{t}-(a^{2}-b^{2}p^{2})A^{2}+2(b^{2}-a^{2}q^{2})|h|\Im[\frac{h_{T}}{h}]B}{|h(z)|^{2\lambda}}$
$\displaystyle\frac{-(b^{2}-a^{2}q^{2})B^{2}+2abpqJ(z,h)}{|h(z)|^{2\lambda}}.$
$None$

Case 1: We take $q=\frac{b}{a}$,
$p=\frac{\frac{a}{|h|^{\lambda-1}}}{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}$,
$A=\frac{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}{at}|h|^{\lambda}$,
and (3.1) becomes

$\displaystyle\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2\lambda}}$
$\displaystyle\geq$
$\displaystyle\frac{2(a^{2}-b^{2}p^{2})At\frac{|h|_{N}}{t}+2abpqJ(z,h)-(a^{2}-b^{2}p^{2})A^{2}}{|h(z)|^{2\lambda}}$
$\displaystyle=$
$\displaystyle\frac{2a\alpha}{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}\frac{|h|_{N}}{t}\frac{1}{|h|^{\lambda}}+2b^{2}\frac{\frac{a}{|h|^{3\lambda-1}}}{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}J(z,h)-\frac{\alpha}{t^{2}}.$
$None$

Integrating both sides, we obtain

$\displaystyle\iint_{\mathbb{A}_{1}}\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2\lambda}}dz$
$\displaystyle\geq$
$\displaystyle\iint_{\mathbb{A}_{1}}\frac{2a\alpha}{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}\frac{|h|_{N}}{t}\frac{1}{|h|^{\lambda}}dz+2b^{2}\iint_{\mathbb{A}_{1}}\frac{\frac{a}{|h|^{3\lambda-1}}J(z,h)}{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}dz-\iint_{\mathbb{A}_{1}}\frac{\alpha}{t^{2}}dz$
$\displaystyle=$ $\displaystyle 4\pi
a\alpha\int_{1}^{r}\frac{\frac{1}{|h|^{\lambda}}}{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}\frac{\partial|h|}{\partial
t}dt+4\pi
ab\int_{1}^{R}\frac{\frac{1}{\tau^{3\lambda-2}}}{\sqrt{\frac{1}{\tau^{2(\lambda-1)}}+\frac{\alpha}{b^{2}}}}d\tau-2\pi\alpha\int_{1}^{r}\frac{1}{t}dt$
$\displaystyle=$ $\displaystyle\frac{4\pi
a\alpha}{b(1-\lambda)}\ln\frac{\frac{b}{R^{{}^{\lambda-1}}}+\sqrt{\frac{b^{{}^{2}}}{R^{2(\lambda-1)}}+\alpha}}{b+\sqrt{b^{2}+\alpha}}+\frac{2\pi
a\alpha}{b(1-\lambda)}\ln\frac{b+\sqrt{b^{2}+\alpha}}{\frac{b}{R^{\lambda-1}}+\sqrt{\frac{b^{{}^{2}}}{R^{{}^{2(\lambda-1)}}}+\alpha}}$
$\displaystyle+\frac{2\pi
a}{(1-\lambda)b}\left[\frac{b}{R^{\lambda-1}}\sqrt{\frac{b^{2}}{R^{2(\lambda-1)}}+\alpha}-b\sqrt{b^{2}+\alpha}\right]-2\pi\alpha\ln
r.$ $None$

It is easy to see from (2.23), (2.25) and (3.3) that

$\displaystyle\mathbb{E}_{\lambda}[h]$ $\displaystyle=$
$\displaystyle\iint_{\mathbb{A}_{1}}\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2\lambda}}dz$
$\displaystyle\geq$ $\displaystyle\frac{2\pi
ab}{\lambda-1}\left(\sqrt{1+\frac{\alpha}{b^{2}}}-\frac{1}{R^{{}^{\lambda-1}}}\sqrt{\frac{1}{R^{2(\lambda-1)}}+\frac{\alpha}{b^{2}}}\right)$
$\displaystyle=$
$\displaystyle\iint_{\mathbb{A}_{1}}\frac{a^{2}|h^{\ast}_{\lambda
N}|^{2}+b^{2}|h^{\ast}_{\lambda
T}|^{2}}{|h^{\ast}_{\lambda}(z)|^{2\lambda}}dz$ $\displaystyle=$
$\displaystyle\mathbb{E}_{\lambda}[h^{\ast}_{\lambda}].$ $None$

$\mathbf{Case}$ $\mathbf{2:}$ We take $p=\frac{a}{b}$,
$q=\frac{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}{\frac{a}{|h|^{\lambda-1}}}$,
$B=\frac{|h|}{t}$, and (3.1) reduces to

$\displaystyle\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2\lambda}}$
$\displaystyle\geq$
$\displaystyle\frac{2(b^{2}-a^{2}q^{2})B|h|\Im\frac{h_{T}}{h}+2abpqJ(z,h)-(b^{2}-a^{2}q^{2})B^{2}}{|h(z)|^{2\lambda}}$
$\displaystyle=$
$\displaystyle\frac{2\left(b^{2}-a^{2}\frac{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}{\frac{a^{2}}{|h|^{2(\lambda-1)}}}\right)|h|\frac{|h|}{t}\Im\frac{h_{T}}{h}+2a^{2}\frac{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}{\frac{a}{|h|^{\lambda-1}}}J(z,h)}{|h|^{2\lambda}}$
$\displaystyle\frac{-\left(b^{2}-a^{2}\frac{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}{\frac{a^{2}}{|h|^{2(\lambda-1)}}}\right)\frac{|h|^{2}}{t^{2}}}{|h|^{2\lambda}}$
$\displaystyle=$
$\displaystyle-2\alpha\frac{\Im\frac{h_{T}}{h}}{t}+2a\frac{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}{|h|^{\lambda+1}}J(z,h)+\frac{\alpha}{t^{2}}.$
$None$

Integrating this inequality (3.5) over $\mathbb{A}_{1}$, we have

$\displaystyle\iint_{\mathbb{A}_{1}}\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2\lambda}}dz$
$\displaystyle\geq$
$\displaystyle-2\alpha\iint_{\mathbb{A}_{1}}\frac{\Im\frac{h_{T}}{h}}{t}dz+2a\iint_{\mathbb{A}_{1}}\frac{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}{|h|^{\lambda+1}}J(z,h)dz+\iint_{\mathbb{A}_{1}}\frac{\alpha}{t^{2}}dz$
$\displaystyle=$ $\displaystyle-4\pi\alpha\int_{1}^{r}\frac{1}{t}dt+\frac{4\pi
ab}{1-\lambda}\int_{1}^{R}\sqrt{\frac{1}{\tau^{{}^{2(\lambda-1)}}}+\frac{\alpha}{b^{2}}}d\left(\frac{1}{\tau^{{}^{\lambda-1}}}\right)+2\pi\alpha\int_{1}^{r}\frac{1}{t}dt$
$\displaystyle=$ $\displaystyle-2\pi\alpha\ln r+\frac{2\pi
ab}{1-\lambda}\left(\frac{1}{R^{{}^{\lambda-1}}}\sqrt{\frac{1}{R^{2(\lambda-1)}}+\frac{\alpha}{b^{2}}}-\sqrt{1+\frac{\alpha}{b^{2}}}\right)$
$\displaystyle+\frac{2\pi
a\alpha}{(1-\lambda)b}\ln\frac{\frac{1}{R^{\lambda-1}}+\sqrt{\frac{1}{R^{2(\lambda-1)}}+\frac{\alpha}{b^{2}}}}{1+\sqrt{1+\frac{\alpha}{b^{2}}}}.$
$None$

By means of (2.23), (2.25) and (3.6), we obtain (3.4) again.

$\mathbf{Uniqueness}$ Let

$h(te^{i\theta})=\rho(t,\theta)e^{i\varphi(t,\theta)}.$ $None$

Firstly, we prove the uniqueness in Case 1. Now suppose the equality in (3.4)
holds. Then all the equalities (3.1) and (3.2) must hold, which implies that

$J(z,h)=\Im(\overline{h_{N}}h_{T})=|h_{N}||h_{T}|,$ $None$ $|h_{N}|=|h|_{N},$
$None$ $bp|h_{N}|=aq|h_{T}|,$ $None$
$A=|h_{N}|=\frac{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}{at}|h|^{\lambda},$
$None$ $|h_{T}|=|h|\Im[\frac{h_{T}}{h}].$ $None$

By (3.9), we have

$\rho_{t}=|h|_{N}=|h_{N}|=|h_{t}|=\sqrt{\rho_{t}^{2}+\rho^{2}\varphi_{t}^{2}},$
$None$

then $\varphi_{t}=0$. Thus $\varphi(t,\theta)=\varphi(\theta)$. By
$q=\frac{b}{a}$,
$p=\frac{\frac{a}{|h|^{\lambda-1}}}{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}$
and (3.10), we know that $bp|h_{N}|=aq|h_{T}|=b|h_{T}|$, then

$p|h_{N}|=|h_{T}|.$ $None$

By means of (3.11) and (3.14), we get

$\frac{\frac{a}{|h|^{\lambda-1}}}{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}\frac{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}{at}|h|^{\lambda}=|h_{T}|$

which reduces to

$\frac{|h|}{t}=|h_{T}|.$ $None$

Thus

$\frac{\rho}{t}=|h_{T}|=|\frac{\rho_{\theta}e^{i\varphi}+\rho
e^{i\varphi}i\varphi_{\theta}}{t}|=\frac{\sqrt{\rho^{2}_{\theta}+\rho^{2}\varphi^{2}_{\theta}}}{t},$
$None$

then

$\rho=\sqrt{\rho^{2}_{\theta}+\rho^{2}\varphi^{2}_{\theta}}.$ $None$

By (3.12) and (3.15-3.17), we obtain

$\frac{\sqrt{\rho^{2}_{\theta}+\rho^{2}\varphi^{2}_{\theta}}}{t}=\rho\frac{\varphi_{\theta}}{t}=\sqrt{\rho^{2}_{\theta}+\rho^{2}\varphi^{2}_{\theta}}\frac{\varphi_{\theta}}{t},$

then

$\varphi_{\theta}=1.$ $None$

According to $\varphi_{t}=0$, we have $\varphi=\theta+\beta$, where $\beta$ is
a real constant. By (3.17) and (3.18) we infer that
$\rho=\sqrt{\rho^{2}_{\theta}+\rho^{2}}$, then $\rho_{\theta}=0$. Thus
$\rho(t,\theta)=\rho(t)$. By (3.10) and (3.12) we have
$p|h_{N}|=|h_{T}|=|h|\Im[\frac{h_{T}}{h}],$ and then
$p\rho_{t}=\rho\frac{\varphi_{\theta}}{t}=\frac{\rho}{t},$ that is
$\frac{\frac{a}{|h|^{\lambda-1}}}{\sqrt{\frac{b^{2}}{|h|^{2(\lambda-1)}}+\alpha}}\rho_{t}=\frac{\rho}{t}.$
We have

$\frac{\frac{a}{(1-\lambda)b}d\left(\frac{b}{\rho^{\lambda-1}}\right)}{\sqrt{\frac{b^{2}}{\rho^{2(\lambda-1)}}+\alpha}}=\frac{dt}{t}.$
$None$

After integrating the equality (3.19) and using that $|h|=1$ for $|z|=1$, we
arrive at

$\rho=\frac{2^{\frac{1}{\lambda-1}}(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{\frac{1}{\lambda-1}}t^{\frac{b}{a}}}{\left[(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{2}-t^{\frac{2(\lambda-1)b}{a}}\frac{\alpha}{b^{2}}\right]^{\frac{1}{\lambda-1}}}.$
$None$

Thus

$h(z)=h^{\ast}_{\lambda}(z)=\frac{2^{\frac{1}{\lambda-1}}(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{\frac{1}{\lambda-1}}|z|^{\frac{b}{a}}}{\left[(1+\sqrt{1+\frac{\alpha}{b^{2}}})^{2}-|z|^{\frac{2(\lambda-1)b}{a}}\frac{\alpha}{b^{2}}\right]^{\frac{1}{\lambda-1}}}e^{i\theta+\beta},$
$None$

where $\beta$ is a real constant.

The uniqueness in case 2 is obtained in a similar way.

###  3.2. When $\lambda=1$

Obviously, for $p_{\ast}$, $q_{\ast}\in\mathbb{R}$, we have the following
inequality

$\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2}}\geq\frac{(a^{2}-b^{2}p_{\ast}^{2})|h_{N}|^{2}+(b^{2}-a^{2}q_{\ast}^{2})|h_{T}|^{2}+2p_{\ast}q_{\ast}ab|h_{N}|h_{T}|}{|h(z)|^{2}}.$
$None$

We also consider two cases.

Case 1: Let $q_{\ast}=\frac{b}{a}$, and (3.22) reduces to

$\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2}}\geq\frac{(a^{2}-b^{2}p_{\ast}^{2})|h_{N}|^{2}+2p_{\ast}b^{2}|h_{N}||h_{T}|}{|h(z)|^{2}}.$
$None$

Since $|h_{N}||h_{T}|\geq\Im(\overline{h_{N}}h_{T})=J(z,h)$ and
$|h_{N}|\geq|h|_{N},$ by (3.23) we have

$\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2}}\geq\frac{(a^{2}-b^{2}p_{\ast}^{2})|h|_{N}^{2}+2p_{\ast}b^{2}J(z,h)}{|h(z)|^{2}}.$
$None$

By integrating over $\mathbb{A}_{1}$, we can get

$\iint_{\mathbb{A}_{1}}\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2}}dz\geq(a^{2}-b^{2}p_{\ast}^{2})\iint_{\mathbb{A}_{1}}\frac{|h|_{N}^{2}}{|h(z)|^{2}}dz+2p_{\ast}b^{2}\iint_{\mathbb{A}_{1}}\frac{J(z,h)}{|h(z)|^{2}}dz.$
$None$

By H$\ddot{{\rm o}}$lder inequality, we have

$\displaystyle\Big{(}\iint_{\mathbb{A}_{1}}\frac{|h|_{N}}{|z||h(z)|}dz\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\iint_{\mathbb{A}_{1}}\frac{|h|_{N}^{2}}{|h(z)|^{2}}dz\iint_{\mathbb{A}_{1}}\frac{1}{|z|^{2}}dz$
$\displaystyle=$ $\displaystyle 2\pi\ln
r\iint_{\mathbb{A}_{1}}\frac{|h|_{N}^{2}}{|h(z)|^{2}}dz.$ $None$

By Fubini’s theorem

$\displaystyle\iint_{\mathbb{A}_{1}}\frac{|h|_{N}}{|z||h(z)|}dz$
$\displaystyle=$
$\displaystyle\int^{2\pi}_{0}\left(\int_{1}^{r}\frac{1}{|h(z)|}\frac{\partial|h(z)|}{\partial
t}dt\right)d\theta$ $\displaystyle=$
$\displaystyle\int^{2\pi}_{0}\left(\int_{1}^{R}\frac{d\varrho}{\varrho}\right)d\theta=2\pi\ln
R.$ $None$

According to (3.26) and (3.27), we know

$\iint_{\mathbb{A}_{1}}\frac{|h|_{N}^{2}}{|h(z)|^{2}}dz\geq
2\pi\frac{\ln^{2}R}{\ln r}.$ $None$

Moreover,

$\iint_{\mathbb{A}_{1}}\frac{J(z,h)}{|h(z)|^{2}}dz=\iint_{\mathbb{A}_{2}}\frac{1}{|w|^{2}}dw=2\pi\ln
R.$ $None$

According to (3.25), (3.28) and (3.29), we obtain

$\iint_{\mathbb{A}_{1}}\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2}}dz\geq
2\pi(a^{2}-b^{2}p_{\ast}^{2})\frac{\ln^{2}R}{\ln r}+2p_{\ast}b^{2}2\pi\ln R.$
$None$

Let $p_{\ast}=\frac{\ln r}{\ln R}$, and by (2.30), then (3.30) reduces to

$\displaystyle\mathbb{E}_{1}[h]$ $\displaystyle=$
$\displaystyle\iint_{\mathbb{A}_{1}}\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2}}dz$
$\displaystyle\geq$ $\displaystyle 2\pi\ln R\Big{(}a^{2}\frac{\ln R}{\ln
r}+b^{2}\frac{\ln r}{\ln R}\Big{)}$ $\displaystyle=$
$\displaystyle\iint_{\mathbb{A}_{1}}\frac{a^{2}|h^{*}_{N}|^{2}+b^{2}|h^{*}_{T}|^{2}}{|h_{1}^{*}(z)|^{2}}dz$
$\displaystyle=$ $\displaystyle\mathbb{E}_{1}[h^{\ast}_{1}].$

Case 2: We take $p_{\ast}=\frac{a}{b}$ in (3.22) and obtain

$\displaystyle\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2}}$
$\displaystyle\geq$
$\displaystyle\frac{(b^{2}-a^{2}q_{\ast}^{2})|h_{T}|^{2}+2a^{2}q_{\ast}|h_{N}||h_{T}|}{|h(z)|^{2}}$
$\displaystyle\geq$
$\displaystyle\frac{(b^{2}-a^{2}q_{\ast}^{2})|h_{T}|^{2}+2a^{2}q_{\ast}J(z,h)}{|h(z)|^{2}}.$
$None$

Integrating over $\mathbb{A}_{1}$ in (3.31), we get

$\iint_{\mathbb{A}_{1}}\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2}}dz\geq(b^{2}-a^{2}q_{\ast}^{2})\iint_{\mathbb{A}_{1}}\frac{|h_{T}|^{2}}{|h(z)|^{2}}dz+2a^{2}q_{\ast}\iint_{\mathbb{A}_{1}}\frac{J(z,h)}{|h(z)|^{2}}dz.$
$None$

By means of H$\ddot{{\rm o}}$lder inequality, we have

$\displaystyle\Big{(}\iint_{\mathbb{A}_{1}}\frac{|h_{\theta}|}{|z|^{2}|h(z)|}dz\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\iint_{\mathbb{A}_{1}}\frac{|h_{\theta}|^{2}}{|z|^{2}|h(z)|^{2}}dz\iint_{\mathbb{A}_{1}}\frac{1}{|z|^{2}}dz$
$\displaystyle=$ $\displaystyle 2\pi\ln
r\iint_{\mathbb{A}_{1}}\frac{|h_{\theta}|^{2}}{|z|^{2}|h(z)|^{2}}dz.$ $None$

Since

$2\pi\ln
r=2\pi\int_{1}^{r}\frac{dt}{t}=\iint_{\mathbb{A}_{1}}\frac{1}{|z|}\Im\Big{(}\frac{h_{T}}{h}\Big{)}dz\leq\iint_{\mathbb{A}_{1}}\frac{|h_{\theta}|}{|z|^{2}|h(z)|}dz.$
$None$

According to (3.33) and (3.34), we get

$\iint_{\mathbb{A}_{1}}\frac{|h_{T}|^{2}}{|h(z)|^{2}}dz=\iint_{\mathbb{A}_{1}}\frac{|h_{\theta}|^{2}}{|z|^{2}|h(z)|^{2}}dz\geq
2\pi\ln r.$ $None$

Therefore, (3.32) reduces to

$\iint_{\mathbb{A}_{1}}\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2}}dz\geq(b^{2}-a^{2}q_{\ast}^{2})2\pi\ln
r+2a^{2}q_{\ast}2\pi\ln R.$ $None$

Let $q_{\ast}=\frac{\ln R}{\ln r}$, by (3.36) we get

$\displaystyle\mathbb{E}_{1}[h]$ $\displaystyle=$
$\displaystyle\iint_{\mathbb{A}_{1}}\frac{a^{2}|h_{N}|^{2}+b^{2}|h_{T}|^{2}}{|h(z)|^{2}}dz$
$\displaystyle\geq$ $\displaystyle 2\pi\ln R\Big{(}a^{2}\frac{\ln R}{\ln
r}+b^{2}\frac{\ln r}{\ln R}\Big{)}$ $\displaystyle\geq$
$\displaystyle\iint_{\mathbb{A}_{1}}\frac{a^{2}|h^{*}_{N}|^{2}+b^{2}|h^{*}_{T}|^{2}}{|h_{1}^{*}(z)|^{2}}dz$
$\displaystyle=$ $\displaystyle\mathbb{E}_{1}[h^{\ast}_{1}].$

As in the proof of $\lambda\neq 1$, the uniqueness is obtained in a similar
way. The proof of Theorem 1.1 is completed.

##  4\. The extremal problem for weighted combined distortion

In this section, we will consider the extremal problem for weighted combined
distortion. Certainly, the extremal problem for integrable distortion is
connected with the extremal problem for harmonic energy. For the related
approaches to the extremal problem for integrable distortion we refer to the
papers ([1],[2], [3], [9], [15] etc.).

At first, we give the definition of the weighted combined distortion. For
$a,b>0$ and $h\in\mathfrak{F}(\mathbb{A}_{1},\mathbb{A}_{2})$, we define the
weighted combined distortion by

$\mathbb{K}_{\lambda}[h]=\iint_{\mathbb{A}_{1}}\frac{a^{2}\rho^{2}|\nabla\Theta|^{2}+b^{2}|\nabla\rho|^{2}}{J(z,h)|z|^{2\lambda}}dz.$
$None$

where $h(z)=\rho e^{i\Theta}$.

Next, by similar reasoning of in proof Theorem 4.1 in [19], we obtain the
relationship between weighted combined energy and weighted combined
distortion.

######  Lemma 4.1.

Assume that $h$ is a map of $\mathbb{A}_{1}$ to $\mathbb{A}_{2}$, and let
$f=h^{-1}$. If

$E_{\lambda}[f]=\iint_{\mathbb{A}_{2}}(a^{2}|f_{N}|^{2}+b^{2}|f_{T}|^{2})\frac{1}{|f(w)|^{2\lambda}}dw,$
$None$

then

$E_{\lambda}[f]=\mathbb{K}_{\lambda}[h].$ $None$

where $\mathbb{K}_{\lambda}[h]$ is defined in (4.1).

Finally, we can obtain the following result from Theorem 1.1 and Lemma 4.1.

######  Theorem 4.2.

(I) When $\lambda\neq 1$, under the Nitsche type condition (1.16), for
$f\in\mathfrak{H}(\mathbb{A}_{2},\mathbb{A}_{1})$, the weighted combined
distortion $\mathbb{K}_{\lambda}[f]$ attains its minimum for a radial mapping
$f^{\ast}_{\lambda}=h_{\lambda}^{\ast-1}$ (as defined (1.17)). That is to say,

$\mathbb{K}_{\lambda}[f]\geq\mathbb{K}[f^{\ast}_{\lambda}].$

The minimizer is unique up to a rotation of the annuli.

(II) When $\lambda=1$, for $f\in\mathfrak{H}(\mathbb{A}_{2},\mathbb{A}_{1})$,
the weighted combined distortion $\mathbb{K}_{\lambda}[f]$ attains its minimum
for a radial mapping $f^{\ast}_{1}=h_{1}^{\ast-1}$ (as defined (1.20)). The
minimizer is unique up to a rotation of the annuli.

Acknowledgements The authors would like to thank to the referee for a very
careful reading of manuscript.

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